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From mathematics learner to mathematics teacher : preservice teachers’ growth of understanding of teaching… Borgen, Katharine Louise 2006

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From Mathematics Learner to Mathematics Teacher: Preservice Teachers' Growth of Understanding of Teaching and Learning Mathematics by Katharine Louise Borgen B .Ed . , University of Alberta, 1968 M.A., University of British Co lumb ia , 1998 A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in The Faculty of Graduate Studies (Curriculum Studies) University of British Co lumbia June 2006 © Katharine Lou ise Borgen, 2006 11 Abstract This study was des igned to determine whether using the Pir ie-Kieren Dynamica l Theory for the Growth of Mathemat ical Understanding as a model enhanced the growth of preservice teachers ' understanding of teaching and learning mathemat ics. The study a lso investigated the eff icacy of using the theory as a framework with which to ana lyze that growth. The study charts the growth of understanding of four preservice, secondary, mathemat ics teachers during the integrated portion of their teacher educat ion program during which they were introduced to the Pir ie-Kieren theory and encouraged to cons ider it during their reflections on their learning. This portion of the program concentrated on mathemat ics teaching and learning and was des igned to help them develop an understanding and practice of teaching and learning which reflects the present-day concept ions of mathemat ics and mathemat ics educat ion as dynamic p rocesses . The Pir ie-Kieren Dynamica l theory was modeled for the preservice teachers because of its al ignment with these views. V ideo data on the four preservice teachers was col lected during the integrated portion of their teacher educat ion program and during their pract icum exper ience. Ana lys is of this data resulted in a portrait of each of the four individuals. T h e s e portraits were then used as the data for the analys is of developing understanding of teaching and learning mathemat ics. The definitions of the terms used in the Pir ie-Kieren theory to descr ibe the different levels of understanding of mathematics were modif ied, retaining the integrity of the Ill original, to determine whether the theory could be used as an analytic tool to d i scuss the growth of understanding of teaching and learning mathemat ics. A modif ied model that reflected these definitions was a lso deve loped. The analys is indicated that the modified definitions provided an effective structure for d iscuss ing the growth of understanding of teaching and learning mathemat ics, both for the preservice teachers and the researcher. It a lso indicated that, as with the learning of mathemat ics, the developing understanding of the activity of teaching and of what it means to learn mathemat ics is an individual exper ience and is based on one 's own background, or one 's own Primitive Knowing. Th is development is a dynamic process which involves Folding Back to previously held Images to examine them in light of newly acquired concepts. Table of Contents Abstract ii Tab le of Contents iv List of Tab les xi List of F igures xii Acknowledgements xiii Chapter 1 Introduction to the Study 1 1.1. Rat ionale for the study 1 1.2. Teacher educat ion 3 1.2.1. Attitudes and beliefs 3 1.2.2. The use of theories 5 1.2.3. The use of imagery in teacher educat ion 7 1.2.4. Requis i tes of teacher educat ion 8 1.3. Images of mathematics educat ion 9 1.3.1. Requis i tes of mathemat ics educat ion 11 1.4. Pu rpose of this study 12 1.5. The organizat ion of the chapters 15 Chapter 2 The Pir ie-Kieren Dynamica l Theory for the Growth of Mathemat ical Understanding 18 2.1. Background to the Pir ie-Kieren theory 19 2.2. A model for the Pir ie-Kieren Dynamica l Theory for the Growth of Mathemat ical Understanding 20 2.2.1. Mapp ings 21 V 2.2.2. Transcendenta l recurs iveness 23 2.3. The layers of understanding in the Pir ie-Kieren theory 25 2.3.1. Layer 1: Primitive Knowing 26 2.3.2. Layer 2: Image Making 26 2.3.3. Layer 3: Image Having 28 2.3.4. Layer 4: Property Noticing 29 2.3.5. Layer 5: Formal is ing 30 2.3.6. Layer 6: Observ ing 32 2.3.7. Level 7: Structuring 33 2.3.8. Level 8: Inventising 34 2.4. Features of the Pir ie-Kieren theory 35 2.4.1. Feature 1: Folding Back 35 2.4. 2. Feature 2: Interventions 42 2.4.2.1. T h e m e s of Interventions 42 2.4.2.2. Teach ing styles 44 2.4.2.3. Towers ' observat ions on the use of Interventions 46 2.4.3. Feature 3: Don't Need Boundar ies 47 2.4.3.1. The importance of the crossover from Property Noticing to Formal is ing 48 2.4.4. Feature 4 : Complementar i t ies 50 2.5. Extens ions and elaborat ions 52 2.5.1. Elaborat ing the enactivist nature of the theory 53 2.5.2. Alternative forms of Mapping 54 2.5.3. Col lect ive understanding and shared understanding 58 2.5.4. Prospect ive teachers ' growth of mathematical understanding 60 2.6. Modif ication of the Pir ie-Kieren theory to a new context 62 2.6.1. The modified definitions 63 2.6.2. The modified Pir ie-Kieren model 69 Chapter 3 Rev iew of Literature - Teacher Educat ion 3.1. A context for d iscuss ion of preservice teacher educat ion 71 3.2. A d iscuss ion about the scholarship of teaching and teacher educat ion. . 76 3.3. What is teaching? 78 3.3.1. The role of theory in teaching 80 3.3.2. The role of reflection in teaching 82 3.3.3. The role of knowledge in teaching 83 3.3.3.1. The role of subject content knowledge in teaching 83 3.3.3.2. The role of mathematical content knowledge in the teaching of mathemat ics 85 3.3.3.3. The role of pedagogical content knowledge 86 3.3.3.4. The role of pedagogica l content knowledge in mathemat ics teaching 88 3.3.3.5. The role of practical knowledge in teaching 89 3.3.4. The role of attitudes and beliefs in teaching 90 3.3.4.1. The importance of changing attitudes and belief about the nature of mathemat ics 92 3.4. What is effective mathematics teaching? 93 3.5. What is learning to teach? 94 3.5.1. The effect of prior school ing on beliefs and Images of teaching 95 3.5.2. Effects of previous mathematical exper iences on beliefs and Images of teaching mathemat ics 98 3.5.3. The effect of teacher educat ion programs on beliefs and Images of teaching 99 3.5.4. The effects of the practicum exper ience on beliefs and Images of teaching 100 3.6. What can teacher educat ion programs d o ? 102 3.7. Mathemat ics teacher educat ion programs 105 3.8. Conclud ing remarks on teacher educat ion 108 Chapter 4 Rev iew of the literature: V ideo data and Portraiture 110 4 .1 . A context for v ideo as a source of data collection 110 4.1 .1 . Col lect ing video data 111 4.1.2. Advan tages of v ideo data 114 4.1.3. A process for analyzing video data 118 4.1.4. Validity in v ideo research 120 4.2. Consider ing c a s e study and portraiture ..: 121 4.2 .1 . C a s e study 121 4.2.2. Portraiture 122 4.2.3. Choos ing c a s e study and portraiture 123 4.3. Writing of a portrait 125 4.3 .1 . Rigor in portraiture 127 v i i i 4.3.2. Trust in portraiture 127 4.3.3. Authentif ication in portraiture 128 4.4. Conc lud ing remarks on v ideo data col lect ion, c a s e study and portraiture 129 Chapter 5 Methodology - Outl ine of Study 130 5.1. The setting 130 5.2. Framework under which the study was be carried out 132 5.2.1. Presentat ion of the Pir ie-Kieren theory 133 5.2.2. Data collection '.: 136 5.2.3. Initial quest ionnaire 136 5.2.4. Outl ine of v ideo data collection 137 5.2.5. Genera l data collection 139 5.2.6. Pract icum data collection 141 5.2.7. Reflect ion v ideos 142 5.2.8. Debriefing 143 5.2.9. Field notes and c lass ass ignments 143 5.2.10. D iscuss ion of the collection of the data 144 5.3. A n outline of the data analys is 144 5.4. The drawing of the portraits 147 5.5. Ana lys is of the portraits 154 Chapter 6 Presentat ion of the Portraits 160 6.1. Introduction 160 6.2. A portrait of Soph ia 161 ix 6.3. A portrait of Lance 182 6.4. A portrait of Ell ie 205 6.5. A portrait of W a y n e 232 Chapter 7 Ana lys is of the Portraits 262 7.1. Introduction 262 7.2. Ana lys is of Soph ia 's portrait 264 7.2.1. Classi fy ing Soph ia 's levels 264 7.2.2. Chart ing Soph ia 's growth 276 7.2.3. A d iscuss ion of Soph ia 's growth 278 7.3. Ana lys is of Lance 's portrait 282 7.3.1. Classi fy ing Lance ' s levels 282 7.3.2. Chart ing Lance 's growth 294 7.2.3. A d iscuss ion of Lance 's growth 296 7.4. Ana lys is of El l ie 's portrait 300 7.4.1. Classi fy ing El l ie 's levels .... 