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Re-rooting the learning space : minding where children’s mathematics grow Thom, Jennifer S. 2004

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7le~tootin$tykejdeatnlnf Space /hlndln$ TOkete dkitdrens /hatkematicg <3f*ow> bv  Jennifer S. Thorn B. Ed. University of Victoria, 1993 M . A. The University of British Columbia, 1999 A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The Faculty of Graduate Studies (Department of Curriculum Studies) We accept this dissertation as conforming to the required standard  The University of British Columbia July 2004 © Jennifer S. Thorn,  2004  Abstract This disquisition presents a qualitative study that investigated the complicit nature of theory and practice in mathematics teaching. Situated within an ecological perspective, this research interrogates the role that theory plays as a cognizing domain in which one's pedagogy of teaching mathematics co-exists and co-evolves. A systemic exploration of mathematics and the teaching and learning of it is conducted and assessed against tenets of complexity, sustainability, languaging, co-emergence, integration, and recursion. This study reveals the impact that theoretical discourses have on the kind of place and the forms of mathematics that are enabled and disabled through the metaphors, perceptions of mathematical understanding, and conceptions of time that are embodied and enacted by the teacher and her students. This research involved the explication of the teacher's assumed theoretical and practical patterns of teaching mathematics. The expressive forms in which this disquisition is written provide interpretive snapshots that document the teacher's conceptual journey from that of a heavily mechanistic, linear, and hierarchical mindset towards the development of an ecologically coherent theoretical domain for teaching. The classroom vignettes of the teacher, another teacher with whom she collaborated, and the second and third grade students span a course of two and half school years. These vignettes focus on the teacher's work in occasioning ecological forms of teaching, learning, and mathematics in the classroom. The analysis of these episodes revealed stark differences from that of her previous teaching practice not only in the nature of the students' understandings, their ways of acting and being mathematical but also, in the kinds of mathematics that arose during the lessons.  Table of Contents Abstract  .  ii  Table of Contents  iii  List of Figures  vi  Acknowledgments A Fore-word For The Reader  viii ix  We A r e Connected T o T h i s Earth  1  Significances i n the Flow of Interaction  4  Space Wanted  9  Encircling O u r Perceptions of the W o r l d and the Mathematics Classroom Our Individual Ways of Being Our Social Ways of Being Our Cultural Ways of Being  11 13 19 23  H e l p Wanted  30  Settling In  31  Beginning W i t h Place  34  Reflections Revealing the Fragility of Classroom Mathematics A Student's Place Back on the Strip: M y University Years From the Side of Pedagogy T h i r d Time A r o u n d C o m m o n Space Vacancy Turning the Soil: The Beginning Years of Teaching  37 39 40 42 44 45 46 47  Just H a n d T h e m D o w n The Mathematics... O r Not?!  53  Part 2: Settling In(to AnOther Space)  55  Surfacing and Noticing  59  Even Deeper  72  Metaphorical Mattering of Mathematics  74  iii  Does the way on portrays mathematical understanding matter? Theoretical Portraits of Mathematical Understanding 78 Examining Mathematical Understanding from a Constructivist Perspective... 81 Emphasizing the Dynamic Nature of Mathematical Understanding 116 Rethinking Mathematical Understanding from an Enactive Perspective 138 New Furniture  148  Scripting an Unscripted Lesson 158 Mathematical Language, Languaging, and Residues of Learning 162 Arisings (A continuation of Mathematical Language, Languaging & Residues of Learning 167 D o the metaphors with which we describe beginnings and ends really matter?  169  Reflecting on Place as Atmosphere and Ecological Metaphors  172  Embedding and Rooting an Ecological Classroom  Sense of Place for Mathematics i n the 184  Attending to the Physical Space i n the Classroom Stepping Into Jennifer's Classroom Materials Matter  186 188 191  Conceptions of T i m e Absolute Time Linear Time Wholring Time  192 193 195 208  Enacting Reconceptualizations of Time 209 The 4 Rs of Recursion: Reflecting "Revisiting •Reintegrating* Renewal.... 218 M a k i n g Three Spaces for Recursion 222 A Snowflake of a Different K i n d 227 Re-viewing and Seeing Differently 237 Recursion as Relations: When Triangles Become Square 245 Creating Patterns That Connect Interactional Spaces for Mathematics In the Classroom Meeting with Mac: A Study of Opposites and Relatedness Connecting as Shape-shifting Opening Spaces O f Their O w n  250 255 257 287 303  Mathematics Beyond the Classroom Keeping O u r Classes Connected Mathematical Gifts Celebrating Together  321 325 325 330  Beyond Spaces of Inter-action Connecting M e , Connecting Us Connecting Us to It  339 340 358  iv  Spaces for Unpredictable Mathematics Is A Half O f A Half Really A Half? In Search O f Mermaids  364 367 369  Help Wanted  378  O f f The Beaten Track  379  References  383  v  List of Figures  1  Figure 1.  Skemp's Two-level Cybernetic M o d e l of Intelligence  Figure 2.  Summary of Hieberfs Three Sites for Linking Symbolic Mathematics with Understanding 87  Figure 3.  Analytic Framework of Mathematical Processes  94  Figure 4.  S O L O Taxonomy  97  Figure 6.  Mathematical Actions, Processes, Objects, and Schemas  110  Figure 7.  Genetic Decomposition of the Concept of Limit  113  Figure 8.  Process, Implementation, Decomposition  Figure 9.  Algebraic Problem and V i s u a l Solution  117  Figure 10.  Algebraic Problem and Analytic Solution  118  Figure 11.  Visualization/Analysis M o d e l  119  Figure 12.  and  Revision  82  Process  of  Leveled Analysis  Genetic 114  124  Figure 13.  Example of a Mathematical Problem  125  Figure 14.  Partial Inventory of an Individual's Resources  126  Figure 15.  Conceptualization of the Structure of Memory  127  Figure 16.  Time-Line G r a p h of a Typical Student  128  Figure 17.  Time-Line Graph of a Mathematician  129  Figure 18.  Time-Line Graph Instruction  Figure 19.  Configuration of Three Stages of Mathematical Conception  Figure 20.  Integration of Communication  of  Student's  Mathematical  Problem  Solving  Conception  and  Actions  After 130 134  Mathematical 141  vi  Figure 21.  Model of a Dynamical Theory of Understanding  Figure 22.  Definitions of Terms and Features of the Pirie-Kieren Model For the Growth of Mathematical Understanding 143  Figure 23.  Primitive Knowing as Knowledge  the  Source  the Growth of Mathematical 142  for  All  Other Mathematical 145  vii  There are many people who have supported me and contributed to this work. I owe each of them a great deal of gratitude and thanks. Without the sincere interest and enthusiasm of my committee members, this disquisition would not have been possible. I am grateful to my advisor Susan for the tremendous amount of time, support, and energy that she devoted to my research. The dedication and sheer joy she has for teaching and learning is infectious. Karen deserves warm thanks for inviting me into the discourse of ecology and helping me to find a place of resonance within it. To Tom, for his incredible energy, thoughtfulness, and direction in the development of my work. And to Ann for her help during the beginning stages of this research. Thank you to you all. I have learned so much and been truly inspired! My colleagues at the University of Victoria and beyond have been extremely supportive of my studies at The University of British Columbia- especially Werner and Chetwho on several occasions interrupted my rhythm of thinking and provoked me to see things differently. 1 am thankful to Donna for teaching with me and to each and every one of my students for seeding and feeding my curiosity to explore and seek a better understanding of what it means to teach and learn mathematics. I would also like to acknowledge the generous grant I received from The University of British Columbia. It greatly aided in my researching and writing of this work. I am lucky to have such a supportive family as I do. Their kindness and guidance have provided me with the confidence to pursue my studies. I thank Patsy for spending many a conversation helping me to craft my ideas and get them down on paper. I extend immense gratitude to Lucas for his care, patience, and constant encouragement during my years of graduate work. And to our little one whose gentle, calm nature afforded me many uninterrupted hours in which to write. Without them, this Journey would not have meant so much or been as wonderful as it has. BI 1  viii  So understand a living system such as a tree, in an ecologically system mean& that it is, not poAaiMe to- examine the twee By simply teducing it d individual parts or analyzing it from pant to- whale. Slather, it mean& that indeed need ta study, the tree's leaves, Branches,found,mat system, and its with the environment Out puun many different vantage points ta mahe sen each part exists in dynamic relationship, with the others OA an integrated sy She Aome can Be Aaid about the purpose of tPu& acquisition. Jt is not m Be a vecipe fax haw ta teach mathematicA well OK ta serve OA dimply a desc account of. a teaching practice. Jt is, in eAAence, a systemic exploration into em&eddedness and the emergence of. theory and practice in mathematicA teach Qiven the nature of. thia, research and the theoretical tealm in wh situated, it WOA important far the work, ta Be expressed in a form that poAAeAAed an ecological sensibility. Upon first glance, it appears ta Be a coll of separate compositions. Ctnd although each piece is an entity unto itself, was not to vender the research as a piecing, together of theory, data, and an instead, to Bring a multiversal perspective to it and expose the co^existen evolution, of theory^and-practice. Shus, the emtadiment of this disquisition' also evident in the organization of the text as a whole and the diversity of structures within it. She organic way in which this disquisition is organized can Be liken tree: that any one leaf is neither directly connected ta the other, leaves, n need to view them in any particular order yet at the same time, all are inte as eAsential partA of the tree By way of its Blanches, and therefore, are necessa making the tree a coherent whale, Mere, the compositions that were sourc and audio taped classroom sessions, journal entries, students' work, and field notes are not necesAarily directly linked to one another or sequentia Each piece is considered ta Be a smaller yet integral system of thinking tha forms larger conceptual clusters within an ecological mind-space. CLnd to these interrelated knowledge systems seek to inform the mathematical space. SJhe different analytic viewpoints are communicated through the fo expressive structures:  Metaphors and visuals have Been used ta reflect how it was that 3, WO conceptualizing the theory and my teaching.  ix  featured quote* OK questions positioned on a Blank page are the theore artifacts that served aa, provocations, fox my further research. Jn this ac they, one, intendedtointerrupt the xeadex's flow and signal a shift OK op another conceptual Apace.  juxtaposition af text with other teat visually expresses the recursive an emergent layers of thinking that arose dwdng the research. Sometimes, th af juxtaposition was u&edtoset ideas with ax against each athefc in order what funds, af theoretical tension ox generative spaces, would arise for m consideration. While other times,, the juxtaposing af ideas, enabled the de af relationships Between one author's, thinking with that af another's,.  She use af Much and white or colour for certain texts, and images, empha theoretical underpinnings, that J, consideredtobe clearly, defined as, appose J, perceivedtobe evex-changitig and indeterminate. Sox example, 3, ma Much and white in the visual-text collage an constxuctivist notions,toconv conceived as, theory, that was, "clear-cut" whereas, colour was, used in visual-text collagetoexpress, theory, that 3, understood as. imbued with qualities, that were unpredictable, ever-changing, and not so clear cut.  She actual figuring af text such as, a newspaper article, conversation, free poem, as, well as, whether it was organized in a lefttonight,toptobottom, forth, sporadic, or circular manner sought to capture the conceptual an metaphorical essence af the ideas, being discussed  flmdikafy and balding af text functions, as, the bringing forth af ideas in  development af theory while still presexving the contextual background fro these ideas emerged. Shis form af writing enables onetosee the "double that was present in my thinking.  3t was critical that each piece af writing in some way highlighted the in personal particularities, af this research. So da this,, different "characters developed. She characters, in this disquisition axe my students,, a teaching and myself. 3n order to analyze and interpret my teaching from multiple perspectives, my character takes, an several different "personalities". Jn s the compositions, J, am the main character and describe events as, J, pexc expeuence(d) them either 'in the moment' aa, they unfolded ox by, ta reflective stance. Jn other instances, J, am another character altogether o x  present in the piece at ail. SJhis allowed me to interpret the research connected yet more distanced or 'outside' perspective. Jn still others, the H find me in conversation with another character. SJhese vignettes reveal th questioning and assessing of the theoretical coherence concerning my resear Jn addition to the stales, of writing, characters, and personalities, the fonts of the text help to visually distinguish between the different 'tone perspectives taken on in the analysis and interpretation of the work. Jina disquisition need not Be a front to Back, left to right, top to Bottom re  recognizing the various parts of this work here and in the table of conte hoped that the reader will engage with the same spirit as one would expl living tree— perhaps, examining its integrated and integral being from its climbing up,, hanging from a branch and gating around, leaping from one  another, or even peering down at the always emerging whole from a distan  xi  1  WE ARE CONNECTED IQ iniv EHnlll As we continue t o e x i s t e n c e . Simply put, this We are pour chemical cock- not independent be- means abandoning the t a i l s i n t o the e n v i - i n g s . We are p a r t o f , re-production of our ronment and move f a s t connected to, and ' C a r t e s i a n - s e l f - a s and furiously from " j u s t one p a r t i c u l a r s e r t i v e - N e w t o n i a n ' one t e c h n o l o g i c a l ad- strand i n the web of ways of being i n o r venture to another, l i f e " a explains Ca- der t o c u l t i v a t e it i s no longer a p r a . more i n t e g r a t i v e exmatter of choice but istence on this a matter of f a c t t h a t earth. I t e n t a i l s r e i n order f o r l i v i n g r o o t i n g our t h i n k i n g systems on the e a r t h so that we may comto s u r v i v e , we must prehend the world not l i v e within i t s limj u s t i n l i n e a r , anaits of sustainabillytical, rational, ity. and reductionist terms, but i n ways Upon our clumsy that are n o n l i n e a r , awakening t o the enconnected, i n t u i t i v e , vironmental crisis, and holistic. As we are presented with w e l l , value needs t o the rude r e a l i z a t i o n Deep,integral chan- be placed on cooperathat the impact of ges will not take t i o n , quality, and our a c t i o n s cannot be place i f our a c t i o n s conservation i n s t e a d 'contained' and the reduce p o l l u t i o n of anthropocentric, e f f e c t s of them reach to and decrease stress exploitive, or comfarther and deeper on the earth's natup e t i t i v e a c ts of than we ever anticir a l systems remain domination and mindpated . The ongoing on devastation of the rooted i n our d e s i r e s sets that focus to improve human 'the bottom d o l l a r ' . world's n a t u r a l and h e a l t h and maximize This i s not a simple cultural systems of 'exchangmakes this p o i n t p r o f i t a t the c o s t of matter a l l other forms of i n g ' our c u r r e n t ways c l e a r : The r e s u l t s of living f o r ecohow we l i v e are not nature. I f we are t o of ones. The only f e l t by the l o - prevent f u r t h e r dam- l o g i c a l anthrocal human community age t o the e n v i r o n - mechanistic, traditions and our neighboring ment i t i s c r i t i c a l p o c e n t r i c we embody in change our that communities, but what that we culture today percep- our we do a f f e c t s a l l mechanistic evolving that i s on t h i s e a r t h t i o n s of the world t o have been since the with us—the land, ones that are eco- s t e a d i l y I n d u s t r i a l Revolution water, a i r , and l o g i c a l . and so too w i l l i t every living being take take time f o r that depends on these (continued on page 2) 2 sources for their 2  3  1  'Yes but, we recycle!"  6  4  5  ecocentric practices to become taken-forgranted patterns w i t h i n our t h i n k i n g , a c t i o n s , and i d e n t i ties .  QUESTION: But what does t h i s have t o do with the teaching and learning of mathematics i n the c l a s s room?  "Okay, b u t what does this mean and where would one start?"  5. Bowers 1997, 2003; Capra, 1993, 1996; Snyder, 1990; 6. Capra, 1996; Naess 1985, 1988; O r r , 1992, 1994; S u z u k i & D r e s s e l , 1999. 7. As c i t e d i n S u z u k i , 1997, p. 199.  What is a Tree?  place might be t o look f o r spaces i n which t o propagate A tree, we might say, is not n o n l i n e a r , connected, fluid, and h o l i s t i c so much a thing as a rhythm of patterns of t h i n k i n g exchange, or perhaps a centre of mathematically. A t organizational forces. Transpirathe same time, mathe- tion induces the upward flow of matics educators water and dissolved materials, could begin the proc- facilitating an inflow from the ess o f assessing the soil. If we were aware of this embedded and taken- rather than the appearance of a for-granted l i n e a r tree-form, we might regard the and mechanistic r i t u - tree as a centre of a force-field to a l s that are pracwhich water is drawn....The obtised within the EVERYTHING! field o f mathemat- ject to which we attach signifii c s education and i n - cance is the configuration of the forces necessary to being a For e c o c e n t r i c s i d e classrooms. thinking to bring By not excusing tree....rigid attention to boundaabout a paradigmatic ourselves from t h i s ries can obscure the act of being turn that possesses task, we can begin itself. l o n g e v i t y , depth, and working towards r e -Neil Evernden, The Natural i n t e g r i t y , i t cannot connecting and d e v e l Alien be restricted t o oping mathematically the domain of e c o l o g i c a l ways o f This redefinition of some'environmental c l e a n - being. thing as familiar as a tree ae at l i n e s s '. I t has t o first rings strange. But we can become our n a t u r a l , recognize the more-than-treeeveryday way o f beNotes form it describes, just as we i n g . Learning t o be know that a forest is more than ecologically mindful 1. M. C. Bateson,1994, just the trees that grow there, cannot be t r e a t e d as 1996; Bowers, 1993, an ' a d d i t i o n a l compo- 1995,2000; S u z u k i , 1999. and that our intercourse with nent' i n c h i l d r e n ' s 2. M. C. Bateson, 1994, the world extends beyond the education but an i n - 1996; Bowers, 1993, edges of our skin. Our language t e g r a l p a r t o f each 1995, 2000, 2001; falls short of our apprehension and every classroom. S u z u k i , 1999. because of the way we have been In short, t h i s means 3. Capra, 1993, 1996, p. taught to identify the world. We that teachers and 7. belong to, are made of, that students of mathe- 4. Fox, 1990; Naess, matics may not be 1985; 1986; 1996; O r r ,world that surrounds us, and we respond to it in ways beyond excusedI 1992, S u z u k i , 1997. knowing. Perhaps a s t a r t i n g 3  ANSWER:  1.  We axe constantly, engaged in the flam o interacting But often, itfanot until much latex, that we ap^Meciate the significance of it.  Reflection A mirror reflects an image seen as the image is seen. It does not change the looking. To reflect on what we do, or are, is something else. It reveals what we could not see.'  It happened years ago. Jennifer could not remember when exactly, but at some point as a very young child she was drawn into the enchantment of the "enveloping and sensuous earth". Eyes wide and bright, Jennifer giggles as she stories-out a cluster of her treasures to me. She speaks of wild landscapes just beyond her grandparents' orchard; tunneling on her stomach and disappearing into the tall, sweet grass; lying on her back and watching the night sky for cascading meteorites; and feeling the cool dampness underfoot as she creeps silendy and listens hard to find that mysterious chirping cricket. Among these treasures are many more: ones of forests, of the ocean, others storied inside her ba-chan [Japanese for grandmother] tanka poems, as well as Chinese proverbs told to her by her ba-ba [Cantonese for father]. "I suppose" Jennifer reflects, "because my family life was rooted in a kind of living that looked to nature for metaphors and life lessons, that I also seek to understand the world as a living system that is interrelated to everything else. And I guess that is why I wonder how ecological forms of thinking might help us to better understand this place we call the mathematics classroom." "Classroom mathematics and ecology? Interrelated? Please, tell me you're kidding!" I gasped. "I know, I K N O W , " she replied and then paused. "But listen" she urged. Anticipating a long, winding, twisting, turning kind of response, I prepared myself. Straightening my posture, I took a deep breath as Jennifer began taking me down her explanatory path. "You see," she began, "ecology and classroom mathematics have everything to do with each other." Jennifer then told me that the word, ecology had come from the ancient Greek word, oikos. It meant "the family household" and "the maintaining of its daily operations." Eventually, oikos was integrated into the term, oecologia, coined by Ernst Haeckel in 1866. Described as "the study of the environmental conditions of existence," oecologie was eventually shortened to what we know today as ecology* "I remember exacdy what was going on in my mind when I first learned about the history of the word" said Jennifer. "To be quite honest, I hadn't given it much thought at the time because I was preoccupied thinking about something else. Even though I listened to what was being said, it was similar to having to attend to a different matter when you already have your hands full with something else! You see, I was taking a summer graduate course and the professor was explaining to the class how the word ecology came to be. I remember smiling and nodding as I listened and then quickly switching back to my previous thought. It wasn't until much later that I realized the significance of the encounter." 1  6  Jennifer explained that her conundrum had been trying to communicate to other people the importance of being ecologically mindful in the mathematics classroom. 'You see, it was easier for me to say what it didn't mean. Being ecologically mindful wasn't necessarily about taking environmental issues and making them into mathematical problems. It wasn't about conducting scientific inquiries- you know, taking ecological ways of thinking and using them as a magnifying lens to examine the field of mathematics education and then perform experiments in the classroom. It wasn't about forming a hypothesis, replicating procedures, generating conclusions or formulating a unifying theory that could be transplanted into every classroom. What I was finding it extremely difficult however, was how to explain in simple words what being ecological did mean to the mathematics classroom. M y descriptions were cumbersome- that knowing and acting were embodied withand-in one's way of being- or, a mindful comprehension of the integrative, holistic, and nonlinear nature of teacher's practices and children's learning of mathematics. So, while I spent my time trying to sort out the ideas that I viewed as being problematic in developing an ecological sensibility for teaching and learning mathematics" said Jennifer, "I was completely ignorant of the fact that I did have a way to express my understanding of ecology and mathematics education! Oikos." Jennifer told me that, upon reflection, it was the term, oikos that captured exactly how she understood her mathematics class to be. Here, she explained that she imagined it to be very much like a family household. As the children's teacher, she saw her role as caring for and sustaining the mathematical relationships and interactions of her students. "So just as environmental thinking focuses on human relationships with nature," Jennifer smiled, "it is a similar focus that I have for my teaching and children's learning of mathematics. It's about examining and assessing the kinds of mathematical relationships, as well as the forms of mathematics that emerge in the classroom, and responding to them in my manners of teaching mathematics". "And your reason for wanting to be ecologically mindful?" "My wanting to be an ecologically responsive mathematics teacher comes from caring for how mathematics exists in the classroom, my teaching, and the children's learning of it. It is about being committed to sustaining relations that are not only ecologically coherent in the classroom but also ones that promote a sense of cohesiveness within the larger educational communities." 3  7  Jennifer then picked up the book she'd been reading before I had arrived. Opening it to page 78, she read aloud: We are living in a time of both creativity and concern about education, and the decisions that are made for the classroom will feed directly into the way graduates  "and children," she added, participate in society and the way they impact on the natural  "and social-cultural,'' she said, systems around them. Bringing our conversation to a close, Jennifer said, "and so you see, the choices we make as mathematics teachers not only affect the kinds of mathematics children learn in school, but equally, the ways in which children are taught to learn and the ways they will interact with mathematics outside of school will affect the world they live in. We, mathematics education, and ecology do not exist in separate households but, rather, we share a common space." 4  Notes  1. Abram, 1996, p. 15. 2. Also, an email correspondence with C . A . Bowers in which we discussed, Donald Worster's (1990) book, Nature's economy: A history of ecological ideas. I particular, the definition of ecology which is described by the author, p. 191-1922. 3. Naess, 1985,1986,1988,1996; Varela, 1999. 4. M . C . Bateson, 1996, p.78.  8  SPACE WANTED  IMPISH*§* "W^  A UTOl'SHWl  "How c a n we b r e a k o u t  Looking to share a space o f o u r c o n v e n t i o n a l with ecology. Interested a p p r o a c h e s a n d i m a in what ecologically co- g i n e more p r o d u c t i v e " Reply herent forms of teaching ta ol t em ra ni altbiovxe:s ?T1I9M9M7S and learning of mathematics could mean for the classroom. Can move in IMMEDIATELY. systems. It also 1  (continued  from  page  36)  THE 3 FACES OF ECOLOGY According to M. C. Bateson , there are three "faces" or realms of ecology: empirical) environmental, and systemic. The author defines empirical ecology as biological, meteorological, and geographical studies that focus on understanding how the planet is changing and how these changes affect the interrelationships of the world's natural systems. The environmental face of ecology is concerned with identifying the level of impact that our ways of living have on the earth's 8  involves the development of solutions for environmental problems that will minimize harmful stress on the earth. It is within the systemic realm of ecology where mathematics teaching and learning can be most radically explored. This is because systemic thinking focuses on seeking "the pattern which connects" a system or systems together as interdependent and interacting wholes. In the field of mathematics education, a "system" could be an individual teacher or a student. It could also be a collective 3  9  V  A  G  A  M  C  Y  Seeking one primary teacher to teach grades 2/3. Separate room. "Shared facilities".  group such as a mathematics class, the school, and so on. The connecting pattern or patterns that interrelate these systems together as a dynamic whole encompass the forms of knowledge, actions, and identities that are brought into being as a result of the ongoing interactions in the system (s) and the ways in which they are sustained by the system (s). By focusing on relational qualities, ecological ways of thinking give rise to viewing the world as an integrated whole; a dynamic and fluid network in which all living and socialcultural systems are interconnected. The (continued  on page  79)  logy that the fol- (continued from page 24) lowing query emerToday we are world is not con- ges: calling on the nations ceptualized as being and. the peoples of the composed of a colIn what ways can world to change lection of separate systemic manners of personal attitudes thinking about entities, but instead, and practices. "To mathematics education as a highly complex adopt the ethic for enable forms of unity i n which all teaching and spaces living sustainably, for children's learning people must reexsystems are interof mathematics to amine their values related and therepossess an ecological fore, interdepenand alter their besensibility? havior. Society must dent. promote values that It makes sense support this ethic then, that when and discourage those looking at maththat are incomematics teaching 1. Stigler Sc Hiebert, patible with the su1997, p. 14. from an ecological stainable way of life" perspective, it would (from Caring for the be conceived as 2. M . C. Bateson, Earth: A Struggle for 1996. similar to that of Sustainable Living. children's mathemaGland, Switzerland: tical learning. M a - 3. 6. Bateson, 1980, IUCN/UNEP/WWF, 1991. thematics teaching 1991). as a fluid, complex process implies that 4. A . B . Davis, 1996; To whom is this call A . B . Davis, it exists always, i n addressed? The ethic Sumara, Sc Lucerelation to the onfor sustainable living Kapler, 2000; going interactions of has always been part Kieren, Pirie, Sc the students, the of our cultures, and. the people lived in line Calvert, 1999; mathematics, and with such a way of the material and M a r t i n , 1999; life. It was the Wesnonmaterial enviTowers, M a r t i n , tern materialisticronment of the classSc Pirie, 2000. consumerist strateroom. gy, considered the esA n d so it is by 5. Thorn, 2008. (continued  from  page  78)  NOTES  4  0  taking a systemic perspective from 6. Kotagama, 1993, p., 120-121. within the con10 ceptual space of eco-  sence of "development," that shattered the foundation of sustainable living that is 6  (continued  on page 80)  ^fes,  but  ecological mathematics  what  $ives  view classroom  rise  ofi  the for  to  a  world? that  systemic, ot  the  matter?  11  o  if  Mi  1  12  13  9 think that If we start by looking at how we are as individuals and then connect this to how we exist as collective groups, you'll be able to appreciate why a systemic understanding of the world and the mathematics classroom is really about 'layers of living'. St creates a conceptual space in which we can make visible what often remains invisible- the co-emergent, complex nature of our  S  fc I N  biological,  structure determined, social, and  CULTURAL  G  Okay, Let's begin! As humans, we exist in the world as what Maturana and Varela would call, "autopoietic," or self-making systems! We possess both "organization" and "structure". 9fs our organization that distinguishes you and me as people and not, say, fish or goats! And it's our structure that can be described as the internal dynamics and relations that enable you and me to develop ways of knowing, acting, and being that are uniquely our own. Simply put, your structure is not the same as my structure and it is because of our structural diversity that we can distinguish you and me as being different people. Y%ut how is it that we are different?  structurally  Maturana and Varela describe "structural coupling" as the process by which our structures evolve. The changes that occur in our knowing, actions, and identities arise from the recursive interactions between two or more living organisms. A nice sounding this mean?  definition, but what does  Well, if we take this idea of structure and think of a person's understanding of mathematics to be his or her mathematical structure, in a way similar to how  14  a person's forms of knowing, acting, and being are impacted by life experiences - a person's mathematical structure too undergoes changes as a result of his or her mathematical interactions. 2  And so, because one's structure is dependent on the kinds of mathematical interactions one has and how they then feed into what the person already understands, these differences in experience and impact create differences between one person's mathematical understanding and that of another person's... hence, diversity in mathematical structures. Ufes. And recursively, how we go on then, to teach or learn mathematics will now be shaped by these structural differences. This also means that as a class engages in mathematics, structural coupling is arising in the structures of the individual students and their teacher. The growth that arises from this process is dynamic and continuous. 9t happens in us moment to moment as we experience human and nonhuman perturbations in the environment. So it's the perturbations structural changes?  that make for  No, not exactly. 9t isn't the perturbation that determines how our structures change. And, perturbations only exist if they are perceived by the person as "perturbatory" Rather, it's the individual based on his or her structure, who specifies what will or won't be a perturbation, whether or not coupling will occur, and if so, the kind of internal changes that will arise in his or her. structure. Knowing this, we can say that we exist in the world as autopoietic and "structurally determined" systems." 3  Would this mean then, that in the mathematics classroom, it is the child who determines based on his or her internal structure, what will and will not serve as occasions for learning mathematics? ijes, and it's the child's mathematical that shapes and is shaped by future  understandings-understandings.  his or her  structure,  5  15  %ut what about the teacher? Ssn't it the teacher who teaches the class what mathematics to learn? Of course it should be expected that a teacher attend to children's mathematical learning in wags that are invocative and provocative However, given a systemic view, we can't assume that the teacher exists as the only source for occasioning children's mathematical perturbations. Engaging in mental reflection about mathematics or taking part in mathematical Interactions with the human and nonhuman environment can also serve as possible sources for structural changes to occur. 6  7  8  9  So, even if a teacher intends to have the class learn, say... a new strategy for adding 3-digit numbers together, it is the child, N O T the teacher who determines if and what kind of learning will arise? Exactly. And when structural changes do take place, new pathways or relationships emerge and Impact on the child's existing mathematical understandings. So it's impossible for us to predetermine how our individual structures will evolve since they are ever-changing because of the coupling process! This is what A. %. Davis, Sumara & Luce-Kapler mean when they say that "learning is D E P E N D E N T O N , but cannot be D E T E R M I N E D BY teaching"! Mathematical learning takes place with the environment: as unpredictable yet recursive growth of one's mathematical structure of understandings. 0  1  Okay, 9 can see how we as individuals are shaped by the interactions we have with the environment but it seems to me to be a very inward, insular way to view the mathematics classroom, don't you think? This kind of thinking moves in only one direction- from the environment to the individual child. Up to this point it has. However, a systemic view does bring forth a 'coemergent worldview', if you will, in that it recognizes the interdependence and complex circularity that exists between the environment and living systems, ^ust as our internal structures are ever-evolving through our interactions with the environment, the environment is also undergoing structural changes. These changes within us and within the larger environment recursively shape  16  what will be possible in terms of future interactions and how each will respond to the other! Cewontin elaborates on this complex circularity when he explains that: The organism [living being] and the environment are not actually separately determined. The environment is not a structure imposed on living beings from the outside but is in facta creation of those beings. The environment is not an autonomous process but a reflection of the biology of the species, ^ust as there is no organism without an environment, so there is no environment without an organism! 2  3  This co-evolution that takes place as we and the environment interact raises an important issue when considering the mathematics classroom. S7f/s not only the environment that shapes the teacher's or a child's mathematical ways of knowing, acting, and being, but it's also the teacher's or child's interactions that affect what future events and responses will take place within the larger classroom. Each needs the other. That's right. Now can you begin to see how the world can be viewed as an integrated whole by recognizing the interdependence of living systems and their environments? Life unfolds by way of "natural drift" ' - as a result of the recursive interactions between living systems and the environment. This coevolutionary view of the world differs from other perspectives that project images of evolution as being a linear process of competitive domination where species and their environments are not interdependent but separate from each other. 4  ijes. A subtle yet important suspect.  difference, 9  What's more, a systemic, ecological view doesn't portray mathematics teaching or children's mathematical learning as being individualistic and linear in nature. They arise fluidly in relation to each other and with that of the larger environment be it a mathematics class, a school, or even the educational system. An ecological perspective brings attention to understanding interrelationships within the mathematics classroom. 9n the beginning of our conversation, you mentioned that we are also social and cultural beings, yes?  17  yes, that's right. Keeping in mind what we've discussed in terms of how we are as individuals and how we and the environment co-emerge, let's move outwards to the broader, social realm. Y}y doing so, we can continue to discuss "the pattern which connects" our living as individuals to our collective actions, identities, and wags of knowing as social systems. 15  18  19  Maturana characterizes social interaction as being: when two or more structure determined systems interact recurrently with each other in a particular medium, they enter in a history of congruent structural changes that follows a course that arises moment after moment contingent on their recurrent interactions, to their own internal structural dynamics, and to their interactions with the medium, and which lasts until... they separate. 9n daily life, such a course of structural change in a system contingent on the sequence of its interactions in the medium in which it conserves organization and adaptation is called 'drift'. 16  9n terms of the classroom, this would mean that social mathematical interactions arise when two or more children work mathematically together, importantly, the learning occasioned from these mathematical interactions not only shapes the further development of each child's understandings but also, the collective understandings of the partner or group and the larger mathematical environment in which the Interaction took place. These collective forms of knowledge, actions, and identities and how they're created through social interactions are whatMaturana refers to as "drift". 9f we understand human social systems to be what Qregory tfateson and Maturana refer to as systems that evolve through the cohesive, collective interactions of the members, then what we know, how we act, and who we are can't be taken as happening only within the realm of the individual. Such growth also needs to be recognized as emerging from our collective manners of living— the relations that are created through ongoing interactions and that which connects us as interdependent, social beings. And are social phenomena, like our individual structures, unpredictable too? yes. ^ust as we can't predetermine the evolution of an organism or its environment because they are dynamically interactive, we can't predetermine the collective mathematical activity that will take place in a mathematics classroom. 9n terms of a class' mathematical learning, it's naturally unpredictable because children's internal and collective structures are constantly changing from moment to moment! 9  9 see. So a child isn't only a "structurally determined" learner but he or she is also a member of larger social systems... such as a mathematics class? 18  20  Exactly! And In the classroom it's not only the child but also the children and their teacher interacting at the same time... as individual and collective wholes, responding to environmental perturbationsshaping and being shaped by the mathematical learning that emerges! Like our individual understandings, collective forms of learning aren't thought of as being transmitted from an external entity. They are constantly emerging and coemerging through social interactions, because of this, children's mathematical growth can be described as being "much like paths that exist only as they are laid down in walking." 9  20  Can you explain in more nature of social interactions forms they can take on?  detail, the and what  Sn terms of their nature, 9 think of them as Maturana does... like "conversations in progress". Maturana explains that social interactions can be brief, withdrawn from and then re-entered again or, they can be continuous. 9t is these "conversations" within social systems that he considers necessary in how it is we come to know and be in the world. Social dynamics are what bind us as a pair or group of living beings together as a collective, social system. Co-emergence takes place as we interact with and in relation to one another... we are able to coordinate and re-coordinate how we think, our actions, 'how we are' basically, in order to maintain cohesive ways of being with one another. Sn this way, social relationships that keep a collective unity intact can be seen as similar and just as critical to the co-evolution that takes place with individual organisms and their environments. 21  Maturana's idea of "languaging" is useful because it describes the process by which social systems function and evolve as collective unities. Now, it's important that you don't think of languaging as simply individuals engaged in verbal conversation with themselves or others. Languaging involves the physical, verbal, and mental ways we humans think and interact among one another, but it is also the understandings that arise from such linguistic interactions. St's how we are able to coordinate and recoordinate our ways of being so that we can continue to interact within groups and develop collective forms of knowing. Sn other words, "languaging" in the mathematics classroom entails the mathematical understandings that emerge from the different ways in which members of the class think and engage mathematically with one another, because we exist in language and have the. potential to be languaging agents, it is possible for new understandings to arise. Knowledge systems evolve then, as a result of our social activity. 22  21  So does this mean that it's through the interactions of the class that collective mathematical understandings which are different from personal ones come into being? ijes. Now, let's talk about our cultural wags of being.  22  The practices of teaching and learning school mathematics are two examples of "cultural behaviours" In our Society. (generally speaking, cultural behaviours are social patterns that span generations. St's as a result of our living in the languaging process of "cultural drift" that we establish collective wags of being that are passed on and evolve from generation to generation. So cultural wags of being are phenomena that continue generations?  social over  yes. And the historical transformation that happens is a result of the recurrent interactions and languaging between the older and gounger members of a cultural group. 9n the same manner that drift is explained bg Maturana and Varela as necessary for us to evolve with the environment and socially with others, so too is cultural drift necessary for the continuity and evolution of cultural systems. 23  Okay. 9 see how cultural drift provides a systemic way for us to understand say, how human-centred and mechanistic social patterns established in the industrial Revolution have continued into today's culture. But what 9 don't yet understand is what ecological thinkers such as powers, Capra, Naess, and Orr '' mean when they say that our cultural ways of being shape how we perceive and therefore, exist in the world. 2  We do much more than simply live in the world— remember our about coupling?  conversation  yes.  24  Well, 9 believe that connected with this understanding is the fact that THE WORLD we bring forth is not only created through our individual and social existence. 9t is also SPECIFIED through our cultural knowledge, actions, and identities. Cultural phenomena, when CONSERVED, seamlessly coemerge from one generation to the next and create a world from our living WTTHIN CULTURAL "COGNITIVE CIRCLES."25 These cognitive circles arise from our cultural ways of living and we justify the patterns that they occasion as a matter "OF TRADITION" OR, less reflectively and more acceptingly expressed, we say it is simply "JUST THE WAY THINGS ARE." 9n this way, cultural patterns are embodied in our thinking, THEY BECOME US,  DISAPPEARING FROM  THE SURFACE OF OUR CONSCIOUSNESS.  26  Surely, our cultural actions, beliefs, and identities aren't that invisibly specific! Hmm... consider the images we as members of Western culture attach to the idea of what it means for a person to be an 'individual'. When we think about what makes a person an individual, often embedded within this is the notion of independence'. As a teacher, 9 find that parents often express to me that it's important for their children as "individuals", to be self-sufficient, able to think for themselves, make independent decisions, and be their own people. 9n valuing these qualities, we teachers and members of older generations encourage younger generations to develop their independence by providing learning opportunities that focus on the "individual" or "autonomous" child. Within our culture, independence and individuality serve as distinguishing qualities of being successful. They engender a sense of freedom, self-reliance and "standing out from the crowd". Well, isn't that what we should be doing? encouraging students to be independent individuals? Hold on for a moment. Let's contrast this image with what it means be an individual in Japanese and Chinese cultures. Traditionally, within these two cultural circles, the image of an "independent" individual is not the image that comes to one's mind. This is because in Japanese and Chinese cultures, the younger members are taught by the older members that an individual is not defined in terms of self-reliance or self-sufficiency but more in how the individual contributes to the well-being of his or her family, ijou see, a person's identity exists in the collective sense of the family. This can be seen in how people address one another. Unlike in Western society where we are distinguished on a first name basis such as "Jennifer" or in a first-name-lastname order as "Jennifer Thorn", people in Japanese and Chinese 25  cultures are addressed by their last name or In a last-name-first-name order such as 'Thom-san" or "Thorn-Jennifer", identity isn't derived from the validation of one's self but from the respect for the family as a collective whole. Conservation of these relations is carried forth through one's values and actions that foster the well being of the family as a collective whole. Qlven instances such as these, we can betterunderstandwhat Maturana '''speaks of when he says that it's "in the implicit or explicit accepted premises under which their different kinds of discourse, actions, and justification for actions take place" ...that cultures create taken-for-granted and, hence, invisible yet distinctive cognitive circles. 2  Okay... yes... how cultural beliefs create blinders... that shape how we experience the world... but, if we are truly blind to our cultural ways of being, is it even possible for us to become aware of them? One way for us to examine just how culturally embedded our lives are is to consider the cultural experiences, beliefs, and values that emerge from the recurrent interactions of a group and metaphorically become what Qregory Bateson and Bowers refer to as cultural "maps". 28  29  A map?! A map. Simply put, a culture's map identifies what its members will and won't perceive as having significance by rooting these features within the culture's temporal, spatial, spoken, written, and symbolic language. Because these cultural distinctions permeate the group's languaging, a culture's map as Neil Postman would say, "does much more than construct concepts about the events and things in the world; it tells us what sorts of concepts we ought to construct"* Can you give me an example of a feature or concept that we create or recreate in our living out of these metaphorical maps? Just look back at how the idea of the individual is distinguished and played out as a feature of Western and Asian cultural maps. The former very much influences a person to value ways of thinking and actions that enable an identity of independence while the latter, emphasizes a person's connection to her or his family and imbues a sense of interdependence. Same concept- "the individual", but completely different cultural conceptualizations. 26  yes, which lead to totally different of interacting in the world.  ways  Speaking from a systemic, ecological thinking space, it's here in the envelopment of the cultural realm that we live... nested within our individual and collective layers of knowing and being. And it's here that we dwell in our practices of teaching and learning mathematics. Jor me, the mathematics classroom is imagined as an integrated space where living, social, and cultural systems co-exist and co-evolve. 9n an ontological manner, we are living systems within social systems within cultural systems. Encircled once more to include all other living and natural systems on the earth, it is how we humans come to exist as "just one particular strand in the web of life" 9t's in this way that the world isn't perceived as a collection of separate "parts", but as a dynamic whole; a complex unity of all living and social-cultural systems that are fluidly interconnected and, therefore, necessarily interdependent. And it's here in this conceptual space of knowing that a systemically ecological view of the world and the mathematics classroom resides. 31  27  JMotes 1. Maturana,  1987;Maturana and Varela. 1980,1987.  2. 3. 4. 5. 6.  Capra, 1996; Jleischaker, 1990. Kieren and Pine, 1992. Jor example, see Maturana, 1988b, p. 36. Kieren and Pine, 1992; Pine and Kieren, 1992. A. %. Davis and Sumara, 1997; Martin, 1999; Pirie and Kieren, 1994b; Simmt and A. %. Davis, 1998. 7. Snvocative interventions are those which engender folding back— a recursive form of revisiting to earlier layers of understanding. See Kieren and Pirie 1992; Pirie and Kieren 1994b; Towers 1998. 8. Provocative interventions are those which enable learners to move outwards to new and deeper understandings. See Kieren and Pirie 1992; Pirie and Kieren 1994b; Towers 1998. 9. Simmt and A. S- Davis, 1998; Martin, 1999. 10. Maturana, 1988a, 1988b, 2001; Maturana and Varela, 1987. 11. Emphasis added. A. 1$. Davis, Sumara and Luce-Kapler2000, page 64; Pirie and Kieren, 1992. 12. Lewontin, 1983. 13. Lewontin, 1983, as cited in Varela, Thompson, and Pnsch, 1996, p. 198. 14. Maturana and Varela, 1987.  15. Q. Sateson, 1980, p. 9; 1991. 16. Maturana, 1988b, p. 46. 17. Pirie and Kieren, 1989,1994a. 18. Bowers and Jlinders, 1991; A. 8- Davis, 1995. 19. Kieren and Pirie, 1992; Steffe and Tzur, 1994. 20. Varela et ai, 1996, p. 205. 21. Maturana, 1988b; 1991; 1995; 1997a, 1997b, 1998:2001. 22. Maturana 1997a, 1997b, 1998. 23. Maturana, 1988b, 1995,1997a, 1997b, 1998; Maturana and Varela, 1987 24. Sowers'1998 graduate seminar; Capra, 1996; Naess, 1985,1988; Orr, 1992,1994. 25. Maturana and Varela, 1987. 26. Maturana, 1988b. 1997a, 1997b, 1998; Ruiz, 1996. 27. Maturana, 1988b, p. 41. 28. Q. mteson, 1972,1980,1991. 29. Sowers, 1993,1995,1997a, 1997b, 2000.  30. Postman 1995, p. 181. 31. Capra, 1996, p. 7.  29  SPACE WANTEB  HELP WANTED  "How can we break out Looking to share a space of our conventional with ecology. Interested approaches and imain what ecologically co- gine more productive herent forms of teaching a l t e r n a t i v e s ? " Reply and learning of math- to mailbox: T1I9M9M7S ematics could mean for the classroom. Can move in IMMEDIATELY. systems, It also 1  VACANCY Seeking one primary teacher to teach grades 2/3. Separate room. "Shared facilities".  group such as a mathematics class, involves the de- the school, and so velopment o f solu- on. (continued from page 36} tions for environThe connecting mental problems pattern or patterns O F EeOEdOT that will minimize that interrelate harmful s t r e s s on these systems toAccording t o M .the earth. It is with- gether as a dynamic C. Bateson , there in t h e systemic whole encompass are three "faces" or realm of ecology the forms of knowrealms o f ecology: where mathematics ledge, actions, and empirical, environ- t e a c h i n g a n d learn- identities that are mental, a n d system- ing can be most brought into being as ic. The author de- radically explored. a result of the onfines empirical eco- This is because sy- going interactions in logy as biological, stemic ttiisMiig fo- the system Cs) and m e t e o r o l o g i c a l ^ a n d cuses on s e e k i n g the ways in which geographical studies "the pattern which they are sustained that focus on un- connects" a system by the system (s). derstanding how the or systems together By focusing on planet is changing a s i a t e r d e p e a d e a t relational qualities, and how these and interacting ecological ways of changes affect the wholes. thinking give rise to interrelationships of In the field of viewing the world as the w o r l d s n a t u r a l mathematics educa- a a integrated whole; systems. The en- tion, a fluid, "system" a dynamic and vironmental face of could be an i n - network i n which a l l ecology is concerned dividual teacher or a living and. socialwith identifying the student. It could cultural systems are level of impact that a l s o be a e o l l e c t i w e interconnected. The our ways of living have om the earth's (continued on page 79) THE  3  F  A  C  E  S  8  3  ?  30  Settling In Thinking about how I (it's me, Jennifer!) might respond to this help wanted ad, I thought it best to 'bring it home as it were and invite Stigler and Hiebert's question into this ecological thinking space of mine. However, once inside, I soon realized that although this question certainly belonged in the realm of classroom mathematics, it was going to be difficult if not impossible for me to have an open conversation with it! Explained another way, it is like when you spot THE sofa in a furniture store but as soon as you get it home and put it into your living room, the sofa does not look so fabulous anymore. Instead, it is clearly out-of-place because it does not go with any of your existing furniture. For me, (and in spite of the authors' good intentions) this seemed to be the case with bringing this question home; neither the question nor the ecological space suited each other. You see, the manner in which the question is posed: How can we break OUt Of our conventional approaches and. imagine more productive alternatives? 1  puts forth for me as a teacher, an end-result' mindset of improving productivity in the mathematics classroom; the need to diSPOSC Of or discard teaching practices that are perceived to be Old or Commonplace and to aCQUire new teaching tOOlS so that we may increase or maximize children's learning of mathematics. Before one is able to think of possible ways to respond to this question, its linear, disconnecting posture has already "mapped" for us that manners of teaching are commoditizable "relationships", ones that we marry into and, if necessary, divorce ourselvesfrom.What is more, is that within the question's reductionistic confines, Stigler and Hiebert's query closes itself offfromthe opportunity for deep changes to  take  root You see, even if we changedfromone teaching style to another, radical shifts in the mathematics classroom would not likely occur if our thinking continued to be fashionedfromthe mechanistic pattern of productivity. The persistence of such a mindset is disabling in that it denies the very possibility of Stigler and Hiebert's query being one that provokes teachers to become more integrative and creative. 31  And because the authors' question docs not allow for an examination of the ecological coherence of the mathematics classroom, it creates the impossibility for mathematics teaching and learning to be conceived as holistic, organic, recursive, and co-emergent.  Determined to reconceptualize Stigler and Hiebert's query within an ecological realm, I set out tofinda question that would make sense in such a space. The question needed to be one that encouraged a systemic, ecological way of knowing: one that would allow for developing an understanding of how the 'layers' of our living are always and necessarily shaping how we teach mathematics. Instead of provoking a knee-jerk reactionfromthe reader or myself, I wanted this question to be taken as an open invitation to look deep and make visible what often remains invisible- the cognitive circles and cultural maps we lay down in our paths of teaching. Once brought to the surface, these experiences, beliefs, and values can be examined and assessed in terms of the forms of mathematics teaching that they enable or disable, the ways in which they become embedded within the mathematical language of the classroom, and the impact they have on how children come to know mathematics. Just then, a similar yet radically differently expressed question invited itself into my thinking:  Mem might we as teachers reconsider our conventional patterns of mathematics teaching? (Und &u doing ao, how can we He-seed learni spaces, that nurture and sustain children's mathematical growth?  Not only does this question share Stigler and Hiebert's concern for how mathematics teaching and learning takes place in the classroom, but it also makes sense within a systemically ecological thinking space. The question provides the necessary focal structure for the reconceptualization of mathematics teaching to occur while the ecological mind-space allows a place for such an exploration to unfold. Together they enable examination into how it is that what we know and who we are emerge and become our manners of teaching mathematics. Feeling as though I was beginning to settle into this new space of mine, I wondered what to do next. Where might one begin to create openings for ecologically minded ways of teaching and learning mathematics in the classroom? I found myself moving back and forth between reading and pondering Stigler and Hiebert's question and considering how to explore my ecocentric one. It was in the midst of this back and forthing that I realized both of these questions spoke of concern for this place we call "the mathematics classroom". It made sense for me then, that the place to begin was to begin with "place". 32  Notes i. Emphasis added, Stigler and Hiebert, 1997, p. 14.  Sense of place complex? We tend not to think so, mainly because our attachments to places, like the ease with which we usually sustain them, are unthinkingly taken for granted. As normally experienced, sense of place quite simply is, as natural and straightforward as our fondness for certain colours and culinary tastes, and the thought that it might be complicated, or even very interesting, seldom crosses our minds... 1  1  Basso, 1996, pp. xiii.  ith Bassos description of the ways in which we create and sustain our relationships with places brings forth just how strong our connections to place are. In doing so, hc reminds the reader how taken-for-granted, forgotten, unnoticed, or ignored the actual textures and patterns that make a place a place become. Primary, basic, and essential, sense of place is undeniably and always a critical part of every mathematics classroom. If we perceive the world as a place in which we live as systems within systems, then the mathematics classroom, as place, is not constituted simply by the presence of four walls, some furniture, a teacher, students, and mathematics. Necessarily, this place includes the relations that evolvefromthe intermingling of teacher, children, their surroundings, and mathematics. If we think about how we come to know places, then our sense of a mathematical place in the classroom emergesfromthe spaces in which we perceive mathematics to arise and the forms it takes on. Put another way, the classroom as a mathematical place and how we connect with it not only comesfromwhat we know and feel, but the kind of place it becomes grows out of the interactions we have with it. Thus, "[w]e do not define places-, they do not define us. Rather, in dynamic interplay, we come to form together" . As an elementary teacher, I have always been committed to a holistic way of thinking about mathematics in the classroom, one that facilitated my development in teaching mathematics and fostered children's mathematical learning. But it was not until recently after reading Basso's book, Wisdom sits in places, that I began to think ecologically about the mathematics classroom as "place". Moreover, as ecological ways of being are not an everyday practice in our society or its educational systems, it is understandable that one would not think of the mathematics classroom in such a manner, much less be able to imagine what an ecologically coherent mathematical place might mean. Sense of place and ideas associated with place do not come about naturally or consciously for us. As a result, they remain hidden or invisibly embedded within our taken-for-granted manners of teaching mathematics. Through my experiences in gaining a deeper understanding for how my teaching shapes the mathematics classroom and trying to create a sense of place that embodies ecological notions such as recursion, co-emergence, andfluidintegration, I have learned that this kind of work cannot be achieved by what we think may be "breaking out of conventional approaches". To do so in the attempt to get rid all that undermines an ecological sense of place in the mathematics classroom would be 1  2  35  naive and superficial. Systemic, ecological changes need to begin by first considering what place means to the mathematics classroom. It involves looking deep and engaging in the complex, recursive process of identifying and questioning one's taken-for-granted conventions of thinking about and teaching mathematicsasking ourselves how they contribute to the sense of place that exists in the mathematics classroom. Why is this so important? Does it REALLY matter? I think so. Let me explain by describing the different senses of place for mathematics that I have come to know as both a learner and a teacher, lliese vignettes chronicle my growing understanding of place. They provide revealing glimpses into how deeply mechanistic, commoditized, linear, and disconnected my common sense of place for the mathematics classroom was and the challenges I faced in making it into a cohesive whole. Notes  1. Basso, 1996; Camus, 1955; A. B. Davis, 1996. 2. A. B. Davis, 1996, p. 132  36  Reflections  Revealing (^lasstoom  the  ^ta$Ultq /Kathematlcs  It [mathematics] requires silence a n d neat rows a n d ramrod postures that imitate its exactitudes. It requires neither joy nor sadness, but a mood of detached inevitability: anyone could be here in my place and things would proceed identically. 1  1  Jardine, 1994, p. 109.  a Student'A  Place  tJjvtaugh my years, of elementary schooling., 3, grew, to believe t  spaces where mathematics existed in the classroom, the forms it assumed relationship, with it were clearly marked out by my teachers and had litt anything to do with myself or my peers. 3, came to know that at schoo teacher who makes, mathematics happen. Just lihe 5fO, programmers,, 3, w to myself, teachers always ensured that mathematics, began, ended, or exactly the same time each day. SJhey were conscientious, not to let mathe into any other programs of study, such as science, socials, art, or language only time when mathematics, did extend past its designated slot was after if we had not completed our exercises, during class. (Sur lessons were similar to that of learning to catch a ball. 5irst we watch the teacher demonstrate how, to do the mathematics, and then Head teacher would throw problems up, onto the chalMoard or to us in the fo texthooh. Scrambling to catch the mathematics, we would madly record th numbers and symbols in their correct linear fashion, practise, practise, prac then hopefully, toss, the mathematics, correctly bach to our teacher. Cn o we would await the moments when timed drills, pop, quizzes, and tests b stage where we performed our proficiency and ability to juggle addition, sub multiplication, and divisor 3, also learned that the teacher liked it best when mathematics, hap with other classmates but rather, silently in our heads, and figuring out s should not involve fingers, drawings, or counting! JVever questioning bu reproducing, this is how my teachers and we students busily created and m a place of anonymity for mathematics in the classroom.  39  Encountering familiar issues in a strange setting is like returning on a second circuit of a Mobius strip and coming to the experience from the opposite side. Seen from a contrasting point of view or seen suddenly through the eyes of an outsider, one's own familiar patterns can become accessible to choice and criticism. With yet another return, what seemed radically different is revealed as part of a common space. 1  M. C. Bateson, 1994, p. 31.  Slack on tAe Strip: My, 'Uniueuuiu tyew  thoujards the end of completing my undergraduate degree in educatio  University of Victoria, 3, was required to identify my teaching area of conc Stased en my interests,, it was a toss up, Between the visual arts, and m education. 3, found it difficult to choose one over the other and so, 3, cho that 3, felt was in the most need of rescuing. She two years of mathe education and mathematics courses that followed made for what 3, consid two more journeys around the Mohius strip,. She first trip,, which 3, hav described, was my childhood experiences, learning mathematics in scho second, puun the perspective of a teacher-to-Be, and the third, from a le mathematics, again is- what 3, describe far you now. during each of these vetums, 3, found myself questioning and shiftin conceptions of what it meant to teach and learn mathematics. 3iowever, it afterwards that 3, realized it was on these journeys that 3, was- visi revisiting the notion of mathematics' place in the classroom  41  Swwi 3JU Side of tfedageyy,  JVsow enrolled in mathematics education and approaching it not puu  of. a mathematicA Atudent but coming, at it puun the opposite side— one -- provided me with the contrasting point of view necessary to reveal ta granted conceptions J, held about classroom mathematics. Jt began in my first mathematics education doss. Jn this course whe focus was an the teaching of mathematicA, the professor engaged UA teachers through modeling possible ways to develop, children'A conceptual unders actually experiencing hands-on minds-en activities for ourselves, and ass student understanding through video analysis,. Jn doing so, this professor d many of my tahen-for-granted assumptions, about school mathematicA; ones included it as being an activity of simply "doing tasks, and solving pmB quickly in one's head" and that "mathematics can be best (earned in isola What became apparent was that as teachers we needed to be cre thinking about our manners of teaching but also in thinking creatively ab mathematicA itself. Storms, of teaching that communicated to students there ways to solve a problem, avoiding what he called "heavy-handed" teach implied teachiny-By-telling, and developing open-ended activities, such as games or riddles that draw children into the complexity of the mathematicA ins of repelling them puun it? became important foci in my growth as a te Moreover, this professor made me realize that, just as mathematicA shoul brought into being through the teacher, the children, and a variety of settin classroom, it was also important for teachers to enable learners to dev meaningful connections between the mathematics, they study at school mathematics that occurs in their daily activities at home* and in their comm learning puun the opposite side of the Mobius strips- puuntileperspectiv a teacher-to-be-- J, began to understand the impact that teachers have on of mathematics that arise in the classroom. (Ltd it was here that J, began co how J, might enact a pedagogy that embodied a sense of connectedne mathematics with the classroom 1  1  Motes 1. £iedthe,1995,p,.52. 2. £iedtke, 1995, p. 56. 42  3. fiedthe, 1996-7; £iedtae etal., 1998. 4. £kdtRe,2CCV. 5. SUnaldi, 1989.  SJhvuL JJme (hound AJJU  following September, 3, found myself taking, get another trip,  the Motius strip,. SJhis time again, from the perspective of a learner. Mere i my mathematics, courses, 3, experienced first hand, what it actually felt mathematics, took place in the dynamic ways, that the first professor had SJhese professors made it clear to us that we would neither be given nor memorize formulas or procedures. Seeling panicked, my first reaction wa had registered for the wrong mathematics classes— how on earth was 3, game if these professors were not going to show us which mathematics w toss back and forth? fortunately 3, persevered, continued to attend the classes, and for t time in all of my years of learning 3, became convinced that "teal" mathem not a game of catch but wither a something is brought into being. Suddenl mathematics no longer took place in an anonymous world but with the w human and natural contexts. We spent our time examining, questioning watching mathematical patterns emerge in different areas such as bio economics,, and everyday life, learning in this manner provoked and enabl explore, devise, and create self-generated methods and mathematical form in place.  44  Common Space  g encountering, familiar issues as M. C SkUeson describes and com  them from the opposite side(s,), we experience them as different or new recursively, upon examination and questioning what we think to be distin lihe the Mobius strip, we come to realize that the apparently disparate issue exist on separate planes but rather, exist within a common space. Sor me, 3, realized as a result of moving along what 3, perceived t completely different planes— My- experiences, of elementary mathematics mathematics education classes, and university mathematics courses, was existed within a common conceptual space. Whether my experiences were learner or teacher-to-be, they were all situated within the realm of sense of mathematics. Cis a beginning teacher, 3 did not enter the classroom with a fixed i my mind of mathematics, as an activity that consisted of teacher demonstra student reproductions, but instead, an image of mathematics as an on engagement in which children and their teacher "adventure" in a classroom knowing together. 5  45  SPACE WANTED  HELP WANTED  Looking to share a space "How c a n we b r e a k o u t conventional with ecology. Interested oa fp p r ooauc r hes and imain what ecologically co- g i n e more p r o d u c t i v e herent forms of teaching a l t e r n a t i v e s ? " R e p l y and learning of math- t o m a i l b o x : T1I9M9M7S ematics could mean for the classroom. Can move in IMMEDIATELY. systems. I t also 1  (continued  from page 36)  involves the velopment o f t i o n s  T  H  E  3  F  A  C  E  S  O F ECOLOGY  f o r  s o l u -  e n v i r o n -  mental t h a t  de-  p r o b l e m s  w i l l  m i n i m i z e  harmful stress on According to M . the earth. It is withG. Bateson , there in the systemic are three "faces" or r e a l m of ecology realms o f ecology: where mathematics empirical, environ- teaching and learnmental, and s y s t e m - i n g can be most ic. The author d e - radically explored. lines empirical eco- This i s because syl o g y as biological, stemic thinking f o meteorological, a n d c u s e s on seeking g e o g r a p h i c a l s t u d i e s "the pattern which w h i c h f o c u s on unconnects" a s y s t e m derstanding how the or systems t o g e t h e r planet is changing as interdependent and how these a n d interacting changes affect the w h o l e s . interrelationships o f f In the field of the world's natural mathematics educasystems. The en- tion, a "system" vironmental face of could be an inecology is concerned dividual teacher or a with identifying the student. It could level of impact that also be a collective our ways of living 3  8  h a v e  on  th©  e a r t h ' s  46  VACANCY Seeking one primary teacher to teach grades 2/3. Separate room. "Shared facilities".  group such as a mathematics class, the school, and so on. The connecting pattern or patterns t h a t interrelate these systems together as a dynamic whole encompass the forms of knowledge, actions, and identities that are brought into being as a  r e s u l t  of  the  on-  going interactions in the system (s) and the ways in which they are sustained by the system Cs). By focusing on relational qualities, ecological ways of thinking give rise to viewing the world as an integrated whole; a d y n a m i c a n d fluid network in which all living and socialcultural systems are interconnected. The (continued  on page 79)  Many, and perhaps most teachers begin their careers with the conviction that they will avoid those teaching practices that they found unhelpful or inappropriate when they were students However, ... most beginning teachers quickly find themselves settling into patterns of teaching that are strikingly similar to the ones they intended to avoid. 1  1  A. B. Davis, Sumara & Luce-Kapler, 2000, p. 41.  5 A v u u n #  JJU Sail:  *X Began teaching second and third grade children in ^Richmond, S Columbia. When £ entered the teaching field, £ distinctly, remember being one hand, to inspire a more connected sense of place for mathematics, classroom but on the other hand, careful not to become another "Mrs,. P Mrs,. Fibonacci (a storybook character) is, an elementary school teacher w and lives, math to such an extreme that she makes learning an unbearable for the children in her class, because everything turns, into a mathematica for them to solve, for the main character, learning mathematics, becomes he cannot escape:  "What if this keeps up for a whole year? How many minutes of math madness would that be?" "What's your problem" says my sister. "365 days x 24 hours x 60 minutes," I snarl. 2  £ihe all beginning teachers,, £ devoted enormous, amounts, of time to p my lessons,. £ Head journals, for mathematics, teachers,, went to workshop for new ideas,, collected 'Heal life' materials to connect the children's, ma with familiar contexts,, and designed interactive mathematical tasks, th engage every child in my class (all the while, being careful not to cast curses!). Skd even though the children, their parents, and my colleagues s be pleased with my efforts, £ did not feel as if £ was accomplishing what set out to do. My teaching and the children's mathematical learning still disconnected Motes. 1. Scies/dka and Smith, 1995. 2. Scieszka, and Smith, 1995, p~ 27.  48  What w e conserve, to conserve in  what w e w i s h owe Ivings, is  Insight, I believe, refers to that depth of understanding that comes by setting experiences, yours and mine, familiar and exotic, new and old, side by side, learning by letting them speak to one another. 1  what what  1  2  determines what can and cannot change i n o u r l i v e s 2  M . C . Bateson, 1994, p. 14. Maturana, 1997b, p. 5.  49  In the months that followed, I searched and scrutinized my mathematics program to find the source of my unease. I poured over the curriculum guides to be certain that I was teaching the correct concepts and the skills for the different grade levels. I revised the order and adjusted instructional sequences so that they moved more efficiently. I continued to tweak or elaborate the content of my lessons, depending on the needs of my students. Looking at the program as a whole, I felt that I was engaging the children in mathematical work that enabled their learning to be both "hands-on" and "minds-on" and that I was opening spaces where mathematics could be integrated with other subject areas. Unable to find any obvious problems, I continued to proceed along the current course. Then, several months later, I started to question the kinds of relationships that existed between my teaching and the children's learning of mathematics. I took a reflective step back and examined my mathematics program for a pattern or patterns that connected the children's mathematical learning spaces together as a whole. In doing so, taken-for-granted ways of teaching began to emerge. I discovered that these were not only rituals unique to myself, but surprisingly, they were matter-of-fact ways of being for my colleagues too... even those of my schoolteachers! For me, these teaching practices had simply become THE way to facilitate children's learning in the mathematics classroom. Intrigued with this discover}^, I decided to write my conventional manners of being down on paper. As I did this, it became apparent to me just how incredibly matter-of-fact they were and how deeply embedded in my teaching these "shared facilities" had become. 2  3  50  • B E F O R E  BEGINNING  THEIR  GRADE  G R O U P  A  •  DIFFERENT  W H E N  DIVIDE  P L A N N I N G T H E  T E N EQUAL  T H E  T E N  S C H O O L  •  A L W A Y S  •  MAKE  EVERYDAY  INFORMAL,  T E A C H I N G CHILD)  THAT  LESSONS. B E T W E E N  FOR T H E  SO,  A N D  TEACH  S C H O O L  SKILLS  TO  EACH  YEAR,  FOR  "ADDITION",  SIMPLY  EACH  Y O U C A N N O W ALLOCATE  TEACHING  O N E  TO "MULTIPLICATION",  S H O U L D  CONSISTENT S C H E D U L E  RECESS  CONCEPTS  CONCRETE  O F  TO RELATIONAL  BREAK  S H O U L D  STAGE  FACILITATE  CONSTRUCTION  PROCEDURAL  TO  MONTH  ACCORDING  TO  GRADE O N E O F  MONTH  A N D  SO  TO ON,  J U N E .  SURE  MATHEMATICAL  AN  M O N T H S  MATHEMATICS  (E.G.,  DOING  PROCEED  L E S S O N .  CONCEPTS  B Y  T H E CHILDREN  T H E N  T H E C U R R I C U L U M  A N O T H E R  T H E E N D O F  SORT  DONE,  MATHEMATICS  PARTS.  "SUBTRACTION",  TO  O N C E  MATHEMATICAL  INTO  UNTIL  A N Y L E S S O N ,  LEVEL.  TO T H E  A M O U N T S IT  IN  O F TIME  R E G U L A R L Y  A N D L U N C H  B E T A U G H T MORE  IN  EACH  ABSTRACT  ( T H E A  GIVEN DAY  HOUR).  SEQUENTIALLY;  FORMAL,  STUDENT'S  K N O W L E D G E  A R E  FROM ONES.  A U T O N O M O U S  C O N C E P T U A L  TO  ORDER.  After reading this "must do list", it was apparent that the sensibility of wholeness and flow that I desired for the mathematics classroom did not exist. Instead, was one that embodied rigid, mechanistic, and disconnected qualities. These could be seen, enacted in my scheduling of lessons at same time everyday, my "taking inventory" of the mathematics curricula and then "packaging" them up into discrete "units" of instruction, teaching separate grade-specific lessons, and my always doing so in a manner that proceeded from the concrete to the abstract. The kind of place that I had intended to root and the one that had actually become embedded were in contradiction to each other. What served as tried and true rituals for teaching mathematics had unthinkingly become that which was furthering the "cultivation of discrete parts without respect or responsibility for the whole" . M y teaching actions not only dismembered mathematics for the children but, on another level, I had also dismembered mathematics from itself. I say this because one might argue that my efforts to teach for the students' conceptual then procedural then relational knowledge could be viewed as in keeping with facilitating connected understandings. However, despite the fact that I did this in my teaching within each of the concepts and procedures, I was still teaching the concepts as separate "parts" and attention was not paid to enabling the students' connections among concepts, procedures or mathematical topics. One might alsoargue that put together, these individual "units" of instruction came to form a complete mathematics program. This might be true; however, the "units" still 4  51  were not fluid or dynamic but rather, discrete and static. In taking this reflective step back, I could see that it was not enough for me to design a well put-together mathematics program and I began wondering what I might do in order to engender a clear sense of flow in the mathematics classroom. Even though I could see how some of my invisible or assumed ways of teaching were undermining this, I did not know what kinds of "re-rooting" (conceptual or otherwise) were necessary. What I had learned however, was just as Basso describes, place is not something that can be taken-for-granted- not even in the mathematics classroom. Place is primary and basic yet at the same time, it is far more complex than had originally crossed my mind! If places are indeed created and sustained through interaction, then the mathematics classroom as place, only exists in being. Further, it can be said that what distinguishes one mathematics classroom from another is its sense of place. Together, it is the kinds of mathematics that emerge from one's teaching and from children's learning that become the defining textures and tones of a mathematics classroom.  Notes 1. Liedtke, 1995, p. 51. 2. G . Bateson, 1980,1991. 3. See "Vacancy" advertisement, p. 43. 4. Berry, 1983, p. 34.  52  EDUCATION  "JUST HAND THEM DOWN THE M A T H E MATICS" . . . OR NOT?! WHEN MATHEMATICS is imagined and. enacted as objectified, static knowledge that is to be traditionally passed down from one generation to the next, the teaching and learning of mathematics is disabled from ever becoming anything else. Under the air of "hand-me-downs", it is easy to understand why mathematics is taught and learned out of a sense of obligation or contempt rather than a sense of open desire or wonder and why, mathematics is all too often considered as that which is to be mastered rather than that which is to be understood. In com-  moditizing mathematics, we make absurd the possibility for us as teachers and to those who we teach mathematics to perceive it as anything else but a fixed and ina.nirn.ate entity. In this way of conceiving mathematics, we make it inconceivable for school mathematics to become something else other than just a collection of hand-me-downs. The embeddedness of these images within one's taken for granted ways of thinking about mathematics not only make it natural for us to assume mathematics to be an inanimate "thing", but in doing so, displaces mathematics as that which exists "out there". Given this mindset, it is not surprising why a teacher would feel impelled to set the class onto a straight and narrow, one-way course so that the students too, become collectors of mathematics. Given 53  this mindset, it makes sense to ingrain the ritualistic practice of "acquiring" mathematics into school unit and lesson plans, methods of assessment, and enact it in the classroom; product oriented practices which focus on "desired", "expected", or even "measurable" outcomes of instructionthat after instruction, the student will have "mastered" the mathematics taught in the lesson before "moving on" to the next part of the curricular course. Of course, the ways in which children are instructed to take possession of their mathematical handme-downs of concepts, skills, and even attitudes may vary. Still, "teaching by telling", engaging students in "hunting for", having them "seek out" "hidden" mathematics within "real" world contexts, and even "explorations" "designed" for children's discovery (continued on page 84)  (continued from page 8)  of mathematics are all examples of teaching and learning forms that keep alive, this tradition of "handing down" of mathematics. Moreover, when product-oriented ways of thinking about school mathematics are coupled with a "back to basics" mentality, the teaching and learning of mathematics become subjected to the weigh scale of "how much" in regard to the amount of mathematical facts and skills that children are to learn, and little or no emphasis is placed on such things as their mathematical thinking or understanding. Given this mindset, mathematical processes such as those identified by the National Council of Teachers of Mathematics as problem solving, reasoning, communicating, connecting, and representing would likely he deemed "not essential" by most teachers. If viewed as 1  "additional" knowledge, teaching that attends to children's development of mathematical processes would then depend on whether or not the children have acquired first, the prespecified mathematical facts and skills with which to "process" the mathematics. The point here is that when children are taught to learn mathematics in the tradition of hand-medowns and as a product oriented matter of collecting, hunting down, or retrieving pieces of knowledge, it creates the impossibility for mathematics to be taught and learned in ways that enable it to arise as living and animate. Now, identifying the limitations of how mathematics exists in the classroom and the possibility of it becoming something else is all fine and good. But doing so means that the conversation does not 2  end here. Rather, it opens up a whole host of questions that require further interrogation such as: • How can an ecological way of thinking help us to reconsider such taken-for-granted perceptions of classroom mathematics and reimagine a more responsive view for the teaching and learning of it to exist in the classroom? • What shifts in thinking become necessary in order to reimagine classroom mathematics as being something other than a line of hand-me-downs from teacher to child? • What could it mean if we assumed mathematics to be "embodied"? • How could mathematical problem solving, reasoning, communicating, connecting, and expressing be understood as something other than additional knowledge?  Notes  54  1. NCTM, 2000. 8. Baroody, 1993.  Part 2: Settling In(to AnOTHER Space) Jennifer picked up the newspaper and quickly leafed through it. Slowing down as she came towards the "Letters to the Editor" section, she saw that someone had responded to the "JUST HAND THEM MATHEMATICS ... OR NOT?!" article she had been reading.  DOWN  THE  MOVING THINKING SPACES AND REASSESSING OLD FURNITURE  IN RESPONSE TO  LAST  WEEK'S  ARTICLE: I TOTALLY  agree with the author's arguments and the questions are important ones in making positive changes to the math classroom. My concern though, is that real changes can't happen if this job of rethinking and "re-imagining" mathematics in the classroom is approached with the attitude of 'getting rid of or simply 'adding onto' what's already there! What the author didn't say was that it's not about taking ecological ways of thinking and coordinating them like new pieces of furniture into a tired and run down Uving room so that we can update our mind-spaces and have them look more current. It's about moving from invisible and mechanistic places of knowing to ecological ones. It's about recUscovering and assessing the all too familiar furnishings that have been set about (classroom) mathematics, and asking ourselves, "how well do these furnishings go with this space?" All for opening new spaces, Joel  "...getting rid of..." mumbled Jennifer as she read Joel's letter "...adding on to what already exists... no. Definitely not." And so she continued on, reading bits of the letter silently in her head and every so often, sputtering out particular words or phrases. "Precisely!" Jennifer said with matter of fact certainty. Enabling deep changes in her teaching was not about changing out of certain "approaches" and slipping into 55  new ones. She agreed with Joel that what she needed to do was to examine her mathematics teaching from where she was now (conceptually) standing. Jennifer wondered what she might see and see differently from an ecological perspective. What kinds of furnishings had become so comfortable and such an integral part of her mathematics teaching that they were now permanent and perhaps, invisible? Jennifer questioned whether they would even suit an ecological mind-space. And more to the point, she was anxious to know what kind of place, what kind of oikos she was "mapping" out for her students' mathematics. But where to start? Jennifer pondered for several days about the specific direction or vantage point she should position her thinking in order to examine these issues. It was only in doing so that she realized there was one theme that kept emerging. It was recursive in the sense that it was the "place" in her thinking, if you will, where Jennifer found herself returning again and again. Moreover, it was not until this moment that she recognized the ever-presence of this location. Here was the place where she existed both in and away from the classroom. When she described it to me, I immediately named it Jennifer's "in-between space". N o t because it was a space of indecision for her but, rather, the in-betweeness had to do with how her teaching and her research co-emerged and co-evolved. For Jennifer, teaching and research neither existed as separate entities nor did they move sequentially from one to the other as she had previously thought. "As a pre-service and a beginning teacher I understood research to be something that was done at the University that produced theory and in turn, became a tool that I could use in my practice of teaching mathematics." she explained. "But now, what comes to mind is an image of teaching and research as continually interacting with one another... they flow into and give rise to one another. This to me is R E A L teaching. It's praxis and not simply establishing and maintaining of one's teaching practice." As I listened to Jennifer describe how her view of mathematics teaching and research had changed, I realized that to characterize her in-between space as the location where the two met or intertwined would be to miss the meaning altogether. They did not meet. They were each other. It was clear that for Jennifer teaching mathematics and her study of it were inseparably interconnected. In a complex yet circular manner, Jennifer considered them to be interacting, coevolving systems- necessary parts of each other. Furthermore, the distinction she made regarding her shift from teaching as a practice to teaching as praxis reveals that mathematics teaching as praxis is not simply a routine that one performs but instead, requires active engagement with; it implies a way of being that is critically reflective and reflexively responsive. A n d evoked from within an ecological realm is the importance of being ever-mindful of how one's knowing, actions, and identity in 1  56  teaching mathematics arc firmly rooted in what wc have already lived and have become embodied in how we are living, and what we will live. In other words, teaching as praxis acknowledges the simultaneity and the complex circularity of that which unfolds from one's teaching is also necessarily enfolded with all that interacts with it. This space certainly was not an "in-between" space. It was not a conceptual space located somewhere in the middle of teaching and research. Really, it was a n O T H E R space. Jennifer added, "I see this kind of reflexivity as being key in attempting to understand how it is that my teaching and research give rise to each other" I then asked her, "If you had to describe in your own words, the 'guts' of it all in a nutshell, how might you do so?" "Simplifying the complex?! Hmm... let's see. I suppose I would have to say that in a nutshell... for me that is, teaching learners mathematics and learning what it means to teach mathematics flow together." And it was in this spirit and in this other place that Jennifer began the process of bringing her teaching into the foreground and encircling it within ecology.  Notes 1. The term "research" is meant to encompass both theoretical work and work done " i n the field/'  57  Jennifer naively assumed that by putting her teaching out in the open and inside the circle of ecology, immediate answers to her questions would be revealed. However, as the days passed, it was only her impatience that became apparent. Frustrated, she picked up a book that she had been reading and turned the page. There in black and white print was the reminder she needed. Letting myself get written by a place. Bodily scars as  the agelines in the droopy skin on the backs of my hands betray. Legibilities of having, once again, lived-through.  Sitting squat. Spending time. Waiting. Reserve. Quiet. Composure. Patience. Letting the boredom arrive. Wasting time. Doing nothing with great deliberateness. Collecting dry bones. Boredom: this is one great little demon we have banished from the discourse of authorship and expression and selfannunciation. Deliberately spending time in the old place, feeling through moist weaknesses: Perception of opportunities requires a sensitivity given through one's own wounds. Here, weakness provides the kind of hermetic, secret perception critical for adaptation to situations. The weak place serves to open US to what is in the air. We feel through our pores which way the wind blows. We turn with the wind; trimmers. An opportunity requires... a sense... which reveals the daimon of a situation. The daimon of a place in antiquity supposedly revealed what the place was good for, its special qualities and dangers. The daimon was thought to be a famiUaris of the  place. To know a situation, one needs to sense what lurks in it. (Hillman, 1987, p. .161)'  Although Jardine's description details how he readies himself to write, his practice of dwelling and "keeping watch" was exactly what Jennifer needed to do. It was obvious to her now that she did not know what aspects of her mathematics teaching needed interrogation and so to go searching for something that you do not know became aridiculousendeavour. Jennifer decided it best if she let her mind wander back to that "other" place. Dwelling there- in that place she described where teaching and learning and what it means to teach mathematics flowed together- she waited patiendy, all the while, keeping watch for what "lurked" in it.  1  Emphasis added. Jardine, 1999, p. 35.  58  Surfacing & Noticing So Jennifer, what came out? 9 wouldn't say that anything really CAME OUT... it was more like bubbles making their way to the surface of the water- and then bursting in the moment of recognition. you know, like, there's one! There's another one! Once you've caught sight of one of them all of a sudden, there they are! And you see them for the first time not because they magically appeared. They'd been there the whole time and were only invisible because you hadn't ever been able to notice them before. What do you mean? you hadn't ever been able to notice them before?! Well, even though my focus was on ecology and the mathematics up until this point, it wasn't specifically about MY teaching.  classroom,  Does it need to be? 9 think at some point it has to be. you see, it was only when 9 began questioning the kind of place 9 was creating for my students' learning of mathematics that 9 saw the need to move deeper— when 9 say deeper, 9 mean delving into the inner layers of my teaching- not just being attentive to what's developing in my present teaching, but what's already been developed, what's become its inner core or, the roots of my mathematics teaching. Had 9 not realized this, any growth 9 made would most likely be superficial because 9 wouldn't have been considering the whole of my teaching- 9 wouldn't have dwelt long enough to notice what was there. So like Joel had mentioned, my ecological ways of being would have been at best, "addons". 1  Okay. Just a minute, you figured out you needed to look at the layers of your teaching but you still hadn't figured out what you needed to be noticing, right? That's right. MORE dwelling! What 9 did know was that by exploring the layers of my teaching, 9 might be able to articulate why 9 was so uneasy about the sense of place 9 was creating for my students in the mathematics classroom... why it didn't feel right. tSut still 9 had no idea what 9 should be examining in my teaching. 59  So what did you do? Well, this time, 9 took the whole of my mathematics 9'd been- you know, into the space of language and  teaching back to where languaging.  Then what? As 9 surrounded my thinking with theoretical literature on languaging and the pervasive nature of language, 9 began by asking myself, So how do these concepts Inform my understanding of teaching mathematics? And... ? Certain key ideas began to pool together. 9n fact, they were direct from the books and articles 9'd been reading.  guotations  Such as? Well, like:  "Language T H I N K S US as we think within language."  2  "METAPH01 IS NOT A MERE EMBELLISHMENT; IT I S THE BASIC MEANS BY WHICH ABSTRACT THOUGHT IS MADE POSSIBLE." 3  Th e map is N O f tlie territory.  4  "...[language] does much more than construct concepts about the events and things in the world: it TELLS us what sorts of concepts we ought to construct." 5  And these "pooled" they were all... They all had to do with the metaphorical  together  because  nature of how we think. 60  So it was just works?  these  specific  authors'  yes and no. 9t was these four quotations that kept making their wag into mg mind-space of language and languaging but they weren't the only ideas 9 was thinking with. They'd emerged from a background of other authors'work such as Sfard , Abram , Capra ,M. C. bateson , van Manen , Jardine , Orr , Maturana andVarela , Lakoffand Johnson'".... but yes, it was these specific metavoices that really helped to pinpoint my position of noticing- so that 9 could begin to examine my mathematics teaching. 6  7  8  9  10  11  12  13  And so what was your position or perspective of noticing? Can you explain it tome? Let me see. Well for starters, it was directed towards the way in which metaphors become embodied in our forms of knowing, our actions, and our identities. Remember when we were talking about language?  Ufes. By "metaphorical language" 9 mean, the spoken, written, spatial (how phenomena are portrayed to exist), temporal (how time is conceptualized), and symbolic forms of communication that distinctively structure one's teaching. So briefly, there it is. My point of noticing was to examine the metaphorical "furnishings" if you will, of teaching mathematics and directly related to this, the kind of place 9 was bringing forth in the classroom. Do you think that metaphorical forms of communication can really impact one's mathematics teaching in such a profound manner? yes 9 do. Just take a moment to think about it: Theoretical "FOUNDATIONS", instructional "UNITS", conceptual "FRAMEWORKS", "NETWORKS", learning "JIGSAWS", cognitive "STRUCTURES", "SCAFFOLDING", 'BUILDING" knowledge... and so on. Metaphors. We are constantly reading them, hearing them, using them, and thinking with them. So usual they become like what Joel wrote, permanent fixtures in one's mind and over time, we no longer notice their presence. Unquestioned, these metaphors become embedded in our takenfor-granted languagelanguage used to conceptualize mathematics teaching and learning. Language that was directly impacting my teaching, my students' learning, and the kind of mathematics that was emerging. 61  Wait a minute. Qolng back to what you just said... uou claimed that metaphors were affecting the KIND of mathematics that was emerging. Surelg, math is math! Metaphors have nothing to do with the nature of mathematics. Oh but theg do! Mg previous mind-structure predetermined the mathematics content that was to be taught and learned. Because of this, 9 wasn't aware of or didn't pag ang attention to emergent kinds of mathematics , integrative kinds of mathematics , or individual-within-collective mathematics. 15  16  17  So what? Well, it does matter. 9f gou're not aware of how metaphorical language is thinking gou as gou think with it, it's difficult to understand how it functions and becomes an enabling and disabling feature in our manners of thinking and wags of being. Okag. So make sense of it for me at the classroom level. 9 still want to know what surfaced for gou when gou interrogated gour teaching. Let's go back to mg original guestion, what came out? What surfaced as a result of gour dwelling in this theoretical realm and In this other space of gours?! What came to the surface of my consciousness were metaphors that carried with them very vivid meanings of how 9 thought, taught, and Identified my role in the mathematics classroom. Jor the first time, 9 began to understand how UNconscious these metaphors were. What do gou mean? Well, because 9 hadn't realized how pervasive and taken-for-granted they were, the metaphors existed below the surface of my consciousness. They were definitely there but up until this point, 9 didn't have a theoretical way to examine my teaching and so, it was impossible for me to notice these metaphors. Theg were simply matter-of-fact ways of conceptualizing and enacting my teaching.  62  Jor example, how you MECHANISTICALLY separated and organized the mathematics content for each grade level into teachable UNITS of instruction?  ijes. And so it was these metaphors that you began rising to the surface?  embedded to notice  Exactly! And how it happened was just as 9 described for you at the start of our conversation— like bubbles making their way to the surface of the water and punctuated by them bursting as soon as 9 spotted them. What 9 also learned as 9searched to identify my taken-for-granted metaphors was that they arose from the theoretical languages of my undergraduate mathematics education and Ministry documents , teacher texts, and mathematics literature 9 was working with! Ujou see, it was here that the relationship between my activities in reading and writing and how 9 envisioned my work in the classroom... that is, planning, teaching, and assessing children's understanding of mathematics became clear. Almost instantly, the metaphors that'd been totally invisible were now so obvious. Because they were visible, 9 could see the metaphorical images embedded in everything from the way 9 imagined the mathematics class to my conceptualization of mathematics curricula. 18  9  What kinds of metaphors? examples.  Qive me some  63  "the  AUTONOMOUS child:  "CHILD-CFNTRFD approach"  to\e  is to  tea  CONSTRUCTED"  knowledge is  IS t  h  e  "V/T7ATOR-  unM»  ate  e  Hi 'it,  '4?  "PROCEDURAL understanding" "CONCEPTUAL understanding" "SCAFFOLDING" "MATHEMATICAL POWER"  u  Stage 1: Child's Language The nalural language a child uses lo describe the concepl tn a familiar situation, often a real-world story Modeling — | - -  Creating —|~* Shanng  <•>  CO  <  o  "INSTRUMENTA1 u  ^ri  hunderstanding"  F  !_LJ  u z O N A L  un J  ders Ending'  " F O R M A L understa  nding"  Stage 2: Material Language The new language thai might be used with concrete or ptctonal materials as a child acts out or represents the real-world slory Modeling --j-*-  Creating —j-—  Stage 3: Mathematical Language The use ot a tew words to record Ihe language that describes the action ol the materials. This stage leads to using more specific mathematical language Mode<>ng  Creating  O  Stage i  u  ^noMe< way of recording acnon  E o  Shanng  ~ p " Shanng~  Symbolic Language  The use of mathematical symbols as an even  MooV'nq  j-  Creat»ng —j—  Shanng  (Ft Reuille Irons & C J Irons, 1989)  64  mmmzfiics elms  ^or SB" €5 —  MATHEMATICS itself existed as a CONNECTED YET FIXED bodg of knowledge. 9t was made up of separate STRANDS of algebra, geometry, numbers and operations, measurement and so on.  instance,  MATHEMATICS  take  CLASS  As  the 9  imagined it, the class was composed of mgself and the students as AUTONOMOUS INDIVIDUALS Angthing outside the individual was considered to be part of the EXTERNAL environment.  1 "Sift* J-*  I VIANDS"  65  9 envisioned meaningful MATHEMATICAL LEARNING to occur in  a UNHHRECTIONAL and HIERARCHICAL MANNER  beginning at an informal, concrete stage and then moving towards HISHER levels of more formal, abstract stages of understanding.  MATHEMATICAL UNDERSTANDINGS were knowledge  STRUCTURES or FRAMEWORKS, Jor me, the bigger and more elaborately constructed the structure was, the better the individual's understanding.  mathematical understanding  66  Based on the premise that if knowledge was about building structures,TEACHING mathematics for me became an activity of directing mg students' thinking towards PREDETERMINED learning outcomes in regards to what they SHOULD know and facilitating the ways in which they sto^CONSTRUCT such knowledge.  teaching mathematics mm  fTUDINTS OM A COfflSE towards PRHEfBUHNEB learning OUTCOMES  curricula And as well CURRICULA were JIGSAW PUZZLES for teachers to assemble by piecing together concepts and skills set out by the Ministry and other STANDARD mathematics documents.  ty  e  jigsaw puzzles  67  9 agree that these are very strong metaphors... mechanistic, linear, and hierarchical ones to be sure. But did they really affect the way you taught mathematics? and if so, how? Having made them visible, 9 also asked myself this same question: Were these metaphors simply figures of speech or were they more than that? Were these metaphors truly powerful forms of language? Language that not only shapes how one perceives mathematics teaching and learning to be but also, profoundly impacts how such events COME to be. So 9 turned my attention— a little apprehensively, 9 must admit, towards examining if and how, these metaphors 9 had identified existed in my forms of teaching. And....?! Rather abruptly 9 came face to face with the notion that all knowing really is doing and all doing really is being! And what's more, how unthinkingly natural it all is. Ljou see, because 9 viewed mathematics to be a connected yet fixed body of knowledge and curricula were puzzles to be assembled, my goal in creating an integrated maths program was to connect the different pieces of mathematics together to produce a "logical" and "coherent" picture for the students. Sin thinking so, it made sense for me to insert their lessons in- in a piecemeal fashion for an hour each day between recess and lunch. And in keeping with the view that mathematical learning was sequential and hierarchical, because 9 taught a multi-age class, it made it necessary to sort the children according to their grade level and teach two different lessons. My role as their teacher was to guide each student's learning in a manner that enabled them to construct sturdy frameworks of understanding; ones that began with concrete foundations of experiential knowings upon which more formal, symbolic representations were built. 