300 7.4.2. Chart ing El l ie 's growth 312 7.4.3. A d iscuss ion of El l ie 's growth 314 7.5 Ana lys is of W a y n e ' s portrait 317 7.5.1. Classi fy ing W a y n e ' s levels 318 7.5.2. Chart ing W a y n e ' s growth 331 7.5.3. A d iscuss ion of W a y n e ' s growth 333 Chapter 8 Conc lus ion 337 8.1. Introduction 337 8.2. Main f indings related to growth based on the use of the Pir ie-Kieren theory 338 8.2.1. Soph ia 338 8.2.2. Lance 340 8.2.3. Ell ie 343 8.2:4. W a y n e 344 8.2.5. Genera l f indings with respect to the Pir ie-Kieren theory 346 8.3. Implications for the Pir ie-Kieren theory 348 8.4. Implications for teacher educat ion 349 8.5. Other suggest ions for future research 351 8.6. Conc lus ion 353 References 355 Append ix A Ethics Permiss ion 367 Append ix B Cou rse Outl ine 368 Append ix C Images Quest ionnaire 370 Append ix D V iews Quest ionnaire 374 Append ix E S T U D E N T I N F O R M E D C O N S E N T F O R M 375 Append ix F Letter of Permiss ion 377 Append ix G Permiss ion from the Vancouve r Schoo l Board 378 Append ix H A D M I N I S T R A T O R I N F O R M E D C O N S E N T F O R M 379 Append ix I T E A C H E R I N F O R M E D C O N S E N T F O R M 381 x i List of Tables Tab le 1: The framework for descr ib ing folding back (adapted from Pirie & Martin, 2000; p. 143) , 37 Tab le 2: Towers ' intervention styles and related teaching strategies 45 List of F igures F i g u r e l : A Mode l for the Pir ie-Kieren Dynamica l Theory for the Growth of Mathemat ical Understanding 21 Figure 2: Samp le mapping of student understanding using the Pir ie-Kieren model '. 22 Figure 3: The Pir ie-Kieren model incorporating the Complementar i t ies 50 Figure 4: Pirie and Martin's Complexi ty Drawing (Pirie & Martin, 2000) 56 Figure 5: Towers ' L inear Mode l (Towers, 1998) 57 Figure 6: Borgen and Manu 's Clustered Confus ions Mapping . 5 8 Figure 7: Borgen 's Modif ied Dual Mode l for the Mapping of the Growth of Understanding of Teach ing and Learning 70 Figure 8: The Mapping of Soph ia 's Growth of Understanding of Teach ing and Learning 278 Figure 9: The Mapping of Lance 's Growth of Understanding of Teach ing and Learning 296 Figure 10: The Mapping of El l ie 's Growth of Understanding of Teach ing and Learning 314 Figure 11: The Mapping of W a y n e ' s Growth of Understanding of Teach ing and Learning 333 Xll l A c k n o w l e d g e m e n t s Writing this P h D dissertation has been a long and arduous process, not only for me, the author, but a lso for my family and fr iends. A s the author, I had the enjoyment of being immersed in the data and analys is. The others, however, supporting and consol ing me, have to put up with my mood shifts and behavioral quirks. Al though it is impossible to descr ibe how much all the support and tolerance has meant to me, I must try. First and foremost, I want to thank my family. For us it has been a trying few years with the loss of severa l family members , but we have held each other up through it all and hopefully we are stronger for it. Specif ical ly, then: To my husband, Bill: Thank you for supporting me (emotionally and financially) so that I could attain my goal . A n d thank you for all the trips that gave me a break so that I could come back f resh, and with new ideas to work on my paper. I know that I was not a lways the eas iest person to be around and that I left many things undone over the past few years /put t ing more pressure on you. It's now my turn to make it up to you. To my chi ldren, Jeff, Jennifer and Cra ig : Thank you for a lways telling me how proud they were of me and asking me how things were going. E a c h of you offered support in your own way: Jeff, even when your life turned upside down, you never forgot to ask about my work and to assure me that I was doing a good job. Jennifer, you didn't let me forget that I w a s a "silly mom" and you did all the cross-check ing for me. Cra ig , whenever I was getting depressed , your big bear hugs and k isses a lways made me feel better and inspired me to continue. I want you all to know how much your support w a s appreciated. I couldn't have done this without you. To my committee, Dr. S u s a n Pir ie, Dr. Lyndon Martin, Dr. Anthony C larke and Dr. Cynth ia Nicol : E a c h of you was there for me when I needed you most, even if you were continents and oceans away. To S u s a n , my mentor, my advisor, my friend: Al though you were living in France, we were able to keep in touch as easi ly as if we lived next door. Thank you for your support, suggest ions and advice. Many t imes I thought you were just creating work for me - but once done, I real ized that the suggest ions helped me do better research and write a better paper. To Lyndon: Thank you for the quiet support and short notes of encouragement . Somehow, once you moved to Eng land, you s e e m e d to sense just when I needed a bit of a "Hel lo, how's it go ing?" To Tony: Thank you for being so positive and encouraging. Y o u were a lways prompt with replies, even when in Austral ia on sabbat ical . Thank you for the adv ice on reorganizing the paper to make it read better. To Cynth ia: Y o u once said that you didn't do much, but you must real ize how important it was for me to have someone here to talk to. I want you to know XIV how much I appreciated the fact that you a lways found time for me. Y o u helped me through some hard and lonely t imes. < To my examiners , Dr. Sa rah Berenson , Dr. J a m e s Gask i l l , and Dr. Car l Leggo, and the chair of my oral defense, Dr. Geo rge B luman: Thank you for the effort and attention you out in to reading the paper, for your thoughtful quest ions and for catching my spell ing and typing mistakes. I feel honored to have had the four of you as part of this process. Lastly, there are many people who supported me throughout this writing. I cannot thank them all individually, but there are some who need specia l mention. To the rest of my family - parents, brothers, sisters, in-laws, n ieces, nephews, etc. - including those who are no longer with us: Thank you. Y o u were my models and my inspiration. To Donna Jenner and Dr. Jenni fer Thorn: My fr iends, I thank you for your encouragement throughout. To Bob Hapke: Thank you for all the computer expert ise. Y o u were a lways able to figure out what I was talking about even when I didn't know! Thank you for being able to make sense of my drawings to create the table for charting the growth of the preservice teachers ' understandings. Thank you for being my friend and a lways giving me words of encouragement . A spec ia l thank you for retrieving the "corrupted dissertation". I don't know what I would have done if it had had to be retyped! To Nathan Hapke: Thank you for, so quickly, creating the computer ized version of the modified model from my rough sketches. To Dr. J a m e s Sherri l l : Thank you for the advice you gave me when I was about to defend my M A : "Remember , Katharine. This is your research and you know more about it than anyone." This thought stayed with me through my P h D research and I have passed it on to many others. 