9 even remember being asked on several occasions as to how 9 defined myself as teacher! And, how did you describe yourself? 9 was the children's FACILITATOR... THE initiator of learning opportunities stay with the metaphors, the provider of building materials.  or to  After all of this, what was your reaction in realizing that your teaching was indeed enactions of the mechanistic linear, and 68  hierarchical metaphors thinking with?  that you  were  9f you'd asked me this question before finding this all out 9 would've said that 9'd probably be shocked, disappointed... even horrified if 9 were to discover that my teaching contradicted the ecological perspective that 9 thought 9 was embracing. What do you mean? Ljou weren't shocked, disappointed, or horrified? IZeally. 9'd think that burying one's head in the sand so to speak, would be a common reaction. Strangely enough, it was more of an affirmation... finally being able to recognize the metaphors that had existed for so long beneath the surface of my consciousness and then to see the embodiment of them in my teachingrelief through affirmation... yes, that's what it was. Jorsome time, 9'd had a hunch... a gut feeling that the connected sense of place 9 was trying to create in the math classroom wasn't quite there... but 9 couldn't put my finger on it as to why. But you had, hadn't you? Ljou had identified rituals in your teaching that were linear, mechanistic, and hierarchical? Sure, 9 was able to point to teaching actions that 9'd unthinkingly inherited and see them as problematic... such as planning a program by dividing and ordering the mathematical concepts and skills for each grade into a September through June sequence! But doing so only indicated forms of teaching 9 deemed as undesirable. 9t still didn't provide me with any kind of understanding as to what was giving rise to them or how 9 might go about creating a more ecological sense of place for my teaching and my students' learning. ijes, that's right. 9 agree. 9t was only when 9 moved deeper into my teaching and examined metaphors... Hmm.... how can 9 describe the process to you....  my  Jor me, this process was very much like "[fjingering the contaminated wound" — explicating my metaphors and then watching them fester —how the metaphors were being enacted in my mathematics teaching, ijes, that's an accurate image. 20  69  How horrible that sounds! The image certainly impresses mg experience as incredibly uncomfortableeven painful. And in some ways, it was. finding out that you re doing exactlg the opposite of what gou are trying to do is definitely a distressing, uncomfortable mind-space to be in... but at the same time and in a different way, 9 by no means considered the study of teaching as pathologic. Although that's how 9 think most people would interpret your description. 9 know... but no. To come to this place in my thinking was critical. The uncomfortableness of it all was not a prompt for identifging and remedying a problem so much as it was an opening for me to arrive at a new place of understanding. Ljou see, 9 considered the dis-ease of these events to be integral and vital to mg growth in teaching mathematics. 9t was because of dwelling in this mind-space that 9 was able to grab hold of what 9 could only before express as being a hunch or a gut feeling and now, 9 was able to actuallg put words to it and finallg sag THERE it is! Making the invisible visible! When you describe your metaphors and explain how they gave rise to your forms of teaching, it really elucidates the point you've been trying to make; that the metaphors and metaphorical patterns with which we think have everything to do with one's teaching of mathematics and the sense of place that is created in the classroom. What also becomes clear is that even though you wanted to create a connected sense of place through creating an integrated mathematics program, the metaphors gou unconsciously rooted in your mind critically disabled the possibility for a more organic or ecological kind of integrative mathematics to emerge. The metaphors onlg allowed for mechanical piecemeal forms of teaching and learningdefinitely not those that are dynamic, flowing, or unpredictably open. 9t's exactly as you expressed earlier in our last conversationthat 70  teaching learners mathematics and learning what it means to teach mathematics really do flow together. That all said, going back to Joel's question, now that you'd figured out what furniture" didn't suit your ecological "living room", how did you go about finding furnishings that would?  /Votes 1. Aoki, 1983; 1993; MacDonald and Purpel, 1987; lAhrmacher, 1997. 2. Qraduate seminar by C. A. Bowers, July, 1998. Paraphrased Heidegger, 1962, p. 188-203. 3. Emphasis added, Lakoff and Nunez, 2000, p. 39.  from  4. q. Bateson, 1972.  5. Emphasis added, Postman, 1995, p. 181. 6. Sfard, 1997,1998 7. Abram, 1996, 8. eapra, 1993; 1996. 9. M. C. Bateson, 1994,1996,2000. 10. vanManen, 1986,1991. 11. Jardine, 1994,1999,2000. 12. Orr, 1992,1994. 13. Maturana 1988a, 1988b, 1991,1995, 1997a, 1997b, 1998,2001; Maturana and Varela 1980, l987;Varela, 1999. 74. Cakoff and Johnson, 1980,1999. 15. for example, see page 220,233, and 322. 16. jor example, see page 236 and 250. 17. Jor examples, see page 155 and280. 18. Jor example, Ministry of Education, 1990. As well, up until 1995, each subject area was published by the Ministry as a separate curriculum document. 19. That is, texts and literature which 9 considered as being situated within "constructivism". 20. Jardine, 1999, p. 37.  71  Even Deeper  Enactions exposed, Jennifer realized that the metaphors she had rooted in her thinking had become her mathematics teaching. It was impossible for her to consider them as merely figures of theoretical speech. Differently, she now understood them to be the "consensual domains" in which her patterns of thinking and forms of teaching mathematics were specified. In a very real way, these metaphors and metaphorical manners were her rituals for placemaking in the classroom. Jennifer knew that her taken-for-granted ways of teaching mathematics were not engendering the ecological and fluid forms she wished to enact. Even so, she still felt a sense of awkwardness. Jennifer had arrived at a new place of knowing in the other space. It was in these moments of making sense of the limitations of her metaphors and knowing that she wanted to enact ecologically coherent ones that she was also confronted with the fact that one cannot simply change by "exchanging" what one is thinking or doing in the classroom for something else.  72  I  AGAIN 3'm reminded that learning what it means ta teach mathematics is not an automatic process. 3t's not smooth, it's not straightforward, and it certainly doesn't apypcar on demand. KEEPING WATCH WHILE DWELLING REQUIRES PATIENCE.  Before Jennifer could begin to re-imagine metaphors that were ecologicaly sound and work towards rooting them within her classroom praxis, not only did she need to exercise a mindful kind patience but she also needed to move even deeper into that oth space. She had to criticaly question, assess, and then provoke shifts in her thinking. This included an inquiry into mathematics and mathematical understanding.  73  sHow is mathematics conceived? jAnd in doin$ so, what kinds oj> bein$ does it become?  74  PROCLAIMED THE QM££fA£ 0?SCfflHpES PRISTINE IN ITS SACRED CUSTOMS OF PRECISENESS' AND LINEARITY... MATHEMATICS IS ABSTRACTLY ELOQUENT AND REFINED.. IT OFFERS US TRUTHS SO ABSOLUTE SO PURE SO UNAMBIGUOUS.. IN ITS PREDICTABILITY AND CERTAINTY AND EXPLICIT INFALLIBILITY. IT IS NOT T O BE QUESTIONED NOR HELD ACCOUNTABLE FOR ANYTHING EXCEPT ITSELF. TREASURED HEIRLOOMS OF LOGIC AND RATIONAL KNOWLEDGE... UNIVERSAL AND TRANSCENDENT... MATHEMATICS EXISTS IN THE REALM OF OBJECTIVITY... IT IS NEUTRAL AND LIVES WITHOUT REGARD FOR US... OUR BELIEFS... OUR VALUES... OUR ACTIONS... OUR CULTURAL WAYS . . . IT LIVES "OUT THERE" AS THE FOUNDATIONS OF THE UNIVERSE  75  ORDERING AND STRUCTURING THE UNIVERSE  the flowers the snowflakes the ferns and the trees aeatlve  and  beautiful  the stars and the planets the shells of snails the orbits us  looted in 044/1  76  Lakoft & Nunez, 2000; Wheeler, 1967. ' Devlin, 1994,2000. Devlin, 2000, p. 92. * Bunnell, 2001; Maturana, 1988b. A. B. Davis, 1995,1996, 2001; Jardine, 1994; Lakoff & Nunez, 2000. 1  3  5  77  ^Z^oes  the  understanding shape learners  ones to  waif  one  matter? perception understand  potttaxfs ^And ofi  mathematical i-fc so,  what  how it  mathematics?  does  means  it -foot  Theoretical Portraits of IMathematical Understanding  79  HP i h e topic of mathematical understanding continues to be one of critical focus for mathematics educators. As a result, there exists an array of models and interpretations that address aspects of mathematical understanding from the very general to the very specific. Two themes inherent in this particular collection of works are that of cohesion and tension. Interestingly, there is a general agreement among mathematics educators of what "good" mathematical understanding entails, while at the same time, the ways in which educators portray the nature of mathematical understanding, how it comes to be or should be developed, and the forms that arise create a contrast against one another. First, several works from perspectives situated within what can be considered to be part of a constructivist realm are showcased. Here, one will get a senseforwhat it means toframemathematical thinking and learning within tills theoretical discourse and how it portrays understanding as the building and rebuilding of mental schemas. Second, research that seeks to move away from linear or constructivist minded frameworks in order to interpret children's mathematical understanding as more holistic and dynamic are explored. Finally, works that are located within an enactive realm and that strive to illuminate mathematical understanding as being a coemergently complex phenomenon are examined and discussed.  80  EXAMINING MATHEMATICAL UNDERSTANDING  Model of Intelligence and Forms of Mathematical Understanding Skemp's (1979) model of intelligence offers a qualitative means for describing individuals' mathematical understanding. The two-level, cybernetic model (see Figure 1) consists of two internal systems: delta-one and delta-two. Delta-one is defined by Skemp as a sensori-motor system that directs an individual's physical mathematical actions based on information received from the external environment. Delta-two serves as the site where construction and reconstruction of an individual's mental mathematical schemas take place. It is this process of schema construction and reconstruction that allows for the mathematicalfimctioningof delta-one to occur. Thus, it is "the construction and testing by delta-two within delta-one of the schemas and plans that delta-one must have to do its job" (Skemp, 1979, p. 44). It is here in delta-two where Skemp identifies mathematical understanding as developing. The specific ways in which these internal systems function together is described by Skemp (1978, 1979) as evidenced through one's "instrumental", "relational", and "formal" or, "logical" forms of mathematical understanding.  E  N ACTION  2  INFORMATION  ] ACTION  '  /  *• —— L  li  \1  INFORMATION  •——  N  ~—M  E H T  Figure 1. Skemp's two-level cybernetic model of intelligence.  82  Instrumental Understanding Skemp explains instrumental understanding as being a function of delta-one. This form of mathematical understanding enables a person to correctly apply previously learned procedures to the solving of mathematical problems. Instrumental understanding in essence, is the learning of "what to do" with the mathematics. It does not however, enable the learner to develop conceptual grasps for interpreting why a method works or what the symbols might mean. (Skemp, 1978,1979). For example, by remembering the words and the order of the letters in the acronym, "BODMAS" (Brackets O f Division, Multiplication, Addition, Subtraction), a person can carry out the correct sequence of numerical operations for solving complicated calculations. The way in which learners are able to develop instrumental mathematical understanding is through rote methods of demonstration and further practising of a particular procedure or set of skills until they become routine. Although the cognitive structures in delta-two that result from instrumental learning enable a person to manipulate mathematical symbols and rules, the persons actions remain restricted because the connections that are formed in the delta-two schemas exist only as relationships between symbols and rules, not among mathematical concepts. The extent to which one is able to apply one's instrumental understanding to different mathematical contexts then remains limited to combining and performing procedures in the prescribed sequence that they were learned. Relational  Understanding  Relational understanding is evidenced by a person who is able to generate appropriate strategies for solving mathematical problems. This form of understanding involves the individual making sense of why particular methods of mathematics may work and why others may not be effective when solving certain problems (Skemp, 1978,  1979). In other words, relational  understanding implies the learners knowing of "what to do" and "why" certain mathematical actions prove to be effective. The manipulation of mathematical concepts and schemas is described by Skemp as a function of delta-one while the individual's conscious or unconscious  83  reflection of these concepts and schemas takes place in delta-two. Unlike instrumental understanding, relational understanding gives rise to schemas that connect mathematical concepts with procedures. This form of learning is thought to develop as an individual alternates between activities of interacting mathematically in the external environment and mentally reflecting on these experiences. As relational understanding can only be achieved through the individual's conceptual integration of mathematics, tins process requires more time than does instrumental learning through rote methods. However once acquired, relational mathematical understanding is seen to be more flexible because such knowledge is connected to mathematical concepts and not to specific contexts, it can continue to develop. And unlike instrumental learning where an individual recognizes a mathematical problem and then applies and performs a prescribed procedure to solve for it, an individual with relational understanding can derive mathematical procedures through conceptualizing or comprehending the task at hand. Mathematical symbols do not exist simply as abstract objects on which an individual performs actions but rather, they carry meaning for the individual in that the symbols are objects to which conceptual understanding can be attached and enable the construction of connected schemas of concepts and skills.  Formal or Logical Understanding Skemp (1979) characterizes this form of understanding to be present when an individual consciously connects symbolic mathematical language together with meaningful ideas and logical reasoning. This can occur as either a delta-one activity or in both delta-one and delta-two. If a person possesses delta-one logical understanding, the learner is able to reflect on his or her mathematical actions through an "if... then" type of rationalization; that is, "if I perform the correct methods to solve a given problem, then the result should be correct" type of thinking. On the other hand, Skemp describes logical understanding that takes place in both deltas as being when an individual is able to show through formal mathematical demonstration or proof that the mathematics that has been applied makes sense through inferences that connect the given  84  premises of the problems to established mathematical axioms or theorems. This type of formal functioning that occurs within delta-two enables the individual to become aware of the connections between delta-one and delta-two activities and establishes consistency between the individual's mathematical schemas and solutions. Last, within each of these three forms of understanding- instrumental, relational, and logical-formal- there can be "intuitive" and "reflective" dimensions to the learners mental functioning (Skemp, 1979). Intuitive mathematical functions are characterized as spontaneous processes that occur in delta-one and do not necessarily include the delta-two system. When intuition occurs in both deltas, this gives rise to the unconsciousreflectionof die individual. However, it is only when the individual is consciously aware of his or her activities in both the first and second deltas that this process can be considered reflective.  Hiebert's Views On Mathematical Understanding Hiebert and Wearne (1992,1996) apply a constructivist definition found within cognitive psychology (Brownell, 1935-, R. B. Davis, 1984; Hiebert & Carpenter, 1992; Lesh, Post, & Behr, 1987) to define their view of what they consider to be mathematical understanding. They refer to it as the learners development of mental connections and formation of networks that serve as representations of mathematical ideas. For example, Hiebert and Wearne would consider a wellconnected understanding of multi-digit addition to be a network that consists of the child's connected knowledge of concepts regarding place value, basic facts, and the ability to generate effective procedures to deal with the task at hand. They believe the process by which understanding of mathematical ideas occurs is an unpredictable, recurrent, and nonlinear progression. Furthermore, the flexibility of an individual's mathematical understanding is seen as an indication that the learner has constructed mental networks that have many points for external information to enter and to trigger the individual's successful adaptation, acquisition, and retrieval regarding appropriate strategies to solve mathematical problems.  85  Building Bridges to Connect Informal and Symbolic Mathematics with Student Understanding Hiebert's independent and collaborative research seeks to understand the relationships that exist between children's conceptual understanding of mathematical concepts and their external abilities to recall and modify existing procedures, construct suitable methods, adopt prescribed rules, and use symbolic mathematical language with understanding (Hiebert, 1989; Hiebert & Carpenter, 1992; Hiebert et al., 1996; Hiebert & Wearne, 1993, 1996). A common thread that runs through Hiebert's research concerns itself with previous studies (Carpenter, Hiebert, & Moser, 1983; Lindquist, Carpenter, Silver, & Matthews, 1983; Hiebert & Wearne, 1986) and similar arguments that are raised by other mathematics educators such as Usiskin (T996),  Pimm (1987), and Carraher, Carraher, & Schliemann (1987) regarding students' lack of  connection between symbolic mathematics found in the classroom and that which occurs in their everyday life. His work emphasizes the need for less formal mathematical representations to serve as a means by which meaningful connections for children's understanding and application of symbolic mathematics can be developed. As well, Hiebert asserts that in order for students to be able to use the symbolic language of mathematics to their advantage, essential connections regarding their informal, experiential knowledge must be recognized by teachers and made explicit to their students (Hiebert, 1989). Hiebert's framework (1989) (see Figure 2), which identifies three critical sites for linking written symbols with understanding, highlights the necessity for children to develop meanings for the ways in which symbolic mathematics can be used as a powerful language in solving problems. This model makes clear the need for students to "make the symbols work for them" instead of "working with the symbols". Stressing the importance for the learner's utilization and integration of out-of-school mathematical behaviours" with school mathematics, Hiebert's model clearly identifies that the final stage of mathematical understanding should not be the learner's ability to perform symbolic mathematics but rather, the child's meaningful understanding for and their ability to reintegrate their use of symbolic mathematics into a variety of settings. 1  For example, interpreting, judging, devising, estimating, and evaluating.  86  IJnkirig Written Symbols with Understandings Site i: IntEtpretarion and Development ofMeaningforSymbols What do these symbols mean? What am I being asked to do or find? This site focuses attention on the written symbols and the ideas or objects that they represent: • numerals as representing quantities (e.g.. 5 km or 5 apples) • operations (ie. addition, subtraction, multiplication, and division) as actions on quantities in the natural world • signs as describing relationships between or among quantities (e.g.. =, <, and>)  SitE z: Developing MeaningforRules Establishing what to do and why This site includes the use of manipulatives when introducing rules or procedures, as an important step in illustrating how a rule works and connects the symbolic answer to the concrete solution.  Site 3: Prefacing an Answer Making an estimation Taking action (ie. applying the chosen procedure) Examining the solution based on previous estimation Apply or relate the symbolic problem back into an informal context (e.g.. "Would your answer hold true when put back into a real-world context?)  Figure 2. Summary of Hiebert's (1989) three sites for linking symbolic mathematics with understanding.  Problemetizing Children's Learning of Mathematics Teaching methods considered to foster rich, connected schemas of mathematical understanding are ones that enable children to "problemetize" their mathematics (Hiebert et al., 1996,  1997). Hiebert et al. distinguish problemetizing mathematics as being different from  problem solving approaches to learning that imply teacher demonstration and children's imitation of identifying key words in a problem, selecting an appropriate method, and performing prescribed calculations to solve a task. Rather, problemetizing of mathematics is viewed as facilitating deep mathematical understanding because it focuses on students making sense of and developing meaningful relationships between their mathematical ideas and  87  mathematical actions. Hiebert et al. (1997) explain this approach to learning mathematics as elucidating: ...reflective inquiry as the key to integrating ideas and actions. Problematic situations, and methods of inquiry used to resolve them, elicit ideas and actions. This is what distinguishes problemetizing from traditional problem solving in wliich an acquired procedure is applied, (p. 24) Within these types of mathematics lessons, the teacher strives to structure learning opportunities that are not only interesting to the students but also introduce to them important mathematics. Students are expected to make sense of the mathematics and methods they employ through discussions led by the teacher that interrogate the effectiveness of particular methods, as well as exploring different ways of representing their understandings through written, verbal, objects, pictorial, symbolic, and informal means of mathematical language (Hiebert et al., 1996, Hiebert & Wearne, 1992, 1993, 1996). As students seek to resolve problematic situations such as determining the difference between 72 and 39, the teachers facilitating them into into actively generating, adopting, or reflecting on mathematical strategies and ideas allows the students' learning to be "tasks, and discussions... [which] connect with where students are and that are likely to leave an important mathematical residue" (Hiebert et al., 1996, p. 17).  The Construction of Mathematical Understanding R. B. Davis (1984) explains mathematical understanding in a manner similar to Minsky and Papert's (1972) view-, that mathematical understanding is present when an individual is able to integrate a new idea into a larger structure of previously constructed ideas. R. B. Davis (1992) uses the metaphor of assembling a jigsaw puzzle to illustrate his view: ...that one assembles ideas in one's mind much as one assembles a jig-saw puzzle.  88  Each new candidate piece, like each new idea, can be used only if it fits into the aggregate of pieces that have previously been assembled, (p. 228) R. B . Davis (1992) states that if we consider mathematical proofs or even Skemp's (1978, 1979) "reflective, logical" understanding to be the results of mathematical activity, then mathematical understanding must be taken to be the result of children's working with mathematics. In this way, R. B . Davis (1992) considers mathematics to be a result of children's understandings: Instead of starting with mathematical ideas, and then applying them, [teachers] should start with problems or tasks, and as a result of working on these problems the children would be left with a residue of mathematics... that mathematics is what you have left over after you have worked on problems, (p. 237)  The Teaching of Mathematics R. B . Davis poses a similar argument to one found in Hiebert et al.'s works (1996; 1997) that stresses that rather than teaching children mathematics through methods of showing and telling, connected understanding can only develop when children have established for themselves a reason for doing mathematics. Solving tasks in this manner provides opportunities for students to decide whether they will employ already established methods or construct mathematical procedures on their own. This is explained below: Instead of telling students what to do, and leaving them wondering about why one does it this way, the new approach helps students understand the task or the goal, and gives students the responsibility for inventing ways to solve the problem. (R. B . Davis, 1992, p. 238) In examining the role of the mathematics teacher, R. B . Davis and Vinner (1986) claim that if we believe students "build up" their mathematical schemas through constructing and reconstructing ideas based on their previous experiences, then mathematics teachers play an integral role in the students' learning. O n the other hand, the teacher's instructional actions  89  cannot be viewed as those that ultimately determine the ways in which students form their mathematical schemas. Mathematical schemas according, to their view, are assumed to be constructed from and always influenced by the child's previous mathematical experiences. R, B. Davis and Vinner (1986) raise another issue with respect to student errors in mathematics. They assert that student errors should not necessarily be considered an indicator of lack of understanding but could be, in fact, the student's retrieval or selection of an inappropriate mathematical idea. Since teachers cannot determine what mathematics a child will or will not choose to retrieve, this becomes a responsibility of students to be aware of their mathematical understanding. R. B. Davis and Vinner encourage learning settings that engage students in nonroutine mathematical problems (Schoenfeld, 1985; Silver, 1994) as a way for learners to further construct their mathematical understanding and learn skills in monitoring their mathematical actions (R. B. Davis, 1984; R. B. Davis & Vinner, 1986).  Mathematical Ambiguities R. B. Davis and Vinner (1986) identify five sources within students' school and out-ofschool experiences that can obscure students' conceptual understanding in mathematics. These are as follows: the language of mathematics, assembling mathematical representationsfrompremathematicalfragments,building mathematical concepts, the impact of specific examples, and children's misinterpretation of mathematical experiences. They argue that teachers should not try to exclude ideas from contexts outside of mathematics because these, as all other mental representations, serve as a necessary parts in children's assembly of pre-mathematical ideas (R. B . Davis, 1984; Lewin, 1986). So in a manner similar to Sierpinska's argument for understanding the importance of epistemological obstacles and Dubinsky's method of genetic decomposition for mathematics instruction, R B . Davis and Vinner advise that we should not attempt to prevent children from developing mathematical misconceptions but rather, enable them to  90  become aware of misconceptions in their thinking and how overcoming them is necessary in being able to make sense of mathematics.  Mathematics as a, Language In North America, the English language is viewed as the linguistic base with which students enter school with and from which they begin to build mathematical ideas. The English language is also identified as a source of many difficulties in terms of students' mathematical understanding (R. B . Davis, 1984-, R. B . Davis & Vinner, 1986). Pimm (1987) explores the possible reasons for this confusion. Pimm (1987) puts forth the notion that just as English has specific ways in which it functions as a language, mathematics too possesses its own linguistic register and has "a set of meanings that is appropriate to a particular function of language, together with the words and structures that express these meanings" (p. 75). H e explains that within the mathematics register there are "specialist terms" or, words that hold specific meanings in the context or discipline in which they are functioning. Durkin and Shire (1991) refer to these specialist terms as "lexical ambiguities" of mathematical language and make further distinctions between these words by classifying them into four subcategories- "homonymy" \ "polysemy" , "homophony" , and "shifts 3  4  in applications".  5  A n example illustrating this difference between English and mathematics linguistic registers is observed in the use of the word "any". In ordinary everyday contexts, this word is Homonymy describes words that have the same form as in English but imply different meanings in mathematics. For example, the word "leaves" does not signify "leaves* on a tree or the verb "to leave", but rather, describes die subtractive action in mathematics. Polysemy characterizes mathematical words that may have two or more different but related meanings to their English definitions. For example, die word "product" in English, can be denned as "something that has been made", and in mathematics, takes on a similar meaning, "a quantity obtained by multiplication". Homophony is defined as two or more distinct words that have identical pronunciation but entirely different meanings- as observed in the words "two", "too", and "to", or "sum" and "some", or "pi* and "pie". Shifts in applications are similar to what Pimm (1987) describes as "notational metaphors* and these are mathematical symbols that in combination with other symbols, convey particular meanings. Here we can see that the number "5* can be applied in mathematics to communicate the nominal meaning of "the numberfive",the ordinal meaning of "the fifth number*, the cardinal meaning of "1,2,3,4,5*, or the visual representation of "5". For an in depth discussion regarding these lexical ambiguities, please see Durkin & Shire (1991). 2  3  4  5  91  most often taken to mean some' yet in mathematics, this word implies 'every' such as, "is any odd number prime?" Interpeting this question in a nonmathematical manner, one's answer would most certainly be yes, as seen in the case of the number five or seven. However, comprehending the word as meaning the latter, one would have to answer no, as not all odd numbers such as nine, have only factors of one and itself. A second example is located in our use of number words, that function in mathematics not only as adjectives as in "one" house, but can also exist as nouns such as when we speak of prime "numbers", implying that numbers have distinct qualities. Moreover, given the mathematical fact that "four fours are sixteen", number words also operate as adjectives and nouns.  Building Mathematical Concepts, Specific Examples, and Students' Mismterpretations Metaphorical usage of English words in mathematics is evident in elementary school when children learn to "cany" when regrouping numbers in addition, to "boiTOw" when renaming numbers in subtraction, or making reference to the "face" when identifying surfaces of 3-D objects. These metaphors serve as tools for students to think and build images about mathematical ideas and concepts (Pimm, 1985, 1987). In other words, they are "functioning images... which [can] 6  connect the ideas of mathematics with objects and processes that [students] feel they know and understand" (Pimm, 1987, p.97). Pimm cautions teachers that while metaphors are valuable tools in helping children conceptualize mathematical ideas, it is necessary for teachers to help students to define the usefulness of a metaphor by exploring it in many different contexts. By doing so students can develop an understanding of where and when their use of metaphors is appropriate and when it may be a mathematical act of over generalization. In the same sense this does not mean that given linguistic ambiguities in mathematics, teachers should try to teach for all possible meanings or misconceptions that may arise when Pimm (1985,1987) identifies three types of metaphors existing in mathematics: "structural", "idiosyncratic", and "standard* or "conventional* metaphors. Structural metaphors refer to the ways in which symbols are arranged together and take on different meanings. Idiosyncratic metaphors are metaphors invented by the user to make sense of a mathematical concept or idea. Standard or conventional metaphors are used and understood by many people. Examples include "having" for positive numbers, "owing* for negative numbers, a mathematical function is a "machine", or an equation is a "balance* 6  92  students learn a particular mathematical concept (R. B. Davis & Vinner, 1986). Rather, it is to be expected that at different stages of a child's learning, some aspects of mathematical concepts will be fully developed while other aspects may be partially explored or not at all. Besides metaphorical overgeneralization, there will be other times when children take specific instances and construct generalizations from them. This phenomenon, that R. B. Davis and Vinner (1986) see as impacting on students' mathematical understanding, can be observed when students learn about the multiplication of whole numbers and form the conclusion that multiplication of any numbers always produces a greater number. While they argue that it is fine for teachers to let learners maintain their partial understanding of a concept as long as the contexts in which the students are applying it is appropriate, R. B. Davis and Vinner also stress the need for teachers to provide settings that provoke children to engage in reconstructing their understanding of a concept. In the example of multiplication, reconstruction of the concept would become necessary when the children begin to work with decimals, fractions, and negative numbers. Finally, R. B. Davis and Vinner (1986) explain that because of the ambiguities that exist in mathematics students may misinterpret what mathematics is being taught and thus, teachers need to be cognizant that children's focus on unnecessary or extraneous aspects of a given concept can also lead to mathematical misconceptions.  Herscovics and Bergeron's Analytic Framework In their effort towards enabling teachers to teach for children's mathematical understanding, Herscovics and Bergeron (1981, 1982, 1988a, 1988b; Ilerscovics, 1989) developed their analytical framework (see Figure 3), that has been used to describe key characteristics of particular mathematical concepts such as 'number' (Herscovics & Bergeron, 1988b;  Herscovics, Bergeron, & Bergeron, 1986a, 1986b), length' and 'surface area' (Heraud,  1988),  and algebraic concepts such as 'slope' (Dionne & Boukhssimi, 1988). Herscovics and  Bergeron (1988a) assert that the development of individuals' conceptual understanding  93  should always begin in their physical, concrete world. They advocate for teacher practices to be those that value not only children's written answers but place an equal emphasis on children's thinking processes. Herscovics and Bergeron consider their framework to be a tool that can aid in epistemological analysis of mathematical concepts. Moreover, by accounting for the different components of Herscovics and Bergeron's model when planning instruction for a particular mathematical concept, teachers can design and provide richer mathematical learning settings.  UNDERSTANDING OF PRELIMINARY PHYSICAL CONCEPT imuflivB  L Logico-physical  undQfstandirtgT P '  o c e d  " ' r a  oglco-physical " ^abstraction L  UNDERSTANDING OF EMERGING MATHEMATICAL CONCEPT Figure 3. Analytic framework of mathematical processes (Herscovics, 1989). The model is divided into two partially sequential but non-hierarchical tiers. Herscovics and Bergeron's two-tiered model conceptualizes mathematical understanding as being a partially sequential process that begins first with the individual's intuitive understanding of physical concepts and then develops through a series of levels into an abstract, mathematical concept. The arrows within the model indicate that forms of logico-physical and logico-mathematical abstraction are generated from the individual's preliminary physical concepts. The other arrows show that an individual's understanding of a mathematical concept does not require all three parts within the first tier. This assertion is supported by Herscovics and Bergeron's (1988b) observations of young kindergarten children who are seen to have mastered counting procedures and the formalization of the concept of number but have not yet comprehended all the invariancesregarding quantity and rank. 94  The first tier is identified as "understanding of preliminary physical concepts" and consists of three distinct components of understanding; "intuitive", logico-physical procedures", and "logico-physical abstraction". Here, intuitive understanding is based on the individuals visual perception,  that  provides  non-numerical  approximations.  Logico-physical procedural  understanding is evidenced by an individual's ability to relate his or her intuitive knowledge through 'physically acting out' mathematical concepts. They describe logico-physical abstraction to be when an individual synthesizes and constructs meaningful relationships such as reversibility or generalizations between physical mathematical concepts (Herscovics, 1989; Herscovics & Bergeron, 1981,1982,1983,1984,1988a). The second tier encompasses another three components- logico-mathematical procedural understanding",  "logico-mathematical abstraction",  and "formalization", that  Herscovics and Bergeron consider to be integral parts of comprehending mathematical concepts (Herscovics, 1989; Herscovics & Bergeron, 1981, 1982, 1983, 1984, 1988a). Herscovics and Bergeron (1988a) define procedural understanding to be when a learner relates preliminary physical concepts that underpin logico-mathematical procedures, such as counting methods for determining quantity or rank by using them appropriately in a given context (Bergeron, Herscovics, Bergeron, 1986; Herscovics et. al, 1986a). Logico-mathematical abstraction refers to the individual's construction of connecting logico-mathematical invariants together with related logico-physical invariants to form generalizations, such as coming to know that the commutativity of addition as a property applies to all pairs of natural numbers (e.g., 4+3 and 3+4 both equal 7) (I Ierscovics, Bergeron, & Bergeron, 1986b). Finally, formalization is characterized by Herscovics and Bergeron as an individual's activity of axiomatizing and producing mathematical proofs. At an elementary level, children's discovery of axioms andfindinglogical mathematical justifications would be taken as indicative of formalization. They also consider formalization to include the enclosing of a mathematical notion into a formal definition as  95  well as the use of mathematical symbolization for such notations. This type of formalization of procedural understanding such as counting can be observed when a child writes out a sequence of digits.  Mathematical Understanding as a Taxonomy Mathematics educators Peggand Currie(i998) agree with Piagetian views that assume older children learn in a qualitatively better way than do younger children because they have more developed mental structures. A t the same time however, Pegg and Currie take a different stance with respect to observing and analyzing students' mathematical understanding. They support the view put forth by Biggs and Collis (1982) as well as other researchers (Blake, 1978; Hallam, 1967) that different methods rather than ones that generalize students' academic performances based on Piagetian cognitive developmental stages are necessary in order to provide detailed descriptions regarding students' learning within discipline-specific contexts. The prestructural, unistructural, multistructural, relational, and extended abstract levels in Biggs and Collis' S O L O (Structure of the Observed Learning Outcome) taxonomy are described as being "isomorphic to, but logically distinct from, the stages of preoperational, early concrete, middle concrete, concrete generalization, and formal operational, respectively" (Biggs & Collis, 1982, p. 31). There are also four dimensions within each of the five levels that are used to further categorize student responses. They are as follows: working memory capacity, operations relating task content with cue or question and response, and general, overall structure (see Figure 4). In keeping with Piagetian models, that focus on hypothetical cognitive structures ( H C S ) , S O L O also forms a concrete to abstract framework. In contrast, unlike H C S , that characterizes the individual in terms of age and stage of development, the S O L O taxonomy does not attempt to describe the learner, but rather, the quality of the learner's  response(s)  within a specific context and, in terms of the theory's levels and dimensions. B y adapting elements from Biggs and Collis' (1982) theoretical framework of the S O L O taxonomy,  96  Pegg and Currie assert that in this way, they are able to analyze students' mathematical understandings in a more detailed manner than allowed by Piaget's developmental stages.  AppUcarion of the SOLO Taxonomy to the Analysis of Students' Mathematical I Tnderstanding Just as Biggs and Collis created the S O L O taxonomy because they deemed Piaget's developmental stages as not appropriate for looking at children's understandings, mathematics educators Pegg and Currie (1998) integrate elements of S O L O to further elaborate on the van Hiele theory (van Hiele, 1986; van Hiele-Geldof, 1984) (see Figure 5) in order to analyze children's geometric understanding. Pegg and Curries main criticism concerning the van Hiele model of geometric thought is that the model "cannot address questions posed outside of the direct notions of properties of figures, class inclusion, and deduction about which the theory is explicit" (1998, p. 334-335). What is common to the S O L O taxonomy and the van Hiele model is that they both derive from Piagetian roots. The difference between the two models is  4 Rfspttnxv Structure Developmental base Mayo with minimal  SOLO  1  Relating  Consistency  aye  description  Capacity  operation  and closure  Extended Abstract  l-ormal Operations (16+  years!  Maximal: cue +  Dcduclton and  Inconsistencies  relevant data -t-  induction. Can  resolved. No felt  interrelations  generalize 10  need to give closed  situations not  decisions—conclusions  experienced  held open, or qualified  + hypotheses  Cut  Res/wnsc X X  ^  to allow logically possible alternatives. ( R „ R . . or R.)  Concrete Generalization (1.1-15 years)  Relational  High: cue +  Induciion. Can  No inconsistency within  relevant data -r  generalize within  the given system, but since  intcrrelations  given or exper-  closure is unique so incon-  ienced context  sistencies may occur when  using related  he goes outside the system  X X  aspects  0 0 0  97  Middle Concrete  Multistructural  (10-12 years)  Medium: cue +  Can "generalize"  X  Although has a feeling  isolated  only in terms of  for consistency can be  X  relevant data  a few limited  inconsistent because  X  and independent  closes too soon on basis  aspects  of isolated fixations on different conclusions  •  with same data  O  data, and so can come to  Early Concrete  .Unistructural  (7-9 years)  Ltw: cue +  Can "generalize"  one relevant  only in terms  datum  of one aspect  o o X  No felt need for consistency, thus  X  closes too quickly:  X  •  jumps to conclusions on one aspect, and inconsistent  j,  Denial, tautology,  No felt need for  ~  and response  transduction.  consistency.  confused  Bound to  Closes without even  specifics  seeing the problem  so can be very  Prc-opcralional  Preslructural  (4-6 years)  Minimal: cue  •• o  R  c o  X  •• o o o  " Kinds of data used: X = irrelevant or inappropriate: •  = related and given in display: O = related and hypothetical, not given.  Figure 4. Biggs and Collis' (1982) five levels and brief descriptions of each level of the S O L O Taxonomy and the taxonomy's loose correspondence with Piaget's stages of cognitive development. that by integrating S O L O into the van Hiele model, geometric understanding changes from being conceptualized as whether or not student has "mastered" levels of understanding (Pegg & *Davey,  1998), to categorizing  students' responses in a polychotomous manner whereby answers  can be grouped with similar characteristics and reflect various stages of cognitive growth (Pegg & Currie,  1998).  Given the descriptions above, Pegg incorporates notions of response levels drawn from the S O L O taxonomy (Biggs & Collis, 1982)  to the van Hiele theory and, by doing so,  elaborates on the level descriptors to enable more inclusive criteria against which to compare and 98  Level 1: Figures are identified according to their overall appearance. Properties play no explicit role in this identification process. Level 2: Figures are identified in terms of properties and are considered to be independent of one another. *Level 2 A : Figures are identified in terms of one single property, such as the length of sides of a figure. *Level 2B: Several properties are identified but exist in isolation of one another. Level 3: Relationship between previously identified properties of a geometric figure are now established. *The student is able to order the properties so that one or more properties give rise or imply other properties. Level 4: Deduction is understood and students can develop mathematical proofs.  Figure 5 Pegg's (1997) description of four levels of the van Hiele theory. (*) indicate S O L O derived adaptations made to the van Hiele theory. 7  analyze students' geometric understanding. Specifically, Pegg  (1998)  (1997,1998) and Pegg  and Davey  focus on the elaboration of the second and third levels regarding the van Hiele model.  There is a confusing linguistic mismatching of the levels between the van Hiele model and Pegg's (1997) revision of it. The van Hiele model begins with the basic level of geometric understanding and continues onto levels one through four. Pegg's level one are assumed to be an elaboration to that ofvan Hide's basic level. Pegg's level 2A and 2B are taken to be complementary to level one in the van Hiele model. As well, Pegg's level 3 and level 4 are understood as being elaborations of the second and third levels of the van Hiele model. 7  99  Here, unistructural responses and multistructural responses are considered to be within the 9  concrete symbolic mode are associated with level 2 thinking (Pegg & Currie, 1998). Level 3 10  responses are evidenced when students can generate an overview of, or identify the important elements within a task in order to form an appropriate generalization. Here, relational" responses of concrete symbolic mode are deemed by Pegg and Currie (1998) as characteristic of level 3 thinking. This particular model of geometric thought has been used with a variety of learners that ranges from primary students looking at basic two-dimensional shapes (Whitland & Pegg,  1999), to secondary (Currie & Pegg, 1997) as well as pre-service primary teachers (Lawrie, 1996).  Mathematical Understanding as Overcoming Epistemologjcal Obstacles Understanding as Both a Process and an Act Sierpinska (1990) examines what it means for mathematical understanding to be  conceptualized as both a process and an act. She agrees that "[understanding is achieved slowly along with the accumulation of properties of objects, examples and development of concepts concerning relations between classes of concepts" cited (Lindsay & Norman, 1984, p. 438). In this manner, Sierpinska views mathematical understanding to be the process by which an individual's constant construction and reconstruction of ideas and meanings results in the Next to prestructural, unistructural responses are described by Biggs and Collis (1991) as being the second most concrete form of understandings. A unistructural response involves the learner only having to comprehend the given task or question by relating the question with a response that incorporates one of the conceptsfoundwithin the problem. For example, given a picture of two equilateral triangles and asked to respond in terms of what is the same about the twofigures,a unistructural response could be that bothfigureshave three corners, three corners being one of many possible similarities. 8  Multistructural responses are defined by Biggs and Collis (1991) as when an individual is able to focus on two or more relevant concepts at one time. While considered to be a more sophisticated level of understanding than that of unistructural responses, a second characteristic of multistructural responses is evidenced by the individual's comprehension of the concepts as being separate and not as related ideas. An example found within this level of understanding could be a student who identifies, given a picture of the two equilateral triangles, that both triangles have three comers each and that each of the triangles' three sides are of equal length. 9  Described by Biggs and Collis (1982) as being when individuals are capable of using and learning symbol systems. Typically, this level of functioning takes place in late primary through secondary school years and requires the individual to be able to internalize and generate representations of objects and events as words or images. "The relational response requires, in addition (to accessing a number of concepts), an overview of relevant concepts while being able to monitor the process or task from beginning to end, thus allowing for a logically complete conclusion* (Pegg & Currie, 1998, p. 337-338) 10  11  100  establishment of connections between mathematical concepts. Sierpinska (1990) also draws on Ricoeur's (1989) notion regarding the dialectic nature of the process of understanding and as acts within a process. Sierpinska (1990) adds that although she agrees with Ricoeur's general idea of the dialectic between an individual's understanding and explaining as "starting with a guess and developing through consecutive validations and modification of the guess" (p. 26), it is difficult to directly apply this model to the comprehension of mathematical concepts. To do so would necessitate the individuals experience in working through a variety of situations, "because the understanding of a concept is not normally reached through reading a single text. It demands being involved in certain activities, problem situations, dialogues and discussions, and the interpretation of many different texts" (Sierpinska, 1990, p. 26). Sierpinska integrates the ideas of both Lindsay and Norman, and Ricoeur, to develop the notion for mathematical understanding to exist as a process and act of constant consttTiction, generalization, and resynthesis of ideas and relationships between concepts through a spiraling "process" of dialectic interpretation.  Processes of understanding are seen as lattices of acts of understanding linked by various reasonings  (explanations, validations) and  a  (relatively) 'good'  understanding of a given mathematical situation (concept, theory, problem) is said to be achieved if the process of understanding contained a certain number of especially significant acts, namely acts of overcoming obstacles specific to that mathematical situation. (Emphasis added, Sierpinska, 1994, xiv)  A HistDricf>empirical Approach Sierpinska (1987,1990,1994) explains that in the act of understanding mathematics, new ways of knowing are established. Distinguishing her theoretical work as being different from other models of mathematical understanding that focus on "levels of understanding" (Herscovics & Bergeron, 1988b; Pirie & Kieren, 1989; van Hiele, 1986), "cognitive structures" (Dubinsky & Lewin, 1986; Lesh, Landau, & Hamilton, 1983), or the "dialectic coupling of procedural 101  and relational forms of understanding" (Sfard, 2000; Skemp, 1978,1979), Sierpinska classifies her research as being that of a historico-empirical approach. This approach examines students' understanding of mathematics from a perspective that focuses on the "obstacles to understanding encountered both in the history of the development of mathematics and in today's students." (Sierpinska, 1994, p. 120): [F]rom the point of view of mathematics education, what is interesting are [sic)]exactly these 'accelerations and regressions' and 'epistemological gaps', as well as epistemological obstacles' and difficulties because it is assumed that to learn is to overcome a difficulty. That an equilibrium has to be finally attainedthis [sic] is taken as a banality; the problem is that without first destabilizing the student's cognitive structures no process of equilibration will ever occur, i.e., no learning of something radically new will ever occur. (Sierpinska, 1994, p. 121) Sierpinska argues that in order to improve students' mathematical understanding, teaching should focus on intervenrtions that help students overcome epistemological obstacles. So we must introduce the students into new problem situations and expect all kinds of difficulties, misunderstandings and obstacles to emerge and it is our main task as teachers to help the students in overcoming these, in becoming aware of the differences; then the students will perhaps be able to make the necessary reorganizations. (Sierpinska, 1994, p. 122) Below are specific forms of knowing that Sierpinska sees as impacting on children's mathematical understanding.  Epistemological Obstacles Sierpinska asserts that it is through the examination of students' acts of understanding that we can interpret thinking processes and epistemological obstacles' that are involved in 1  students' construction of meaning regarding mathematical concepts. She makes the point that Sierpinska applies Bachelard's (1975,1983) notion of "epistemological obstacles" to describe an individual's unconscious ways of knowing or understanding that constrain their ability to think about mathematical concepts in general, elaborated, or more abstract ways. 12  102  although specific methods of measuring students' acts of understanding need to be developed, strategies for teachers to engage students in confronting and overcoming epistemological obstacles also need to be generated. [Ijnstead of trying to replace the students' 'wrong' knowledge by the 'correct' one, the teacher's effort should be invested into negotiations of meanings with the students, invention of special challenging problems in which a student would experience a mental conflict that would bring to his or her awareness that his or her way of understanding is probably not the only possible one, that it is not universal. (Sierpinska, 1994, xii) Furthermore, the partial ordering of a learner's acts of understanding would enable the student's depth of mathematical understanding to be compared against criteria and it could be measured in terms of the number and quality of the acts of understanding demonstrated. As well, the number of epistemological obstacles an individual may need to overcome could then be identified. Sierpinska (1990) argues that unlike intuitive knowledge, that she describes as "irresistible and certain", rational knowledge in mathematics is acquired through the individual's exercise of rigour and attention. Interrogating and synthesizing r^rspectives of understanding from Locke  (1985), Dewey (1988), and Hoyles (1986), Sierpinska (1990) generates four categories or acts of conceptual mathematical understanding that she deems as necessary for students to experience and use in their studies of mathematics. They are as follows: "identification", "discrimination", "generalization", and "synthesis". Her subsequent work (Sierpinska, 1994) deals with the elaboration of these categories whereby she integrates Vygotski's (1987) theory of intellectual operations. By doing so, Sierpinska forms a more detailed framework that provides descriptions for the acts and processes involved in students' development of mathematical concepts and the types of epistemological obstacles that may occur.  103  The genesis of concepts in a child, according to Vygotski, is the genesis of his or her intellectual operations such as generalization, identification of features of objects, their comparison and differentiation, and synthesis of thoughts in the form of systems. The very same operations lie at the foundations of understanding.. .. The various geneticformsof these operations, discovered and described by Vygotski, seemed to provide, almost immediately, the possible genetic forms of understanding. Moreover, the theory can be used to explain some of the curious ways in which students understand mathematical notions, and why, at certain stages of their construction of these notions, they simply cannot understand in a different or more elaborate or more abstract way. (Sierpinska, 1994, p. 142-143)  Sierpinska (1994) distinguishes two key tenets within this idea of epistemological obstacles and how they affect student understanding of mathematics. First, cognition is not seen as an accumulative process but, instead, requires the individual's reflection on past mathematical actions in order for their reconstruction of understanding to occur. It is assumed then, that some form of integration and reorganization is required by the individual in order for his or her way of knowing or understanding to move from one level to another. The second assumption is that an individual must rebuild fundamental understandings that give rise to different philosophical considerations in order to overcome an epistemological obstacle. With this process of rebuilding, Sierpinska adds that new knowings can give rise to future epistemological obstacles through our awareness that an obstacle or obstacles exist in our mathematical understanding or, as a result of the resolution of differences. Therefore, obstacles can be viewed as being positive in the sense that we are able to overcome them or, negative in the sense that we acquire them. [W]e must note that something (a belief, a scheme of thinking) functions as an obstacle often only because either one is unaware of it, or because one does not  104  question it, treating it as dogma. Overcoming an obstacle does not mean switching to another system of beliefs or another persistent and believed universal scheme of thinking but rather in changing the status of these things to 'one possible way of seeing things', 'one possible attitude', or 'a locally valid method of approaching problems' etc. (Sierpinska, 1994, p.  125)  Sierpinska (1994) makes it clear to the reader that, unlike Vygotski's genetic forms of intellectual operations that are chronologically developmental stages, she distinguishes the four categories as coexisting with one another and to be thought of as stages that one progresses through in childhood and adulthood. So even if an adolescent or an adult was confronted with a new mathematical concept, it would be likely that the individual would be working with a low level of conceptual understanding of generalization and synthesis or perhaps with a vague discrimination between the relevant and the irrelevant features of that particular concept. Moreover, Sierpinska (1994) makes the argument that: It seems that one cannot sensibly speak of epistemological obstacles in children before they reach the age of conceptual thinking. Things went easier with the younger children because they did not have to  overcome  epistemological  obstacles. The epistemological obstacles still remained to be constructed, (p. 158)  Mental Operations: Identification and Discrimination When an individual begins to identify features of objects and can distinguish them as being either more or less significant in view of some generalization, this can be considered to be a more elaborate form of mental operation (Sierpinska, 1990, 1994). This is illustrated in the following example: ... at some point in the process of understanding the topic of equations at the highschool level, the student must identify the simultaneous occurrence of variables  105  and the equal sign as features characteristic of equations before he or she starts to conceptually think of equations as equality conditions on variables. (Sierpinska,  1994, p. 151) Chai n-Complexes Sierpinska uses Vygotski's term, "chain-complexes" to describe naive generalizations that precede one's development of conceptual mathematical understanding. This type of understanding can be observed in settings that involve actions of sorting or categorizing. Sierpinska (1994) explains that a chain-complex occurs when "a child... adding objects or pictures of objects to a given model, focuses on the last object added and is satisfied with any link between the new object and this last one, disrespectful of any contradiction that may occur with regard to the previously added objects" (p. 147). She uses an example from Vygotski's research to elucidate this for the reader: ... the child may select several objects having corners or angles when a yellow triangle is presented as a model. Then, at some point, a blue object is selected and we find that the child subsequently begins to select other blue objects that may be circles or semicircles. The child then moves on to a new feature and begins to select more circular objects. In the formation of the chained complex, we find these kinds of transitions from one feature to another. (Vygotski, 1987, p. 139)  A second characteristic of chain-complexes is that they usually take place when an individual is developing an understanding for a mathematical concept that involves the notion of equity. In this case, it is not that the individual considers all the attributes of an object as being of equal significance, but rather, that the individual is not able to stay focused on one particular feature for any considerable length of time. "At one moment it can be, for example, the colour, at another, the shape" (Sierpinska, 1994, p. 149). Here it is not possible for a student to abstract common features  identified from  different  contexts or to synthesize them into a  106  mathematical concept because the individuals' processes of understanding and the actual object of their understanding is constantly undergoing change.  Pseudo-concept of generalization When an individual becomes "aware of the non-essentiality of some assumption, or of the possibility of extending the range of applications" (Sierpinska, 1990, p. T50), this act of mathematical understanding is described by Sierpinska as "generalization". An obstacle that can occur within this category is a "pseudo-concept of generalization" (Sierpinska, 1993, 1994; Sierpinska & Viwegier, 1989), such as when a child identifies geometric shapes according to arbitrary colours so that any object resembling green pattern blocks would be considered squares. This is taken to be a pseudo-concept of generalization because the child is making a generalization but not the mathematical one- that all squares are four-sided polygons. This epistemological obstacle is different from that of a chain-complex because the learner's way of understanding serves as a more holistic or general manner of thinking about mathematical concepts and does not change from situation to situation. Furthermore, due to the pervasive nature of epistemological obstacles, they cannot be easily abandoned nor replaced without considerable reorganization of one's mathematical understanding.  On Abstraction When one is able to maintain one's thinking about the same single feature in order to move beyond "complexization" (Vygotski, 1987) and towards the stage of generalization, one also moves closer towards what Sierpinska refers to as true conceptualization. This is preceded by an intermediary phase that she identifies as "potential concepts" (Sierpinska, 1994). Potential concepts are considered as such because it is possible for the individual to develop an abstract understanding of a mathematical concept once the individual is able to abstract the underlying idea or ideas that is at the core of the concrete, factual, or contextual situations.  The Operation of Synthesis: Conceptual Thinking The formation of a mathematical concept requires the individual to be able to synthesize 107  features of that concept into a coherent whole. Being able to do so implies that the student can construct mathematical relations between two or more properties, facts, or objects (Sierpinska,  1990, 1994). Sierpinska's research into epistemological obstacles and the role that they play in students' struggle to generate and understand mathematical concepts makes it clear to the reader that students cannot achieve or demonstrate conceptual, mathematical thinking through methods that assume learning by telling. Moreover: Concepts cannot be given to the child, ready made, in the verbalized form or symbolic representation. The child has to construct them as generalizations of his or her previous generalizations and it is quite natural diat the adolescent's first concepts may bear little resemblance to the fully fledge ones developed by generalizations made by mathematicians in their adult, mature, and often genius lives. And thus they become obstacles to understanding the theories. (Sierpinska, 1994, p. 159)  Mathematical Understanding as APOS via Reflective Abstraction and Genetic Decomposition The three main areas of research that Dubinsky has explored concern the manners in which individuals construct mental schemas (Cottrill et al., 1996; Dubinsky, 1992a, 1992b), the role of reflective abstraction (Cottrill et al., 1996; Dubinsky, 1992b), and the interrelationship between visual and analytic strategies (Zazkis, Dubinsky, & Dautermann, 1996) in students' development of mathematical concepts. In his collaboration with Cottrill et al. (1996), Dubinsky maintains a view that deep mathematical understanding is characterized by "an individual's tendency to respond, in a social context, to a perceived problem situation by constructing, reconstructing, and organizing in her or his mind, mathematical processes and objects that deal with the situation.* Cotrill et al. (1996) argue that mathematics cannot be regarded as a set of static concepts that can be passively acquired by students but rather, sound mathematical understanding necessitates students' active struggle in constructing and reconstructing their own mathematical thinking- their schemas. They make the contention that by not addressing In keeping with Piagetian views, Cotrill et al. characterize effective mathematical knowledge as being the successful adaptation and accommodation of an individual's schemas. * (p. 171). 1 3  108  students' incorrect conceptions, teachers reinforce by further embedding the students' misconceptions into their mathematical schemas. Cotrill et al. (1996) describe mathematical knowledge as being a spiraling cycle in which an individual's reflection on mathematical actions, processes, and objects are integrated together to produce mental schemas or networks. Dubinsky (1992b) and Cottrill et al. (T996) refer to this cyclical process as the APOS" theory (see 4  Figure 6).  APOS Theory Actions Dubinsky (1992b) and Cottrill et al. (1996) explain mathematical actions as "any physical  5  or mental transformation of [mathematical] objects to obtain other [mathematical] objects" (Cottrill et al., 1996, p. 171). These actions can consist of one response or a sequence of connected responses that occur when an individual reacts to a perceived external event. Further still, it is when the individual reflects on his or her mathematical action(s) that their action(s) become a process.  Processes Cotrill et al. (1996) define a mathematical process to be: . . . a transformation of an object (or objects) that has the important characteristic that the individual is in control of... in the sense that he or she is able to describe, or reflect on, all of the steps in the transformation without necessarily performing them. (p. 171) Once constructed, a process can be manipulated and combined with other mathematical processes. For example, once an individual understands that "three add four makes seven", this understanding can be reversed and connected to the process of subtraction, "seven take away four makes three". It is these manipulations together with the individual's reflecting on the APOS is an acronym for "actions, processes, objects, schemas*. An example of a physical action could be a student recording or manipulating a mathematical calculation onto paper. A mental action on the other hand, could be a student recalling some mathematical fact such as 6 + 6 = 12 from memory. 1 4  15  109  REFLECTIVE  ABSTRACTION  fnteriorlzaiiori  OBJECTS  PROCESSES  / ' /  ./Coordination Reversal  Generalization Figure 6. Dubinskys (1992) model for the cyclical nature of mathematical actions, processes, objects, and schemas. mathematics at hand that give rise to new processes and can foster the development of relationships between other process constructs to form a schema or, a mathematical object. Objects Dubinsky (1992b) and Cottrill et al. (1996) describe mathematical objects as being "constructed through the encapsulation of a process. This encapsulation is achieved when the individual becomes aware of the totality of the process, realizes that transformations can act on it, and is able to construct such transformation" (Cottrill et al., 1996, p. 171). The student is able to flexibly move their thinking back and forth between objects and processes of a mathematical idea. Mathematical objects exist as dense and symbolic mathematical schemas with which an individual is able to respond to many different contexts by de-encapsulating a mathematical concept in order to retrieve the appropriate processes or actions.  110  Schemas As mentioned above, Dubinsky (1992b) and Cottrill et al. (1996) explain schemas to be coherent mental networks made up of mathematical representations of actions, processes, and objects. Once formed, these networks can also be interrelated with other schemas. Moreover, it is through continual, reflective constructing and reconstructing of mathematical schemas that an individual is able to make sense of, and deal with problematic situations by modifying or developing new mathematical processes, objects, or schemas.  The Role of Reflective Abstraction in Mathematical Understanding Dubinsky (1992b) regards Piaget's notion of reflective abstraction as playing a critical role in the development of students' mathematical thinking. Moreover, Dubinsky (1992b) supports Piaget's (Beth & Piaget, 1966; Piaget, 1985) view that "first... reflective abstraction has no absolute beginning structure and second, that it continues up on through higher mathematics (Beth & Piaget, 1966, p. 203-208). By keeping in one's mind the A P O S model (see Figure 6) while reading the following descriptions regarding the different forms ofreflectiveabstraction, one is able to understand how these types of mathematical abstraction play integral parts in an individual's movement from one stage to the next in developing their mathematical thinking from actions to processes, operations, and schemas (Cottrill, 1996; Dubinsky, 1992a, 1992b). In contrast to empirical and pseudo-empirical forms of abstraction , Dubinsky (1992a, 16  1992b) considers reflective abstraction to be the most sophisticated. He views reflective abstraction to be necessary for advanced mathematical understanding because it is the process by which students are able to internally coordinate their mathematical actions and form mental mathematical generalizations about external objects or events. Within reflective abstraction, Dubinsky further distinguishes five specific stages of thinking that enable deep mathematical Based on Beth and Piaget (1966), Dubinsky (1992a, 1992b) defines empirical abstraction as being the least advanced form of abstract thinking; as one's ability to make generalizations) regarding the common properties of a collection of objects such as, "all the blocks are blue*. Pseudo-empirical abstraction is considered to be a more advanced form of thinking than that of empirical abstraction but not as sophisticated as reflective abstraction. This type of internal construction enables an individual to sort out mathematical properties that are being acted out on a set of objects. For example, a person's action of aligning two sets of objects demonstrates that that person has an understanding of a oneto-one correspondence among the sets of objects. 1 6  Ill  understanding: "interiorization", "coordination", "encapsulation", "generalization", and "reversal" (see Figure 6).  Interiorization  occurs when the learner is able to consciously reflect on mathematical  actions and combine them with other actions. Through reflection and  coordinating two or more  mathematical processes together, the individual is able to construct new mathematical objects through  encapsulating  or converting the process(es) into a concept. Once an abstract  mathematical object or concept is formed, the learner is able to generalize this knowledge and apply it to many different contexts. Further still, this understanding of relationships that exist among mathematical processes, objects, or schemas allows the learner to thinkflexiblyand thus, reverse their thinking.'  7  Genetic Decomposition Dubinsky (1992a, 1992b) and Cottrill et al., (1996) propose "genetic decomposition" as a possible means by which eflective teaching methods can foster students' mathematical understanding as framed by the APOS model. Genetic decomposition attempts to identify the elements of thinking that are necessary for students' construction of schemas regarding a particular math concept of say, 'limit'. Dubinsky and his collaborators (Cottrill et al.) characterize genetic decomposition as a cyclical tool that begins with an analysis of the mathematics for a particular concept and then compares it with students' understanding in order to develop a specific sequence for the learning of the mathematical concept. According to Dubinsky, before any instruction takes place, analytic decomposition is necessary. This approach involves breaking down into smaller chunks and then sequencing mathematical problems into specific steps so that students will be led to construct the particular concept (see Figure 7). Methods for instruction are then designed and implemented, and observations are collected regarding the mathematical activities of the students. These observations are then compared against the first genetic decomposition and necessary revisions regarding the sequencing of the mathematical problems  For instance, by understanding multiplication as the complimentary operation of division we can think about 4x3 = 12 as being reverse expression of 12 + 3 = 4.  17  112  or instructional strategies are made. The entire cycle of decomposition, implementation, and comparison is then repeated until no more revisions are needed (see Figure 8).  Preliminary Genetic Decomposition Our description of what might occur is organized in six steps that occur only very roughly in the given order and with a great deal of "backing and filling" as the  student constructs the concept of limit. 1.  The action of evaluating the function/at a few points, each successive point  closer to a than was the previous point. 2. 3.  4.  5.  6.  Interiorization of the action o f Step I to a single process in which fix) approaches L as x approaches a. Encapsulate the process of Step 2 so that, for example, in talking about combination properties of limits, the limit process becomes an object to which actions (e.g., determine if a certain property holds) can be applied. Reconstruct the process of Step 2 in terms of intervals and inequalities. This is done by introducing numerical estimates of the closeness of approach, in symbols, 0 < \x - a\ < 5 and \f(x) - L\ < e. A p p l y a quantification schema to connect the reconstructed process of the previous step to obtain the formal definition o f limit. A s we indicated in our comments on the literature, applying this definition is a process in which one imagines iterating through all positive numbers and, for each one called e, visiting every positive number, c a l l i n g each 8 this time, considering each value, called x in the appropriate interval, and checking the inequalities. The implication and the quantification lead to a decision as to whether the definition is satisfied. A completed e-8 conception applied to specific situations.  Figure 7. Cottrill et al.'s (1996) example of the genetic decomposition of the concept of limit.  113  Figure 8. Cottrill et al.'s (1996) diagram of the cycl& regarding" the process of genetic decomposition, its implementation, and its revision.  114  CONSTRUCT RECONSTRUCT FIT CONNECT BUILDUP PROMBLEMETtZE INVENT ADAPT EQUILIBRIUM OVERCOME ACQUIRE RETRIEVI ELICITED BY THE FACILITATE - TEACHER DESTABILIZING CONFRONTING INDIVIDUAL! I N T E R N A L M E N T A L P R O C E D U R A L INTUITIVE REFLECTIVE INFORMAL FORMAL! UNPREDICTABLE LINEAR NONLINEAR CONCRETE TO-ABSTRACT RECURRENT IHIERARCHIGAL ACTIONS PROCESSES •CTS CONCEPTUAL TOOLS NETWORKS IHEMAS REPRESENTATIONS JIGSAt PUZZLES OBSTACLES RESIDUE! IUENTIAL MEASURABLE GATEGORIZABLE TAXONOMiCAL GENETIC DECONSTRUCTS  115  E m p h a s i z i n g the D y n a m i c  Nature  of M a t h e m a t i c a l U n d e r s t a n d i n g  116  Visual and Analytic Strategies as Interrelated Qualities of Mathematical Understanding Taking a different approach from that of investigating mathematical understanding through the development of stages and sequences, Zazkis, Dubinsky, and Dautermann's study (1996) examines the possible relationship(s) between the visual and analytic strategies that undergraduate students employed in solving algebraic problems. The students in this study were given mathematical problems that could be solved using either a "visual" (see Figure 9) or "analytic" approach (see Figure 10). In terms of defining these two different strategies, Zazkis et al. (1996) describe \dsualization as occurring when one forms a relationship between what one sees in one's mind as a mental construct and that which is experienced through one's senses in the physical environment. Analytic thinking is characterized as being: any mental manipulation of objects or processes with or without the aid of symbols. [For example,] a biologist who analyzes the nature of a plant through decomposing it into its parts, as well as thinking about the relationships among those parts and synthesizing them into various other wholes such as leaves, flowers, and seeds. Thus we include the naming of parts in our view of analysis, but we also include intellectualizing them into various new wholes. (Zazkis et al., 1996, p. 442) Their definition of mathematical analysis also includes the five previously mentioned forms of reflective abstractions' that enable the individual to construct mental representations. Zazkis et al.'s (1996) research explores previous claims that attribute visual strategies as being less sophisticated (Eisenberg & Dreyfus, 1991; Gollwitzer, 1991; Presmeg, 1986a; Vinner, 1989) and even restrictive to students' mathematical abilities in connecting visual representations to symbolic forms of mathematics (Kruteskii, 1976; Presmeg, 1986a, 1986b). Interestingly, Zazkis et al. (1996) found that the students did not use one or the other of visual or analytic  1nteriorization, coordination, encapsulation, generalization, and reversal.  1  117  Our specific situation deals with the dihedral group of order four, denoted D , and we will consider students' thinking about two problems: List the elements of this group, and calculate the products, according to the group operation, of pairs of elements. We chose to observe students working with these D problems because (a) each of the interpretations described below represents roughly the same level of mathematical sophistication, (b) both processes are simple enough to be carried out quickly and are therefore manageable during a clinical interview, and (c) the situation itself is complex enough to bring out distinctions in the students' understanding. The group D can be modeled in two ways. The approach that we take to be highly related to visual thinking is expressed in terms of the symmetries of a square. In this view, the elements ot the group are the four rotations of the square around its center—in 0, 90, 180, and 270 degrees, together with four reflections or "flips" (across lines connecting the midpoints of the opposite sides and the two diagonals). The group operation between two symmetries consists of performing one symmetry on a square and then performing the other on the result. Using this approach, a mathematics student might employ a physical model of the square to achieve an understanding of its various rotations and flips. Figure 1 illustrates this method of calculating the product of two symmetries. The student performs a vertical flip followed by a 90degree clockwise rotation to arrive at the reflection or flip acros?the diagonal with positive slope. 4  4  4  Figure 9. Sample algebraic problem and visual solution (Zazkis et al.,  1996).  118  A second approach to D , which we take to be more representative of analytic thinking, expresses the group in terms of permutations of four objects. The group operation in this case consists of applying a specific algorithm to multiply these objects and produce a permutation product. Thus a vertical flip of the square might be represented by the permutation 4  ^1234^ 2143  and a 90-degree rotation by the permutation '1234^ ,234lJ  After multiplying these permutations we get  Figure 10. Zazkis et al. s (1996) example of analytic solution for the same algebraic problem. strategies but employed a combination of the two approaches. Consequently, they developed an alternative model that does not dichotomize visualization and analysis but puts forth the idea of the two as interrelated. Thus, visual and analytic thinking are "interacting and mutually supporting modes of thinking, rather than as two sides of a coin or as a dichotomy or continuum". (Zazkis et al., 1996, p. 454). Their \isualizer-analyzer (VA) model conceptualizes the two problem solving approaches to be interdependent and challenges Piaget's (1977) claim that "some people are particularly visual, others mainly motor, auditory, etc." (p. 684). Moreover, Zazkis et al. align their view to be more in keeping with Clements' (1982a, 1982b) work that explores learners as not only being "visualizers" and "verbalizers" but also, "mixers"- those individuals who "do not have a tendency one way or the other" (Clements, 1982b, p. 34). Instead of describing student visualization and analysis as being located on opposite ends of a continuum, the V A model (see Figure 11) makes the assumption that: [i]t could be that a preference for, and difficulties with, visualization (Bishop, 1986; Goldenberg, 1991; Tall, 1991; Vinner, 1989)  is no more than an individual's  tendency to dwell on one side or another of the triangle, for example, when 119  communicating her or his thinking an individual might be more comfortable drawing pictures or writing formulas, but that does not change the fact that he or she needs analytic thinking in determining what to draw, and he or she eventually constructs a rich mental picture that determines what symbols to write. (Zazkis et al., 1996, p. 453) As Zazkis et al. assume visualization and analysis to be intertwined and dierefore inseparable, they state that it is not possible to make claims that strive to categorize students' mathematical thinking or prioritize one method over the other. This model serves to encourage mathematics educators to conceptualize visual and analytic thinking as being equally important in students' development of fluid and rich mathematical understanding.  A  V5 V4 V3 V2 V1  (frf  st-—  J—  v.  __V>  — -  A5  _V>  A4  j£> A2  V  A1  J-  Figure 1 1 . Visualization/Analysis Model (Zazkis et al., 1996).  Schoenfeld's Views on Mathematical Understanding Schoenfeld (1989a) argues against linear, hierarchical frameworks that characterize children's mathematical knowledge as existing as a series of stages that begin with naive understandings and move progressively towards formal mathematical knowledge. This is because the structure of such frameworks does not highlight the fragmented and unstable nature of children's knowledge structures. Second, Schoenfeld does not conceptualize mathematical understanding as being unidirectional but views it as occurring through a back and forth or 120  bidirectional manner. Thirdly and in similar vein to Hiebert (1989) and Pimm (1987), Schoenfeld (1989a, 1991) advocates for models that focus on children's mathematical understanding as continuous and not those that "trap" it in structures of linearity. Thus, by describing mathematical understanding as being circular in nature emphasizes mathematical ways of knowing as being constantly reintegrated and giving rise to more complexformsof knowledge. Moreover, in order for educators to gain an understanding regarding children's development of mathematical concepts, teachers must focus on examining the dynamics of children's mathematical actions as well as how they evolve through language and social interactions (Schoenfeld, 1989a, 1991,1992,1996). Mathematics as Schoenfeld describes it, is all about deep, connected understandings; that is, understandings that occur not only within an individual's mind, but also collaboratively and socially through the interacting with others: One often thinks of the stereotype, the isolate mathematician alone in his office, struggling to prove a new theorem. This is certainly a part of mathematics, but there is a social aspect of it as well, an aspect that Diaconis captured perfectly. Coming to grips with mathematics involves "talking and explaining, false starts, and the interaction of personalities. "All of it, not the least of which is the challenge of the false starts, is indeed a great joy. (Schoenfeld, 1991, p. 328) Schoenfeld (1992) applies Ryle's (1949) description to distinguish instrumental knowledge as being "knowledge that" and relational knowledge as "knowing how". He also supports Hiebert's  (1985) view that there is not a distinct line or boundary that separates these two forms of mathematical knowledge but rather, each informs and gives rise to the other. According to Schoenfeld, mathematical understanding that is to be conceptualized as fluid and dynamic cannot be explored through models that assume mathematical development to be a monitonic process that entails the adding on of more knowledge to their knowledge base. Instead, research that aims to describe mathematical understanding as fluid and dynamic demands multiple perspectives that not only address specific issues of mathematical  121  understanding but also connect individual theories to larger realms within mathematics education: ... we need focus and pluralism, and an occasional step back to look at the big picture.... there should be broad diversity in what we look at, and the methods we use to do the looking- 1 don't believe unified theories or methodologies are around the corner-.... We need to work on our descriptions both of the forest and of the trees within. (Schoenfeld, 1989a, p. 116)  A ModelforAnalyzing Students' Mathematical Understanding Schoenfeld's (1989a) video analysis examines students' conceptual understanding of mathematics through different levels of detail as it unfolds during problem solving situations. His research explores the possibilities of computer-based learning for enhancing mathematical understanding as it pertains to students' graphing of straight lines (see Figure 12). Schoenfeld's (1989a) analytic model enables comparisons to be made between students' mathematical structures (the right-hand column) and preestablished mathematical forms (the middle column) as well as defining a student's level of complexity with respect to their conceptual schema(s). As seen in the Figure 12, the left-hand column represents the particular "lens" through which the researcher is examining student interactions; that is, the macro level focuses on the student's mathematical schemas, the middle level examines the entailments of larger schemas, the micro level moves closer in to the connections associated with the entailments of schemas, and the fourth level zooms in to the contexts that give, rise to the student's micro level of understanding. Schoenfeld's leveled structure of analysis enables the researcher  to not only deconstruct  mathematical concepts and students' conceptual  understanding but also to tease out the relationships that exist amongst them. Consequently, Schoenfeld (1989a, 1991) conceives deep mathematical understanding as being that which allows for flexibility and proficiency because of its rich, well developed micro  122  level of a knowledge base. Unstable mathematical understandings and misconceptions on the other hand are taken to be a result of an individual's ill-grounded connections at the micro level. Furthermore, Schoenfeld, like R. B. Davis (1992) views formal mathematics as a product or residue of well-connected mathematical understanding. He emphasizes these points below: If you understand how things fit together in mathematics, there is very little to memorize. That is, the important thing in mathematics is seeing the connections, seeing what makes things tick and how they fit together. Doing the mathematics is putting together the connections and making sense of the structure. Writing down the results- the formal statements that codify your understanding- is die end product, rather than the starting place. (Schoenfeld, 1991, p. 328)  123  Graphs of Straight Lines: Levels of Analysis and Structure* Levels:  t. Macro-organization of knowledge, at the schema level, e.g  Traditional View of Subject matter  2-slot schema slope  2. Compiled knowledge macro-entities and entailments, e.g.:  y-intercept  •Jr  m Is the slops of L m>0: L rises large |m| : I steep (and more.) The point (o.b) is the y* Intercept ol L... (etc.)  3. Fine-grained superstructure supporting domain knowledge: conceptual atoms (nodes) and connections, e.g.:  2 1  and x„  are directed line segments, eo their ratio indicates direction (e.g. + indicating "up, right') and steepness (so much y (or so much x)  when x-0 in y - mx+b, y - Ox+b - b. so the point (O.b) is (a) on the line, and (b) on the y-axis. Hence it Is the y-lntercept. (We call this level ol structure the "Cartesian connection.] 4. The limited applications contexts out of and across which individuals construct the conceptual atoms that are seen at level 3  Our understanding of IN'i cognitive structure  Void. (In traditional analyses -• and in people who have well developed domain knowledge - these traces have vanished and the relevant conceptual atoms are the nodes at level 3.)  slope  x-lntercepty-lntercepi  4  m is the slope of I m>0: L falls large |m| : L flat... the x-intercept has a place In the equation and on the graph... the y-lntercept has a place In the equation and on the graph... N  "2 "1 but this knowledge is nominal and. while It Is used to compute slope, the computation has no graphical entailments. not clear or stable •'  IN knows that "b Is the y-lntercept." but her understanding is nominal and is not tied to the underlying structure as In the Cartesian connection.  m is fuzzy, neither stable across contexts nor consistently evoked. Its meanings evolve. The role ot the xintercept evolves over the sessions. In 4 slightly dillereni contexts. IN has 4 different meanings and interpretations for the y-intercept - ie, the meaning ot the term is context- dependent.  •The use of the term "levels" in this context does not presume that the structures discussed are hierarchical, or that they have the customary entailments of structures.  Figure i i . An example demonstrating Sehoenfeld's (1989a) leveled analysis of regarding the graphing of straight lines and the comparison between that of established mathematics and that of student mathematical understanding.  124  Problem Solvingforthe Development of Mathematical Understanding Schoenfeld (1992) further elaborates on his analytic model (1989a) by taking a step closer and looking at individuals' mathematical knowledge bases. Here he examines what relevant information students draw on during mathematical problem solving, and the ways in which they retrieve and employ this information. For example, given the problem: You are given two intersecting straight lines and a point P marked on one of them, as in the figure below. Show how to construct, using a straightedge and compass, a circle that is tangent to both lines and that as the point P as its point of tangency to one of the lines.  Figure 13. Example of a mathematical problem and its solution used to analyze student problem solving actions (Schoenfeld, 1992).  125  a student's mathematical understanding can be assessed against the table below:  Degree of Knowledge  of  facts  and  procedures  Does the student  a. know nothing about b. know about the existence of, but nothing about the details of c. partially recall or suspect the details, but with little certainty d. confidently believe  the tangent to a circle is perpendicular to the radius drawn to the point of tangency (true) any two constructive loci suffice to determine the location of a point (true with qualifications) the center of an inscribed circle in a triangle lies at the intersection of the medians (false)  a (correct) procedure for bisecting an angle a (correct) procedure for dropping a perpendicular to a line from a point an (incorrect) procedure for erecting a perpendicular to a line through a given point on that line  Figure 14. Partial inventory of an individual's resources for working out the construction problem as described in Figure 13 (Schoenfeld (1992). Here, informal knowledge is defined as that knowledge that a student brings to bear on a particular problem such as a students mathematical intuitions and their more formal knowledge consists of mathematical facts, definitions, or algorithmic procedures. The ways in which these forms of knowledge are expressed may vary depending on the individual's confidence or certainty of them.  The Structure of Memory: Access to Resources Schoenfeld (1992) proposes another model that oudines his conceptualization of our memory system and how the contents of memory arc organized, accessed, and processed in a sequential yet somewhat circular manner (see Figure 15). Schoenfeld explains that visual, auditory, and tactile information is received through what he calls as "sensory buffers" or, short term memory. Short term memory is described as the location where the thinking gets done. If 1 2 6  sensory information is attended to within one's short term memory, then it is converted into forms that are further developed through the working and long-term memory systems. Working memory is different from that of short and long term memory because not only does one's working memory take in information from these other two sites but it is also where metalevel processes occur and enable one to construct mental representations. In addition to this, within one's working memory, one is able to structure planning, monitor, and evaluate one's  Problem  Sensory Buffer  Working Memory Metalevel processes:  Task  planning monitoring evaluation Mental Representations  Environment  Long-term Memory Math knowledge Metacognitive knowledge Beliefs about: math self Real-world knowledge  _L OUTPUT  Figure 15. Sehoenfeld's (1992) conceptualization of the structure of memory. mathematical actions. It is this activity that takes place in our working memory that enables us to externalize our mathematical thinking through various physical, written, or verbal forms of expression. Long-term memory system is considered as a "permanent knowledge repository" (Schoenfeld, 1992, p. 351). It is a neural network in which mathematical knowledge functions as nodes and relational knowledge as the strands that connect these nodes together to form the network. 127  Sehoenfeld's (1985, 1987, 1989b, 1992) studies that examined and compared students' executive or control skills (see Figure 16) to those of expert mathematicians (see Figure 17) revealed that unlike the students who spent much of their time exploring the mathematical problems, mathematicians spent the majority of their problem solving time making sense of the situation, analyzing, and structuring their exploration; that is, in thinking through the situation at hand, mathematicians produced solutions through generating and implementing devised methods that demonstrated a high level of control and perseverance. In the context of mathematical problem solving, "control" refers to the way in which an individual selects goals and subgoals, monitors, revises, and assesses their progress of a problem solving activity. Control also includes how one makes use of and sense of given or found information in attempts to solve a problem. The second managerial strategy, "perseverance" refers to an individual's intuitive, experiential sense in knowing when to continue with and not give up too soon on a chosen strategy or action but also, knowing when to abandon a particular strategy or action and search for a more effective or useful one.  Activity  'Read  •  Analyze Explore Han Implement Verify Elapsed Time (Minutes) Figure 16. Time-line graph of a typical student attempting to solve a non-standard problem  (Schoenfeld, 1992,).  128  Activity  Read Analyze Ikplore Plan Implement Verify 5 10 15 Elapsed Time (Mmites) Figure 17. Time-line graph of a mathematician working on a difficult problem (Schoenfeld, 1992). Based on these observations and implementing instruction that focuses on students' development of control and perseverance in their mathematical thinking (see Figure 18), Schoenfeld's (1989b; 1992) work supports other researchers' (Carraher et. al, 1987; Hart, 1989; Hiebert, 1989; Taplin, 1995) contentions for perseverance and control as two critical qualities necessary for well developed mathematical understanding. Schoenfeld advocates teaching methods that engage students in reflecting and routinely explaining their mathematical actions as well as in providing reasons for why their actions make sense within the given context. In doing so, he states that these managerial skills will then become a natural way of thinking about mathematics and enable more complex mathematical understandings to occur.  129  Bapsed T&ie (Minutes) Figure 18. Time-line graph of students problem solving actions after implementation of instruction that focused on development of metacognitive problem solving skills (Schoenfeld,  i99)2  The Metaphorical Nature of Mathematical Understanding: The Work ofAnna Sfard Sfard  (1991,1994,1998,2000) examines mathematical understanding as being  rooted in  and growing from one's use of conceptual metaphors. Metaphors, she explains, not only provide us with a means by which to explain our thinking, but they also shape our ways of understanding and knowing mathematics. This is expressed by Sfard below when she speaks of Reddy's  (1978)  notion of conduit metaphor and connects it to that of mathematician's conceptions of what it means to understand mathematics: Rather than being just tools for a better understanding and memorizing, conceptual metaphors are often the primary source of mathematical concepts. The constitutive role of metaphor has been mentioned explicitiy by the mathematicians whom I have interviewed in one of my studies. (Sfard, 1994)  130  In this way or another, all of them made it clear that without a metaphor, a new concept is not a concept at all. They also repeatedly emphasized the indispensability of the metaphor in the subsequent problem-solving process. (Sfard, 1994, as cited in Sfard, 1997, p. 340) Sfard (1998, 2000) also makes the observation that the two main types of educational metaphors being used today in mathematics education characterize children's mathematical understanding in two different manners; that is, metaphors that describe mathematical understanding as a process of "acquisition" and metaphors that describe mathematical understanding as developed through "participation". Acquistionist metaphors are defined by Sfard as views that describe children's conceptual understanding of mathematics to be a process by which "basic units of knowledge... can be accumulated, gradually refined, and combined to form ever richer cognitive structures." (1998, p. 5). Furthermore, Sfard distinguishes that: The picture is not much different when we talk about the learner as a person who constructs meaning. This approach, which today seems natural and self-evident, brings to mind the activity of accumulating material goods. The language of "knowledgeacquisition"and "concept development" makes us think about the human mind as a container to be filled with certain materials and abut the learner as becoming an owner of these materials. (1998, p. 5) She contrasts the acquisition metaphor with a participation metaphor and states that "the P M [participation metaphor] shifts the focus to the evolving bonds between the individual and others.... Indeed, P M makes salient the dialectic nature of the learning interaction: The whole and the parts affect and inform each other" (1998, p. 5). As well, she also notes that unlike the acquisition metaphor that emphasizes mathematical knowledge as a product of learning and teaching, the participation metaphor amplifies mathematical knowing occurring through ongoing interaction within a mathematical community. Sfard (1998) recognizes the need for metaphors to be flexible and diverse because "...too  131  great a devotion to one particular metaphor and rejection of all others can lead to theoretical distortions and to undesirable practical consequences" (p. 5): We have to accept the fact that the metaphors we use while theorizing may be good enough to fit small areas, but none of them suffice to cover the entire field, ln other words, we must learn to satisfy ourselves with only local sense making. A realistic thinker knows he or she has to give up the hope that the little patches of coherence will eventually combine into a consistent global theory. It seems that the sooner we accept the thought that our work is bound to produce patchwork of metaphors rather than a unified, homogeneous theory of learning, die better for us and for those whose lives are likely to be affected by our work. (Sfard, 1998, p. 12)  Sfard (1998, 2000) does not enter into current (mathematical) educational debates that aim to delineate learning as being conceptualized through either acquistionist or participative metaphors but rather, thinks that we should take the best qualities of both metaphorical ways of thinking and use them not in a divisive manner but in an integrated manner. That is, that discourse should focus on distinguishing contexts in which applications of each approach proves effective. In addition to this, Sfard stresses that even if we wanted to subscribe to framing mathematical understanding as say, solely participatory in nature, due to our cultural embeddedness in acquisitionist language we cannot help but to think acquisitionally, with objects and abstract properties- it is a part of our taken for granted ways of being. Sfard (1997, 1998) states that both metaphorical ways of thinking offer qualities that the other cannot and in doing so, argues that "the most powerful research is the one that stands on more than one metaphorical leg" (Sfard, 1998, p. 11) because these metaphors provide tension from which theories can be interrogated. Sfard identifies one limitation that can occur when only using a participative approach to teaching mathematics, and that is that this way of thinking about mathematical learning can 132  lead to the "complete delegitimatization of instruction that is not problem based or not situated in a real-life context" (1998, p. 11) and "[t]his is difficult, when mathematics at some point exists within the symbolic, abstract realm" (Sfard, 1998, p. 36). Conversely, Sfard explains that when applying a solely acquisitional approach to teaching mathematics together with the assumption that learners build their own conceptual understanding of mathematics, the problem of bridging individual and collective knowledge becomes difficult.  Connecting Mathematical Processes of Knowing with Objects of Knowledge: Operational and Structural Conceptions of Mathematics Instead of describing mathematical understanding as that which exists as either object or action, Sfard's research (1991, 1992, 2000) attempts to bridge this dichotomous gap and establishes the need for the co-existence of both mathematical knowing and knowledge; that is, that "an adequate combination of the A M and the P M would bring to the fore the advantages of them" (1998, p. 11). In describing the conceptual development of mathematics, Sfard (1997) characterizes it as being "a zig-zag movement with our conceptual schemes as constituting an "autopoietic system"; that is, a "system which is continually self-producing" (Maturana & Varela, 1987, p. 355). These qualities regarding mathematical understanding are pervasive elements throughout Sfard's diverse activities of research and reflect the value she holds for both structural or abstract knowledge and operational or context-bound knowings. Sfard seeks to describe the interrelationships that connect mathematical knowledge and knowing by examining the processes that facilitate children's formation of abstract, symbolic, concepts in mathematics. Sfard's (1991) identifies three hierarchical stages of mathematical conceptions. The stages are referred to as: "interiorization", "condensation", and "reification" (see Figure 19). She defines the interiorization as the stage in which a child is developing an operational concept of the mathematics they are using to perform an action on a given problem. For example, "When I fill each of these three boxes up to the top and count the total number of cubes, I can find out how  133  much each container holds". All operational concepts that are formed within the interiori2ation stage are considered to be context-specific knowings.  