1 Chapter 1 Introduction to the Study There is an abundance of literature on the beliefs of teachers about teaching and learning and a lso on the bel iefs that preserv ice teachers hold regarding the nature of mathemat ics and mathemat ical learning. Th is literature s e e m s to indicate that often teachers ' beliefs about teaching and the nature of learning are determined before they enter a teacher educat ion program and that the effect of the program is minimal in changing these beliefs. O n e possibil ity cited for this lack of change is that the length and intensity of the teacher educat ion programs are not sufficient to inf luence the understanding of the prospect ive teachers regarding the p rocess of teaching and learning. Bel iefs, it is felt, often are not founded on fact and therefore need proof and t ime in order to effectively be re-def ined. 1.1. Rationale for the study Being an effective teacher involves having the "ability to transform subject content through a range of techniques, sui table examp les , analog ies, etc., to make the content more a s s e s s a b l e and memorab le to the learner" (Smith, 2001 ; p. 117) or to be able to translate one 's own knowledge into c lass room curricular events that make that knowledge access ib le to students, using the students ' interests and motivations to learn a particular topic (Carter, 1990). A central tenet of teaching is "an understanding of how to recognize, evaluate, and implement activities with pupi ls ' learning in mind" (Carter, 1990; p. 295). However , people tend to become teachers in subject or content areas in which they did wel l and therefore often do not understand the problems that others might exper ience in the learning of the subject matter. A n intermediary step might be to integrate theory of understanding as well as t ime to practice and d iscuss these theories in teacher educat ion programs. However, in teacher educat ion, and thus in teaching, there is a perpetual struggle between integration of theory and pract ice. Whi le theory cannot capture the pract ice/enactment of the situation, "it is in the expl icat ion of why particular roles might be productive that theorizing can be benef ic ia l" (Roth & Tobin , 2002; p. 168). A n d , while awareness of different educat ional theories and research may help develop different forms of praxis that apply to different teaching situations, the "signif icance [is] not just understanding that much of teaching occurs as habitus that generates pract ices but a lso of understanding the critical need to connect theories to the exper ience of teaching" (Roth & Tob in , 2002; p. 296). Present ing preserv ice teachers with one theory that g ives them a way of thinking about students ' developing understanding of mathemat ics, and by expos ing them to situations in which they think about and d i scuss student learning using that theory a s their lens may help them come to a better understanding of ways in which students develop understanding of speci f ic topics and this in turn may help them cons ider different ways in which to present material s o a s to make it more meaningful to their students. 3 1.2. Teacher education Learning to teach is a process that begins poss ib ly before one starts grade schoo l and which cont inues throughout one 's years as a student in the c lass room (Lortie, 1975; Br i tzman, 1991; Smi th, 2001 ; Roth & Tobin , 2002). However , to officially become a teacher in North Amer i ca , or to be certified a s a teacher, one must complete a teacher educat ion program, which, in most situations, will involve at least one university degree. W h e n an individual reaches this point in his/her development a s a teacher, "[l]earning to teach should ... be s e e n as a p rocess of experiential learning and the student teachers ' exper iences shou ld be taken as starting points for learning" (Koster, Kor thagen & Wubbe l s , 1998; p. 77). Thus , the teacher educat ion program and the c lass room exper ience both during and after the teacher educat ion program should provide exper iences that will help further develop one 's ideas and concepts about teaching, using one ' s personal background as the starting point (Roth & Tob in , 2002; Smi th, 2 0 0 1 ; Fe iman-Nemsor , 1990). A teacher educat ion program that takes into considerat ion the backgrounds of the preserv ice teachers a s well as presenting them with further ideas regarding the nature of teaching and learning would s e e m a good program within which a prospect ive teacher is m a d e to feel secure in his/her own understanding while, at the s a m e t ime, al lowing for the learning of new ideas and the development of new understandings. 1.2.1. Attitudes and beliefs The usefu lness of teacher educat ion programs in changing the attitudes and beliefs that one holds about teaching, about the nature of the subject matter 4 being taught and about the manner in which subject matter is learned has been quest ioned (Kagan, 1992; Smi th, 1999). Resea rch s e e m s to indicate that most teacher educat ion programs have had little effect on prospect ive teachers ' images of what it m e a n s to teach and to learn (Fe iman-Nemser , 1990; Pa jares , 1992; C o o n e y & Shea ly , 1997). Therefore, whether or not a teacher educat ion program can help prospect ive teachers develop a phi losophy or image of teaching that is different from that which they have already formed before entering the program and if this new image can be rendered strong enough to maintain when they begin teaching is an important point for considerat ion (Cooney & Shea ly , 1997; Fe iman-Nemsor , 1990). In order to determine the possibi l i ty of a program having the ability to change a prospect ive teacher 's perspect ive on teaching and learning, one must therefore cons ider poss ib le reasons that the teacher educat ion programs are not sufficient to change these perspect ives and understandings. S i n c e bel iefs are formed over t ime, in order to change them, it is likely that pedagog ica l pract ices that support changing them c a n only be constructed by engag ing prospect ive teachers , over t ime, in exper iences that incorporate the des i red changes . A l s o , s ince many attitudes and beliefs are non-rational and intuitive, s o m e of them may be changed only by deal ing with them directly and experiential ly rather than by attempting to inf luence them through rational, analyt ic m e a n s . T o add ress this issue, prospect ive teachers must be exposed to innovative teaching styles throughout their preservice educat ion (Wubbels, Kor thagan & B roekman , 1997). A program that cons iders these findings, a 5 program that presents preserv ice teachers with a theory at the beginning of their teacher educat ion program and which exposes them to situations in which they study student understanding through the lens of that theory, as well as one which encourages them to use the language embedded in the theory in d iscuss ion of their own and students' understanding would s e e m an appropriate way in which to cons ider changes that might occur in the understanding and bel iefs about teaching and learning of prospect ive teachers a s they complete their teacher educat ion program. 1.2.2. The use of theories Learn ing environment research indicates that e a c h theoretical f ramework and/or theory involves a set of constructs and provides only one window through which to v iew learning and the environment. For example , activity theory indicates that we are co-creators of our learning and have the power to act based on the subject, object and tools avai lable in a community guided by rules and regulat ions (Roth & Tob in , 2002). Thus , while e a c h theory provides its own perspect ive on a situation, w e must be active in the construction of our own understanding of a situation. By being presented with another point of v iew, be ing presented with a theory that is used and d i scussed in pract ice, preserv ice teachers may be helped to v iew teaching and learning from a different perspect ive. Having a new perspect ive on and a new way of viewing teaching and learning may a lso help preserv ice teachers develop the essent ia ls of pedagog ica l content knowledge, knowledge which is cons idered necessary in order to 6 effectively teach content knowledge. Pedagog ica l content knowledge goes beyond knowing subject content knowledge by providing ways of present ing that knowledge so as to make it comprehens ib le to others (Shulman, 1986). Pres tage and Perks (2001) refer to this as a distinction between "learner-knowledge in mathemat ics [and] teacher-knowledge in mathemat ics, the former is the knowledge needed to pass examinat ions, to find the solut ions to mathemat ical problems; the latter is the knowledge needed to plan for others to c o m e to learn the mathemat ics" (p. 102) (italics original). In order to des ign appropriate c lass room activit ies, analys is of topics through reflection "to reconsider their own personal understandings of mathemat ics" (Prestage & Perks , 2001 ; p. 108) is necessary to develop the teacher-knowledge with which to inform teaching that will chal lenge and meet the needs of the students. Thus , teaching mathemat ics for understanding requires knowledge of mathemat ics, knowledge of student learning, and knowledge of mathemat ics pedagogy. Th is impl ies the need to have both conceptual knowledge, the "rich relationship, linking new ideas to ideas that are a l ready understood" (Stump, 2001 ; p. 210) and procedural knowledge, the "formal language and symbol sys tems, as well a s a lgebra ic algorithms and rules" (Stump, 2001 ; p. 210) of the content. A n understanding of the manner in which students develop understanding of mathemat ical content would s e e m essent ia l in a teacher educat ion program for prospect ive secondary mathemat ics teachers . 7 1.2.3. The use of imagery in teacher education S c h o n (1988) bel ieves that, in order to develop an understanding of how students come to understand mathematics, preservice teachers ' percept ions about teaching and learning must be sought and that the preserv ice teachers must be helped in finding meaningful metaphors in which to f rame their work so that they can reflect on their own understanding in relation to that of their students. Smith (1999) indicates that imagery is a poss ib le m e a n s of helping prospect ive teachers reconstruct their pre-formed world images of the mean ing of teach ing, and the use of metaphors and/or analogies, such as that of compar ing learning to teach to learning to play a musica l instrument, can be used during teacher educat ion to help preservice teachers real ize that the apprenticeship of observation as descr ibed by Lortie (1975) is inadequate preparat ion for learning to teach. Theory and first hand exper ience are necessary if one is to develop competency a s a teacher (Smith, 1999). In consider ing the use of theory, one must real ize that theory d o e s not tell o n e how activit ies will be manifested or how one 's students will act, but rather that it provides a framework for consider ing different approaches they might take (K ieren, 1997). A w a r e n e s s of different theories and research results may help preserv ice teachers cons ider different forms of praxis, and this knowledge may a l so help them real ize why the different pract ices lead to different productivit ies (Kreber, 2002; Roth & Tob in , 2002). 8 1.2.4. Requisites of teacher education Undergraduates often view learning as "acquisi t ion of knowledge, memor iz ing, utilization and/or use of knowledge [as opposed to] abstraction of meaning ... and an interpretive p rocess a imed at understanding reality" (Hattie & Marsh , 1996; p. 531). However, in order to make s e n s e of how students think and learn, a teacher must be attentive to the ways in which his/her own learning is the s a m e as , or is different f rom the way in wh ich the students think and learn (Schon , 1987). A teacher needs to be able to think as a student would think and to build o n the images that the students might have , not only on his/her own images (Wubbels, et al., 1997). Thus , a requisite of teacher educat ion programs shou ld be to help prospect ive teachers, if necessary , re-adjust their image of teaching and learning by having them reflect on the manner in which they were taught, what they think teaching should entail , and how they think students learn (Wubbels et al., 1997). The program should help prospect ive teachers reflect on their own learning exper iences , realizing that students have different preferences for learning, different learning styles, and that severa l poss ib le explanat ions can be g iven for a specif ic situation (Kreber, 2002) . A n effective teacher educat ion program needs to help prepare teachers to teach a curr iculum that may have different characterist ics from that which they themse lves exper ienced and to teach students who have different learning styles and needs than they themse lves have (Wubbels et al., 1997). The focus of teacher educat ion should, therefore, not be on how o n e teaches , but rather should be on how students learn (Davis, Sumara , & Luce-Kap ler , 2000). 9 During their teacher educat ion program, prospect ive teachers need to learn theory, need to observe others teaching in order to b e c o m e familiar with the task of teaching and the structure and nature of the learning p rocess , a s wel l a s have the opportunity to practice what they have thus learned (Roth & Tob in , 2002) . Whi le this involves further observat ion, and observat ion, it has been indicated, is not sufficient in learning to teach, it should be noted that the observat ions m a d e by students as preservice teachers will be very different from those exper ienced by them a s students in a c lassroom attempting to learn the subject content (Pajares, 1992). The new observat ions, informed by theory and d iscuss ion , should help the preservice teachers notice how different pract ices affect different students and lead to difference ou tcomes. Thus , the teacher educat ion program should provide prospect ive teachers with an opportunity to observe the teaching p rocess and should help them learn what and how to attend to the particulars of pract ice. The observat ions should be fol lowed by d i scuss ion of what is observed in light of theories learned (Kreber, 2002 ; Ro th & Tob in , 2002) and these d iscuss ions must then be fol lowed by pract ice s o that the prospect ive teachers not only learn about, but can a lso exper ience what they have learned to s e e how the theory appl ies in situ. Theory and pract ice are therefore inseparable (Schu lman, 2002). 1.3. Images of mathematics education In the realm of mathemat ics educat ion, Wubbe l s , Kor thagan and B r o e k m a n (1997) indicate that emphas is has shifted from a mechanist ic , structured approach in which mathemat ics is v iewed a s a set of algorithms with 10 rules for applying them in an organized deduct ive sys tem, to a more holistic view whereby individuals construct knowledge and principles from concrete situations. The Platonic view of mathemat ics as 'out there' and ' l inear' has been chal lenged by mathemat ics educators who v iew the subject as dynamic , and who v iew the learning of mathematics a s a to-and-fro p rocess of interrelating ideas whereby individuals construct their own images of what it m e a n s . Thus , the learning of mathemat ics is often seen as unpredictable (Pirie & K ieren, 1994a; C o o n e y & Shea ly , 1997). Mathemat ical learning d o e s not begin and end in the c lass room but is a dynamic whole with the environment (Davis et al., 2000). Sma l l group d iscuss ions are seen as a m e a n s by which to deve lop informal approaches which can later be deve loped into more s tandard ized, formalistic forms if necessary (Wubbels et al, 1997; N C T M Pr inc ip les and Standards, 2000). However , "many teachers ' traditional exper iences with and orientations to mathemat ical pedagogy hinder their ability to conce ive and enact a kind of pract ice that centers on mathemat ical understanding and reasoning and that si tuates skil ls in context" (Bal l , 1993 p. 162 cited in S tump, 2001 ; p. 208). Thus , a s preservice mathemat ics teachers may have been taught by mathemat ics teachers who, even if they did not explicitly state it, thought of mathemat ics a s l inear and cumulat ive, the images and attitudes the preserv ice teachers hold toward the teaching and learning of mathemat ics will have been inf luenced by that approach. This , in turn, will affect their ability to enact a new form of 11 curr iculum unless it is specif ical ly addressed in their teacher educat ion program. 