Figure 19. J. S. Thorn's (2004) diagram that characterizes Sfard's (1991) configuration of the three stages of mathematical conception. The second stage- the condensation stage is described by Sfard as being when learners are able to metaphorically, "stand back" and begin to reintegrate or make generalizations about their mathematical understandings. It is this middle stage that elicits an interplay between the synthesis of the child's previous operational mathematical conceptions and move towards the formation of an abstract, structural concept. Using, again, the example of the container, a child may now think, "Each time I filled and counted the number of cubes each of the three containers held. I wonder if there is a way that I can determine how many cubes the containers hold without having to fill each box and count the cubes by ones?" The final stage of rcification is explained by Sfard as the point at which the learner is able to comprehend the mathematical concept- in this case, the volume of rectangular prisms as an "object" or a "thing" that is symbolic, dense, and versatile. So, being able to understand that, "If I want to know how many cubes any box can hold, all I need to do is to measure (with cubes) and multiply the length of the box by the width of the box by the height of the box." Hence, by condensing one's knowing of operational, context-specific actions through ongoing analysis, one's mathematical knowledge can become reified into a flexible, structural form.  134  The Limitations of Metaphors Sfard  (1997, 1998)  points out that although metaphors enable us to think about  mathematics in powerfully abstract and symbolic ways at the same time, they are also shaped and limited within the confines of our experiential knowledge. Below are ontological obstacles identified by Sfard as they pertain to the integration of metaphors, metaphorical overprojection, and metaphorical confinement. In keeping with Sierpinska and R. B. Davis & Vinner's views, Sfard too considers it necessary for students to overcome these obstacles in order to integrate many different metaphors and develop a sound, stable conceptual knowledge of mathematics.  Integrating Mrtaphora Sfard  (1991, 1992, 1997)  explains that one possible reason for students' difficulties with  integrating metaphors is due to their inability to allow certain characteristics of the metaphor to fade into the background in order to integrate new qualities that will extend their knowledge to new or different mathematical situations. For example, when learning about the concept of division, one must, in a sense "forget" one's previous understanding that the operation of division when applied to whole numbers "makes the quotient smaller" in order to develop an understanding for why division "makes the quotient bigger" when working with fractions and decimals.  Metaphorical Overprojection Instances when metaphorical overprojection take place involve situations where an inconsistency lies in the student's mathematical actions: Without abandonment of certain characteristics there may be a danger of a logical incompatibility with the new context or with metaphors contributing to the construction of the new concept. Appropriate modifications, however, are sometimes difficult to perform. Certain characteristics, being a vital component of the source notion, would refuse to go. (Sfard, 1997, p. 368) She provides the following example of a student who divides the factor 'x - 2' into both sides of  135  the equation, (x-2)*(x + 3) = 2(x-2). This is considered to be anoverprojection of "an equarion is a balance" metaphor because when the student performs this operation on both sides of the equation, the student is thinking that equality has been maintained when in actual fact, the root x = 2 has been lost. Metaphorical overprojection can also occur when a student tries to integrate two incompatible metaphors such as "number as quantity" with the concept of complex numbers. If students cannot exclude or in some way 'forget' the quantitative quality of numbers, there results an incompatibility that limits and proves problematic to their conceptual understanding of complex numbers.  Metaphorical Confinement Metaphorical confinement as a third ontological obstacle occurs when a student's metaphor is not broad enough to allow for the development of different, related metaphors or mathematical concepts. This form of confinement in mathematical understanding is present when students can only visualize fractions as being "part(s) of a whole"; with this image, their conceptual understanding is confined and because it cannot be opened to fractions existing as "object(s) within a larger group of objects", or as another way of expressing the divisive action.  136  constructs frameworks macro middle micro levels schemas short term working long term networks nodes product knowledge object action knowing interiorize condense reintegrate reify visual analytic individual collective social mental instrumental relational operational structural metaphorical hierarchical hierarchical combining intertwining inseparable fragmented unstable continuous deep connected circular somewhat circular distinct dichotomizable categorizable linear sequential sequential prioritizabfe naive to formal informal formal back and forth zigzag bidirectional fluid dynamic adding on correspondences flexible proficient organize manage access process persevere control reflect acquire construct  participate  interact  137  IRetkm&iHf ^Mathematical 1lKde*4ta*cU*$  from  a*  Snactwe penAfieetive  138  Mathematical Understanding as Objects of Personal and Public Forms of Communication In her recent work Sfard (2000) has shifted her perspective to a more enactive one. Mathematical thinking is now regarded by Sfard (2000) as a form of communication; an integral cognitive process which allows individuals to not only express with others how and what they are understanding about the mathematics at hand but also, mathematical thinking as communication shapes how we individually and collectively make sense of mathematics. She contends that our reasons for communicating are not to establish mathematical objects but rather, mathematical objects are brought into being because we need them to develop conceptual understanding in terms of our own internal thinking and in conversations with others. Sfard (2000) explains that mathematical objects (physical, verbal, mental) arise as a product of our need to communicate; not the other way around: We do not start with mathematical objects and then communicate, we communicate and through this dialogic process, mathematical objects come into being. In keeping with this Sfard (2000) makes the assertion: I will argue that the claim of the primacy of communication imposes a literal reversal of this relationship: Instead of being merely helpful in constructing and sharing the knowledge of preexisting mathematical objects, communication and its demands must now be regarded as the primary cause for their existence, (p. 4) For these reasons Sfard's research (2000) and the collaborative work that she has done with Kieran (Sfard & Kieran, 2000) reveal mathematical communication as having positive, neutral, and even detrimental impact on students'conceptual understanding. Together, Sfard, and Sfard and Kieran's research lessens the gap between our conceptions of students' internal, cognitive thinking as being separate from their interactive and communicative ways of acting. As well, their work brings forth the notion of internal and collective mathematical understanding as being conceived together. Sfard (1997-, 2000) supports Maturana and Varela's (1987) view that cognition in its most encompassing sense, co-evolves from our ways of knowing, our actions, and in our individual and  139  collective identities. This is evident when Sfard speaks of mathematical understanding as being shaped and evolving within ourselves and with others. Similarly to that of Gadamer (1989) who describes understanding as being like that of a "conversation"- dynamic, unpredictable, and dependent on the conversants, Sfard  (2000) too  uses this metaphor when she characterizes  mathematical thinking as it occurs when individual students work collectively in a larger group: Thinking, like conversation between two people, involves turn taking, asking questions and giving answers, and building each new utterance-whether audible or silent, whether in words or in other symbols-on previous ones in such a manner that all are interconnected in an essential way. (p. 5) Sfard (2000) argues that rather than simply viewing mathematical understanding as being that which exists either in the objective, public realm or in the individual, private realm, we need to also focus on the relationships that emerge between formal mathematics and that which is considered to be informal and embodied; meaning, the connections which take place within individual and collective conversations and the ways in which these interactions affect mathematical understanding.  Mathematical Conceptualization as Complex Circularity By integrating the latest works of Sfard, which illuminate students' mathematical thinking as being circular and complex with Sfard's model of mathematical conception  (1991),  the latter shifts from that which was linear in structure, to a view of mathematical understanding as being co-emergent and cyclical (see Figure 20). This co-emerging of theory is possible when we examine the definition for "attended" focus. Sfard  (2000) describes  this as being the  mathematics or mathematical object which arises as the individual or group's subject of conversation. Here as in Sfard's previous model of concept formation, this can be seen as corresponding to the condensation stage. In both cases of attended focus and condensation stage, there is an interactivity which involves the weaving together of several foci. Secondly, the form of mathematics or mathematical thinking which resembles the reification stage can be seen 140  in what Sfard (2000) describes as a "pronounced" focus of conversation; when "the learner[s] can flexibly move back and forth if needed to other realms whereby effective communication mediates these transitions" (p. 33). Thirdly, that which is considered to be the "intended" focus of mathematical communication, specified on an individual level as being each persons interpretation of the pronounced and attended foci, fits with Sfard's operational metaphors located in the interiorization stage.  >c|»er£ft&f!ftt concept metap*v&f intended fo cue  s  structural concept-metaphor * prortounoei-j focus  condensation attended focus  Figure 20. J. S. Thorn's (2004) diagram which attempts to integrate Sfard's theories regarding mathematical conception (1991) and mathematical communication (2000).  Mathematical Understanding as Growth: The Pirie-Kieren Model Pirie and Kieren's cognitive mappings of individuals and groups of students identify mathematical understanding as being simultaneously individual and collective, dynamic, occurring on many levels at once, and revealing qualities of transcendence and recursiveness (see Figure 21 and 22). They consider learners to be autopoietic beings who determine what  141  phenomena will be experienced as perturbations and who specify the ways in which they structure their mathematical thinking (Kieren & Pirie, 1992). As well, Pirie and Kieren argue that mathematical understanding does not occur as a result of student interactions with others or the  Figure 21. Model of a dynamical theory of the growth of mathematical understanding (Pirie &  Kieren, 1994a).  1 4 2  Definitions of Levels of Mathematical Growth Primitive doing or knowing: All the knowledge that a learner or group of learners  bring to the particular mathematics and from which all new understandings develop. Image making: making distinctions in previous knowing and using it in new ways. "Image" not only include physical and verbal forms, but mental representations as well. Image having: using a mental construct about a topic without performing the particular activities that brought it about. Property noticing: making note of distinctions, combinations or connections between images, predicting how they might be achieved and recording such relationships. Fonnalizing: abstracting a method or common quality from the noted properties which are not dependent on meaningful images. Observing: reflecting on and coordinating formal activity, expressing coordinations such as theorems. Structuring: explaining or theorizing one's formal observations in terms of a logical structure. Inventising: breaking away from preconceptions that brought about previous understanding and creating new questions which might grown into a completely different concept.  Other Features of the Model • Folding back: moving to an inner level in order to extend one's current, inadequate understanding when faced with problems at any level. • "Don't need" boundaries: indicated by the model's bold rings; conveys the idea that beyond the boundary one does not need the specific inner understanding that gave rise to the outer knowing. • Each level beyond primitive knowing is composed of a complementarity of acting and expressing necessary before one is able move to the next level; acting encompasses all previous understanding, and expressing gives distinct substance to that particular level.  Figure 22. Definitions of terms and features regarding the Pirie-Kieren model for the growth of mathematical understanding. (Adapted from Stoute, 2000). environment but rather, comes to be through the structural changes within, between, and among learners and the environment (Kieren, Gordon Calvert, Reid, & Simmt, 1995; Gordon Calvert, 1999; Simmt, 2000). It is this part of the Pirie and Kieren's view on mathematical understanding which emphasizes the notion that mathematical inter-activity as critically important for mathematical learning to grow. Moreover, Pirie and Kieren define mathematical 143  understanding as the embodiment of all verbal, physical, and written acts: Mathematical understanding can be characterized as leveled but non-linear. It is a recursive phenomenon and recursion is seen to occur when thinking moves between levels of sophistication. Indeed each level of understanding is contained within succeeding levels. Any particular level is dependent on the forms and processes witliin and, further is constrained by those without. (Pirie & Kieren, 1989, p. 8)  A Descriptive Not Prescriptive Model Pirie and Kieren make it clear that their model is not intended to be used to define or prescribe a particular sequence of static levels which constitute students' mathematical learning but rather, a way of conceptualizing the learning of mathematics as unpredictable and complex phenomena. As well, Pirie and Kieren (1994b) do not distinguish mathematical growth as being monological pathways, or privilege one's fluency to use formal language and mathematical symbols as representing formal mathematical understandings. Mathematical understanding is not only assumed to grow in complexity through the learner or collective unity's outward movement, but also from inward movement or, what they call, folding back to previous levels of knowing. Folding back is not a redoing of what has already been done, but moves the learner or group of learners back to inner levels of mathematical knowings where they will reintegrate understandings as a result of the perturbations experienced in previous outer levels before moving on (Kieren & Pirie, 1991; Martin, 1999; l o w e r s , Martin, & Pirie, 2000). I h i s model also reflects the notion of mathematical knowings existing simultaneously as a product, producer, and process (A. B . Davis, 1995, 1996; A . B . Davis & Sumara, 1997, 2000; Kieren, Simmt, Gordon Calvert, & Reid, 1996; Maturana & Varela, 1987). In this way, Pirie and Kieran advocate for learning settings that encourage students' engagement in folding back in order for the cocmcrgcncc of their sclf-rcfcrcncing, remembering, and reintegration of mathematical knowings to occur (Pirie & Kieren, 1992).  144  Thus, the Pirie-Kieren model of mathematical growth  (1989)  reflects an enactive  perspective because it provides a theoretical lens which focuses specifically on the complex, coemergent,  and  unpredictable  nature  of  mathematical  understanding. Mathematical  understanding is viewed as occurring through interrelated, fluid processes and evolving in a fractal-like manner (Kieren, 1990; Pirie & Kieren, 1989; Pirie & Kieren, 1994b). They describe the model's structure as neither liierarcliical nor linear. The realms of mathematical knowings found within this model exist as embedded, unbounded circles which are self-similar and compatible with one another. Moreover, the Pirie-Kieren model reflects Maturana and Varela's  (1987) axiom  of "all doing is knowing, and all knowing is doing" (p.  26) because  it locates  primitive doing or knowing as being the roots from which all other mathematical knowings emerge (see Figure 23).  Figure 23. Model illustrating primitive knowing as the source of all other mathematical knowledge (Kieren, 1990).  145  all  doln$ is knowing, and  simultaneously  individual  personal interrelated dynamic  alt  puttie embodied  static  private  knowing is doing' and internal  co-emergent  mnlti-leveted (mem  collective external complex cample-*  recursive ptvdtetaUe mondogleM inward-and-outward audible silent vialbte tefhtfefee^ jor mat Informal  knowing actions identities  autopoictic structural changes within Set ween among selfrejerencing folding Sack collecting invocations provocation* 0m tkickened embedded unbounded sei^-simdat fractal-tike exists only in being intet-actively intra-activcty co-evolving Ln-retatlon in-converstation with the \ interdependent txttxnal environment growth  ' Maturana & V a r a i a *&87 p 26  146  "Desptie the dv ierstiy of posto ins taken by mathema ctis educators reg ways in whcih mathemactial understandn ig is portrayed and the manne deveo lps, it is evd ient in ths i tilerature coe lcto in that there is a general the varyn ig perspecv ties that "good" mathemactial understandn ig n ivov les the ofn iformal and formal mathemactial knowe ldge, that it is fe lxb iy l fluid, an appe ild to respond to many dfierent stiuato ins. The theoretcial portratis o lcated in construcv tisim emphas zie the bud lin ig mathemactial know eldge as schemas, ones' progresso in through specfic stag as the maneuven rig of ones' mathema ctis over a vare ity of conceptual posto ins taken by the authors in the second set of tilerature are sg ilh those seen in the first as these researchers seek to n iterpret the d mathemactial understandn ig and expo lre o frms of know eldge as ben ig phenomena. And finaly, in the thrid groupn ig of tilerature, the work of educators who share an enacv tie r^rspecv tie was examn ied. more ve iwpon its serve to hg ihg ilht the co-emergent and comp elx nature of understandn ig, how it it is n idv id iuay l and coe lctv iey l brought n ito ben i embod eid o frms in whcih it exs its. These  147  Matting, taken a good look at the metaphorical furnishings J had arranged comfortably around my teaching, and teuealed the diversity how- mathematicA and mathematical understanding might be portraye £ nam faced the took of, deciding whether (and why) they veally tutt (or did not AUU) the ecological mind-space in which £ WOA now. dwelling. Jn order to do this, J, had to consider what fundi of thinkin my metaphorical furnishings enabled. Gbnd if they were not engender ecologically coherent manners of conceptualising my mathematics teaching, what metaphorical furnishings would?  148  So what of my mathematics curricula as, jigsaw puzzles metaphor? She metaphor creates the image of a mathematics curriculum as being set of, pieces... like concepts, skills, domains of mathematics.... Complicated mauve, but as such, are do- and undo-' able pieces and a a whole— clearly, defined and separately visible. So-, when given to students, and the pieces are assembled correctly, they reveal a coher picture from its interlocking parts. (Shay yes, this metaphor is, a ve 'tidy' and 'systematic' way for a teacher to think about mathematic curricula,ffiutwhat this metaphor does not do is reveal the ecologica qualities that distinguish teaching and learning of mathematics as, dynamic and complex,. d. Si. 3)avis,, Samara, and £uce-SCapler make the distinction between complicated and complex thinking when they describe complicated thinking as that which ''aims to reduce phenomena to elemental components, wet causes, and fundamental laws." SJhey us the example of a clock as a complicated mechanism and state that a complicated understanding of it would involve ''detailed knowledge of each of its parts." and how the clock can be disassembled and reassembled. Jn contrast, a complex comprehension of a clock entails not only an understanding of its parts and the ''interdependencies o its, parts," but also, the role that is played by the clock is, necessa ''embedded in [and thus affects, as it is affected by] social and natural environments* (  f  2  3  SJhe authors make the important point that something conceptualized a complicated manner as in the case of the clock, C = A + B which implies, that C (ie., the clock) can be taken apart and put back togeth again. SJunking in a complex way however, assumes that C depends o "A" and "B" but that at the same time, it exists as something other th just A + B. Sake for example a cake, which can be considered to be a complex entity. Once the ingredients, have been mixed together a baked, you cannot take it apart again to get back what went int making it. She cake exists, as a complex form because of the reactio between the ingredients and its environment. 149  So- even if a teacher managed ta design a mathematics program that mathematical topics together in a way that produced a complete pictur the curriculum would temain static. Jt would still be a set of distinct pieces and hence, necessarily a "complicated" NOT "complex" curricular form. She program would only (te a product of its parts and not something that possesses, the potential for possibilities greater th that? or a curriculum that embodies an awareness for the vole it pla in the whole of teaching and learning- mathematics. What 3. needed w a complex view of the curriculum. Something more than a complicated one. CL new metaphor.  150  dn ecologically coherent metaphor of a mathematics cwtriculum needs, ta b creates an image of a flexible, dynamic network that co-eooloes as a resu interactions of the teacher, children, and the environment. Ci mathematics c might then, not be thought of as a commoditVzed "thing"... that which " what teachers are to teach and what children are to leam. b  Q. Skueson's map metaphor works well in that mathematics curricula can be co-emerges wkk m&themmiical understood in light of an ecological perspective. Envisioned as a map,, a teacher can locate mathematical topics, concepts, and skills as important landmarks for the class! learning. Once these locations are marked out, the teacher can then think about how they are connected to one another. Jnherent in this is the idea that what cannot ^kmmf^tke landscape cannot be sketched out in advance are the actual paths on which the children will travel to get to the mathematical locatio understandings they will establish when'and after they come to these si although a teacher might be able to mark out the mathematical landmar impossible to predict the conditions of the ever-changing landscape (ije., t terrain that the class will encounter while engaged in the mathematical stud  cutticuia  7  151  Jn tight of this, children's mathematical understanding and and learning can Be likened to what Varela, SJhompson, and Sto&ch describe as I "paths that are laid down in walking'' . (Ltd like paths, theg. are rarely if. ever straightforward but instead, spread out in several directions and entail twists, turns and simtch-bachs. Mathematics curricula and learning imagined in this ma distinguishes, them as co-emergent phenomena that are brought into being th students, their teacher, and mathematical settings.  mMki.nuitlcai-. MtiMmstmtd&m  8  tf  tette^-  ^ ^ ° ^ r i ^  e  differently, from before, J, have come to envision the mathematics, class as an image teacher and children interacting, withi individual and collective mathematical realms,... nested systems but not at all discrete. She larger human and nonhuman environment is not an external entity but an interdependent one. Once a moot point of earlier thinking, it is now a critical one. Just as children's mathematical knowings impac and shape the classroom environment, the 9  ijS$^  mMkemtltiCS  daSB  etwifUUuneM  9 ' impacts and ** shapes the understandings, that will take place for the children and so on. SJhese realms exist as. revolving systems dynamic and recursively related to one another. cAan  ed  152  'teaching? encompasses  more  than M E as T H E teacher; it includes- the children as uteU as * U j ^ ^jj ^jf^ the material and nonmaterial if^/ |a ^ environment. Gf course, 3 am a ^ . "IT source of provocation and 1^ ^Jr tfMwtcattcw a/! children's ^ ^* mathematical learning but importantly, 3,'m not the only jj^* one. "oihos", teaching and learning, exist within systemic ^ relationships of difference. r^When the class is perceived as a system of individual and \k collective relationships* *»S connected with the material and immaterial environment , occasions for learning can f\ happen from anything and ^ anyone within this web. When 3. think about a metaphor for mathematics, instead of notions of "static integration" or "separate stra algebra, geometry and so on, 3, imagine them to be fluid and co-emerge that make mathematics a dynamic living whole. Jn this way, mathematic conceived as a 'ahap&-&hifter'-- arising and metamorphosing as it interacts contexts of which it is a part. Sake for example, a linear equation as one opTiathematics, %epnesented algebraically and the equation as a graph a another... or a pattern as, a numerical sequence or as a three dimensional st 10  Ganging wtdttskingzhmpe "rmMm" ' — what comes to he,  ,  &  ..  . ... .- '^fcv^'' "V*  a result of childr&i's isttming  v  ' :  ' ^ ^ ^ J ^ ^ "source" "'"  tmpredi^abiilty pkiyf%dness  a beginning open tQpas$it$lJtim  153  Mow imaginable are other equally- critical differences. Qiven this mindmathematics can emerge as a "residue" or a "source" of children's- learn does not have to tie 'kept' as a "product?' that is produced within the sequ synchronicity of predetermined "outputs" fed tiy particular "inputs". Jt framed in a deterministic or a predeterministic view of classroom mathematic Sioh Davis' notion of mathematical "residue" serves, as a useful beginnin for Rlimagining mathematics in the classroom. (Is, residue, it is assumed that n the origins nor the processes by which a particular mathematics, arises ca precisely located. Jt arises from the mathematical languaging of the individua larger collective^&). Mathematical "processes" such as problem solving, con reasoning, and expressing can be understood as "mathematical language" ; t is, they are the physical, verbal, and mental manners in which we can thi (inter )act, and exist mathematically. (Znd, mathematics as a "source" for child learning embeds a sense of unpredictability and playfulness-' a beginning th open to all kinds of possibilities,. 11  12  13  Jn a radically different metaphorical manner, focus is not only on th mathematics at hand, not only on mathematics, as individual and collective kn but also on mathematics as it resides seamlessly and all at once within pa present, and future contexts of children's learning.  Motes  Ci. 31. Davis, Sumara, and £uce-3iapler, 2000. Ci. 3$. Davis, Sumara, and £uce-JCapler, 2000, py. 62 Ci. Si. Davis, Sumara, and £uce-JiapJler, 2000, py. 62 (Z. Si. Davis, Sumara, and £uce~JCapler, 2000,p. 62-63. CL Si. Davis and Sumara, 2000; Cl. 3i. Davis, Sumara, and £uce-3(apler, 2000; Selsand Meyer, 1997; Meyer and Sets, 1997. 6. JMich, 1971. 7. q. Siateson, 1972; Shorn, 2000; in conversation with S. L Si. Sirie, March, 200 8. Varela etal., 1996, p. 205. 9. femontin, 1983; Varela et al., 1996; 10. a. Si. Davis et al., 2000; Martin, 1999; Sirie and JCieren, 1992; Steffe and 1. 2. 3. 4. 5.  SJIUT, 1994.  11. CL further consideration of St. Si. Davis* ( 1992) notion of mathematic "residue". 12. Si. Si. Davis, 1992. 13. Maturana, 1997a, 1997b, 1998. 154  WHEN MATHEMATICS is imagined and enacted as objectified, static knowledge that is to be traditionally passed down from one generation to the next, the teaching and learning of mathematics is disabled from ever becoming anything else. Under the air of "hand-me-downs", it is easy to understand why mathematics is taught and learned out of a sense of obligation or contempt rather than a sense of open desire or wonder, and why, mathematics is all too often considered as that which is to be mastered rather than that which is to be understood. In com-  moditizing mathematics, we make absurd, the possibility for us as teachers and to those who we teach mathematics to perceive it as anything else but a fixed and inanimate entity. In this way of conceiving mathematics, we make it inconceivable for school mathematics to become something else than just a collection of hand-me-downs. The embeddedness of these images within one's taken for granted ways of thinking about mathematics not only make it natural for us to assume mathematics to be an inanimate "thing", but in doing so, displaces mathematics as that which exists "out there". Given this mindset, it is not surprising why a teacher would feel impelled to set the class onto a straight and narrow, one-way course so that the students too, become collectors of mathematics. Given 155  this mindset, it makes sense to in grain the ritualistic practice of "acquiring" mathematics into school mathematics unit and lesson plans, methods of assessment, and enacted in the classroom; product oriented practices that focus on "desired", "expected", or even "measurable" outcomes of instructionthat after instruction, the student will have "mastered" the mathematics taught in the lesson before "moving on" to the next part of the curricular course. Of course, the ways in which children are instructed to take possession of their mathematical handme-downs of concepts, skills, and even attitudes may vary. Still, "teaching by telling", engaging students in "hunting for", having them "seek out" "hidden" mathematics within "real" world contexts, and even "explorations" "designed" for children's discovery (continued on page 84)  (continued from page 8)  of mathematics are - all examples of teaching and learning forms that keep alive, this tradition of "handing down" of mathematics. Moreover, when product-oriented ways of thinking about school mathematics are coupled with a "back to basics" mentality, the teaching and learning of mathematics become subjected to the weigh scale of "how much" in regard to the amount of mathematical facts and skills that children are to learn and little or no emphasis is placed on such things as their mathematical thinking or understanding. Given this mindset, mathematical processes such as those identified by the National Council of Teachers of Mathematics as problem solving, reasoning, communicating, connecting, and representing would likely be deemed "not essential" by most teachers. Viewed as 1  "additional" knowl- end here. Rather, it edge, teaching that opens up a whole attends to children's host of questions development of math- that require further ematical processes interrogation such would depend on as: whether or not the children have acquired first, the pre- • How can an ecological way of thinking help specified mathematius to reconsider such cal facts and skills taken for granted perwith which to ceptions of classroom "process" the mathemathematics and rematics. imagine a more reThe point here, is sponsive view for the teaching and learning that when children of it to exist in the are taught to learn classroom? mathematics in the tradition of hand-me- • What shifts in thinkdowns and as a prod- ing become necessary uct oriented matter in order to reimagine classroom mathemaof collecting, hunting tics as being somedown, or retrieving thing other than a line pieces of knowledge, of hand-me-downs it creates the imposfrom teacher to child? sibility for mathematics to be taught and • What could it mean if learned in ways that we assumed matheenable it to arise as matics to be "embodied"? living and animate. Now, identifying the limitations of how • How could mathemaproblem solving, mathematics exists tical reasoning, communiin the classroom and cating, connecting, and the possibility of it expressing be underbecoming something stood as something else is ail fine and other than additional good. But in doing so, knowledge? means that the conversation does not 8  States  156  1. NCTM, 3000. 3. Baroody, 1993.  ate take seriously, the view-that mathematics is an everchanging entity perceived, created, and embodied as we interact with the world , then it does not make sense for mathematics to be concerned as a fixed, inanimate, disconnected "thing" that exists "out there". When we assume that the only mathematics we know or can ever know emerges from our patterns of living as socialcultural beings then, mathematics is not an objective, universal, transcendental reality but a living system that is necessarily constitutive in nature. 5Ms means that mathematics takes place in the praxis of living in language. Jts coherence is dependent on those who interact with it 2  3  2  3  Lakoff & Nunez, 2000. Bunnell, 2001; Maturana, 1988b.  157  Scripting an Unscripted Lesson 1  The children gathered on the carpet to hear me read the story, Even Steven and Odd Todd. The beginning of this lesson was simple- there was no formal introduction, no preamble, not even a motivational hook. I just opened the book and started to read. This was not because I had not come prepared, I had. When I planned how I might share this story with the class, I decided that I did not want 1  to ask the children to look for, listen for, or think about anything specific. I did not  want to preface the book by telling them that it was a humorous tale about two cousins who are clearly differentfromeach other in one particular way. 1 did not wish to tell them that in my earlier reading of the book I had found out that for Even Steven, life unfolds as patterns of even numbers and for Odd Todd, life is all about odd numbers. It was not that I wanted to "keep" the mathematicsfromthe children but I set out to create a teaching-and-learning space where the class could experience the story as it unfolded, in that particular moment. I wanted to let the mathematics emerge in a different way than a predetermined, predictable one. In not creating a pre-seripted lesson that was about me, "the teacher", identifying "the mathematics'' that the children were tofindin the story, this lesson remained open to the mathematical possibilities that we might bring forth as we listened and responded to the book as a class.  158  Ci moment of digressive thought:  Problem Solving Reasoning and Proof Communication Connections Representation Expressing  3Jf one glance* at the N C T M ' A list o£ the five "process" standards or read through the descriptions that accompany them, it is possible for teachers to in them as five discrete mathematical "skills". Qiven this, it is. understandable teacher might then present them to the class, as five separate topics,. 2  3t is only when carefully Heading through the NCTMV document, df 5Wincip£ea, and Standards, fm, School Jllathetnatics and making note the Counc statement, " [p,]recesses can be teamed within the Content Standards, and c can be learned within the Jhocess Standards,"*, that a nonlinear image mathematics and mathematical processes, comes, into view. Jn contrast t mechanistic forms of thinking that would have us imagine mathematical proc be mechanisms or devices that act as a conduit through which we "tran  mathematics "into" children, the NCTM proposes that mathematics, cont mathematical processes be understood as being reciprocal in nature. She C takes the position that the process standards are not to be conceived as 'additi knowledge'' and certainly not as a "means to a linear end" in mathema classrooms. 3towever, because this point is communicated very briefly in document, it does not make prominent, this, non-linear and co-emergent i mathematics and mathematical processes. Consequently, it remains faint a easily overlooked by the reader who reads from a background of traditiona mechanistic ways, of thinking. ^Differently, if we set this, image within an ecological realm, it immed conjures up, notions such as complexity,, circularity, and recursion, all of whic to bring forth an implicit understanding that:  159  children's mathematics arise can from and be shaped by mathematical processes, mathematical processes can also arise from and be shaped by children's mathematics.  Jf use gaze more deeply into- the languaging apace of ecology,, w brought forth"to.mathematics content and mathematical processes as, envelo co-emergent entities, ecological thinking not only enables, an awareness complex, circularity, that exists, uuth-and-in content and processes, hut mindfulness for the interdependence that defines them as, being inseparab each other. CCnd it is not that they simply exist in a cyclical sense; that on the formation of the other, but in an integral manner, mathematics cont mathematical processes emerge and evolve together in relation with each ot  Mow if we take the NCTM'A process standards and consider them fro ecological space that includes Maturana's definition of "languaging" (an keeping with this, Sfard's notion of mathematical thinking as "communicatio processes identified as: problem solving, communicating, reasoning, express connecting can be understood as being forms, of mathematical "languag explained earlier, Maturana's description of languaging entails how colle unities, evolve through their physical, verbal, and mental linguistic man thinking, acting, and existing . Mathematical processes, as forms of mathem language are the mathematical patterns, of thinking, interacting, and bein enable a class for example, to exist and develop as a collective system. J think of mathematical language and mathematics metaphorically co-existing 6  7  160  seamlessly co-emerging with each other, then each is fundamentalfathe oth haw we teach mathematics and how children (earn mathematics. When classroom mathematics is envisioned as a living system that em with the class' mathematical and social-cultural forms of languaging, it becomes- something more than just an end in itself. Mathematics can be as that which arises as the result of children's learning as well as, that whi as a beginning for their learning to occur. Mathematics exists as both a " and a "source'' . When mathematics arises from children's mental introduc social inter-actions, it can be understood as being a "residue" or, "outcom learning" Shis contrasts utith pedagogical views that assume learning c prespecified as "learning outcomes". On ecological examination does not s  9  what the child SHOULD know but rather, on what mathematics, the child a comes to know. Sor instance, if a child explores how different sets of objec arranged in smaller, equal groups, and arrives at an understanding that " makes smaller", this would be considered to be a mathematical residue of t learning. Motes  1. Qristaldi, 1996. 2. M£SM,2000. 3. M£SM, 2000. Olso,seep.29. 4. MCSM, 2000, p. 30-31. 5. 3$areedy,1993. 6. Maturana, 1998. 7. Maturana, 1997a, 1997b, 1998. 8. Qs previously described from an ecological perspective on page 150 and page 167. Qlso, see Sirie and Mom, 2001. 9. Qs, previously described from an ecological perspective on page 150 an page 167. Qlso, see Si. Si. Davis, 1992; Shorn and Sirie, 2002.  161  Mathematical Language, Languaging, & Residues of Learning (a continuation of Scripting an Unscripted Lesson)  A I continued to read to the story to the class, the children began calling out differences that they were noticing between the two characters: "Even Steven gets up every morning at eight o'clock sharp." " Odd Todd likes to get up at nine o'clock sharp." "Odd Todd rides a tricycle." "Even Steven has four bicycles- they have two wheels each." "Even Steven has six cats, eight gerbils, and ten goldfish." "... and twelve sprinklers in his garden!" "Odd Todd hasfivebuttons on his jacket and Even Steven has six on his shirt." While I recorded the children's observations onto a large piece of chart paper Danica looked at what was being written down, glanced away for a moment, and then announced that "Even Steven only likes things that are two, four, six, eight, ten, twelve, and so on... and Odd Todd only likes things that are one, three,five,seven, nine, and eleven". The whole class nodded and smiled in agreement. Mark then added, "Even Steven likes EVEN numbers". This was immediately followed by Robby's comment, "and Odd Todd only likes ODD numbers". However this time, only some of the class nodded or responded with "yeah!" while other children said nothing, looked puzzled, or exclaimed, "what:!" "Numbers that end in zero, two, four, six, and eight are even numbers and numbers that end with one, three,five,seven, and nine are called odd numbers" stated Mark. Still, the class reacted with a mix of nods, furrowed brows, and a bimch of "what?!s". 1  162  Jumping into the conversation, I agreed with the two boys, "Yes, that's one way of thinking about numbers as being of two different kinds" and then began to push this space that had now been opened by Mark and Robby a little further. Recording the numbers on the chart paper as I spoke, I posed a question to the whole class." If two, four, six, eight" I began, "and numbers that end with zero, two, four, six, or eight, such as ten that is written as one-zero, twelve that is written as one-two, thirty-jfour [recording 34]... sixty-eignt [recording 68] can be described as being even and one, three, five, seven, and nine, [recording 1, 3, 5, 7, 9] as well as numbers that end with one, three, five, seven, or nine such as eighty-one, forty- three, eighty-Zrve, sixty-seven, twenty-nine [recording 81, 43, 85, 67, 29] can be described as being odd... what is it that makes certain numbers even and other numbers odd?!... besides just looking at the digit that they end with?!" Silence. The children said nothing. One second... two seconds... three seconds... waiting. They simply stared back at me, shrugging their shoulders. Michelle leapt into the conversation and conjectured that "even numbers- you add two to them... two, four, six, eight, ten". Moving into her space of thinking, I poked around a bit and said "yes, this is true... but what about one, three, five, seven [pointing to the series that I had already written down on the paper]... aren't you also adding two to one to get three and two tofiveto get seven...?" "Oh yeah" Michelle said, smiling. "As well" I added, "think about Even Steven... did he do everything in a sequence of two then four, then six, then eight...? or Odd Todd, did he do everything in a sequence of one, three,five,seven, and so on?" "Hmmm... no" Michelle replied, shaking her headfromside to side. I then had the class form a large circle on the carpet. As I moved behind them on the outside of the circle, I randomly asked some of the children to use counters and to build one of the odd or even numbers listed on the chart paper with counters and to place them in the middle of the circle. By doing so, we would be able to continue the exploration but thistime,take a closer look at the numbers in their physical form. Although the children did not arrange the numbers they had built in any particular order, except for separating the even numbers from the odd numbers, we soon had physical expressions of 2, 4, 6, 8, 10, 12 and of 1, 3, 5, 7, 9, 11. I asked the class to look at the numbers that had been built, to talk to one another, and to see if there was anything that was "even" about the even numbers or anything that could be considered as "odd" about the odd numbers. After working with the students on either side of her, Shelby raised her hand to speak and offered this: "The four, it's even because it has two and two." I, as well as some of the other students nodded and smiled. 163  "Yes. That makes sense... wc can think of four as being evenly two and two." I said, replying to Shelbys comment. Taking the idea that the group seemed to be embracing as making some kind of sense to them, I encouraged the class to pick it up' and play with it a little more. "Okay, what about the other numbers that we've built here? Can we take Shelby's idea and... Is there anything that we could say about the other numbers, using her idea of 'even twos?'" Without any talk, the children clustered themselves into smaller working groups, reached into the middle of the circle and set to arranging the counters for each of the even numbers into pairs, by twos. When all the children had finished working, I asked them what sense, if any, they were able to make from what they had done. They explained to me that two was an "even two", four was "two even twos", six was "three even twos" and so on. And when I posed questions about larger numbers like "what about thirty-six?", I quickly got responses such as "that'd be eighteen even twos". Noticing that the children had not said or done anything with the set of odd numbers they had built, I pointed to them and asked, "what about these?" This provoked the children who were sitting nearest the odd numbers to reach into the middle of the circle and begin to move the counters about. Soon comments such as "this one doesn't have a partner", "neither does this one", and "none of them do" began to surface in the conversation as the students arranged each of the numbers by twos. Then, just as I had done with the even numbers, I asked the class what they could say about larger numbers. "What about twentynine?" or any odd number. Here, I got replies of "it would have fourteen even twos and one leftover" and "if you put it into partners, one would always be left lonely". As we sat back and looked at the two sets of numbers now arranged evenly or unevenly by twos, I asked the children if there was any sense in looking at these numbers as being two different kinds, and what sense if any, was there in Mark's claim that you could simply look at the last digit to figure out if it was an even or an odd number. The class agreed that any number could be described as being either odd or even and when asked, some students indicated by nodding their heads that yes, you could just look at the last digit of a number and determine whether it was even or odd. What intrigued me were the different ways of thinking that the children had created for making sense of odd and even numbers. For instance, Danica explained, "A number bigger than ten, like forty-eight, the forty is groups of ten... even... and the eight is four even twos... it's an even number." 164  "And forty-nine?" I asked. "The forty is four groups of ten... that's even, and nine is an odd number because it has four even twos and one leftover, so it's an odd number", Danica explained. "Are you saying that you don't need to look at the other digits, only the last one to determine if the number is odd or even?" I asked. 'Yes... because they are always even... tens... hundreds... they are always something-zero" she said. For other children, determining whether a number was even or odd was based on different ways of thinking than Danica's place-value-last-digit notion. Some children concluded that if a number could be "split into equal halves" then it would be an even number and "if it couldn't, then it would be odd". Other children replied that if they could arrive at the number by counting up by twosfromzero, the number was even, and if they counted past the number using this rule, then the number would be considered to be odd. Still other children said that through building or visualizing the number as as a set of counters and then "partnering up" the counters by twos, they would be able to determine if a number was odd or even, depending if there was one as a remainder or not.  165  Reflecting on this, lesson, 3 was, reminded of Sheodore Stos/uuVs Bo Cult of, Jnfm/natkmJ Jn it, he explains, that as, humans,, it is, ideas, a information that we think with; that it is, through the integration of pat emerge as. a remit of our Hoed experience that ideas, come to Be and fr information arises,. dnd so too, the class, integrated their ideas, of Hoed ma experiences to form meaningful information about odd and even numbers. not blindly accept Mark's, mathematical fact that even numbers are numb end in C, 2,4, 6, 8, and odd numbers are ones, that end in 1,3, 5, 7, or 9 conjecture of "even twos" as "truths". Slather, the class jumped in mathematical spaces that had been opened and explored these spaces, in develop, individual and collective mathematics that made sense. Qnother afterthought: Cl curious, space that was, not opened becaus nor the children thought of it at the time (and the story frames, the reader to assume that all numbers are even or odd) was, this: 3s there ever a number that is NOT odd or even?  Gn another occasion this might have arisen naturally and taken the cl different mathematical place. Nevertheless, this lesson served as, an exa how- the mathematics of odd and even numbers, was, brought forth by mathematical languaging of the story through their physical building of nu mental images of numbers as a collection of objects and symbols,, as we powerful metaphors of "even twos", "partners" and "lonely ones". She mathematical residue that came to be as events of the children's were unpredictable and distinctive yet integrated. JVot only was, there a residue of the children's, learning; that numbers, could be thought in term even or odd, but there were also mathematical knowings within this residu also collective and individual in nature. Shese included the children's noti pluce-iwlue-and-toohing-at-th^ as well as the patterns, they established through their actions, of counting and arranging. £ike different shades, of a each mathematical residue was, distinctive while at the same time, each b the others, adding depth and dimension to the children's understanding of even numbers. Motes 1. Stos/uxk, 1994.  166  Differently from mathematical residue, when mathematics Becomes that which occasions children's, curiosities, it exists, as, the source or a place— a Beginning, for their further mathematical growth to occur. 3or example, By, posing the question, "3s, it true that division always makes things smaller?'' enables a source from which many different mathematical directions and spaces for children to take their learning can occur. Mathematics as a source creates, openings for children, to move mathematically and to deepen their understandings. Qiventikis,children could explore and establish ways of knowing for when and why such a meaning of division would Be appropriate and also experience situations in which an understanding of when and why it may not make sense and mark out another place of knowing.  Arisings (A continuation ofMathematical Language, Languaging & Residues ofLearning)  Settling down infrontof me again, the children looked up, teling m they were ready tofinishthe story. 1 read onfromwhere we had left when we came to the part in the story where Even Steven sets aside six his lunch and Odd Todd comes along and eats three pancakes that the ch very excited. They stopped gazing at mc and turned towards one another, in astonishment. "He [Odd Todd] just m ade Even Steven's pancakes odd!" one studen exclaimed. Another child giggled a nd then whispered to herfriend,"Odd Todd is clever!" Even after w e continued andfinishedreading the story, the children's c about how the even number of pancakes had been "turned into" an odd pancakes had not diminished. When I asked them what it was that had intrigued about the pancake incident, a flurry of responses came at me: "I want to see what odd numbers I can make "Can ALL even numbers be made into odd ones?" "I w onder how Odd Todd would eat his cousin's 8 pancakes?. so they'd b "Or Even Steven's ten p ancakes?!" "Or his thirty-six p ancakes?!!" "I w ant tofindout some other ways Even Steven could eat his twelve pan "Does a n odd number and another odd number always make an even nu Here within t he same lesson, the mathematical ideas of odd and numbers emerged again; not as mathematical residue but this tim mathematical "sources". The children's questions created new places for them explore and develop their understanding of odd and even numbers. And for the two days that folowed, the class created smaler working groups and the questions that they had posed. 168  T^o the metaphors with which we describe befinnin^s and ends teatt\f mattet?  169  J, suppose it is possible one could argue, that even in mechanistic pa thinning, mathematics can exist as both a beginning and an end in chi learning. Jor example, rather than being defined in terms of "source" "residue", the metaphors of "input" and "output" could be used. Sly doi might wonder whether in fact this does not mahe both ways of thinking p the same. CLnd if not, what exactly is the difference?  