1.3.1. Requisites of mathematics education Most students who enter a secondary mathemat ics teacher educat ion program do so because they were good at mathemat ics (Roth & Tob in , 2002). A s they enter the teacher educat ion program, their images of what it means to teach and to learn mathemat ics are based on their own ability to have s u c c e e d e d in the sys tem and they may not be aware of different perspect ives and approaches and of the different understandings that students may develop. To teach the variety of students that they will be faced with, and to apply the more interactive, progressive manner of teaching implied by the new phi losophies toward the teaching and learning of mathemat ics will require that materials be prepared differently from those from which they themse lves learned. Preserv ice teachers must be made aware of and must have exper iences with the mathemat ical knowledge that is required for a particular appl icat ion, and of the different learning styles that their students may exhibit (Wubbels et al., 1997). A l though a teacher educat ion program may be des igned to help preserv ice teachers develop an understanding of the need to look at mathemat ics teaching and learning differently, there is no guarantee that this will happen or that the prospect ive teachers will be able to carry the new ideas into pract ice (Hart, 2002) . In order to do so , they need the opportunity to pract ice their newly developing understandings about teaching and learning in a support ive a tmosphere. Th is is supposed to be the purpose of the pract icum exper ience. However , the schoo ls in which preserv ice teachers are p laced often work in the 12 traditional manner and use textbooks that present mathemat ics in the traditional manner. A s these are constraints within which preservice teachers must work, and which govern the daily teaching practice of their in-school exper ience, their ability to incorporate into practice the newly developing theories, which they have been learning in the university c lass room, is quest ionable (Wubbels et al., 1997). A theory base which will help preservice mathemat ics teachers overcome the over -emphas is on conforming to existing pract ices at the cost of overlooking new developments and insights g leaned through research is therefore needed in teacher educat ion programs (Koets ier & Wubbe ls , 1995). The program must be such that it will render strong enough and rich enough the exper iences of the preserv ice teacher so that the new images presented and the progressive attitudes deve loped can be incorporated into his/her daily pract ice. Model ing is a particularly good method by which to teach abstract behaviors (Bandura, 1997; Hart, 2002), and thus, model ing of a theory during the teacher educat ion program may be a good m e a n s by which to encourage preserv ice mathemat ics teachers to integrate new methods into their pract ice. 1.4. Purpose of this study In light of the above research , it would appear that, if a teacher educat ion program for prospect ive secondary mathemat ics teachers is to inf luence their images of mathematics and mathemat ics teaching, and if this program is to create images that are strong enough to be integrated into practice, the program must not only teach the theory assoc ia ted with the image that it w ishes to deve lop, but must a lso model it and provide the opportunity for the preservice 13 teachers to practice their developing understanding in a support ive environment. A program concentrat ing on the manner in which learning deve lops and which p laces this into the context of the preservice teachers ' dai ly c lass room life s e e m s to be needed if prospect ive teachers are to have the opportunity to construct new understandings and to develop new techniques. Prospect ive teachers need , not only to learn about a theory, but they must a lso s e e its appl icat ion as a teaching technique in order to develop a better understanding of the activit ies of teaching and learning mathemat ics. See ing the theory in action would al low for the construct ion of new knowledge en route and in situ and possib ly lead to a greater predisposit ion to consider student learning differently. A l s o , by consider ing their own learning and understanding in light of the theory studied, the prospect ive teachers would be able to develop a better understanding of the p rocess that they themse lves had to go through to come to their understanding of the subject matter. Th is , in turn, could inform their understanding of students' developing understanding which could then lead to a new form of praxis (Schon , 1987). T h e purpose of this study is to address this issue of influencing prospect ive teachers ' understandings of the p rocesses of teaching and learning mathemat ics . In order to address this issue, it w a s felt that the presentat ion of a theory for the learning of mathemat ics to preserv ice secondary mathemat ics teachers , descr ib ing, demonstrat ing and d iscuss ing the theory with them, and having them cons ider their own understanding and learning through the lens of the theory would be an appropriate m e a n s by which to cons ider the usefu lness of the theory in influencing their understanding of student understanding. 14 Many different theories of learning exist, and e a c h theory is useful for its own purposes. A lso , many theories of learning exist which are specif ical ly related to the learning of mathemat ics. Fo r example , there is S k e m p ' s (1976) categorizat ion of mathematical learning as relational vs . instrumental, the van Hiele 's (1986) five leveled model , and the age and topic speci f ic two-tiered model of Herscov ics & Bergerson (1988), e a c h of which can be used to descr ibe students' mathematical understanding in an appropriate manner. There is a lso the Pi r ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding, and it is this theory that I c h o s e as most appropriate for this study. Th is theory is d i scussed in detail in the next chapter, but here I state that it w a s c h o s e n specif ical ly b e c a u s e it is about growth of understanding wh ich is what this research invest igates, not what can be done at any one time, but rather, the changes that occur as one deve lops understanding of a particular topic. S i nce many prospect ive mathemat ics teachers hold a more traditional v iew of mathemat ical teaching and understanding than that implied by the Pir ie-Kieren theory, this theory, it w a s felt, would problemat ize their beliefs more than the other theories would. Th is theory, it w a s hoped, would possibly give them a new way of viewing the teaching and learning of mathemat ics as well as a vocabulary with which to d i scuss it. T h e main , crucial feature, however, for its cho ice as the theoretical bas is for this research w a s the fact that it is the only complete theory that pertains to the growth of understanding. That is, it looks not 15 at the static ex istence of understanding by a learner at any one moment, but at the dynamic process of acquir ing understanding in any learning situation. Specif ical ly, then, this study cons idered the quest ions: Does using the Pir ie-Kieren Dynamica l Theory for the Growth of Mathemat ical Understanding (the Pi r ie-Kieren theory) as a model enhance the development of preserv ice teachers ' understanding of the teaching and learning of mathemat ics? D o e s the Pi r ie-Kieren Dynamica l Theory for the Growth of Mathemat ical Understanding offer an illustrative f ramework with which to ana lyze the growth of preserv ice teachers developing understanding of teaching and learning mathemat ics? In order to answer these quest ions, it w a s necessa ry first to cons ider how prospect ive teachers indicated their understanding of what it m e a n s to teach and how students learn. It w a s then important to cons ider whether statements they made about their understanding of teaching and learning, and about their understanding of the theory of how students deve lop an understanding of mathemat ics, that are in line with the theory, carr ied over to c lass room pract ice during their pract icum. That is, were they able to integrate what they learned about teaching and learning into c lass room pract ice? 1.5. The organization of the chapters Th is chapter has presented an overv iew of the need for this research, indicating s o m e of the shor tcomings in teacher educat ion programs and areas 16 that have been identified as needing to be enhanced and has identified the focus of this research. Chapter 2 presents in detail, the Pir ie-Kieren Dynamica l Theory for the Growth of Mathemat ical Understanding, the theory that was used as a model for the preserv ice secondary mathemat ics teachers in this study and which was modif ied for use in analyz ing the data. Chap te r 3 offers a review of the literature related to teacher educat ion with s o m e considerat ion to mathemat ics teacher educat ion. Chap te r 4 presents a review of the literature related to the method of data col lect ion (video) and the first s tage of analys is (case-study and portraiture). Chapter 5 presents the methodology used for the study from the col lection of the data, the writing of the portraits, through to the development of a method of analys is of the portraits using a modif ied version of the Pir ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding. Chapter 6 presents the portraits of four preserv ice teachers who were involved in an intensive integrated program des igned to help them develop a better understanding of how mathemat ical understanding is deve loped. Chapte r 7 forms the second portion of the analys is by consider ing the growth of the understanding of teaching and learning mathemat ics of the preserv ice teachers. The Pi r ie-Kieren theory w a s modif ied to use a s a tool of ana lys is for this chapter. 17 Chapter 8 provides a summary of the f indings of the study based on the analys is of the portraits of the four individual preserv ice teachers, answers the research quest ions, and makes suggest ion for further research. 18 Chapter 2 The Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding T h e Pir ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding (the Pir ie-Kieren theory) is a theory in which the growth of mathemat ical understanding is v iewed as "a whole, dynamic , leveled but non-linear, t ranscendental ly recursive process" (Pirie & K ieren, 1991b; p. 1). The theory is in agreement with the biologically based , self-referencing sys tems a s descr ibed by Maturana & Va re la and by T o m m and the work of Vitale and M a r g e n e a u , who d iscuss the recurs iveness involved in the learning p rocess , whereby recurs iveness impl ies self-defined levels that are different from the previous and in which the new knowing "frees one from the act ions of the prior knowing" (Pir ie & K ieren, 1989; p. 8). Thus , the Pir ie-Kieren theory a lso reflects the more dynamic v iew of mathemat ics teaching and learning as descr ibed by the Nat ional Counc i l for the Teach ing of Mathemat ics, and in consider ing the learning of mathemat ics as a non-l inear, dynamic to-and-fro f low of ideas, it goes further than S ierp inska 's (1990) mode l of understanding a s a ser ies of overcoming obstac les by real izing that these obstac les often force one to fold back s o a s to revisit and re- image previous understandings. The Pir ie-Kieren Theory is thus in acco rdance with the latest thinking on the nature of mathemat ics and the way it c a n best be learned. 19 2.1. Background to the Pirie-Kieren theory Having its foundation in construct iv ism, the Pi r ie-Kieren theory had its concept ion in the research of two mathemat ics educat ion theorists and practit ioners, Dr. S u s a n Pir ie and Dr. T h o m a s Kieren. Al though working in different countries and involved in different a reas of mathemat ics educat ion research at the time, they were brought together in 1988 by a shared belief that learning is a dynamical p rocess and that one never completely understands, but is a lways in the process of developing understanding (Pirie & K ieren, 1991a). Pir ie and Kieren have extended the constructivist v iew of understanding to incorporate an enactivist perspect ive, whereby understanding is individual but is inf luenced by external p rocesses (Kieren, 1992; K ieren, Re id & Pir ie, 1995). A s new research has. deve loped and expanded different aspec ts of the theory, it has been adapted and modif ied to increase its power as a research tool (Pirie & K ieren, 1994a; Kieren et al, 1995). Thus , the Pi r ie-Kieren theory has been in the p rocess of developing for a lmost twenty years , a p rocess of growth and development that can be paral leled with and is consistent with the belief that Pir ie and Kieren have regarding the growth of mathemat ical understanding. For the purposes of this study, it is important to note that the Pi r ie-Kieren theory is a theory for, not a theory of, the growth of mathemat ical understanding. Thus , it provides one way to cons ider the growth of mathemat ical understanding and provides one framework within which to d i scuss the p rocess of learning (Kieren, 1992). A l so , being a theory about growth of understanding, it is not used to determine what is understood at this t ime, a s in assessmen t , but rather, it is 20 used to descr ibe the p rocess by which a student continually reorganizes knowledge into a more meaningful whole. Fo r the purpose of this study this is an important aspect in that the theory can be used to descr ibe the growth of understanding of any mathemat ical concept and at any level. T h e theory provides a language with which to d i scuss these differing understandings and d o e s not put a hierarchical va lue on that understanding. In this study, preserv ice teachers were presented with the theory and had it mode led for them in their university c l asses . In order to use the theory a s an analyt ic tool, the terms which descr ibe the levels of understanding were careful ly re-defined to fit this new context while staying consistent with the original definit ions relating to growth of mathemat ical understanding. 2.2. A model for the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding In order to d i scuss developing understanding using their theory, Pir ie and Kieren deve loped a two d imens iona l representat ion, or a model of their theory (see Figure 1). A s c a n be s e e n , the mode l cons is ts of a set of eight nested circ les which impl ies eight potential layers, or levels in the development of mathemat ical understanding (Pirie and Kieren, 1989). It must be s t ressed that this model is not the theory, but that it can be used to d i scuss and map a particular person 's understanding of a particular topic at a particular t ime. 21 F i g u r e l : A Mode l for the Pi r ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding 2.2.1. Mappings A mapping on the Pi r ie-Kieren mode l is a method whereby an individual 's develop ing understanding, a s it is observed , can be traced on the Pi r ie-Kieren mode l (see Figure 2 as an example) . A mapping d iagram of a student 's 22 developing understanding is ach ieved by placing a student 's observed understanding at the appropriate layer on the Pi r ie-Kieren model and following it through, using solid l ines to indicate connected understanding and broken l ines to indicate d isconnected understanding. Connec ted understanding impl ies that the student has made the necessary transit ions from one layer to the next in developing understanding while d isconnected understanding impl ies that these connect ions have not been made. Disconnected understanding inhibits an Figure 2: S a m p l e mapping of student understanding using the Pi r ie-Kieren mode l individual f rom construct ing further knowledge in a meaningful way (Pirie & K ie ren , 1994a) and may result from the teacher trying to force the student into an outer layer of understanding before he/she has had a chance to work at the inner 23 level to develop his/her understanding. The mapping of a student's developing understanding is a useful p rocess that can help the teacher determine what understanding the student has and can a lso be useful in helping determine the source of s o m e of the student 's misconcept ions which lead to errors. In this study, the model has been modif ied to suit a new context, maintaining the integrity of the original, and will be used in Chapter VII to trace the developing understanding of teaching and learning mathemat ics of four preservice mathemat ics teachers, Soph ia , Lance , Ell ie and W a y n e , as they progress through their year of teacher educat ion in the Secondary Mathemat ics Integrated Program (SMIP) at the University of British Co lumb ia . 