yes of course, mathematics C A N be thought in terms of "inputs" "outputs" of children's learning. Jiowever, there are critical differences these metaphors and the ecological ones of "residue" and "source", you mechanistic metaphor of "output" by its, aery nature, evokes the idea o mathematics, as being a product that is produced or re-produced as a res chain of learning events taking place. "Jnput" conjures, up the notion identifiable, measurable, and even prescribed "ingredients" being used "added to" in order to produce a certain mathematical output, result, o Sogether, these metaphors, embed a sensibility that assumes specific even rise to particular mathematical outcomes can be identified as, such and mathematical outcomes, "fueled!' by mathematical inputs, are predicta perhaps, can even be predetermined. SJhese ways, of thinking about how m exists in children's learning serve only to maintain a confining and reduc view of classroom mathematics. Jn a very different way, the integrity of the ecological metaphor mathematics as, being a "residue" and a "source" for children's learning is do not embed a deterministic or predeternunistic view- of classroom ma Slather, mathematics, as, "residue'' evokes a notion of something that has what is left, or what remains as a result of children's learning. Jt is not a thinking that engenders a aeterministic stance that presupposes that the mathematical residue can 6e precisely located or that the process(es) o languaging acts by which it came to be can be identified, specified, or repl metaphors focus attention on children's mathematics, at hand-- the mathem mathematical understanding that is emerging. Just as important, is, the e metaphor of mathematics as a "source" for children's learning. Jt contrasts mechanistic image of mathematics as, "input' in that the metaphor of " brings, with it, a sense of beginning; a beginning that is not concerne predetermitiing or predicting what "should" follow, but instead, highlight to be mindful and open toward the possibilities that "could!' unfold" antic not. Jmagining mathematics, to be both residue and source infuses, a sens is, open (as when the ground opens, and water springs forth) and ever-cha how one can never step into the same river twice). Jt offers- different 170  and meanings to consider when thinking about how- mathematics might classroom... images and meanings that mechanistic ones cannot.  171  r \ f t e r having spent considerable time moving deeper into that other mindspace, Jennifer emerged with a new understanding of what it meant to teach mathematics. Her work in re-imagining metaphors that possessed an ecological sensibility was occasioned only because she now knew the two systems of thinking to be what Maturana calls consensual domains . She realized how her previous way of making sense of her teaching and her new ecological one highlighted and diminished particular issues or concepts through their different metaphorical languages. For Jennifer, it was impossible to conceive language as simply a tool for communication. 1  2  We exist in language and it is through our Being in language and languaging that ate Bring forth metaphors, that invisibly and powerfully Become our ways of thinning, how we teach, our ways of researching, and ultimately, the kinds, of places that are created in the mathematics classroom.  Notes 1. Maturana, 1988b. 2. "Different" here, does not necessarily mean "incompatible" or "disparate" but rather, "diverse".  172  "Atmosphere," as the word suggests, is a vaporlike sphere which envelopes and affects everything.... The sense of mood or atmosphere is a profound part of our existence. By it we know the character of the world around us. Mood is a way of knowing and being in the world.... the way in which space is lived and experienced. 1  1  van Manen, 1986, p. 32  classroom, every school contains a certain atmosphere [sense of place]. The question is not whether there should be a pervasive atmosphere in the [classroom or the] school, but rather what kind is proper for it, worthy of it. [E]very  2  Emphasis added, van Manen, 1986, p. 31.  Sense of place. Place-making. van Marten's thinning about atmosphere echoes, the very, notions, being explored here of, ecological thinking and of place.  Worthiness She author moves, the conversation into the realm of education and highlights, the taken for grantedness, of atmosphere. Me identifies, it as, being that which profoundly, shapes, teaching and (learning, van Manen makes, important, the need to be aware of it, to consider the qualities, that make up, a particular atmosphere, and to exercise a mindfulness and care for the sense of place we bring forth for and together with our learners. Shis, certainly implicates the examination of one's metaphors, and the mathematics teaching and learning occasioned from the embodiment of them. Sn doing so, van Manen hopes to provoke us, to question the worthiness, of our actions, in relation to the hind of place such rituals, in teaching mathematics create.  Because this tacit knowledge influences both what and howlearning occurs- for all participants.... In some instances, the teacher must keep up with the social reconstituting of takenfor-granted knowledge, and in other instances... the teacher should take a leadership role that can only be fulfilled by modeling and not simply by substituting a new set of takenfor-granted beliefs for the older ones. 3  3  Emphasis added, Bowers & Flinders, 1991, p. 11.  She "leadership, rale" that Siawers and Minders speak of is, important in developing- one's teaching of mathematics. Jtelating with earlier sentiments, growth as- a teacher is not about breaking away from one's entire teaching and replacing it with something else. Mat only is the possibility of this questionable, such thinking only exacerbates 'thisor-that', 'either-or' attitudes and reactions, differently, what Sfowers and blinders argue for is in keeping with what van Manen too seeks. Situated in the mathematics classroom,tikisentails a teacher's MINDFUL consideration for the KIND of oikos, being created for learners through what one chooses ta conserve in one's ways, of knowing and actions as, a mathematics teacher. Jor me, this, has meant assessing the worthiness of the metaphors embedded in my teaching of mathematics, identifying which ones did not engender an ecological sensibility and re-imagining ones that would. (Sne might assume that doing so should make the enactment of these conceptual shifts natural and effortless given that J, was now thinking within a different theoretical system. Jiowever, each day that J, stepped into the classroom, J, was confronted by the fact that...  i m  r e m ii nnda ee d that  l e a r n i n g  what i t  even though 3, have consciously created "differences" in horn 3, conceiv 1  means  to  teach  m a t h e m a t i c s  is  not  an  mathematics teaching, in the classroom-- as part of this place, 3, am still a i  ,  a u t o m a t i c process, i t s n o t  .  1  s m o o t h ,  •„  y  it s not  always will he unconsciously embedded in a web of taken for granted  straightforward,  and  it  certainly  doesn't  relationships-- taken for granted language in which and by, which 3, tea  >ear o n a e m a n  and children learn mathematics.  2  SJhe challenge then becomes, which relationships need to be a critical of one's teaching consciousness?  cHow mi$kt one $o about catvln$ out difrfietent spacesfiotteachi ^staUish new telatlonsktys?  Expose a difrfjetent kind crfi place -foot mathematics to y  178  (Zssuming teaching to- he praxis and net a practice imparts, the understanding that one can never Be separate from the students themselves (and the metaphors theg. Bring tc their understanding of. place), the events that unfold, or the relationships that exist within the classroom. Oil coemerge and co-evolve as living, systems WITHIN systems dynamically responding to one another. One's, teaching then is not identified in so much WHAT it is But instead, HOW it is— horn it is, an interdependent part of the whole. So for me, my work is, concerned with how my mathematics, teaching relates to the larger whole of embedding an ecological sense of place for mathematics, in the classroom. (Is well, there is not the anxious, temptation to systemATically take my new found metaphors, and fit them into the classroom. 5iy situating my study of mathematics teaching within the systemic realm, the focus. Becomes grounded in understanding how my metaphors think me (in ecologically coherent ways) as J, think within them. Once again mooing eff the mental line enables me to head towards that "other" space. 3  oikos... relationships... relationships as patterns... ... as patterns of difference. So in the same way that Maturana and Varela speak of knowledge as constitutive in nature; that is, as dynamic structural relationships within a living system and that which distinguishes one living system from another, Q. Skdeson's* concept ef mind as, connecting patterns ef differentiation is, also what is, at the heart of this, work. Hocusing on difference in an ecological manner means recognizing difference as a relationship, and not as, a thing.  "Hps  A? knowing  is doina is 6eiH4  TEfe.  tMtulktitu'  7  K 1 ^ I  ^3  r- -  iii  180  She difference that makes a difference in ma teaching of mathematic as teaming, place centres on the impact that ma metaphorical patter of thinhing have on the oikos of it. So what are the patterns of difference in ma new metaphors?  Jn contrast to the metaphors embedded in my previous conceptualisations of the mathematics class, mathematics,, and curricula as a collection of individual "parts", mathematics teaching as a linear chain of events, and mathematical learning and understanding, as building structures, these new metaphors, of maps, paths, living systems, residue, and source evoke notions of nonlinearity and unpredictability. SJhey speak of mathematics teaching and learning a complex and recursive growth. SJhey create the image of everythin existing in fluid relationship to everything else. SJhey enable the integration of mathematics to be understood not as a product of teaching and learning but as that which happens in the flow of the two. Qnd, that neither mathematics teaching nor the learning of it n mathematics itself for that matter, exist as "things" but all arise in th midst of dynatnic interplay... varying in form and being occasioned in different ways.  dhese, J weald say are the patterns of difference in my metaphors.  182  Metes  1. Q.fBatesen, 1972,1975. 2. (Sf course, this, also, includes the tahen-ppc-gxanted language and langu society and particularly, horn this has influenced the students.. 3. engaging in the examination of hew- "language thinks us as we think language." 4. Q. fBatesen, 1972, p. 318.  183  S-mkeddLnf And TlootlHf an oj  ^blace jot  /Katkematlcs  £Lcolo$ical  In the  Sense C^Lasstoom  What we conserve, what we wish to conserve in our living, is what Knowing the ground on which I walk Minding the paths that unfold.  determines what can and what cannot change in our lives. 1  1  (Maturana, 1997b, p. 5)  185  [c]hildren must feel that... the space, materials, and projects, values and sustains (sic) their interaction and communication. 2  2  Rinaldi, 1990 as cited in Edwards, 1993, p. 137.  ^tten<iln$ to the  Space In the C^tasstoom  187  Stepping into a classroom is not unlike studying a painting. Both tell of relationships- the assumptions, ways of knowing, and the kind of experiences that have come to form a classroom's sense of place or the subject matter of a painting. In this manner, albeit through different methods, teachers and artists engage in the act of portraiture. In creating a painting, these relationships are expressed through the media and design of the work. The feeling of coldness and tension for example, might be told through the painter's use of colour and texture whereas coldness and tension in the mathematics classroom is often conveyed through the austere and rigid manner in which the subject is presented by the teacher. Connected to this is how a teacher structures the actual physical space for learning and the types of materials made available for students' use that define the kinds of teaching-andlearning patterns that exist.  Stepping Into Jennifer s Classroom What  is most apparent upon entering Jennifer's classroom is that there are many different areas that make up the space. Like a house with many rooms but without the walls and partitions, this class too has different "rooms". Each of these rooms or what she considers to be physical spaces for learning, are distinct in their purpose and the kinds of mathematical interactions they enable.  188  Small e r o u p w o r k a t t h e i r d e s k s  Large eroup work a t the round table  The children's desks are arranged in 2 x 2 groupings in the middle of the room from which the chalkboard, chart board, and overhead projector screen can be seen. Although the desks store student belongings and each bares a child's name, the childrenfrequendymove about and use one another's desks. This area allows the students room to work on their own, with a partner, or in small groups. When in larger groups, the children often choose to work at the large round table located acrossfromthese desks.  Meeting place  O n the  floor  Two carpeted spaces can be found on either side of the desk area. The larger of the two is a meeting place of sorts, where Jennifer and her students gather in the morning and at the close of each school day. This is also the place where the class meets to share and examine the mathematics they have been working on. Like the table groupings, the carpet allows the students to work on mathematics on their own, in pairs, as a small group, but also, as one large class. This space, the round 189  tabic, and the cluster of desks provide useful places for the students to work with mathematical manipulatives.  Physical mathematical worlc at the chalkboard On the other side of the desk area is the smaller of the two carpeted spaces. This one was created by Jennifer so that she and her students could use the chalkboard, overhead projector, and pocket chart during mathematics lessons. All of the equipment is positioned so that they are at appropriate height for the children to access. This place of gathering is not so much for active, physical investigations of  mathematics, but for the class to come and record, display, or bring key mathematical ideas into focus forfartherstudies. Finally, Jennifer uses still other physical spaces to extend the children's mathematical learning beyond the walls of the classroom. They include an adjoining classroom that provides a large, completely open area in which to work, the children's homes, the natural environment, and a neighboring elementary school. The latter three offer contexts where the students can take the mathematics that they have developed in the classroom and explore the relationships that Me within these other settings.  Materials Matter  Along one of the walls in Jennifer's classroom are three large shelving units:  one with blue drawers, one with yellow, and one with red. Each shelving unit contains sixteen drawers, that hold a variety of materials used for mathematics. The blue drawers contain mathematical games and puzzles. The yellow drawers contain calculators and different kinds of manipulatives for mathematical investigations. They are used by the class as physical objects with which to think about particular mathematics or with which to express their mathematical thinking. They include such items as: as number cubes, tangrams, calculators, zaks, geoboards, multi-link cubes, pattern blocks, attribute blocks, base ten blocks, and double-sided counters. There are also many found items that the children have collected their homes and neighbourhood such as different kinds of rice, pasta, beans, seeds, stones, chestnuts, and buttons. Here, the students make use of these materials for purposes of sorting and classifying, estimating and measuring (i.e., linear, area, volume, mass), computations, as well as number concepts and counting strategies. The third set of red drawers contain papers of all types: plain, lined, graph, dot, and construction. These are used by the class to build or record their mathematics. Together, the materials and organization of spaces for learning in this classroom create a portrait that tells of an open, accessible place that nurtures the' growth of children's mathematical thinking. The classroom as a physical and expressiveformof the relationships that are embedded here, communicates the importance for mathematical interactions to be flexible and diverse. 1  from  Notes 1. These are interlocking polygons used for building 3-D structures. 191  An environment is a living, changing system. More than the physical space, it includes the way time is structured and the roles we are expected to play. It conditions how we feel, think, and behave. 1  1  Emphasis added, Greenman, 1988, p. 5.  "RBSQLUTE, true and mathematical TIME, of itself and from its own nature, flows equably without regard to anything external, remains similar and immouable...." 1  sure in its certainty TIME lives its own independent existence... SEPARATE  and  d i s c o n n e c t e d . . .  created and affected by no person no space  no-thing... TIME does not exist with us or because of us... TIME is just "out there". ... and even though... TIME is undetectable through human perception ... 2  1 2  Emphasis added, Newton, 1687 as cited in Koyre, 1958, p. 161-162. Abram, 1996.  193  it is REAL. and even though... TIME is empty... spaceless... invisible... and...  h  a  s pe  less....  it is REAL.  TIME i s e x a c t l y i n f i n i t e . . . i t can't be taken a-p-a-r-t or pressed together i t can't be disposed <* i t just I S  trust it is there.  "Time is just this: the number of a motion with respect to  195  the prior and the posterior" Time as relative — as linear — as 1  1  Aristotle, 1969. 196  straight as an arrow — flatly fuses with events — figuring the  197  spaces — delineating relationships — defining moments —  198  stringing beads — one after another after  another  after  199  another — each as a middle that unfolds from a beginning  200  and towards an end — forward moving — this kind of time  201  — these kinds of spaces — can be measured — numbered —  202  separated — sorted — sequenced —  fixed —  located —  203  walking the temporal tightrope —  PAST behind us —  that  204  was then — PRESENT — this is now — FUTURE AHEAD —  205  how  much  further  ?  206  keep going  207  T. C McLunan, l^fl/, p. }2->5. Abram, Bowers, 15^7; Maturana, ' Kabat-Zinn, \99+, p. +. Maturana, ZOOl. ' Maturana, 2001. ' HekJe^=r, 1?24-, p. H-E. Abram, \9?6; Hdde^er,1?24-,p. 22E. 1  1  2001.  +  7  208  £na.ctin$  *Reconceftt utilizations  of  ^Z-ime.  209  Creating a classroom that engenders a systemic sense of place fm mathematics requires consideration of horn time is understood, horn it is enacted through one's teaching, and the impact it has on shaping the spaces, where mathematics occurs. SJhe languaging effect that my unconscious linear image of time had on the structuring ef my, mathematics program and the kinds of learning spaces that were then actualized is a clear example of this. Distinguished by grade levels, set within clearly marked Boundaries of a Monday through Friday, 11:20 am to 12:00 pm and September through June frameworks, J, conserved and delineated a place for mathematics in the classroom as a means to an end. She mathematical content ef each lesson picked up, from where the last one left off and moved the students forward in a concrete to abstract fashion to the next preplanned lesson. Sjogether, these lessons formed instructional "units" that in turn became the year's, mathematics program. Sn the larger scheme of things, each of these programs served as component parts within the SC-12 mathematics curriculum My linear differencing of time "mapped" in my responsibility as teacher to provide mathematics programs that functioned as links in a curricular chain and progressed the children in their mathematics from one year to the next. Sn doing so, a straightforward linear timespace of mathematics teaching and learning was created and maintained. Consequently, what my conceptualization of time did not map in was the possibility for time-spaces to embody fluidity, nonlinearity, and recursion-- critical qualities inherent in my new metaphors.  XOhat if time was enacted as nonlinear, flowing, and recursive?  cHow might it be occasioned in the mathematics classroom? ®  TO hat role could it play in re-placing mathematics teaching and learning  TO hat kinds of mathematics might then emerge?  211  'Raising Questions and Questioning the /Instcets jSlneAt  tyltne-Spaces  fot  /Hatkematlcs  In  the dLassioom  212  My engagement in wondering, WHAT IF? has, revealed ali hinds of unnoticed linear, and static time-spaces with which 3, have furnished my mathematics program. ^Becoming aware of them and (kinging them into question has, not been unlike my experience in confronting and re-imagining new metaphors. ASSESSING THIS  FURNITURE:  Why do 3 organize my  mathematics teaching into two separate programs? Slonestly, 3 could not come up, with any meaningful reasons for why 3, separated and taught two one year mathematics programs. Sure, 3, could sag, that organizing time in this way clearly defined what students were to learn in second and third grade mathematics. 3, could also say, that it served as the base from which 3, could plan the program overview, units of instruction, and individual lesson plans, Siowever, given the metaphors with which 3, was now working, none of these explanations of cut and dried efficiency made sense to me anymore. Categorically complicated! and fixed, they were devoid of anything that assumes complexity or strives to be life-giving. Up until now-, the issue of time merging with space had never been an important consideration of my teaching. 3t was not a part of my map of what it means to teach mathematics. £inear ways of structuring time and spaces in which to teach mathematics was what was taken for granted and ritualized by everybody-- administrators, teachers, parents, and the students, themselves. Siut now. the issue of time was emerging as a difference of critical importance that would not go away and demanded my attention. JVow visible through confronting my temporal ignorance, contemplating its incoherence in my teaching and compounding this with my desire to embed an ecological sense of place in tile classroom for my students' mathematics, rethinking my enactment of time proved to occasion new patterns, of difference for the coming school year...  I continued to teach second and third grade students but began preparing for the upcoming year of teaching by considering the kinds of temporal patterns necessary for opening teaching and learning spaces that were organic and generative. No longer wanting to teach mathematics in a grade to grade manner but in away that focused on nurturing an ecological sense of place and the dynamic growth of the children's mathematical understanding, I made the decision to teach a two year program as opposed to two separate one year programs or a single grade class. By doing so, I was able to expand the time-spacefromten to twenty months. In contrast to my image of teaching and learning as a linear time-space, I envisioned the first year of the program as being an enveloping and co-emergent layer that grew out of the children's previous inner layers of individual and collective mathematical activity. The second year was too conceived as a living curricular system and one that would create further layering of the children's mathematics. The Pirie-Kieren model of growth of mathematical understanding was a critical part of the structuring of the program as it locates children's embodied mathematical knowings as being where all new knowledge develops and thus, the place where the two year program would begin. Iftimeis to be enacted as nonlinear and recursive, then the space in which it is a part must also emerge as such. Thinking systemically about the growth of mathematical understanding necessitates teaching to be a dynamic and responsive activity. Teaching and learning as living systems unfold moment to moment, coexisting and co-evolving in relation to each other. Although the two are viewed as activities that cannot be prescripted, this does not assume that responsive teaching does not require anticipative preparation on the teachers' part or that it is a random activity. As praxis, teaching responsively means being attentive and mindful towards how one'steachingimpacts and is impacted by the class' mathematical work. 2  3  A mathematics curriculum envisioned as a map, enables teachers to locate I identified important learning aims that included the goals of the Ministry mathematical topics, concepts, and skills that are considered to be important and other documents (e.g., NCTM, BCAMT) forthe second and third grades but landmarksforthe class learning... what cannot be sketched out in advance, are the instead of sequencing them or categorizing them according to grade levels, I actual paths that the children will travel... to get to the mathematical locations... or marked these aims on my two year map as important mathematical sites to be 4  5  214  the understandings they will establish when and after they come to these sites... the explored. As I did this, I also considered contexts that would be open to many map... cannot possibly show the diversity of the landscape... [children's mathematical different kinds of mathematical investigations and ones that the class might revisit paths are rarely if ever, straightforward but instead, spread out in several directions later on in the year or even again in die second year. While I mapped out a and entail twists, turns and switch-backs... Mathematics curricula and learning a curriculum, I was ever mindful that it was just that- a map; distinguishing imagined in this manner distinguishes them as co-emergent phenomena that are pedagogical and mathematical features for teaching, and that the actual forms brought into being through children, their teacher, and mathematical, settings, that my teaching and the students' learning would take on were yet to unfold. 6  Because it was my intention for the program not to have a prescribed teaching sequence, it was important for the learning spaces to be ones that not only had an open flow in terms of mathematical content but also, ones that would encourage the reintegration and renewal of the children's mathematics. To do this, provocative themes were created by myself, with the children themselves, and with other teachers with whom I collaborated. These themes arose from the children's ongoing work and usually came in the form of curious questions or specific topics such as Who are these things we call numbers?, SnowQakes, Numb and Mathematics About Me. So instead of instructional units organized by particular mathematics such as addition, subtraction, or geometry, these themes focused my teaching and the students'work on exploring mathematics as a diverse and interconnected whole. I It was here that I realized how the integrity of this program would be compromised if I continued theritualof scheduling mathematics into 40 minute daily intervals. Affirming my disbelief that children only have short attention spans, I was inspired by the stories of the Reggio Emilia schools and looked for ways in which I might expand and enable flexibletime-spacesto exist in the classroom. I seeded the program with the idea of establishing a place that focused on students' mathematical growth. It made sense then, for these themes to occur in time-space that not only allowed for the children to work on ongoing projects, but also ones which enabled the mathematics to shar^-shift into different forms of mathematical languaging, to branch off, intersect, and flow from realfifecontexts into purely mathematical ones and vise versa. Instead of cutting up and inserting mathematics lessons into 40 minutetimesslots, I opened up larger spaces of time within the day and even entire school days to allow for this. 7  8  215  Ser the past two- months, the children and 3 have been working on a variety of. projects and investigations that focus on their meaning^mahing af numbers within different contexts. Shis week, the class watched the film Motes on a SMangie? Steviewing the film on several occasions, and from different mathematical perspectives piqued the children's curiosities. She vantage points they, marked out— all of which happened to revolve around the idea of 'three', established the specific mathematical spaces for the children's explorations. Snside these learning spaces, the students worked to develop, understandings for the threeness of a triangle, the threeness of particular numbers, why three is considered odd and when it becomes even, the rhythmic pattern of three-four time, and the aesthetic value of three in creating visual art. 3tere, concepts of addition, multiplication, number theory, patterns, and geometry were brought into being through the class' engagement with the film.  Shese time-spaces contrast with previous lessons in which 3, clearly marked beginnings, middles,, and ends (ie.., introduction, development, and conclusion). SJhey serve as examples of how mathematics curricula can arise in forms that are generative and embody a sense of flow... where time and space are taken to be inseparable and give rise to one another within to the contextual boundaries of the class' mathematical experiences. Qlso highlighted is how mathematical concepts emerge from the class in a way that assumes mathematics to be a living system. 3n these lessons, the mathematics taken up, were not explorations in the practice of dissecting and dismembering but quite the opposite. Studying the film as an entity— as a mathematical form in itself allowed it to be viewed as a source from which further studies could 6e investigated and connected to one another*~ keeping mathematics intact and a dynamic whole.  Mates  1. a. Si. S)uvis et al., 2000. 2. Shis"ideaof an inside cane parallels, the JUrie-Jiieren (1994b) dynamical of. growth of, mathematical understanding which defines primitive kn Being all the knowledge that a child or group, of children bring ta the pa mathematics and from which all new understandings, develop* Jt is this mathematical knowledge that J, assumed ma students, possessed upon en two gear program. 3. M. C. Siateson, 1994. 4. Ministry, of education, 1995. 5. Star example, resource material and educational literature published Mxdional Council for Seachers of Mathematics, and Slritish Columbia CLss for Mathematics, Seachers. 6. Jn the same manner as the themes in this program, the mathemati investigations, were "open ended" in structure, emphasis was placed students' active engagement in problem solving and problem posing "nonHautine" settings. Jor example, see Shawn and Walter, 1983; Qo 1994; Jtesh, 1981; Schoenfeld, 1985; Silver, 1994; Walter and JSrown, 19 7.2ardine,1999. 8. Mew, 1993. 9. Motional Mm Shard, 1969.  217  Me 4 3t& of SUcwuiott Jtef£ectiay  • SUtU&Uing. • ^Reintegrating • ^Renewal  M. C. Jkuesan's description of learning as traveling along a Alotuus strip, continues to be a powerful image that prompts me to examine and assess the relationships, embedded in my teaching. Jt also challenges me to consider how J, might consciously enable recursive forms of learning in the classroom. JJf mathematical growth is viewed from an enactive perspective, tike other forms of (knowing, it is an embodied phenomenon that develops through relating and ^experiencing mathematics from an "opposite Aide... a contrasting point of view... or seen suddenly, through the eyes of an outsider". Jiecause of this, children's learning cannot be achieved through repetitious acts of reproduction or sequential assembly lines of tasks. Woing so implies learning is a matter of practising by redoing what one already knows or taking what one knows and adding to it in a piecemeal manner. 1  Mathematical growth as a recursive event? connotes the actual changing of one's mathematical understanding in ways that are complex and co-emergent. While its evolution possesses qualities of self-similarity based on primitive knowings or inner layers,, what it becomes and what it occasions upon each recursion is something qualitatively rather than quantitatively different.  Jiecursion in this, sense is. net just a better understanding, it isa new understanding that is more than the sum of. its parts. Mathematics teaching that provokes recursive forms of (earning, is more than C=A+B. Jnstead of understanding as grouting through an additive action, it is seeing the mathematics in a different wag,. Similar to the change in potential possibilities that occur as a result of structural coupling or drift, this is represented in the 3*irie and JCieren model by. the "don't need boundaries''; where once beyond a don't need boundary, one does not need the specific inner understanding that gave rise to the outer knowing. Jor example, a child who takes nine counters, arranges them into pairs, finds out that one is leftover, learns, that this distinguishes, the number as. being "odd" and then repeats these actions, to find out that eleven is also an odd number is, a qualitatively, different hind of understanding than that of a child who works, with various numbers and then explains that "any number is odd if you divide it in two equal groups, and there is one [whole] leftover". Shrough making meaningful images for specific numbers, the first child has added to or thickened her understanding by performing the strategy of pairing up, the counters and looking for one that is leftover to find out that eleven too is, an odd number. She second child however, has explored different numbers and identified a pattern or relationship, that can be applied to ANY number to determine whether (or not) it is odd. Shis child is not making sense of the mathematics in a repetitive manner or on a situation specific basis but conceptualising the idea of odd numbers in a different and general way— for ANY instance. Shis child's thinking comes, from a place of formalising that is located on the JUrie-3iieren model beyond the don't need boundary which separates it from primitive knowing, image making, image having, and property noticing.  So, what does this conception of teaming* mean fm teaching in the classroom? Jt places importance on making space fm children to reflect on their mathematical patterns of thinking and to revisit their mathematics inside different contexts so that they may critique what they understand from their current place of knowing. Opening teaming spaces tike this not only allow fm students to relate to the mathematics in multiple ways, hut it also creates the possibility for occasions in which children can reintegrate and thus renew their mathematical understandings kg, conceptualizing them differently. Jn wanting my students' learning to 6e recursive , one of my aims, fm the two year program is to establish opportunities for the 43ts, to take place; to engage the children to reflect on, revisit, reintegrate, and renew thevc mathematical understandings. Shis means that learning is not just an event which happens when one encounters something for the first time. Jt occurs when one comes at that something from an opposite side. Sahing this image of traveling along a mobius strip, and situating it in the context of classroom mathematics where that "something'' is mathematics, J, am curious as to what an opposite side (or, sides?l) might look tike in the mathematics, classroom, what forms, they might take, and what hinds ef understandings will arise. 3  Motes 1. M. C SkUeson, 1994, p. 14. 2. Siieren and Side, 1991. 3. Doll, 1993.  220  Jt places importance on making space for children to reflect an their mathematical patterns, of thinking and to revisit their mathematics inside different contexts so that then may critique what they understand from their current place of knowing. (Opening learning spaces tike this not only allows for students to relate to the mathematics in multiple ways, but it also creates the possibility for occasions in which children can reintegrate and thus renew their mathematical understandings by conceptualising them differently. Jn wanting my students' learning to be recursive , one of my aims for the two uear program is to establish opportunities for the 4-Jts to take place; to engage the children to reflect on, revisit, reintegrate, and renew their mathematical understandings. SJhis means that learning is not just an event that happens, when one encounters, something for the firsttime.Jt occurs when one comes at that something from an opposite side. Jlaking this image of traveling along a mokius strip, and situating it in the context of classroom mathematics where that "something'' is mathematics, 3, am curious as to what an opposite side (or, sides?!) might look like in the mathematics classroom, what forms they might take, and what kinds of understandings will arise. 3  Making Three Spaces for Recursion Can You Guess Our Mystery Number?!! 'were mathematical "gifts" created and exchanged with the class' math buddies at a different elementary school. This project brought together several weeks of class investigations in which Jennifer focused on the students' development of different images for thinking with and expressing numbers through the language of: manipulative models, pictorial, informal or formal symbols (verbal or written), and real world situations through dramatization and descriptions. The children worked with a partner to craft a set of clues that would become the riddle for their chosen "mystery" number. Accompanying this list of clues, the students had to also provide eight other numbers in addition to the mystery one as possible choices from which their math buddies could identify the mystery number. Playful as this project was, Jennifer had taken care in designing it so that the structure of the activity would encourage the students' to reflect, integrate, and reintegrate their conceptual understanding of number. By engaging the children in both the making and receiving of the mathematical gifts, Jennifer effectively opened three mathematical spaces for her students' recursion- first, to reflect and integrate their understanding of number, second, to create a riddle by reintegrating their thinking and taking on a problem posing perspective, and third, to identify their math buddies' mystery number by using their understanding in a problem solving manner. Before the children set off on the riddle making project, Jennifer gathered the class onto the carpet and asked them to think about the number investigations they had done and what they now understood about numbers. She explained to her students that by collecting these images and recording them as a web on the chart paper, they could then use these to craft their clues. As the children volunteered their ideas about specific numbers, Jennifer wrote them down on the chart paper. For those who found this challenging, Jennifer encouraged them to think about a specific number and then fold back and pick out a particular image that would be true or characteristic of that number. After the class shared a few examples and nonexamples for certain numbers, Jennifer then prompted the children to see if they could take their specific number image and property notice or express it in a more formal manner; as a descriptor for any number. Here, the children produced a variety of images and expressed that any number could be characterized in terms of: the number by which you could skip countup to it from zero or skip count down from it to reach zero, whether or not it 2  222  fell between two given values on the number line, whether the number was divisi by a certain value, if it could be arranged into equal groups, if the number was m than (or, how much more than) or less than (or, how much less th whether it was considered to be even or odd, what the sum of the numbers dig were, and characteristics of the number in terms of the place value of its digits. Once these had been recorded on the chart, Jennifer helped the students to find their partners and a place in the classroom so that they could get started on making the mathematical gift for their buddies.  Entering the second learning space, Mark and Danica left the carpet and  went and found a table nearby to begin working on the project. After a brief discussion about the possible numbers that it could be, the two decided that 80 would be their mystery number. Taking a suggestionfromone of their classmates, Danica and Mark thought it a good idea tofirstbrainstorm and record as many different kinds of clues as they could think of and then choose the "best" ones for their riddle. That afternoon, Jennifer asked the class to share the clues they had been working on and to discuss some of considerations they became aware of while working with their partner. She reminded the students to use the web of ideas that they had generated as a reference to help them critique their peers' clues regarding the variety and kinds of images being used. As well as giving clues that began with The number IS...", some partners had also incorporated clues that started with The number is NOT..." or The number does NOT..." When Jennifer commented that this was a clever thing to do, the class agreed and they began discussing why using "not" in their clues encouraged a different kind of thinking for the person solving theriddle.Here, the idea of having tofirstunderstand the meaning of the clue and then having to be able tofigureout what the opposite of it would be, delighted the children. On returning to their tables, partners giggled to each other as they thought the process but in reverse so that they could craft these clues into their riddles. Now satisfied with their store of clues, Danica and Mark set to making decisions as to which ones would be used for theirriddle.It was here they noticed that some of their clues made it obvious to the receiver what the mystery number was while others were descriptive but not as telling. Mark and Danica sorted their clues and organized in such a way that their riddle opened with general characteristics about the number and then move towards ones that were more specific. The two students also thought it clever to intersperse these with a few that were redundant or unnecessary such as"... is less than 139" after already giving the clue "... is between 2 and I2I". During another whole group discussion, the entire class 223  agreed that the 'sum of the digits in the number clue was the most telling of them all and that it should appear as thefinalone. Finally, Danica and Mark moved onto generating the eight other possible numbers for their riddle. They decided that each of the eight numbers had to fit at least two or more but not all of the given clues and they were careful to select only those that satisfied this criteria. This way", Mark explained, Tommy and Ewan will have to think [as to] why these aren't the mystery number." They might think the mystery number isfifty[because its between 2 and 121, is an even number, is less than 139, does not have a 4 in the ones place, can be put into groups of 5, and it's more than 10] but it's not because its digits don't make eight", added Danica. By the close of the afternoon, the Mark and Danica's gift was wrapped in a riddle and ready to give to their math buddies.  224  iliLn  You  Guess O u r  Mystery  .Number?!  o Stc if ya* ran find our mystery numtar. Here ar<? ttie tJiic-s.  *Jric total of i-.31 lhe difits :r. the number is _ >  Here i i r t  your choices!  _S7f J2  V\0  •  ivi.v  .tM hen. .:lx »w fess M  -VI.J  rk*fcureiKme»f fl.^uvT  Danica and Marie's mystery number riddle for Ewan and Tommy 225  One week later, Danica and Mark and Ewan and Tommy received their mystery numberriddlesfromeach other. This third learning space that Jennifer created in the structuring of this project required the children to take on a problem solving role as opposed to their previous problem posing one. Not only were each set of partners to identify what they thought the mystery number to be, but they also needed to provide justifications for why the other eight numbers could not be the mystery number. We think your mystery number t$ %,Q. because: It couldn't hc  }flfl5  because  it  ill  ii  jflflrtftlflfl ....  It couldn't be _7 5  AWftML  because  _™  ™  _____  :  because it fS Irftflre thafl  Ii couldn't be IMfO J i a _  hJgmiM-  _  __;______,,  It couldn't, be _J£jL_ because It jjnLib.4jmjeL4iifla-_  It couldn't be  '_JU  It couldn't he  _J  _  because it f,flftl_k-Mf  iJ^J^MJ^jSmt.. J i l L _  It couldn't be  jlhift-jo  ™  fcftS (k fg^f .  _  because  TT  itjS iSSiJ&1.  L__ because itj^£S4___ _  Finally, it couldn't be _£iL_ because when you add _5_ and together, you get a total of 5 not Ewan and Tommy's response to Danica and Marie's riddle  Notes 1. This project was adaptedfrom"Find the Secret Number". See Liedtke, 1983, p. 89.  2. Martin, 1999. 226  In a different manner, sometimes recursion came as an occasion of renewal. These were times-spaces in which the students revisited a familiar context, thought about the mathematics they had brought forth, and worked on connecting related but different ways of thinking about the mathematics than they had before.  227  A  Snowflake of a Different K i n d  l\s part of the snowflake studies that the class was investigating outdoors, in photographs, on video, and through stories, the children constructed their own Koch snowflake. Each student was to identify an interesting pattern that they observed while creating the snowflake.  Christina's snowfialce  228  (Xs the students worked on their own to find a pattern in their snowflakes, there woo. definitely, an animate quality in the way the children perceived the mathematical features of the snawflake. SJhrougheut the lesson, students, were heard talking, to themselves about how the patterns in the mowflake were "growing". On observation like this elucidates the conception that the students held toward their mathematics, Jt points to the fact that they did not see the purpose of the task to he one of locating "the" answer or as the retrieval of preexistent static mathematics. Communicated in their verbal descriptions of growing patterns is the understanding that this activity was about "keeping watch" for the mathematics, to emerge.  Excited about the patterns they had found and fascinated that each one continued on "forever", the students decided to compose a short story that captured the mathematics they saw arising from the snowflake.  229  \  »'  /  • rrir'« A<>h»:r ,.1:* ,  /'""'  ^"••""\  taff  Jamuvcy: Julie's storu is about a triangle turned fractal 6u its, ever increasing  . ^ f l mF  -JLi m__€ S.i__e*> T h e pattern o f "sides" in Julie's snowflalce  230  January: Shoaji's story described the pattern he saw in the number of triangles, he added to- each subsequent layer of the fractal.  •  "One.  "One triangle, one triangle, one triangle."  T w o here, here, here, here, here, and here."  "Eight, eight, eight, eight, eight, eight."  4* §n*..M3 j * * ,  ..±tiangles.  The pattern of "triangles" in Shouji's snowflake  231  Januaru-: Clare was fascinated % the "corners" that were emerging in fractal. When ashed to. explain what she considered a "corner" t she defined it as an "elbow-" in the snowflake; where the edge fractal "turned direction." Jhus, the corners that Clare was count were located on the perimeter of the fractal.  "One, one, and one."  "Three, three, three." [three in between each of the three original "corners'*]  f\lf&XWLJ*.aw±  Then it changes to nine corners and seven corners, nine and seven, and nine and seven." [She arrives at a total thirty-six additional corners by subtracting 3 and 9from48]  T h e pattern o f "corners" in Clare's snowflake  232  The class revisited their snowflakes on three other occasions during the school year. For each of these sessions, Shouji, Julie, and Clare considered the total number of triangles, sides, or corners that they had identified previously in each of their fractals. Re-viewing the first and then the subsequent layers of the Koch snowflake, the children worked to develop different ways of thinking and expressing the snowflakes growth incorporating their use of symbols and their knowledge of number operations.  dpjiili  fi-ii.i'.i  i  tar  K - V I I  Julie has symbolically expressed the total number of aides for each stage of growth, (Zs well, embodied in her use of repeated addition, is her visualization of the number of sides as being organized into pairs of sides for the second stage and then as six clusters of eight sides each for the third stage.  'Moot Mere, Julie communicates the "groups of" concept through her use of'everyday* language.  * -  June: SJhe symbols Julie has used here reveal that she has interpreted the snowflahe's growth as being a repetitive subtractive action of one side, two sides, and then eight sides each until the total number of sides have been accounted for.  Three more interpretations by Julie  234  dpril: Sheuji's use of symbols express the total number of the triangles as a multiplicative and repetitively additive process.  May: Me thinks with the groups of" image through 'everyday,' language.  June: Shouji's symbolic notation describes the fractal's growth as being a divisive action where the total number of triangles split into an increasing number of equal groups. Shis was clear when he described the process as the snowflake becoming or turning "into" groups of triangles of which they had an equal number "each"— "One into one triangle, six into one triangle each, eighteen into three triangles each, and sixty-six into eleven triangles each." Three more interpretations by Shouji 235  dpjiit: Clone's written work communicates, her understanding of the fractal's increase af corners, as Being, as process that involves repeated addition.  3-51=3  •  ' Aiay: SJhinhing through the multiplicative idea af "groups af", Clare made use af both symbols and'everyday' language. Ik •* t* •>»•» ikJ "4MU{|  i n *t i l l .  !*i9_diat.aU7=3  J  June: Clare's use af symbols illustrates her conceptualization af the fractal's corners appearing through the operation af division. Mer understanding is viewed as similar to Shouji's explanation but in this context, there are three comers af one each, then twelve corners af which three clusters have three corners each and three clusters have one each, and then forty-eight corners af which three cluster have nine corners each and three clusters have seven each. Three more interpretations by Clare  236  On still other occasions, the class returned to familar contexts but investigated them in completely different mathematical ways than they had before.  237  fRe-uiewitity and Seeing, SUffenenti^  Shis Sa££, the class watched the film, Mates aa a SMang£eJ Ser the retur students now in the thkd grade, this was an opportunity, for them to re-v for the new students in the second grade, it was their first time seeing the the previous year, the children studied the film Gu watching it severaltimes thistime(and, in a very non-linear manner to that of last year) they saw mathematics coming to life. Cls a result, they laid down very different le paths.  Sor example, some of the children noticed that the triangles in th "weren't all the same". SJhey watched the film once more and this prom class to study the triangles more closely. She children used their fingers triangles, they drew them or cut them out of paper, and some went lookin the room gathering them. Slaving a good collection of triangles, the class with a partner or in a small groups to explore what was the same and wh different about them. (Zfterwards, the class shared their methods for doin the carpet. Some students had studied the shapes, by moving them abo desks (ije., sliding, flipping, rotating, and transposing one on top, oftikeot make direct comparisons about the lengths of the sides and the effect tha on the "shape" of the triangle. Some children used rulers to measure the each triangle while others took the shapes in their hands, turning and fee side, surface, and corner. Slaving gotten to know these triangles, so intim children naturally wanted ta give them names!  "I call this triangle a[n] almost  "I call this triangle the deferent  all equel triangle because  sided triangle because it has  only two of it's sides are the  3 different lengths of sides."  same." Ethan  Shane 238  "I call this a triple side triangle because It has the same sides." Steven  "they all have 3 corners [and] have 3 sidse." Madelaine "but they [differ in their] length of the sides." Christina What is the same about the triangles? What malces each o f them different?  The class went on to explore how larger triangles could be composedfromusing smaller triangles...  Annie's triangles 239  ...and what other shapes could be made from triangles.  240  ring the same time that the class was watching the film, five students had been working on another project that involved their construction of many different kinds of "pyramids" made from interlocking cubes. The children were sketching diagrams of the top, side, and bottom views of their structures when they told me that the bottoms of the pyramids looked like triangles:  241  The group shared this with the class and this lead into a study of how numbers too could be considered to be "triangular".  Timothy's diagram o f triangular numbers and "bow mucb" they increase each time  "I think the next number will be H5 because... 36 + 9 = 45 so 45 is the next one." Timothy "I think the next number will be 45 because 28 is 7 more than 2l. 36 is eight more than 28. So 45 is 9 more than 36." Steven "I think the next number will be 45 because the pattern is odd, odd, even, even." Clare 242  'The first has one dot. The second number has two [more] dots. The third you add three, the fourth you add four, the fifth you add five, the sixth you add six, the seventh you add seven, the eighth you add eight. The next one you add nine to it. So thirty-six... eight,  thirty-seven, thirty-  thirty-nine, forty, forty-one, forty-two, forty-three, forty-four,  forty-five."  Clare's diag^m that shows that the triangular numbers increases by a corresponding column of dots each time.  243  Mate*  1. JVxawnat $i£m Slowed, 1969.  244  Recursion as Relations: When Triangles Become Square (Three months later...)  £ast week, 3. introduced the class ta the film, Stance Squared! Since then, we Been re-viewing it and using it as a source from which ta occasion their furthe mathematics^ 3Jhe students have been working together, posing prompts a exploring the geometries of the square. 3he children also identified numbers that "square" (spurned on bq their, interest in triangular numbers). Steflecting an t class' curiosities and making plans for tomorrow's, mathematics,, 3, have decide focus their work en both kinds of numbers* 3his will be a chance for the childre not only, fold back and reflect en what they know about triangular numbers AN square numbers but also, for them (on their own and with the class,) ta consi each in light ef the ether. Jmportantly, had the students' previous work with triangles NOT occasioned their investigation into triangular numbers, this lesso would neither have the same recursive potential nor be appropriate. (The next datj)  Sitting  on the carpet, in the middle of the circle, the children helped one another to build the first five triangular numbers with counters. Underneath this, other students worked together to build a row containing the first five square numbers.  As wc were doing this, Robby whispered to Mark and struck up a very livery conversation- lots of head-tilting back and forth, smiling, and "yeah!" going on.  3, could not hear what they, were lathing about and so- 3. ashed Slobby if he share the conversation with the rest of the class. Slobby flashed a bashful smile a then raised his voice to explain what he and Mark had been discussing. H e pointed with his finger to the second and the third triangular number and then to the third square number and told me that, "that number and that number makes this square if you take it and flip it upside down and put it on top of it".  oo  eie  Robby's explanation of how the third scjuare number is created from the second and third triangular numbers  My "WOW!" and the boys' excitement for how they related the two number serie together in a spatial way drew several other children into the discussion. Soon, other students began trying to make sense of this for themselves by talking with one another and displaying similar hand and body gestures to those of Mark and Robby.  Wanting to maintain the focus, and momentum of this investigation, 3 repeated Slobby had said But this, time, 3 also built the numbers with counters as 3, spok that everyone could see the transformation taking place. "If you take the second and third triangular numbers and put them together like this" I said, demonstrating with the counters just as Robby had indicated earlier, "it is the same as the third square number." 246  Shis prompted the neat of the class to continue the pattern hu- combin triangular number with the one that preceded it in order to produce a squa Mot only- did the class do this visually and verbally- as Slobby and Mark had but also, arithmetically by adding- the two- values together, and physic manipulating the counters of triangular numbers.  Qfter the class produced the second through fifth square numbers from ones, Shutny reflected aloud and this time, he related the class* ma actions together into a connecting pattern. Tt will be this number" Danny said as he pointed to the third triangular number, "but in the square number". Pointing to the third square number, which was nine, he explained that that is the resulting spatial structure and number when the third triangular number is combined with the second triangular number. Hence, six plus three makes nine. I encouraged Danny to continue. Pointing with his finger in a left to right direction, beginning with the first triangular number that was one, then moving to the second triangular number that was three, and then to the second square number that was four, he communicated the relationship between the two series of numbers. T h i s first number and the second triangular number will make the second square number. And the second and the third triangular number makes the third square number. The third one and the fourth one makes the fourth square number and so on and so on..."  Qs, 3, listened and watched how Skinny was, thinking about these number see his understanding as also being recursive-- emerging from the mathem had already unfolded and at the same time, bringing forth another co between these numbers. Marh and Slobby's observation that arose property noticing that the second and third triangular numbers could third square number served as a place from which to begin our investig class' further collective work to apply, this notion provided several mo in which this relationship, exists. Qnd S>anny's understanding revealed quality about the triangular and square numbers. Shis time, a mor generative and predictive relationship between them.  247  J realise now, the impact that one's conceptualisation and enactions of. time have on the place-making of a mathematics classroom. Momtimeis imagined and enacted very much structures horn it is to ve experienced kg both teacher and students. Jt is a powerful undercurrent that directly- shapes- the hinds of mathematical events and relationships that are possible or impossible in the classroom. J am also learning to let my ecological metaphors think me as J. think within them, doing so provides me with direction and focus to map in recursive notions of time into my teaching of mathematics. She forms ef teaching and learning that have emerged so far are a stark contrast with my previous mechanistic ones that created a sense ef place where students were expected to assemble specific forms of mathematics before moving en to more complicated lines ef mathematical re-production. Jnstead, these new spaces for learning embody patterns of a temporal difference where past, present, and future exist all-at-once. 3ime is inseparable from space because it is defined by those spaces in which mathematical experience occurs. 3he two are net distinct but exist as one co-determining entity of "time-space". * 2  3  Mates 1. MxMonal SMm Slowed, 1961. 2. (Imam, 1996; Meidegaex, 1962,1972; MeHlzaurSsmtu, 1968. 3. Meuieaaex, 1972; Matwuma 1995 4. Siekieaaex, 1962,1972.  ... children... are taught at a tender age that the way to define something is by what it supposedly is in itself, not by its relation to other things. 1  G. Bateson, 1980, p. 18.  To know something is to know what that something is in the way it speaks to us, in the way it relates to us and we to it. 2  2  van Manen, 1986, p. 4 4 .  ... the pattern which connects... How are you related to this creature? What pattern connects you to it? 3  3  G . Bateson, 1980, p. 9. 252  /flight that Include this cteatute we call mathematics  253  Iteatlnf ^battetns ^Ckat Connect  3ntetactlotu\t Spaces fot /Katkematlcs in the (Ztasstoofn  255  Insight, I believe, refers to that depth of understanding that comes by setting in thinking ecologically about the mathematics class, It's impossible for me to Imagine my students as "«u.toi^o^vtou.s Individuals" ai/ujw,ore. tt doesn't make sense within a systemic t^tl^-space for each student to exist as a separate entity, active on everything else as tf everything else was part of the e x T C R - N A L environment, s o even though i c-owtlnuce to recognize each child as an Individual, I c-oncelve the children to be Individuals within larger collective and environmental systems. M y efforts to children's mathematical growth Involves continuing to make spaces for them to explore their mathematical thinking as Individuals. B»ut rather than furthering their uv^derstavuilt^Q through just a process of adding on of'new' mathematical experiences, attention Is also cast upon looking deeply and examining the understandings embedded In their mathematics. Ifs making opportunities for students to not only engage In Individual mathematical work but also for them to cov^slder how they are understanding the mathematics by drawing on their mathematical fenowlngs and developing relationships amongst them— In this way, reflecting on what they know and letting these understandings "speak to one another".  experiences... side by side, learning by letting them speak to one another.  256  M e e t i n g with M a c : A  K4ac  Study  o f O p p o s i t e s and Relatedness  a n d I a r e s i t t i n g a t a l a r g e t a b l e , r e a d y t o platj a g a m e . S t a r t i n g  with a n y number  of cubes,  we are t o find a t least two possible ways t o take  a w a y e q u a l g - o u p s o f c u b e s u n t i l n o c u b e s r e m a i n . M a c is t o r e c o r d e a c h o f the stories using whatever kind(s) o f symbolic notation he wishes t o use.  As we play this game, Vm interested In the understandings Mac brills to the taste. As well, since he hasn't had any formal lessons. Involving the operation of division or the symbol, I'm- curious as to how he'll express the mathematical action of repeatedly removing a particular numbe of cubes. I'm looking for occasions In which I can alert Mac to examine his understandings and engage him in tlutttel^g about how he i^clght use what he knows to develop other ways of thinking «bou.t the mathematics at hand. Mac begns the game cubes.  w i t h a p i l e o f 10  "1 started with ten" H e w r i t e s t h e  n u m e r a l ^ 10 o n t o a p a g e i n h i s n o t e b o o k .  i tatee away two cu.bes. M a c r e c o r d s -2 b e s i d e t h e 10 s o t h a t i t reads  10-2. "Because we took two away so it's ten  minus two." I t«tee another two cubes away and repeat this three more times. E a c h time, M a c r e c o r d s a n d t a l k s as he works,  "minus two, minus two, minus  two, minus two! And the answer is... zero!"  257  B-ased o n w h a t  MAG'S  saying  a n d how he's r e c o r d i n g t h e t a k i n g  away  g r o u p s o f t w o , he v i e w s t h i s g a m e a s a g a m e o f repeated s u b t r a c t i o n . c o m f o r t a b l e a n d a b l e to I n d e p e n d e n t l y observes u s i n g  express t h e m a t h e m a t i c a l  both v e r b a l m a t h e m a t i c a l t e r m s a n d  of  He's  actiov^s  recorded s y m b o l s .  As  w e l l , these f o r m s o f m a t h e m a t i c a l l a n g u a g e s e e m to be a n I n t e g r a l p a r t M a c ' s t h i n k i n g b e c a u s e he d o e s n ' t w a t c h t h e e v e n t s f r o m s t a r t to f i n i s h  he  of  and  t h e n f o r m u l a t e h i s e q u a t i o n , he d o e s so I n t a n d e m — a s t h e y are h a p p e n i n g . In this w a y , M a c  "narrates" t h e  mathematical  storij  a s It's  unfolding.  Pushing the ten cubes into the middle of the table, Mac starts again but this time, decides to take a way groups o f five cubes. Ten minus five...", writes 10-5, removes the remaining five cubes, "minus five..." "equals zero" He finishes his number story bu writing; down -5=0. H-avlng ten by  c o m p l e t e d two grouping,  larger  number  selects  twenty  yroutds.  to  remove  I a s k M a c to choose a  for to  ways  the be  next  the  game.  number  He and  to a d d t e n c u b e s to t h e t e n t h a t  are a l r e a d y  In  the m i d d l e o f the table. I  t h e n a s k M a c I f he k n o w s o f a n a m o u n t that  c-ou.lcl  groups  be  taken  f r o m twenty  away  so t h a t  In  zero  ec(ual  cubes  would remain.  "Hmm." Mac looks at the twenty cubes on the table. He places his elbow on the table and rests his head in his hand. Pursing his lips, he thinks out loud. "Maybe you could do... hmm..." Mac takes his elbow o f f the table and rests his hand  258  down beside bis notebook. He looks at his fingers and then stares across the room. Curli ng and uncurling the fingers of his right hand, Mac looks at me, "Kmmm... fives?" I assume Mac's sfei.p-cou.nti.kvg by fives to twenty on his fingers. To be  sure,  I'll asfe hint to explain his response. " A n d how do you. know that fives would  be a good one to cMoostf  "Because you can count by fives to any number." He demonstrates this to me by taking his pencil and touching it down on the table in two spots horizontal to each other with a good size space between them. It appears that Mac's t h i n k i n g with a n I m a g i n a r y horizontal number line along which he sOfzif-c-out^ts,  by fives to twenty. I'm not sure what he means  by "any" number a n d so, I'm going to encourage h i m to continue.  "Like you can count by fives to twenty. Five, ten, fifteen, twenty." Now touching the two points but this time with his index finger, M a c skip counts bt) fives, pointing in a left, right, left, and right fashion. He is also moving his head from side to side in a left to right motion— like a metronome, marking the numbers as he counts and points.  2 5 9  5 15  10 20  Demonstration o f Mac's counting action  Now, It's clear to me that Mac ts N O T counting along a horizontal line but two points that separate the numbers into those that end with 5" those that end with o. Hts counting action ntlght have arisen -from the hundred chart that we use often and hangs in the classroom. T h e numbers I-IOO are organized o n this chart in a 10x10 grid, b e c a u s e o f this, they fall into columns where e a c h number increases b y ten a s y o u move t o the next o n e directly below it a n d so, all o f the numbers within a particular column share t h e same last digit. ifs also possible that Mac's counting method could be a result of the rhythn-uc patten*, that Is generated from skip counting aloutd. "1 started with twenty, minus five, minus five, minus five, minus five, equals zero." M a c records 20-5-5-5-5=0 into bis book.... ... while I t a k e awau, the f o u r g r o u p s o f five. "So, that worte." I g a t h e r the twenty cubes into the middle o f the table ae^in. Mac  begins  the next  number  story.  M o v i n g the c u b e s with his left hand a n d recorAme^ with his right hand, he t a k e s away g r o u p s o f two c u b e s e a c h . "Twenty minus two, minus two, minus two, minus two, minus  260  two, minus two, minus two, minus two, minus two, minus two, equals zero. So that works." In similar  fashion to his other number stories, Mac records 20-2-2-2-2-2-2-2-2-2-2=0.  up until now, Mac's used groups of two and five to divide the cubes of ten and twenty. He's doing this by applying the "opposite" process of repeated subtraction, in other words, through repeated addition Mac's skip counting to arrive at the target number and then simply reverses the process by transforming It Into a repeated subtraction equation, since skip &ou.i*A£i*,Q by multiples of two and five appears to be Mac's strategy of choice, \ wovuier what other thinking he might bring forth If I ask him to co^tlv^ue working with the number twenty. "is there another way?" I place the cubes in front of Mac. "Hmm... you could do it by tens". Mac cjuicldt)  writes 20 down into his notebook.. "We could do it by tens." I take away a group of ten cubes. Mac writes -10 to the right of the numeral 20.  I talce awatj the remaining ten cubes. "Minus ten". He records -10 and then -0. "Equals zero".  I put the cubes back, into the middle of the table. 261  A n attempt to prompt Mac to thlnte of yet another way to evenly divide the twenty cubes. "Oh..." "TVy  by  -fours." ( A n  impulsive  suggestion!) "Okay." M a c beg'ns  t a l k i n g slowly a n d  t a k i n g away g r o u p s o f f o u r c u b e s . E a c h time, be l o o k s t o see bow many  cubes  remain. Mflc appears unsure. T h i s is probably because four is N O T a number by which he u s u a l l y skip counts a n d therefore It Isn't a number he associates with twenty. R e c o r d i n g as he w o r k s , M a c completes  the ecjuation, 20-4-4-4-4-4=0. P o i n t i n g t o what M a c has recorded, I b n n g attention t o two o f his equations. "Now,  looking at what y o u got here for  twenties... loofe at the twenty give away groups of five [20-5-5-5-5=0] a n d look at the twenty give away groups of ten [20-1010=0]. r>o y o u  vwtic-e a n y t h i n g about the  number of groups Mac?"  With his pencil, M a x p o i n t s t o and  20-5-5-5-5=0. That  20-10-10=0  this is just double of  this". "why Is that?"  262  "Because there's two fives in ten so that's four."  Mac then demonstrates this by pointing to each "5" in his equation. "There's FOUR fives there and TWO tens and one twenty. This is FOUR fives... TWO tens and one twenty."  "&y Inviting Mac to stated back and reflect on the two number stories, he Identifies the two equations as being related.  A\  20-10-10=0  20-5-5-5-5=0 M a c ' s e x p l a n a t i o n o f h o w 20-10-10=0 is t h e r e s u l t o f ' d o u b l i n g u p ' f o u r g r o u p s o f five.  Tu.rv<iv<<Q Mac's attention to the other two number stories (I.e., 20-H-H-H-44=0 and 20-2-2-2-2-2-2-2-2-2-2=0), i ask him what he can tell me that Is the same or different about them. "There's just double twos as there are fours... FYetend I didn't write that [points to 20-2-2-2-22-2-2-2-2-2=0]  [points to  down and 1 just counted this 20-4-4-4-4-4=0]—  how many fours.  There are, one, two, three, four, five. There's five fours. So then, I would just have to double that to see how many twos there'd be if 1 did twos. So there's ten twos."  intrlgulngl As Mac explains how he Identifies ten groups of two as being In five groups of four, I get a real sense of the flexibility of his understanding. Taking his "previous Idea of 'doubling' as a process of multiplying by two, Mac's thinking In this context effectively shape-shifts the -process by Inverslng It. in essence, the doubling now becomes 'halving'.  263  20-4-4-4-4  V 20-2-2-2-2-2-2-2-2-2-2=0 Mac's explanation as to why there are twice as many groups of 2s than groups of 4-s  Smiling at Mac, we resume the game. "okay, Lefs see I f we ca IA, utse sow,e of that as we 0 0 along in this game." Mac nods his head. "Yes!" "plcle another number. Twenty-four.'  I add another four cubes to the pile that is already in the middle of the table. Mac jumps in and cjuickly splits the pile of cubes in half or, into two groups of twelve. Twenty-four minus twelve minus twelve."  He records 24-12-12=0. Mac explains that he doesn't need to count the remaining pile because he "knows" that it will also be twelve. Again, I want to prompt Mac- to reflect on his thl^lelivg and. consider how It might impact his understanding now. "... t a k i n g that", I point to 24-12-12=0, "can you think, of another number story 264  that you could tell?" M a c l o o k s a t the equation. "Groups of two."  "And how many twos would you have?" M a c examines t h e equation again a n d then  down t o his hands t h a t  he is  holding palms up. However, they c a n n o t be  seen  because  they  are  hidden  underneath the tabletop. I tell him t o b r i n g them out! M a c s k i p c o u n t s b y twos a n d k e e p s t r a c k b y unfolding o n e finger a t a time until he has six fingers extended. 'Two, four, six, eight, ten, twelve. So double six [i.e., g r o u p s o f h/vo]  is... twelve. So I'd have twelve  twos. Okay." M a c sets t o recording t h e number s t o r y , Twenty-four minus two, minus two, minus two, minus two, minus two, minus two, minus two, minus two, minus two, minus two, minus two, minus two, equals zero... twelve twos." 2H-2-2-2-2-  2-2-2-2-2-2-2-2=0. Mac's carrying -forth his understanding from the last taste and applying it here to help him determine how m.any groups of twos are in twenty-four. "&y making use of the equation that divides twenty-four into half, Mac tatees one of the twelves and skip counts by twos to find out that there are six groups of twos in twelve. He then simply doubles the number of twos to get twelve groups of two. Like his other calculations, none of these are done with paper and pencil but through verbal, mental, and physical forms of computation. 265  ".so, whafs another way that we could  make..." I put the twenty-four cubes back into the centre of the table. "We could do... try fours!" Mac  reaches for the cubes to beg'n taking away groups of four.  \'m not sure why Mac Is reaching -for the cubes, is It because he thinks that since twenty-four Is divisible by two that It'll also be divisible by four since four Is double of two and he wants to double check this with the cubes?  I ask him to guess how many fours he thinks there will be. "Hmm..."  c^lven that Mac doesn't guess or estimate, I take his Idea of groups of fours to be a conjecture and not something that Mac "knows" as a fact. I want to help Mac to develop his thinking by connecting something he does know In orderto solve for what he's trying to figure out. So Instead of letting Mac continue to use the cubes In what i suspect will be a strategy of trlal-anderror, I Interrupt what he's doing and suggest that he reexamine what he's already done.  "How could you take something you know here..."  Mac moves from the cubes to his notes and uses his equation of 2H-2-2-2-2-22-2-2-2-2-2-2=0 as his jumping off point. "1 could say... I would just take half of these off" pointing to the groups of twos, 266  "because I know there's twelve of these." M a c  covers u p six o f t h e g r o u p s o f twos with his h a n d s o that they are  no longer  visible a n d explains t o me, "So take half off twelve twos". T h r o u g h M a c ' s actions,  he  demonstrates the need t o t a k e away six twos in ord&r t o form the g r o u p s o f fours with the other six twos. "... so just count by twos"  Mac  proceeds  to  count  the  remaining twos. H e begins t o s k i p c o u n t them by twos a n d then realizes t h a t he isn't wanting t o find o u t 'how much' remains but rather, 'how many g r o u p s ' o f f o u r there are in twenty-four. H e switches his  strategy.  This  time,  he  t a k e s his pencil a n d partitions o f f two twos at  a  time while he c o u n t s  the  number o f f o u r s . "Okay, that'd be one, two, three, four, five, six . So there'd be six... six fours. So one, two, three, four, five, six." M a c k e e p s t r a c k  o f the  number  of  '-Hs' he's  writing  t o p r o d u c e 24-4-4-4-4-4-4=0 on paper. Finally, M a c double c h e c k s his work by taking  the  pile  of  2+  cubes  and  physically dividing it into six g r o u p s o f four cubes each.  "ofefly, talei-ng what you. know from the •fours,  c a n you. do something new?"  "I could do... hmm..." M a c l o o k s at the c u b e s 267  in front o f him a n d then l o o k s a t me. "Groups  of eight?" M a c s t a r t s  in a n d  implements his d o u b l i n g u p strategy. "So just..."  reaching  f o r the piles  o f four  cubes, M a c p u s h e s two clusters t o g e t h e r until they are all paired u p a n d a r e now in g r o u p s o f eight. "And there'd be three groups."  H e picks u p his pencil a n d r e c o r d s 24-88-8=0. "... minus eight, minus eight, minus eight, equals zero. There we go!"  "oteay. is there any other way we could share twenty-four?" M a c s h a k e s his head. "Hmm. 1 don't thinK so." H e looks over t h e three g r o u p s o f eight c u b e s in the middle o f the table moving his eyes a n d head in a left t o right fashion. "N...nope!"  I p u s h the cubes together s o that thetj  are b a c k  into  a  pile o f  twentxj-fc  rour.  An attempt to enable Mac to not tlunte with the image of groups of eight but to start anew. M a c l o o k s at the cubes a n d s p u t t e r s out, "one" softly.. H e t a k e s single cubes away, "You could do one... minus one... so twenty-  four minus..." U s i n g his left hand t o move the c u b e s awat) a n d his right hand  to  268  record a s he a c t s a n d tells the number story out, M a c monitors t h e number o f o n e s b e i n g recorAeA in his ecjuation. H e p r o d u c e s 24-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-11-1-1-1-1-1-1=0 "And the shortest way of sharing is  one group of twenty four" M a c records 2424=0 down into his boolc.  again, l ask Mac- to review what he's done. "Because he's been working exclusively with the Images of splitting In half and doubling of the vu<.mbers, they don't allow -for six or three to be possible values by which twenty-four can be divided. I try to open another space for him to work. Once  "so If you were to partner up the number stories that you were able to figure out •from, each other... S o this one here" [pointing t o t h e 24-I-I-I-I-I-I-I-I-I-I-I-I-I-I1-1-1-1-1-1-1-1-1-1=0] and then to [ p o i n t i n g t o 24-24=0— t h e number s t o r y M a c h a d written a s t h e reverse o r t h e o p p o s i t e story  t o t h e previous one]... giving  away one whole group... oov^v^tot them with arrows., what other ones did you..." "Well, 1 first did a group of twelve and then from that twelve, I decided to do groups of twos, and from that two, 1 decided to put them together and I did by iours, and then I just added two fours together and I did eight! So those are all connected." M a c  summarizes a n d c o n n e c t s t h e number stories with arrows in that order. H e a l s o c o n n e c t s 24-2-2-2-2-2-2-2-2-2-2-2-2=0  269  and  2H-H-H-H-4-H-H=0 a s b e i n g related  "because... four a n d two... because two twos  make up four." R a t h e r  than  k e e p i n g t o the number stones a b o u t 2 4 , Mac  moves o n t o all o f t h e other  equations he has written. H e relates 2 0 5-5-5-5=0 to 20-10-10=0 and 20-2-2-2-22-2-2-2-2-2=0 Praming  to  20-H-H-4-H-H=0.  t h e equations  in terms o f  d o u b l i n g a n d halving, he explains that he d o e s n ' t see 10-2-2-2-2-2=0 a n d 10-5-5=0 as b e i n g related a n d s o he leaves them as they are. Instead of t r y i n g to bring him back to twenty-four, I shlfb the conversation  to exploring one last number. M a c tells me t o pick a number t h a t c a n be divided into two equal g r o u p s . <^lven what Mac has said a n d my Intent to engage him to thtnfe s t i l l differently, I want to choose a num-ber that'll divide evenly by two but result  In a n OT>T> composite number. I c h o o s e eighteen a n d a s k him if he thinks it is a n a p p r o p r i a t e choice. H e p a u s e s f o r a moment. "Yes". With t h e eighteen c u b e s in front o f him, M a c sets t o work.  "Okay!"  he exclaims. "Let's do...".  M a c s t o p s again. H e places his hand t o his lips a s he l o o k s a t the c u b e s . 270  "what do you. know about eighteen?" A n o t h e r p a u s e . "... How about nines." M a c playfully  s l a p p i n g his hand  down o n the table. " w h a f s nine about eighteen?" "There's two nines in eighteen."  Again, he's, using the Image of doubling or halving of a number. H e writes this,  18-9-9=0  o n t o the page. T o  he a d d s what he's coined, the  "shortest" number s t o r y . "We could also do eighteen minus eighteen equals zero" a n d writes  18-18=0.  H e foil ows this with the "longest"  number s t o r y . "And we could also do eighteen  minus one.... M a c talks o u t loud as  he  records this o n p a p e r a n d eventually produces  I8-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-  1=0. I a s k M a c t o review the three number s t o n e s he's written f o r eighteen. "Now looking at any one of these, can you make up something new?" "I could... this". M a x c h o o s e s  t o rework  I8-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-U0.  271  "How do you,tenowthat?" "Because there's two nines in one eighteen and we could... and this" p o i n t i n g t o 18-9-9=0, "is opposite."  Here, Mac describes, how 9+9=18 and 18-9-9=0 are Inversely or "oppositelly!" related. H e points t o 18-18=0 a n d draws an arrow s o t h a t now 18-9-9=0 is connected t o b o t h 18-18=0 a n d I8-I-I-I-I-I-I-I-I-I-M-I-  1-1-1-1-1-1=0. "So you put those together too". M a c l o o k s a t me a n d concludes, "so... that makes it."  Mac «ppe«rs to be "finished" this game because he's, related all of the number stories to one another. I challenge M a c t o see if he c a n find another  number  story.  F o c u s i n g his  attention o n t h e s t r i n g o f "ones" in t h e ecjuation  I8-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-I-  1=0, I invite him t o , "see If you can find some more groups In there". "We could do groups of two."  Again, putting his doubling strategy to good use! I acknowledge two a s being a possibility for  another  number  story  and  encourage M a c t o make sense o f it.  272  "How do you know you. can do groups of two? what tells you that's possible?" "Because... you can... eighteen is an even number."  "And that n*,eans....?" "As an even number. You can use it and put groups of two in." "oleay."  M a c t a k e s his pencil, "Okay, eighteen minus two, minus two, record  writing  minus  two..." a n d  begins t o  his number story. M a c continues while k e e p i n g t r a c k by  skip  c o u n t i n g the "twos" he has written a l o u d "two, four, six... sixteen, eighteen". "Equals zero" be  s a y s as he finishes t h e equation, 18-2-22-2-2-2-2-2-2=0. M a c tells me t h a t he c a n show that nine twos makes eighteen by "just counting the ones and see how many twos there is "  Mac's using the same strategy he developed to determine how i^any -fours there are in twenty-four. "We can count by two ones at once. One, two, three, four, five, six, seven, eight, nine". M a c c o u n t s the  nine twos t h a t are made by p a i r i n g up two o n e s at a time. "And one, two, three, four, five, six, seven, eight, nine". H e c o u n t s  and  points with his pencil to "prove" that there are indeed, nine twos t o  make 273  eighteen in his equation. "Now, we can put these two together too." H e draws an arrow t o connect  18-2-2-2-2-2-2-2-2-2=0  with  18-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1=0  to  indicate the relationship o f d o u b l i n g the single units o r ones o f this even number in order t o p r o d u c e a number story that has g r o u p s o f twos in it.  Pointing to his first number story,  18-9-  9=0, I a s k him t o look at the "nine and  nine, can nine be told another way? in temts of groups.?" M a c l o o k s down at the equation a n d thinks o u t l o u d . "You can tell it as... three groups of three! Or we can tell it as... groups of nine."  S i n c e he's  already recorded eighteen  s h a r e d by two g r o u p s o f nine, he sets t o work o n determining how many g r o u p s o f three there would be.  "what would that loole Lllee for eighteen?" "For eighteen... that would be... there would be six threes in eighteen because there's two nines there."  M a c points t o  18-9-9=0 in his  notebook.  "So two nines in eighteen. So one of those two [nines]. There would just be... let's just pretend  each of these are three [i.e., three threes] so that would make six" M a c p o i n t s t o the  first  nine a n d then the s e c o n d nine in his 274  equation. "So six groups of three... so six groups  of three!" Mac goes to write this down in his book but stops and looks at me.  mathematical Image Mac ts thinking with— "six groups of thr Isn't the same as how he Imagined his other equations. I ask Mao could reinterpret the story by telling It from the perspective he had expre Here, the  the others.  "Eighteen, minus three minus three, minus three-  equals zero." Mac records 18-3-3-3-3-3-3=0  into his notebook.  Here's an opportunity for Mac to bring together the Image of repeat subtraction that he's been  working  with  and the Image he already had for  multiplication. To do this, I repeat something he's said previously, "NOW Mac, did you hear what  you said? y o u  said,  I  HAve six  G R O U P S O F THR,BB."  "Yes..." Mac continues to look at me,  waiting for me to make mxj point.  operation do you you hear <qROucF\s O F ? " "what  thlnle  of when  "1 think of... 1 think of... like say, 1 wanted groups of four." He takes the eighteen cubes and  arranges them into four groups of four cubes and a group of two. "A group of four, a group of four, a group of four, and a group of four, and a group of two. Those are groups." 275  "oteay." 1 p u s h the counters b a c k into a pile a n d p i c k u p M a c ' s pencil a n d h a n d it t o  him. "How would  you. write  six  groups of three?" H e t a k e s the pencil, a n d s a y s , "six" a n d then s t o p s  and asks  me, "should I use  symbols or write it [i.e., in words]?" I'm curious what symbols (e.g., + orx) or words, (e.g., "and" or "groups of") Mac might use to express how he's t h i n k i n g about six groups of three.  I tell M a c t o  record whatever way  wishes t o e x p r e s s  he  how he is t h i n k i n g  a b o u t it.  "Six times three equals eighteen". M a c writes  6x3=18. H e  t a k e s a l o o k at this o n the  p a p e r a n d then explains t o me t h a t it d o e s n o t make sense with the game we are p l a y i n g o r the other e q u a t i o n s has  recorded; t h a t the object  of  he the  game is t o T A K E A W A Y the number in ecjual g r o u p s until there is n o t h i n g left. "So that wouldn't work because we're trying to make it equal zero." Now  here's a chance for M a c to c-ov^slder the complimentary relationship  between the mathematical forms of ± £ - 3 - 3 - 3 - 3 - 3 - 3 = 0 a n d £>K3=±8. Afttr all, If s this mathematical bacle-and-fbrthlng that Mac's been doing while he's been p l a y i n g the gamel Through his actions of doubling a n d h a l v i n g , he has been ultimately, determining how m a n y ecjual groups w i l l mafee up 276  the given number. Once he's done this, he then ReveRS&s his thinking so that It moves It In the opposite or complimentary direction. He does this by expressing the number as being taken away In equal groups until zero Is reached. M a c studies the two equations again. T h i s time, he does not consider  them as  b e i n g unlike e a c h other, but instead, as both  communicating eighteen  groups  as six  o f three. T h e difference,  explains is t h a t with opposed to  18-3-3-3-3-3-3=0  he as  6x3= 18, "we are not plussing" [the  g r o u p s o f three] "we are minussing them.. When you are taking away it would be subtraction. It's  just  like  this  [pointing  multiplication symbol in  to  6x3=18],  the but it's  subtracting not adding."  in his explanations, Mac not only Identifies the Inverse relationship between repeated subtraction and multiplication but also, how multiplication can be Interpreted as (repeated) addition. "They're opposite but they are also related". M a c  draws a n arrow t o connect t h e two equations together.  "Now, what do you mean by they are opposite B.K.T they are also related?" "Because there are six groups of three. Well, they are not really opposite because there's six times three and there's six groups of three here except for, this is plussing and this is minussing."  277  ".So that li.e., plutsslng and m-i.nwjssi.n0l Is opposite then?" "Yeah but they are also related because there is six groups of three here except we're just taking them away. We're not plussing them."  M a c reaches  for the cubes o n the table a n d shows me a g r o u p o f three b e i n g A D D E D six times to  make  eighteen  "opposite-related"  and  then,  manner,  in  how  an six  g - o u p s o f three can b e T A K E N A W A Y .  Mac's demonstration of 6x^=\8 and 1S-3-3~5-5~5-5~0  " K n o w i n g what you. k n o w about  this  [i.e., the o p p o s i t e relationship o f  16-5-  5-5-5-5-5=0  a n d 6x5=16], c a n you. t h i n k  of any other possibilities?" M a c employs his d o u b l i n g strategy one final time t o solve f o r the last number  278  story. "I could do three groups of six because there's two threes in a six. And I just counted by twos."  H e shows me t h a t b y pairing u p the  threes in  15-3-3-5-3-3-3-0 is  the way he  can c h e c k to make sure t h a t the g r o u p s of  three will double  up  without  any  remainders. M a c counts the number o f g r o u p s o f six t h a t can be made from the six threes. "So one, two, three. So eighteen minus six, minus six, minus six, equals zero".  18-6-6-6=0  H e records  into his n o t e b o o k . ".So you. d i d this, one [pointing t o  3-3-3-3=0]  18-3-3-  a n d then you. f o u n d out the  oppositely-related  one  6x3=18]- s i x  three..."  [pointing  to  related to this one too." M a c draws an arrow to c o n n e c t them  "So, so this one would be  together. "t>o y o u thlnfe y o u could thlnfe o f a m u l t i p l i c a t i o n stv^ttv^t that would be the opposite o f t h i s one?" [pointing to  18-6-6-  6=0] "Hmm..."  M a c p a u s e s f o r a moment. H e  rewrites the ecjuation but replaces subtraction  symbols  with  ones  the of  a d d i t i o n . "Eighteen plus six plus six plus six equals zero.  279  I attempt t o retell a n d a c t o u t the equation that M a c has written u s i n g words  and  my hands  to  show  the  additive a c t i o n . "so if you had eighteen  things, and then you add six and add six and add six, you would end up with  zero." "No. That doesn't make sense."  I suggest t h a t we use the b l o c k s t o try a n d make sense o f it. M a c agrees. I mirror the same actions  that M a c  demonstrated when he showed me how a d d i n g a n d t a k i n g away six g r o u p s o f three were o p p o s i t e l y - r e l a t e d . "So we did  eighteen minus six, so I'm doing the action, right, you talk it out." M a c chimes in a s I t a k e away the g r o u p s o f six"... minus six, minus six .... zero."  Now, Lefs do the opposite/' Still k e e p i n g the eighteen b l o c k s in three g r o u p s o f six  3  I perform the o p p o s i t e action o f  5RJNGING 5 A C K t h e three g r o u p s o f six s o that they form the orig'nal pile o f eighteen, "so tf we go..." "Eighteen plus six..  280  ! s t o p what I am d o i n g a n d return the e r o u p o f six c u b e s b a c k again. "How m u c h are we starting with?" "Zero" M a c narrates the story I'm a c t i n g out. "Zero plus six, plus six, plus six,"  "makes...?* "Um... eighteen! Okay, zero plus six, plus six, plus six, equals  eighteen." M a c writes  0+6+6+6=18. I ask  M a c t o look at  0+6+6+6=18  18-6-6-6=0  and  t o see if he agrees t h a t they  tell o p p o s i t e stories a b o u t eiejiteen. "So they're kind of... So..." M a x draws an arrow  to  connect  the  two  number  stories  togethcler. "can  you, thlnte o f a  kvtu.Ltlplleflti.on  story u s i n g numbers a n d s y m b o l s that's the opposite o f eighteen m i n u s s i x m i n u s s i x m i n u s s i x equals zero?" Mac  looks  at  the  addition- equation  he's just  recorded. "Three groups of six equals eighteen!" H e  then writes  3x6= 18. "So I'll write it down... Three  minus... umm... times six... three times six equals eighteen." finally, M a c R E A L L Y finishes (!)  the session by drawing an arrow from "this one" [  0+6+6+6=18] t o [3x6=18] a n d  then 281  t o "this one also"  [18-6-6-6=0]".  H e explains  that the latter is "related because it's opposite".  18-3-3-3-3-3-3-0 6x3=18 18-6-6-6=0 0+6+6+6=18 3x6=18 Mac's diagram showing trie relationship s between the equations  One of my Intentions In working with Mac has been to enable him. to develop different ways of seeing and expressing the mathematics he brings forth.... not through prespeclfled sequences of I M M E R S I O N but through making space -for, being attentive to, and responding In that moment to the eMER^eNCe of Mac's mathematics. A second focus has been to open learning spaces for Mac to set his mathematics "side by side".... put his Ideas and expressive -fbrms WITH each other and consider the patterns that exist among them. Mac's growth of understanding that arises from these spaces comes about through our  Interactions and his Interplay of making numerical and operational sense of the game. Mac takes numbers and reconfigures them into equal groups  through skip counting and physically re-arranglng cubes... he folds back on what he now knows and extends his thinking to create yet other solutions by decomposing the number again or recombining the groupings... he examines his actions and develops not only compatible but complementary  mathematical forms through making use of different operations. in the end of our session, Mac creates another conceptual layer that integrates  his understanding even more when he effectively brings all five equations together. Importantly, he's not linking one form to another, but 282  comprehending them as Interacting systems of knowing ±g\ By setting his mathematics "side by side", Mac opens a space where he relates and Integrates the educations In ways that most definitely, "speafe tm-athematleallyl to one another".  283  Insight, I  believe, refers to that depth of understanding that comes by setting ... t o h e a r [ a n d b e c o m e  aware  of] m u l t i p l e p o i n t s o f views,  as well as t o express  clarify t h e i r o w n - is n o t seen as c a n c e l i n g t h e i n d i v i d u a l differences, of  identifying  them.., within  the context  o f the g r o u p [or,t h e larger  b u t as a  a n d  means  environment]  4  experiences, yours and mine, familiar and exotic, new and old, side by side, learning by letting them speak to one another. 5  4  s  New, 1993, p. 219. M . C . Bateson, 1994, p. 14.  284  *}junhing systemically about children's mathematical learning raises the Issue that as well as nurturing students' individual growth, emphasis should be placed on how their mathematics relates to larger collective realmsJ 2)oing so allows students get another opportunity to integrate different points of viewing and to gain an understanding for how mathematics exists simultaneously within individual and collective spheres of knowing.  Is a piece of my  on paper. I have lots of these free writings. I call them "free" because I write them very much in the spur of moment. I don't pay attention to the particular words that I'm using or the stru.c-tw.K-e of the sentences. \'m not even writing to anybody. I'm simply letting my t h i n k i n g out and paper is the medium it happens to happen on. They're also free because when ( write each one out I do so on a separate piece of paper and then put it with the rest of them.. I don't use them, to write from, mainly, I j u s t collect them. A n d as I t h i n k out more, I stf: through my free writings and consider how the one I've j u s t written interacts with the others. A n d from time to time, two or more of these ideas will settle together or when Above  ' t h i n k i n g out'  •placed with some other, create a tension that enables new t h i n k i n g to arise.  asking, "what does it mean?" A n d so it goes. Funnily, I'm reminded of my childhood— sitting at the table w i t h my sisters and b r o t h e r w a t c h i n g t h e green tea leaves swirl and sink to the b o t t o m of our cups, creating images and each of us trying to d e c i p h e r t h e inherent meanings. s>o, back to my particular piece of free writing. A s l read It on paper, It makes sense and It's been integral in my current t h i n k i n g and Then I f i n d m y s e l f  teaching. "&ut I've never articulated it any  further... out on paper. So, this Is  the task I've set for myself,  TWO questions that emerge right away are: w h a t  does  "thlnktng-out-on-paper"  this  -fragment  of  mathematics and what might tt look like  In  the  mean  -for classroom  classroom?  immediately comes to mind and highlights the Individual and the collective as necessary parts of the mathematics classroom is the lesson in which I was reading the book, even Steven and odd Todd. Had the lesson only focused on the children's individual ways of t h i n k i n g about odd and even numbers, it is doubtful that they would have An  example that  285  developed the web of understandings that resulted. As Individual knowings, all of the students' responses made some kind of mathematical sense— that Is, rote or conceptual sense. "&ut It wasn't until the students' different ways of thinking and their emerging Ideas were brought together that the mathematical tension necessary to occasion their collective sense-making was possible, it was here In the collective realm, that the children made  collaborative, conceptual sense of why certain numbers, are considered to be "odd" and others to be "even", why even and odd numbers end with particular digits, and what exactly Is or Is not 'two' about them. Had the lessor with each student sharing his or her number Images Instead of bringing them together and exploring how these Ideas were or were not related to one another the children's understanding of odd and even numbers would have most likely existed as a collection of "free" Ideas. H-ad the lesson only focused on the children's, thinking as a class, there wouldn't have been the tension created by their individual understandings, individual and collective understandings are not only necessary farts of the  mathematics classroom,, they are necessary -for the growth of each other. On enables or disables the other.  Notes ±. Cobb, ±^29; A.  Travis and simmt, QD03; In conversation with S. B.  plrle, October ziooz.  286  Connecting as Shape-shifting Hold on for a minute. COM Id you go over that again? 9 suppose uou could say that tine mathematical thinking and emergent ideas that lead to a growth of understanding arise in the dynamic of students simultaneously acting as individuals and as collectives. So to use your favourite word, it's in the flow. Exactly! And through which mathematical shift.  ideas and understandings  shape-  Okay... but can you just show me again, an example of this shape-shifting of ideas and understandings. Well, last week, the children had been given each a small bag of m&ms. During this activity, the students worked on their own: estimating and then finding out how many m&ms there were in the bag, sorting them by colour, graphing, and then determining the divisibility of the coloured sets by whole numbers. okay. As 9 made my way to each of the children while they worked at their desks, 9 got a bird's eye view of the class' results. you did, but your students  didn't.  No. And they weren't looking and comparing their data with one another, instead, they extended their initial analysis to general statements from what they had found in their own bag ofm&ms. Like... Well, like Barnes, for example, observed that for "all colours except green m&ms, there were two or three m&ms." And Holly explained that there were either "lots" of a certain colour or "hardly any" of it. because these were small 287  bags of candy, Holly considered "lots" to be four or more and one or two to be "hardly any." So they were generalizing particular instance.  from their own  yes, and seeing the variations in the children's data, 9 wondered how 9 might bring their thinking to a collective realm so that the students could consider these conjectures within larger sets of data and other people's ideas. So, you weren't just moving their thinking into this realm, but their ideas as well. That's right. ISut how would the larger context do more than create a collection of individual ideas? Well, 9 was interested in what ideas might be born from the ... and how they'd  interaction.  shape-shift.  Exactly. So the next week, the children revisited their work... I ^ i v i d e d into groups o f four o r five, each student studied his or her bar graph' wrote one statement that reflected a quantitative aspect of their bag o f vn&vns and shared it with their small group.  288  ...  —  —  •/L."'.  •••^h  v«5r.  w  1  ™  .'SK-. >^Ajt  f.">  vUt^  .'v.*_.  B  Bin  Green and brown don't have very many. (Marie)  There is no same number of colors. (Robbu)  1 *" '  :M3'\ 9 There is 3 reds and 3 yellows. ( D a n i c a )  "There is Two f>airs of the same numbers and another That's alone." (Michelle)  289  ... "another that's alone'? 9 know! When 9 asked Michelle to tell me more about her statement, she explained that "because there's four here beside each other" and pointed to the four dark brown m&ms and the four green m&ms. "And three that are the same right there", pointing to the three blue m&ms and the three red m&ms, "... there's one that's all alone... it doesn't have a partner of a different colour" she concluded, pointing to the one orange m&m and then to the yellow m&m column of which there wasn't any. She's using the "partner" metaphor ... to describe odd and even numbers the similarities that existed within sample. What happened next? 2  not but her  The students worked in pairs. Michelle with Danica and Mark with Robby. This time, each student took their "true" statement and examined it in relation to their partner's data. To see whether or not it made sense for both sets of data. That's right. tSut what if a statement larger sample?  didn't reflect the  They were to work out how they might revise it so that it made sense and still maintained its original idea. Why didn't theu just throw away the idea if it was incorrect? Because 9 wouldn't let them! That wasn't the point of the lesson. 9t wasn't about right or wrong ideas but to focus on the notion of re-crafting ideas in the face of other evidence so that the integrity of their original thinking could be maintained. Oh okay... focusing on how one's and ideas can shift-shape.  thinking  Exactly. So when 9 went to join the two boys...  290  ^ ^ a r k w a s comparing Robby's graph against trie statement (i.e., green and brown don't have very many.) be bad recorded on bis sheet.  "How are y o u doing, Mark? D o e s it make sense?" I asked. Mark replied "Sort of... because he has three brown and that's not very many and no green." Having qualified three as "not very many", Mark seemed satisfied t o keep his response the same. I then went to see how Danica and Michelle were coming along with their work. While I was doing this, Mark picked up his pencil and underneath his first statement recorded, green dosnt have any in Them. Robby pointed to the sentence and commented that yes, he didn't have ant) green mS-ms in his bag o f candy but Mark did. When I returned, Mark told me that he had changed his statement to that of only green rnSrns. A s this did not reflect his and Robby's sample, I reiterated that they were now looking at both graphs to see if the statement still made sense. Mark immediately shook his head and said no. I then asked them how they might shift it so that it would reflect the larger sample of data. "Keep the same answer." Mark said. "So... keep green and brown don't have very many!" I asked. Mark nodded his head and drew an arrow from his first statement down past the second one to show that he had changed his mind. green and brown don't have very many. \ ^ " e e n dosnt have any in Them. 9n that episode, even though Robby explained very clearly, why Mark's second conclusion didn't fit for both sets of the data. Robby's attempt to shift his partner's thinking didn't have as much impact as 9 thought it might've... until you came back and began working with them. Why do you think this happened? What was it about your interjections that enabled Mark to reflect on his work and to see that his statement still made sense?  291  9 suspect first of all, it's because theu weren't used to working in this way... having to stay with an idea and push how they're thinking of it even when it doesn't immediately appear to resonate with the new context. Mark didn't think that his statement was true of lobby's data so he shifted his idea to only be about green m&ms. tfutwhen he did this, it no longer represented his results! Then, when lobby's pointed out the contraction, it might have been not as effective in prompting Mark to revise his statement because Robby moved from one set of data to the other. My interaction was slightly different in that 9 focused their attention on looking at the graphs together not separately, and asked them how the idea might be altered to accommodate both bags of m&ms. Not only recognizing the contradiction but also, the opportunity to make new sense within this context.  The  boys then examined Robby's statement, There is no same number of colors.  in light of their data. This time Robby moved his head back and forth only to discover that unlike his graph that had different amounts for every colour, Mark's graph had "... lots of the same... same number".  interaction change?  of  the  data  occasioning  ijes. Mark had two each o f blue and red m&rns and one o f each o f brown & green. Mark suggested that they change, the statement to, "... we have the same number o f blue?" Robby looked at the two graphs but did not say anything. "Yes" I said, acknowledging Mark's suggestion but added "... then he's switching what his focus is because he's looking at same amounts of colour, right?" Robby looked at his statement and then beg3n thinking aloud, "...the amount o f colour that...." Mark and I turned to see what Robby was trying to work out. "What do you mean?" I asked Robby. " D o we have t o combine them to get..." Robby asked, taking his pencil and pointing to both o f the graphs. "Yes" I explained, "so you're talking about same number of colours, right?" "Yes" said Robby.  292  Pointing to the statement on his piece of paper, I paraphrased what he and Mark had found. "You said that there's NO same number of colours "I stated, pointing to Robby's graph. "5ut if you were to look at hvobags of m&ms" ! continued and pointed to the two graphs, "you'd find out that that wasn't so." Robby shook his head in agreement. "because here," I pointed to Mark's graph, "Mark got some [i.e., m&ms] that had the same amount o f green and brown and the same amount of blue and red. S o , how could y o u change this idea so that it would make sense f o r both o f the graphs?"  ijou helped them to define the conceptual space they're working in... The mathematical  relationships  of the context. ijes and at the same time, pointed out the need for them to make new sense of that space.  