2.2.2. Transcendental recursiveness Recurs ion is an overriding aspec t of the Pi r ie-Kieren theory and is used throughout the theory in descr ib ing the complex phenomenon in which each layer of understanding occurs as a whole p rocess that is structurally similar to, but not reducible to, any previous state (Kieren & Pir ie, 1991). "The metaphor of recursion is used to highlight the fact that the dynamica l understanding of a person involves states which differ in character but are self-similar" (Pirie, Martin & Kieren, 1996; p. 147) (italics original). L ike Va re la , T h o m p s o n and R o s c h (1991), Pir ie and Kieren bel ieve that individuals are sel f-referencing, autopoiet ic be ings in whom knowing occurs through act ions that are bound in previous act ions. Recurs ion "forms a bas is for looking at the process of developing understanding in which an individual's current understanding acts together with previous understanding and integrates 24 them in the s e n s e that they are cal led into current knowing act ions" (Pirie et al., 1996; p. 147). The transcendental recurs iveness of developing understanding is featured in the eight layers of the Pir ie-Kieren model , firstly, in that they are divided into four sub-sect ions by what are referred to as Don't N e e d Boundar ies . T h e s e boundar ies are indicated by the darker circles in figure 1 and represent a 'crossing over' in understanding so that the individual no longer needs to work with the previous layer of understanding but can work at the new layer. The two layers of e a c h of the four subsect ions are ana logous to those of the other layers. Thus , t ranscendental recursion occurs in that the relationship between Primit ive Knowing and Image Mak ing is similar to the relationship between Image Having and Property Noticing, between Formal is ing and Observ ing , and between Structuring and Inventising where the outer layer involves working on the inner. The Don't Need Boundar ies a lso indicate the t ranscendental ly recursive nature of the Pir ie-Kieren theory in that there is a similar relationship between layers as one c rosses over the boundar ies from Image Mak ing to Image Hav ing, from Property Noticing to Formal is ing, and from Observ ing to Structuring. In e a c h c a s e , the c ross ing over f rees the individual f rom the need to work with speci f ic examp les and enab les him/her to cons ider a more genera l ized situation. Thus , once an individual has an Image, he/she no longer needs to work at Image Mak ing ; once he/she has Formal ised a concept, the need to work with speci f ics no longer exists; and once a Structure is determined, act ions can be 25: independently carr ied out at a meta-mathemat ica l level (Kieren, 1992; Pir ie & Kieren, 1992a). A further example of the t ranscendental recurs iveness in the development of understanding as descr ibed by the Pir ie-Kieren theory is that, within this developing understanding, at any point, there can be a redefinition of the levels. What was Formal ised understanding of one concept may become the Primitive Knowing for another, or the Image Having if the s a m e topic is cons idered at a later t ime. For example , in the study of rational express ions in a lgebra, the Formal ised knowledge that a student has about working with fractions in arithmetic forms the bas ic Images upon which he/she must build the structure for working with the rational express ions . He /she must Fo ld B a c k to this knowledge and use it to construct new Images of the meaning of the operat ions involved when the fractions are no longer numeric in nature. 2.3. The layers of understanding in the Pirie-Kieren theory In this sect ion, the form that learning takes at each layer of understanding a s represented by the circles of the model will be d i scussed . Beginning from an individual 's Primit ive Knowing, the next three layers of understanding a s represented on the Pi r ie-Kieren model demonstrate informal modes of act ion which are local and context dependent , while the second three are formal modes of mathemat ical activity, making them less context dependent and more abstract (Kieren, 1992). If the development of understanding is not d isconnected , the cont inual back and forth flow through these layers of 2 6 understanding can lead to the final s tage, Inventising, at which new mathemat ical concepts can be developed (Kieren etal, 1995). 2.3.1. Layer 1: Primitive Knowing The innermost circle of the Pir ie-Kieren model is referred to as Primitive Knowing. Primitive Knowing is the knowledge that a student brings to the situation, except the knowledge he/she has about the concept that is to be deve loped (Pirie & Kieren, 1991a). Somewhat ana logous to Polany i 's (1983) tacit knowing, it is the background knowledge that a student comes with and from which he/she must work. Primitive, in this context, does not imply a lack of sophist icat ion but simply a starting point, the point at which the student begins the d iscuss ion and, as such , it depends on the background and maturity of the individual (Pirie & Kieren, 1991b, 1992). T h e Primit ive Knowing of a six year o ld, for example , is considerably different from that of a sixteen year old simply b e c a u s e of the life exper iences of e a c h both in and out of the c lass room. A l though a teacher can never be certain of the Primit ive Knowing of a student, it is that which he/she has to cons ider when beginning a mathemat ical d iscuss ion . It is the Primitive Knowing of the student that will inform his/her understanding of the teacher 's presentation of a topic and/or his/her understanding of the terminology that is implicit in that topic. 2.3.2. Layer 2: Image Making T h e second layer of understanding in the Pi r ie-Kieren theory is that of Image Mak ing , an important and active period in the developing of understanding. At this level, the student begins to build on his/her knowledge of 27 the topic under d iscuss ion , working with this knowledge in an attempt to formulate Images that will, at least temporarily, help determine the meaning of the activities. The Images need not be physical models, and the action need not be physical . Either can be enacted entirely in the mind, depending on the situation and the maturity of the student. For example , in working with fractions, a young student might cut a 'pie' or a 'p izza ' in half, then quarters, etc. using a d iagram, whereas an older student will l ikely be able to call upon the abstract Image of 'hal f a s being two equal parts, not having to use physical models . A teacher is re leased from the need to cont inuously individualize instruction by al lowing the students the opportunity, or task -space , to engage in Image Mak ing activit ies as they s e e fit (Kieren et al, 1996). By working at this level, possib ly forming different Images depend ing upon what Primitive Knowing is being recal led until he /she begins to internalize the Images, a student is able to deve lop connected understanding s o that he /she is able to c ross into the next level with new knowledge rooted in exist ing knowledge. Pir ie & Mart in (2002) suggest Gu ided Image Making as a m e a n s of motivating and encouraging weaker students in their developing understanding. For Gu ided Image Mak ing to be effective, it is imperative that the teacher reviews with the students the Primit ive Knowing that he /she wishes them to use , then leads the c lass in Image Mak ing by use of a quest ion that is not immediately so lvab le . In this p rocess , the teacher constantly encourages the students in their work s o that he /she , "almost from the beginning [has the students] strategy hunting del iberately using their mathemat ical knowledge" (Pirie & Mart in, 2002; p. 2 8 16). The students will thus develop an understanding based upon thinking about a problem, not in trying to remember a p rocess . Gu ided Image Mak ing , then, leads the students to a way of thinking, not to a formal ized process . 