Robby and Mark looked at their graphs. Mark decided that the statement could still maintain the notion of 'same amount' but rather than 'colour', he suggested that perhaps they could look at "same number in both [i.e., bags o f m&ms]". Mark began counting the total number o f m&ms in Robby's sample, "three plus five is eight, plus four... is nine" he said as he added the dark brown,  orange, and  yellow m&ms. "... Is twelve, plus two is fourteen" he continued, adding the number o f blue m&ms. "Fifteen" he concluded after adding on the one red m&m. Robby was also totaling up the m&ms in his sample but in his head, nodding while Mark subtotaled aloud and giving one final nod when he reached fifteen. Mark then totaled his number o f m&ms "... and then you'd go two, four, six, eiglit, ten, twelve" he said while counting the dark brown, green, and orange, and yellow m&ms by twos and adding the four remaining blue and red m&ms to end with "... sixteen." "Okay, he's got fifteen and you've got sixteen", ! said and looked back and forth at Robby and Mark. "So what could you do with that idea [i.e., 'same amount']? Could you say something about the number o f m&ms that would make sensse f o r both of those [pointingto the graphs]?" "Maybe, not all b a e s o f m&ms have the same number in them." replied Mark. "What do y o u think, Robby?" I asked.  293  Robby nodded his bead and said "yes". He then wrote Not all m and m bags have the same amount of m and ms. f o r his second statement. Ljou can see how the students' thinking progressed in the way they developed the statement. Staying with the notion of 'same amount created the need for Mark and Robby to reanalyze their data and form a related yet different interpretation. And the result wasn't the effort of just one individual. What came about was a consequence of mathematical interaction amongst the three of us, all thinking around the idea of 'same amount'. What were Danica and Michelle doing? The girls had taken Michelle's idea and "tested" it against both of their graphs. They were smiling and calling me over to see what they'd found out. "How'd you do?" I asked. "Good!" Danica beamed. S h e picked up her g r a p h and showed that "because I have the same as hers... like two two", Michelle's statement remained "true". Danica had the same number o f red and yellow mS-ms and the same number o f blue and dark brown mS-ms in her bag. Michelle supported what Danica had explained, reiterating that "we have two pairs that are the same" and also added, "and then we both have one that's different." This latter statement was in reference to what Michelle had identified as "one that's all alone," which she described as "one t h a f s all alone... it doesn't have a partner o f a different colour." Like Mark, Michelle drew an arrow underneath the first statement t o indicate that the statement remained the same when compared against both sets o f data. A n d j u s t as in Robby and Mark's case, Danica and Michelle were also challenged when they came upon examining their graphs in relation to Danica's statement. Danica commented that "Michelle has three reds, no yellows, and three blues." She then wrote: Michelle has 3 reds but insted of 3 yellows she has 3 blues.  294  Both gf'rls agreed that this only made sense f o r Michelle's graph and that it did not apply to both o f tbe graphs.  So then, what? Well, just like Robby and Mark, they could say what it wasn't but they weren't sure how to "shift-shape" the idea. Bringing forth Danfca's notion o f the 'number o f red and yellow m&ms', I repeated what she had already said but focused their attention on the red and yellow m&ms. "And y o u found out that Michelle has no yellows but the same amount o f reds... could y o u d o something... with the i d e a — looking at the idea o f reds and yellows? What could you say? 5 o t h graphs..." "... have reds in them.... Both graphs have 5 reds." Danica replied. Michelle and Danica double checked the statement to make sure that it reflected both sets o f data. T h e two girls nodded and concluded "yes!" Danica then recorded: both gragphs have 3 reds.  Having examined and revised their ideas against the children form a working group of four.  their partner's  data, 9 had  ( r e a d o u t the statements that theu were t o now focus o n : • green and brown don't have very many. • Not all m and m bags have the same amount of m and ms. • both gragphs have 3 reds. • There is Two pairs of the same numbers and another That's alone.  A s a small group, we reflected on the statements and talked about how two o f them had undergone changes while the other two remained the same. T h e group then began to "test" the statements to see whether they would satisfy all four sets o f data. T h e children laid their graphs out in a row and examined them, beginning with Michelle's claim, There is Two pairs of the same numbers and another That's alone.  Mark quickly pointed out that it no longer made sense because it did not reflect his o r Robby's graphs. T h e four children proceeded t o study the graphs, saying nothing. T o help them, 1 made the suggestion that they might look to make sense o f three o f the graphs rather than all four at once. 2 9 5  Robby looked at tbe girls' g-aphs, then Mark's, and then his. Here, he corrected Mark by pointing out that Mark's graph was coherent with Michelle's pattern because it too had two pairs o f colour sets that have the same number o f m&ms. He had two o f each blue and red m&ms and one o f each o f dark brown and green m&ms. Robby swept his hand across his eraph and explained that it is when they come to his data that the claim does not hold true. "Mine is the hardest because it doesn't have any pairs. S o , it has nothing to do with that" Robby said, pointing to Michelle's statement. Looking from a different perspective, Danica made the observation that Mark's, Robby's, and her bag o f m&ms all have two blue candies. However, she dismisses it because she realizes that Michelle's b a g o f m&m has three blue m&ms. Michelle pi pes in. "Each o f us has four of... well, not o f the same colour... but four o f whatever m&ms." "Four of any colour" clarifies Robby. "Yes." Michelle agrees. "Everybody has four o f every colour..." Danica concludes. Michelle mulls this over, "four of...." "Pour o f any colour" Robby says again, correcting the language o f the conclusion so that it would be true for all of the samples. The eroup agrees with this and makes the conjecture that a bag o f m&ms will have four m&ms o f some colour. Michelle is not sure how certain they really are that every b a g of m&ms will have four candies o f a particular colour and hesitates in committing this to paper, s h e and the other three children decide that they are not completely certain and that the term, "most" needs to be included in their statement. Michelle picks up her pencil and writes: any M o s t bags have four of the some colours.  9n  that  episode,  you  weren't  interacting  with the students,  there, but  the children  ijou  seemed  directly were to be  working on their own... on their own, as a collective entity. Their thinking and work here is a nice example of what 9'd consider to be a collaboratively collective form of shape-shifting. 296  9 think 9 know what LJOU mean. Each time a new aspect was proposed, it was examined and assessed by the children in terms of whether or not it helped to further define the idea before it was integrated with what they had. Mathematical thinking and mathematics in this collective, shape-shifting manner, is really, a recursive and co-evolving process... emerging thoughts, melding or not with that which already exists and through revision, there's renewal of thinking, of ideas... The students moved on to Danica's idea o f three reds. Scanning the four graphs, Danica shoolc her head and answered, "no". Neither Marie or Robby had three red candies in their bags o f m&rns. Danica adjusted her statement in response t o the new data and announced that "everyone has reds". Robby, Mark, and Michelle all nod and unanimously agree with an enthusiastic "yes!" Still keeping the idea of red m&ms... But no longeron The new statement  now satisfies all four  a specific quantity...  samples.  Danica crafts it a bit more and records, every body has atleast I red. Robby read out his statement, "Not all m and m bags have the same amount of m and ms." Marie, Robby, and Danica skip count by a combination o f twos, ones, and threes to find out that Danica has fourteen mS-ms and Michelle has fifteen candies in total. T h e boys no longer think that the statement makes sense. Robby justifies this "because Michelle and mine have the same amount." I reread Robby's statement, emphasizing the language in it, "It says that N O T A L L bags o f m&ms have the same amount. Does that mean that some bags C O U L D have the same amount?" 297  "Yes" replied Robby and Marie. Robby then drew a downward arrow from his second statement to show that the third statement remained the same.  1  Not all m and m bags have the same amount of m and ms.  A s he did this, 1 prompted the other children to thinlc about the meaning o f the statement and t o articulate why the use o f the word, "not" made it worle. "It wouldn't malee sense if Robby said... what?" "Allthe bags o f m&ms have the same amount in them." said Marie. With this, the group moved on to testing the fourth and final conjecture that was Marie's claim that green and brown don't have very many. Immediately, the g/"oup saw that it held true for Marie, Robby, and Danica's graphs but not for Michelle's. She had four o f each o f green and brown candies and they agreed that g v e n the small pacleage o f m&ms, four was too many to be considered "not very many". Wanting t o integrate the language used in Robby's statement, Marie thought aloud, "Not all bags o f m&ms..." Danica interjected, "all bags don't have the equal amount o f browns in each column. Lilee... nobody has the same amount in browns, 'cause Marie has one, I have two, Michelle has four, and Robby has three." Marie agreed with what Danica had said and was about to write "not all the bags have the same number of brown." when Robby spolee up. "I have another one that he might want t o use" he said. "It is... not all bags have very many... not all bags have very many greens." "5ut when you loole at Michelle's, she has four." argued Marie. "I said not all" responded Robby. "So  ?" I said.  Marie paused for a moment. "I would say that not all bags have the same number o f green and brown in them!" He recorded the revised conclusion: not all m and m bags have the same number of green and brown.  Another recursion in his thinking that effectively integrated the notions of "not all" and "sameness" together while still keeping the focus on green and brown m&ms. 298  A d d i n g on one final layer to their analysis, I gave the children two more sets of data to consider:  -  '0.  M •FUJ  Mm ..- i  JET  "  s  Lara's g - a p r  'Tc;!"  •  io•• it  •  Um  |  Amand a s  grapr  Now with six graphs to consider, the eroup found that Michelle's statement remained the same. A s Lara had four dark, brown candies and Amanda had five dark brown and four red m&ms, both samples satisfied the conjecture that most bags would contain four candies of a particular colour. Danica smiled as she nodded to  indicate that her statement was also  satisfactory. According to the two new g-aphs, Amanda and Lara had at least one red m&m. Robby read his statement aloud, "Not all m and m bags have the same amount o f m and ms." Following this, he totaled the number of mS-ms for each of the two samples in his head. Robby concluded Lara and Amanda's bags each contained sixteen m&ms. "Does that hold true still?" I asked. "Yeah, a bit" he replied. "Why are y o u saying a bv'tnow. What have y o u found?!" I asked. "Fifteen goes with mine and Michelle's and... those ones" he explained as he pointed to Lara and Amanda'sg-aphs, "goes with Mark's". "So would y o u want to change y o u r idea or [shift]shape it a little bit?" I asked.  299  "Yes. Most o f the bags have fifteen or sixteen" he said and then recorded this as his final statement. Do you see that? Do you see what's emerging from Robby's  thinking?  yes! He's looking at a different, more specific pattern in the data. He didn't identify it as "average" number of m&ms per bag, but that's the distinction he's making what he's getting at- that of the bags of m&ms, more of them had a total of either fifteen or sixteen candies than not. Mark read out his statement and quickly nodded "yes"; that it still made sense for the six sets o f data. A c c o r d i n g to the graphs, both Amanda and Lara's bags o f m&ms had different quantities o f green and brown when compared with the other four graphs. Having tested and revised their conjectures across the six samples, the children told me that although they were "pretty sure", they were not completely certain that the four conclusions would hold true f o r even larger samples o f m&ms. S7 wonder what would make them very certain of their conjectures. Did you ask them? yes, and... Robby figured that "you'd have to look at as many bags as possible— twenty to forty" while Michelle was still not convinced that this would be enough samples to be really sure o f her statement. She said that she would need to look at many more— "about twenty-five bags every day for twenty-five days!" Danica ag"eed with Robby and explained that a twenty bag data sample would suffice in testing the certainty o f whether o r not every bag o f m&ms had at least one red candy. A n d Mark said that he would need to test between fifty and seventy-five bags o f m&ms to be certain o f his conjecture. bringing this session t o a close, I asked the children to look at their statements and to consider if and how they changed as the number o f samples increased. 300  "Some o f them changed and some stayed the same." said M a r L "Did anyone have an idea that stayed the same f o r the whole lot?" ! asked. All four students shook their heads, "no". Having established that all of their conjectures went through some kind o f revision,! was interested in their thinking on why this happened. "So everyone's changed at some point. O k a y . Why did they change?" I asked. The group explained that each time they looked at the results o f another b a g o f m&ms, they realized that their conjectures were not always consistent with the new data "because not all m&m bags are the same." Mark, Michelle, and Danica all noticed subtle shifts in the shape o f their conjectures over the course o f examining the six samples whereas Robby nodded his head and exclaimed that his work underwent "HUGE  changes. ! started with colours o f  brown and gyeen and then with all colours (i.e., total number o f m&ms in a bag)."  By taking the mathematical patterns that the students identified from each of their graphs and situating them within larger, collective realms Mark, Danica, Michelle, and Robby got firsthand experience in how the mathematical ideas shape-shifted... and how they evolved as a result of the interaction of their individual and group thinking. And, the integrity of the final conclusions that were born from these interactions... they were really, much more than the sum of the parts that fed into them. 9 mean, the specific aspects generated by the children as individual and collective agents came together not in a piecemeal, jigsaw puzzle way but in a co-emergent manner. What they gave rise to were very sophisticated ways of thinking about the data. More specifically and pervasive in all of the conclusions is the understanding of what you can generalize from the data and what you cannot As well, that growth of understanding isn't always a matter of discarding and replacing ideas, but rather, seeking out patterns... relationships.... and through shapeshifting deeper, more comprehensive notions within a concept can be developed... developing thinking into a greater whole.  301  JVotes 1. This graph was adapted from ASMS, 1987, p. 70. 2. This metaphor first emerged while the students and odd numbers. See page 161  were  investigating  .striking a balance between those spaces In which the children and I are Involved In direct mathematical Interactions and spaces where I am not part of their Immediate work Is another Important aspect of place-making In the classroom., when I am working with a child or group, I am In the midst of, and therefore, part of the co-em.ergence of mathematics taking place. On the other hand, when I am not directly Interacting with the students, I become part of the extended environment. Thus, -providing opportunities of a different kind for myself and the children. For the children, they become responsible for opening and exploring their own mathematical spaces. For me, i am able to attend to and learn about their understandings from a more distant vantage point. The s-paces that are created as a result of the children's mathematical Intra or Inter-actlons not only provide me with glimpses Into their understandings but also, sources from which to further Investigate their thinking and perhaps engage them In folding back or provoke them to extend the mathematics through direct Interaction.  Opening Mathematical Spaces Of Their Own W i t h a blue pencil crayon in hand and a piece of paper to record their mathematics, Sammy makes his way to the round table and pulls up a chair for his best friend, Sam, so that they can work together. Sam sits down and looks around the classroom. " S a m " Sammy taps Sam on the shoulder. "Let's do it!"  Sammy asks Sam what he  knows about the number, 72. "How about..." "What equals that number?!" Sammy  right hand in the air. "Oh!  raises his  I Know!"  "How  'bout...  divided!"  Sam leans  towards  Sammy and clasping his hands together smiles, "... will make itecjual..." "How  about times?  W e could say something that  equals that number." "Yeah!" "What is it? Don't say anything! Just let me think."  With his chin resting in his hand and his elbow on the table, Sammy looks in the opposite direction from Sam. Sammy then records  8x9=72, which catches Sam  by surprise. "How do you Know that?"  304  "I Know the times-table!''  .Sam and sammy art searching for something they already know to be "true" about the number, yo.. They don't Intend to develop a mathematical  description for the number but to re-member or fold back and collect an image— to search through their mathematical understandings and locate something they know without question, to be a fact about the number. Although sammy and sam specify that they are wanting a multiplication fact, they aren't concerned about locating a specific one. .Sammy's Image of 2x^—yQ. satisfies this because he simply retrieves It. He hasn't had to engage In any mathematical activity to com,e up with this fact, he just knows this to be true because of "the times-table!" " M y turn! Seventy-two plus seventy-two..." "that  number plus that number divided by two  equals that number.... Two, two..."  Sammy then  proceeds t o a d d the two sets o f twos in 72.+ 72. together. "It's one hundred forty-four! So it's one-hundred forty-four divided by two!"  Sammy attempts t o  move S a m ' s thinking along. Sam pleads with Sammy. "Let me count., Seventy..." "Multiplied by two.' "... eighty, ninety, one hundred...." Sam  first  aloud  a n d then  counts o n ,  whispering  to  himself, k e e p i n g track o f t h e number o f tens he is a d d i n g b y unfolding a finger each time. O n c e done, he then writes 144-2=72. 305  here? Sam's. announces that Its his t u r n to describe an the boys return, to s a i l ' s first idea to Include "dividedl". They set to worte o n developing a mathematical expression that uses this operation a n d now, different aspects of their understanding are being revealed, sam o n the on hand, approaches the taste by using repeated addition to find the s u m ofyo to f-Q. and records ±44-+3.=j?-3,. From, his worteing with specific numbers and his insistence that S a m m y let h i m steip count on his own to determine what the s u m foryo. Is, I'm a s s u m i n g that sam is using what Images he has to come up with the mathematical expression. He doesn't appear to "tenow" this already nor do his actions Indicate that he has a more forma understanding of the pattern he is worteing with... litee, ' i f y o u add a number to itself or double it and then divide It by two, you'll end up with the original number.' whats going  Now S a m m y o n the other hand, is explaining that, "THAT  trying to  complete S a m ' s thinteing by  N U M B E R plus T H A T N U M B E R divided by two equals  formal conceptualization, wher numbers, Sammy's considering the mathematics  THAT  N U M B E R " . H i s thinteing is a more  sam  wortes with specific  from a more distant perspective. He's thinteing about how the actions of repeated addition and dividing by two, work for ANY number and then h a p p l y i n g this understanding to the specific context of yo.. A s well, Sammy's suggestion that S a m "multiply  understands repeated  addition to be  by two" implies  related to  that he  multiplication.  Pause. Giggles. Pause. Sam suggests t a k i n g JL away from  a  ridiculously large number.  More giggles. Pause. 306  "Aa-she-waa!"  Sammy rests his head o n  his h a n d . "Oh! Let's say something about it Let's not do any of the sums." "It is between eight and something" 5 a m  extends  his arms outwards a n d moves his b o d y in a left t o right motion.  .sam's gestures tell me that he's thinking with the Image of numbers existing along a horizontal line. He ex-presses ya as being located between eight and another supposedly, larger number. "It's  between  seventy  and eighty."  Sammy  r e c o r d s It's between seventy and eighty. "That's a good one " S a m I s it odd or even?!" whispers  smiles a t Sammy.  Sammy. Sam t a k e s the pencil a n d writes It is a even number, a n d then states, "It's a two digit number."  "It's  a  two digit  number."  Sammy, smiles  back at Sam. Sam writes It is a two digit number.  Sammy's prompt for them NOT to just continue to perform operations on ya but offer other ways of thinking about ya reveals that these two students  associate numbers as -possessing other properties rather than only being a product of arithmetic actions.  307  /  3 Its bdwieft 70  SO  Sam and Sammu's written work.  I approach  the two  boys,  "saw.  and S a m m y ,  m a y I j o i n you?" Sammy  moves  the  piece  of  paper  between t h e three o f u s a n d begins reading t h e e q u a t i o n s t h a t  they  have  w r i t t e n a b o u t JL. H e tells me w h a t h e h a s already told S a m — that he knows 8 x $ t o b e JL b e c a u s e h e k n o w s h i s t i m e s table. This confirms, my -previous assessment that he's s i m p l y folded bach and collected an image. Now, ! want to know more about the t h i n k i n g that gave rise to what s>am has written for #2..  "ttow did you t h i n k of doing one hundred fortyfour divided by two equals seventy-two?"  308  "Because seventy-two plus seventy-two equals one hundred and forty-four. If we divided by two, we'd be cutting one hundred and forty-four in half."  in  this one sentence, s a w ,  that  reveals  are embodied In the  mathematical expression. Not o n l y  numerically a n d operationally talk  reflectively  m a n y aspects about the understandings  "correct",  Is the  equation  but he's also able to step back a n d  about the relationships a n d distinctions he's made about the  •property noticing. He considers repeated ad and possibly, multiplication by "doubling" yo. as being the opposite or Inverse action of dlv/lslon. He relates "cutting one hundred and forty-fo In half" as being the same as dividing the number by two. Because s a m Is focusing o n specific values— yo., ±-q»q>, a n d 2, It's not possible to say whether he's got a generalized understanding -for what happens when the combination mathematics at hand—  of operations are applied to A N y or A L L . numbers. I ask. S a m a n d  Sammy if they've recorded  any  descriptions other than symbols. "Yeah" S a m p o i n t s t o #5 o n their sheet.  "It's between 70 and 80". "Yeah. W e used words."  "can we move onto that a n d look at that?" [reads aloud] "It is  between seventy and  eighty."  I a s k S a m a n d Sammy why this makes s e n s e t o them. "Because it is." "Because it's higher than seventy and lower than eighty."  309  .Sammy's image ofyz as "higher" than yo and "lower" than so is simila to Sam's previous bodily gestures. He too is visualizing a number line. Sammy's number line however, is a vertical one on which yo. is located "between" yo and so and "ranked" higher than yo and lower than so. 5 where did this image come from? is it simply a linguistic one that he learned by rote? Or is it an expression of conceptual number sense? I suspe that Ifs connected to the game we play in class where the children try to make the highest (or lowest) possible number by placing randomly drawn digits— O through in either the hundreds', tens', or ones' column. In this game, the class creates a V E R T I C A L , number line by recording all of the possible numbers that could have been made with the drawn digits on the chalkboard. T h i s part of the game requires them to Identify and describe to their peers whether or not and sometimes, justify why the number is considered to be "higher" or "lower" In relation to the other numbers already recorded. Moving on, I  read  out  their next d e s c r i p t i o n - " I t  an even number" a n d ask  ts  the b o y s t o talk a b o u t it.  "Because if it [72] was a seventy-three, it would be odd because the seven and the three are odd....". Sammy in a more confident tone of voice, "It's [72] even because seventy-two can be divided by nine... it can be... divided into nine, into nine groups." Sammy then points to 8 x 9 = 72 on their paper.  "okay, so you're saying tf seventy-two Is even, It can be divided..."  "Because eight times nine means eight groups of nine, equals seventy-two."  "okay..." I ask the boys what they think of 73. They immediately in unison, tell me that 73 Is an odd number.  310  "It wouldn't] be divided into equal groups.... Because it can't be divided  by nine, it can't be divided  into  nine groups."  \'m Interested to see whether S a m m y will elaborate on  this notion of eojual groups by applying his thinking to other numbers, I pose the question, "what if it Ithe number] was seventy-one or seventy-four?". No resfonsel... "what makes an even number... even? l" "You can divide it into equal groups." 'You can count by twos. Two's an even  [unfolds  two o f  his fingers],  number  four  [unfolds f o u r o f his fingers], six [lays his right hand over his left hand s o as t o malce more' fingers], ... all the even numbers." Based on the fact that  sam recorded yo. as an even number and sammy  didn't disagree, I might have assumed that they shared the same understanding for why yo. is an even number. However, now that they've  revealed what they each mean by "even", It's clear that sammy and sam have two coMPL-BTBLy different understandings.  sammy understands "even" numbers to be those numbers that can be "divided Into ec\ual groups", in this case, yo. Is considered to be an eve number because It can be shared ev^Nly Into eight groups of nine without any remainders, of course, this Is true of even numbers, but It's not this quality that defines them as such. S a m m y also explains yz as being an odd number that "can't be divided levenlyl Into groups of nine". Because y3 Is both an odd number and a -prime number, and as well, Its digits—y  and 3 are odd and prime. It's difficult to distinguish what he means by "odd".  1  311  contrast, s«m.'s t h i n k i n g of even numbers comes from a mathematically different place of knowing than Sammy's. Sam's understanding of what makes a number even or not Is In fact, the appropriate one. As he demonstrates for Sammy and me, a number can be distinguished as being even (or not) by whether you can 'arrive' at the number by skip counting by twos. IIA,  [reading  #5] "ifs a two  d i g i t number, what do you  mean by that?" "Because there's two numbers." "It's got tens and ones."  "okay, what's a ten and what's a one?" "There's seven tens and..." "... and there's two ones."  "what does seven tens mean?" Sam explains that if y o u t a k e t h e base ten b l o c k s a n d u s e five ten r o d s a n d o n e unit cube, t h e resulting number is ^1. H e draws this with his finger.  Sam's drawing  Sammy a n d S a m explain t o me that seven tens c a n also be t h o u g h t o f a s "seven g r o u p s o f ten" o r "seven times ten".  The boys are demonstrating that they know numbers can be Identified having "digits" and In the case ofjo,, the digits "J-" and the "o." mak number consisting of a total of two digits. A t the same time, they are also communicating their understanding of place value. According to Sammy, 312  yo can be considered as having "tens and ones", .saw, -points out that yo has "seven tens" and s>ammy finishes by that it also has "two ones", when .saw. acts out that s "tens" bloctes and ± "units" block, would be s±, he reveals the understanding that the first digit of a two digit number refers to the grou-ps of ten In that number and the second digit refers to the number of ones or units It has. The boys' further discussion of how seven tens can also mean "seven groups of ten" or "seven ten" tells me that theyfenowthe left-hand digit In the two digit number as having a value that Is ten greater than the digit on the right. I ask. S a m a n d Sammy if t h e r e is a n y t h i n g else t h a t they know a b o u t 72. I leave the table s o that they can work o n their own again. "Let's just draw one more. Groups. Groups of what?... I Know! Let's draw a groups story and then..."  Sammy g r a b s the pencil hand  from  Sam's  a n d moves the p a p e r in front o f  him. "Let's draw... Let's draw... There's seven tens, right? We can draw seven tens." Sam p a u s e s . "Oh... we draw ten groups of seven and one group of two." "SEVEN groups of ten!"  "TEN groups of seven!" "It's backwards" Tut ten circles, with each..." S a m draws the  diagram with his finger t o try a n d explain his thinking t o Sammy.  313  "It's seven strokes for seven tens, remember?" "Oh yeah!!" "One, two, three, four, -five, six, seven."  draws  and  counts  out  seven  Sammy large  circles. "One, two, three, four, five, six, seven, eight, nine, ten."  Sam draws ten X s in the first circle. Sammy's  persistence that  S a m ' s image of ten groups of seven and two ones  recording a diagram of seven tens and two ones suggests that S a m m y ' s t h i n k i n g is firmly grounded i n its placevalue meaning and thus, doesn't consider the former as being an equ valid way to describe JQ. sam's flexibility to move from one Image to th other however, Indicates that he does relate the two Images as meanin is backwards and that they should be  T h e two children continue t a k i n g turns at drawing the diagram until all seven circles are filled with ten  X s each and  the last  circle has  two ^  s in it.  I return to Sammy and S a m ' s table. Sammy's telling Sam t h a t he's t h o u g h t o f a n o t h e r way to describe 72.  314  J u s t when Sammy explains t h a t seven tens a n d two o n e s malce 72, exclaims, "groups!" a n d writes 7  Sam  groups of 10 and I group of 2 makes this number. Vm curious how they might express their t h i n k i n g  with  symbols and so I ask Sam a n d  Sammy i f  they c a n t h i n k of Another way to describe it. Sammy a n d S a m tallc aloud a s Sammy records 10+10+10+10+10+10+10+2=72 o n t o the paper. 'Ten plus ten plus ten plus ten plus ten plus... one, two, three, four, five... plus ten plus ten equals 7 2 . "  The  students' work demonstrates  their fluency  i n moving  from one  mathematical form, of language to another— spoken, written, diagrammatic, and symbolic.  Sammy offers another s u g g e s t i o n . "How about minus?" "How about repeated subtraction?!"  "Oh. I think I know one." Sammy  144+2=72 hundred  and forty-five  seventy-two?  says, minus  points  to  "How about seventy-three  one  equals  Just make this one higher" a s he  points t o 144.  "Maybe we'll make a repeated subtraction." "How will that work?"  315  "One hundred forty-four minus ten..." S a m begins  t o record 144-10. "Oh! I Know!" minus ten, minus ten..  "You need four." Sammy k e e p s t r a c k o f the number o f tens S a m has  subtracted  from 144. "minus ten" "five" Viinus ten"  six "minus ten"  seven "minus two" equals...  Sam finishes t h e ecjuation. 144-10-10-10-10-10-10-10-2=72. point t o #9. "How'd y o u come up with this?" Sam points t o  10+10+10+10+10+10+10+2=72  a n d tells me "I can read the plus."  316  Sammy a d d s : "Forty-two (sic) [presumably, 72] is multiplied twice— two seventy-twos, [points t o 72 in 144+2=72] And if you minus one of the, take away one of the seventy-twos, but actually don't say something [i.e., 72], do it another way, like  this  one [sweeps  bis finger over  10+10+10+10+10+10+10+2=72]. You get seventytwo. Equals another seventy -two [points t o =72  in 144 -10-10-10-10-10-10-10-2=72.] B y asking s a m m y to explain his t h i n k i n g reveals yet other sophisticated aspects o f h i s mathematical understanding— once again, equalities that aren't visible •from, only looking at whafs been written, s a m m y explains that Instead o f w r i t i n g : 144 - 72 = 72, he'd w r i t t e n 144 - (I0+I0+I0+I0+I0+I0+I0+2) =72. The reason why he d i d n ' t record it in this form. Is s i m p l y because he hasn't learned how to use brackets yet. Regardless of t h i s It s t i l l doesn't maize It Impossible for h i m to t h i n k conceptually about the mathematics I n this  m.anner\ Together w i t h  his  physical  finger  0+10+10+10+10+10+10+2=72 a n d verbal reference to "take  pointing  away ONE  to o f the  seventy-twos but don't a c t u a l l y s a y t l t L do It another way, l i k e this one" Illustrates that S>amm.y's very m u c h ' s t a n d i n g back' a n d reflecting on or observing  his formal  understanding of the mathematics.  He's  NOT  s p e a k i n g about the repeated subtraction o f t e n but t s u s i n g a specific case of the distributive law over a d d i t i o n : - (±0+±0+±0 + ±0+±0+±0 + ±0+3.) = -±O-±0-±0-±0-±0-±0-±0-Q. Even though s a m m y hasn't generated a 'theorem.' as such, he Is In fact, u s i n g a formalized u n d e r s t a n d i n g of addition I n t h i s context. I w a n t to k n o w what s a w / s t h i n k i n g a n d so, I don't react to S a m m y ' s explanation b u t continue to listen. "Seventy-two plus seventy-two equals one hundred and  forty-four." A t t b e same  time, S a m  stretches his thumb a n d fingers a r o u n d either e n d o f t h e -I0-I0-I0-I0-I0-I0-I0-2 317  a n d says, "Seventy-two." "Sam just took another seventy-two by not saying it." I r e p e a t what Sammy has j u s t s a i d but in a louder voice t o try a n d b r i n g S a m into the conversation, "sam just  took another seventy-two by not  tt?" "By not., writing it [72] sweeps  his  fingers  down." Sammy in a  circle  a r o u n d IHH-10-10-10-10-10-10-10-2=72. "By not... like... one hundred and forty-four minus... you minus one hundred and forty-four by seventy-two." Sam  also  sweeps  his  finger  across  IHH-10-10-10-10-10-10-10-2=72, "and you get seventy-two. Because... because..." a n d points t o 144+2=72. "But you also minussed! ... Sam ... Sam also minussed seventy-two but he also... one hundred forty-four minus seventy-two equals seventytwo. He also minussed the seventy-two." B-y encouraging them continue, l see that saint's also t h i n k i n g ofjQ i n the same manner as S a m m y . Although the boys don't t a l k i n general terms but repeatedly use #$? a n d #9 as specific examples i n their explanation of how they "minussed. seventy-two", neither s a m orsammy speak of "±-f-f minus ±0, minus ±0, minus ten." and so on. It's clear to me that B O T H students t h i n k of subtracting JQ  in the form ±o+±o+±o+±o+±o+±o+±o+Q  as  they have expressed in #2. M o v i n g to their n i n t h description, I wonder if what S a m m y a n d S a m have recorded here could be true for a n y number, i n other words, can they extend their t h i n k i n g to a more general understanding of this concept? 318  T d say yes, most of them..." "Most probably."  Sam turns t o Sammy. T i c k a  number. Pick a number and I'll try."  referring  "Okay. Try the first  number we did",  to the equation  11+11+11+11+11+11+11+7+1=85  that they h a d written earlier. "Eighty-five and..." "Just pick any number. Just pick a number."  .Sflw. and Sflium-y art engaging in taking what  they know and applying  it to a new context. Vm interested in how they might use the mathematics as a "pAttem" and not, in a situation-specific manner, interrupting them, \ shift the conversation by bringing Sam and Sammy back to their previous conjecture. ' Sammy, you said that you thought it would worfe for any number it W ASflLm-ostopposite, whflt  did you mean by that?"  'It's  subtracting  instead  of adding."  refers me hack, t o what he'd  Sammy  recorAed as  11+11+11+11+11+11+11+7+1=85. "... to do this... maybe the number you pick., you add i t , you add it with the same number that you... picked... you get the big number..." "You  times  it by two! Minus  it"  (i.e., the  number). "Like eighty-five plus eighty-five equals one hundred  319  9  and seventy... and you minus it." Sam p o i n t s t o  ll+ll+ll+ll+ll+ll+ll+7+l. T o get rid of the extra half."  Affcer Listening to Sam and Sflm,wiij explain their thinking again butt this time In more general terms, they are definitely demonstrating that what they know Isn't specific to particular numbers but a formalized pattern that they've reflected on. Watching from the edge, I was able to observe S a m a n d Sammy as they  opened  up and  explored their own mathematical  space.  Their  mathematical activity o c c a s i o n e d opportunities f o r me to g e t a close u p view o f the complex understandings that Sammy a n d S a m have developed for whole positive numbers. Their work was not concerned with the generation o f as  many  arithmetic facts as they could think of, but instead, t h i n k i n g creatively a b o u t the number, Jl. 5 y f o l d i n g back, they developed a n d extended their mathematical ways in which t o express it. A n d as S a m a n d Sammy did this, o n e c a n see how their understandings f o r whole positive numbers existed as an integrated system o f knowing.  Notes In hindsight, had I aslced him to explain whether 55, J?, 77, or 7? was even or o d d , this might have given me the insight into interpreting his understanding.  320  /Katkematlcs T^etfond the (Ztassiootn  3, have been working en defining another direction from which to come at Q, Sktteson's concept of patterns which connect''. 3, would (the to teach collaboratively who shared, a similar interest and vision for teaching mathematics,, ds well, because mg students, will already, be working with other students who are younger and older than them and, there is not another second and third grade class, in my school, it would be great if the teacher also taught children of the same ages,. 3, would like this, collaboration to bean active, ongoing engagement for the teachers, and students,. 3. see this, as, an opportunity not only for sustained cooperative learning-- where long term mathematical relationship can develop for the children's, mathematics, to take root, connect, and grow but also, an occasion for co-emergent and collaborative teaching tf  1  Two weeks kter.oo Slow can 3, make this, happen? More specifically,, with whom?! 3t has, to be with a colleague who not only has, a passion for teaching mathematics but also someone who envisions children's mathematical learning in a similar manner as, 3, do A N D someone who wants to take on the challenge! 3, have made a list of teachers 3, have teamed with in the district but none seem appropriate for this, particular prefect.  That afternoon... I scanned my I N box of emails and noticed that I had received something from my friend, Donna. I double clicked to open the message and found that she had written me a letter asking how my year of teaching had been.  322  Donna and J. first met at the Vniversity white completing owe Masters degre mathematics education. Jt had teen two gears since we graduated and we alw talked about working together but the opportunity had not presented itself.. now! Met only were S)onna and J, seeking to teach in ways, that focused o growth of children's mathematical understanding, but upon reading the rest email, 3, discovered that she would also be teaching second and third grade c in the coming school year. I double clicked on the R E P L Y icon. Typing a response back to her as fast as I could, I told Donna what I had been up to in terms of my teaching and briefly explained my thinking behind the collaborative project. I then asked her if she was interested in working together and clicked on the S E N D icon.  That summer and into the beginning of the school year... D onna and I began planning the "math buddies" program for our two classes. Using the theme of connecting our students with their mathematics, we focused on establishing interactive learning spaces that not only emphasized the children's conceptual connections with mathematics but also, connections that would grow between the two classes and amongst the students as they worked together as one collective group.  Since the program centred on the children's mathematical interactions, we sa a natural context for dtudent-getterated problem posing and problem s projects. Over the course of two years and despite the fact that Donna and I worked in different school districts, we co-created and taught mathematics classes simultaneously in our own classrooms and together as a group. We communicated to each other through e-mail, phone calls, and meeting each other in person. During these meetings, we reflected on lessons taught and took a closer, critical look at the kinds of teaching and learning that were arising. This gave us a chance to study the students' work and assess their mathematical understandings jointly, thereby, gaining insight from each others perspective. It also allowed us to flesh out possible lessons and consider the impact that the teaching we were proposing might have on our classes' learning. From here, we sketched out teaching-and-learning settings. Lessons were not prescriptive nor planned far in advance but evolved from the mathematics that surfaced in the classroom(s). 323  i. See G. Bateson, 1980, p. 9.  Keeping Our Glasses Connected H/ven when it was not possible for Donna and I to physically teach together, we took it as an opportunity to find creative ways to keep the children connected. One way we did this was by teaching similar lessons within similar time frames. This provided occasions for sharing and examining the mathematics that was being developed in our classrooms. For example, as both of our classes were working on developing strategies for solving two and three digit computations, Donna and I presented work that children in the other class had produced. Our students were invited to consider how the methods generated by their buddies were the same or different from theirs and whether (or, not) and why the proposed strategies made mathematical sense. N o t only did this enable the saidents to share what they were learning with the other class but it created spaces for them to critique each others mathematics, to justify their thinking, and to offer new thoughts for their peers to consider.  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  Mathematical Gifts T h e math buddies program was where the students' making and exchanging 325  exchanging of mathematical gifts began. These gifts contained the conceptual ideas that the children happened to be exploring in class and by wrapping them up in imaginative contexts, the students' offerings turned into riddles, creative problems, and puzzles for their buddies to "play" with. Most often, the children worked with a partner during the posing and the solving of these projects.  Sienna and 3. felt this te tie important in order to maximize mathematical comnumication and collaboration.  SJhe children's first gift ta their buddies was. a letter introducing themselve class, the students had been investigating number concepts and decided that it be fitting if theg. introduced themselves in mathematically- imaginative term  the concepts theg. were learning about, such as "I am in the grade that comes befor four and two after kindergarten" and "My favourite num6er is 12 less  Irian  20 and 5  more than 3". SJhe children spent a good part of the next dag measuring, collectin and analyzing the numerical data about themselves such as their height, weight shoe size, how many- members in their family, favourite number, etc, and developed mathematical descriptions for them. SXonnaJs students were delig receive the letters. Shis in turn, opened up similar investigations ef data an analysis for her students who responded with similar letters for my students read •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  "Reflect on trie reflection"  Danica drawing reflective images in the "water"  326  R o b b y : "Using a mirror helps me to check what I've drawn."  "Reflect on the reflection" was a gift that came out of the classes' ongoing studies of symmetry. Here, Donna's students painted pictures on one half of a piece of paper. They then gave them to my students who had to complete the other half of the image by drawing its reflection in the water.  5JO solve this, puzzle, the students needed to locate the horizontal l and then with the aid of a mirror., 'flip' each of the images and pain ether side of the line. •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  "Can y o u build my design?  "Hower  M  pattern bloclc design from Lara, Gregory, Marie, and Srian  327  m/m  fie*r<b>* fop  Umi  fif  "fee  f*$e-  ftewer Jo M m a k e s * W W  fabw  kT+fc* - m o L « 3 t «  foi'n^  tne„ind«?»tof -HkJ W  Lara, Gregory, Mark, and brian's clues for their block design  your rny*Hty DifWvtefAdoi^I/w8«ac)r bdcatrtC youKt/uti  v***  <^«q*  to.saJ+Wtt amy  A letter from the group's math buddies.  "Can you build my design" was a gift created by my students using pattern blocks to make a geometric design (e.g., aflower).Working in pairs or small groups, the children composed a written set of instructions telling their math buddies how to build the block design and included a picture of what it looked like so that they could compare it with what they had built.  My students applied their understanding of geometry and spatial sense to desc name, and interpret relative positions in space, 3or ^Donna's class who receive challenge, this puzzle required them to visualize, identify, and locate the desc positions in space in order to build the design with pattern blocks. Once they done so, the children compared what they had built with the picture of the probl posers' intended design and wrote back to their buddies explaining what pa 328  of the instructions were clear te understand, what steps were difficult te figur and why. •  •  •  •  •  •  5  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  y  \\ \y  Extraordinary Equations For You 1  { V r.-ii:/ rM--:r'3  v  ••.*.}  Carter's number stories about Y).  "Let it roll" was a game my students made for their math buddies. It consisted of a page full of dice showing values of 1-6 and a "target number". Donna's students worked with a partner using as many of the dice to create different number stories about the target number.  She exchanging of gifts and sharing of werh gave rise te mathematics that flow bach and forth, connecting our students and bringing our classrooms together  329  this, way, the teaching-and-learning that emerged in one classroom naturally a part of- what was taught-and-learned in the other classroom and so on much like a conversation' in progress, the mathematics that emerged existe interaction between our two classes and provided an ecological space for grow  Motes 1. Qadamer, 1989. •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  Celebrating Together  Cls far as 3)onna and 3 were concerned, teaching mathematics was neith exclusively independent activity nor did it need be restricted within the con one's classroom. Jn the math buddies program, one of our aims was to blur times, ignore or remove such boundaries we found to be limiting. Jn this w sought to relocate classroom mathematics by planning and teaching lesson tandem through sharing of students' worh, and in the children's exchangin mathematical gifts, dnother way in which we accomplished this was thro "celebration days".  Celebration days were when we brought our classes together and entire school days were devoted to engaging in mathematics as a collective group.  SJhis was a scavenger hunt designed for the children ta team about their, (kiddies attitudes and thoughts regarding mathematics. She children ashed st in the other class questions, Me, "What do you like about mathematics?", "What do you find challenging about mathematics?", and "Can you think of another way to describe the number,  10?". She collection of questions were generated by Donna and myself  well as Both classes of students. •  • • • • • • • • • • • • • • • • • • • • • •  Teaching our classes together gave Donna and I the opportunity to examine the children's mathematics as it emerged and discuss how to plan for subsequent activities. Because we shared similar views on mathematical understanding and considered it to be nonlinear, constandy growing, and ever fluid in nature, the mathematical explorations that we planned needed to be open and responsive to our students' thinking. This meant paying attention to the conversations and questions initiated by the children as they watched a film, read a story, or participated in an activity. Donna and I would then integrate the ideas brought forth by the children into their mathematics for the rest of the day. 1  Jn the morning the children had watched a mathematics film called SUmce squa and Donna had read a non-mathematical booh tilled, Selina and the Bear pa guilt. Donna and J, were curious as to what would happen if we presented the f and the story one after the other and used them as springboards for the studen explorations that afternoon. 3  Sharing; trie storu of Selina and the bear paw quilt 331  Would the children worn and produce something with each of their math b Would our classes engage in some kind of collective, whole group mathem What would the mathematics entail? Qeometric compositions of a squ tessellations? Quilts?] The day before, Donna and I collected materials that we thought the children might find useful such as: wallpaper, rulers, scissors, pattern blocks, plastic 2-D shapes of the ones that appeared in the film and the story, chart paper, large pieces of felt material, glue, scrap paper, pencils, a class set of chalkboards, and coloured paper.  SJhe questions, and discussion that arose after watching the film and listenin story, created the context that shaped the afternoon's investigation. (Sve Donna and S assessed what we thought to be the main themes, coming from children's reactions, to the film and the story. Sheir comments centred arou possible geometric shapes that could be used in composing a square and the s  needed to make the designs and pictures inside the quilt squares, Ser instance shapes you would need to make the bear's claw". Other students spoke about the  transformations of two-dimensional figures that would be necessary— "how  need to turn the shape" in order to make a particular geometric design or picture fo  quilt square, Und given that the entire discussion was, directed toward that and quilt squares, this, naturally became the setting for our students' project. That afternoon, the group decided diat each of them would work with their math buddy from the other class to plan and produce one square that would then form a quilt made by the entire group. Donna and I explained the two themes that we had drawn from the groups discussion and asked the students how these might be worked into their quilt squares. "The partners can decide what they want to do... they can do a design or a picture on the quilt square" said one student. "But they have to do that... they can't just put shapes all over the square... that don't make sense" added another student. "And, we can use the pattern blocks to figure out how to turn them... as tracers too"  explained a third student. The children dispersed from the middle of the gymnasium floor to find their math buddy and to gather the materials that they needed. 332  J)onna and 3. made our way aw und te aii af the students either individu together. Watching, listening, and ashing the children to explain what thei entailed, revealed three geometric and spatial strategies that the children utilizing to plan and make their quilt squares. c  Some students used paper and a pencil to sketch their picture and then u pattern blocks by moving them about. 9$y rotating, flipping, or using the combination with other blocks, they were able to "fill" the regions af their im appropriate shapes,. Other children chose not to draw their design but used pattern blocks as manipalatives with which to plan out and then trace them t their quilt square.  Danny and Jerry's pattern bloclc design  Still, other students relied an their mental image or physical use of a single p block to perform transpormations.  333  Clare explains,  "If we use the triangle and keep turning it this way again and again until it  gets back to the top, it'll make the petals of the flower.  By the end of the day, the children had created a quilt that integrated the designs and patterns they had experienced through the film, the storybook, and their play with manipulative materials.  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  •  "Number a i m nasties"  ring one of our other celebration days, the children helped to create and participate in "number gymnastics''. The students selected one of their "favourite" numbers and used it to work through a series of tasks that challenged the students' conceptual thinking and provided Donna and I a chance to observe their flexibility of number sense regarding concepts we had taught during the year.  She mathematical prompts came puun a journal that £ had kept oaring the cou oftileschool year. Jn it contained a collection of. curious questions and observ made 6y Donna, the ch