2.3.3. Layer 3: Image Having The move from the Image Mak ing to the Image Having layer is an important step in a student 's developing mathemat ical understanding. Th is is the step at which he/she c rosses the first Don't N e e d Boundary. It is often cons idered a s the "Ah ha" cross ing in that the student, for the first t ime, may internalize the understanding of the concept and feel that he /she really understands it. He /she will no longer have to work on the Images of the concept , but will have an Image or mental object with which to work. Th is c rossover f rees him/her from the need for particular act ions and provides an understanding of the concept that is often metaphor ic (Pirie & K ie ren , 1994a). T h e student may think that the equat ion is the ba lance, or that three-quarters is three out of four p ieces . It must be s t ressed that, although the student has now formed an Image of the concept , this Image is not complete and may even be incorrect (Pirie & Kieren, 1994a). Incomplete or incorrect Images may create problems later on, obstac les which may interfere with the student 's developing understanding of a topic. Wha t is meant by an incomplete Image is, for example , concern ing fractions, a student might think of 'hal f a s being one of two equa l parts of a whole, but not consider ing it as a separat ion of a number of objects into two equal s ized groups. That is, he/she could have the Image of 'hal f as an object being cut into two equa l parts, but not that three people out of s ix people form one half of the group. 2 9 A n incorrect Image of 'hal f could be that it implies cutting an object into two parts without realizing that the parts have to be equal . Preserv ice mathemat ics teachers may be unaware of the incomplete and incorrect Images that students might have. Present ing them with the Pi r ie-Kieren theory early in their teacher educat ion program as well as model ing it throughout the program, it was hoped, would make them more aware of students ' understandings and poss ib le misunderstandings and that this would help them in thinking about this in a more active manner which, in turn, would help them be able to think about and plan more appropriate and/or involving teaching and learning activities. A student working at the Image Having layer of developing understanding has different Images about a topic as well a s Images about different topics and is likely to hold these as separate, often unrelated entities. He /she , at s o m e point, may quest ion this, and when this happens, may begin to s e e relat ionships among them. He /she has passed into the next layer of understanding, Property Not ic ing, as represented on the Pir ie-Kieren model . 2.3.4. Layer 4: Property Noticing A s ment ioned earlier, Property Noticing is somewhat akin to Image Mak ing but the individual is now working at a higher level of sophist icat ion for his/her level of maturity and understanding of the concept under considerat ion. He /she begins to bring together discrete Images, previously seen a s d isconnec ted , and examines them for speci f ic properties. Understanding b e c o m e s a simile rather than a metaphor (Pirie & K ieren, 1994a): T h e equat ion js 30 like a balance, or 3A is like taking three people from four people. At this level, the student's thinking is more abstract and he/she needs to work with his/her different Images, seeing the properties for him/herself until he/she is able to get what might be referred to as 'a feel' for the topics. It is important that the teacher does not try to force him/her to too quickly cross the next Don't Need Boundary into the Formalising level. However, many preservice teachers, like many practicing teachers, because of their background experiences and their own learning situations, are not aware of the need for students to notice relationships themselves, and may have a tendency to lead students too quickly into a Formalised mathematical 'understanding'. Since he/she was likely 'good' at mathematics, he/she may not have had to spend much time at the Property Noticing level. That is, already having connected and Formalised understanding, he/she may not recall the process through which he/she had to go to develop that understandin. Similarly, preservice teachers (and many practicing teachers) may not realize the difference between students developing mathematical understanding and their ability to perform mathematical algorithms, surmising that since the students can do the questions assigned they must understand the mathematics. However, often the student may simply be applying an algorithm and has memorized a number of applications but does not see the relationship among them. 2.3.5. Layer 5: Formalising The fifth layer of the Pirie-Kieren theory, the layer at which formulas or algorithms are often developed, is officially referred to as the Formalising level. 31 At this level general izat ion occurs and the student comes to real ize that a method works 'for al l ' c a s e s in that category. By abstracting, the student begins to use more formal mathematical activities and can work with concepts rather than needing speci f ic instances. To illustrate, in an interesting example cited by Pir ie and Kieren (1994a), a student had been working with paper folding at the Image 1 2 4 Mak ing level e f f rac t ions and had noticed that - = - = - , e t c . (Image Having). He then noticed that the multiplication did not have to be by twos but could be by 1 3 9 threes, yielding — = - = — , etc. (Property Noticing). T h e student then happily 2 6 18 stated: "I'll bet it even works for sevenths!" He had abstracted from having to work with speci f ic numbers as , to him, sevenths were the epi tome of all poss ib le fract ions. For h im, 'even sevenths ' was a 'for all ' statement and he Formal ised his understanding of equivalent fractions. If a l lowed to properly deve lop, a s in this c a s e , Formal isat ions b e c o m e abstract ions or meta-ana lyses of one 's own ideas on which one must work to create a pattern (Pirie & Kieren, 1992a). Formal isat ions do not have to involve formulas, or even be expressed symbolical ly. However , they are usual ly exp ressed verbal ly or in s o m e external manner before they are internal ized. By present ing preserv ice teachers with this aspect of the learning p rocess , it w a s hoped that they would be able to realize the importance of al lowing a student to deve lop his/her own understandings. It was hoped that by making them aware of this aspec t of learning mathemat ics, the preservice teachers , in their p lanning and teaching, would s e e the need to al low time for students to 32 work with their Images and Formal izat ions, that is, al low time for Image Making and Property Noticing, in order to deve lop their own understandings rather than leading them directly into a Formal isat ion or trying to force their own Images onto the students. 2.3.6. Layer 6: Observing The sixth level of developing understanding in the Pi r ie-Kieren theory involves an individual's consc ious effort to further his/her understanding. A l ready having a Formal ised understanding of different aspec ts of mathemat ics, he/she actively sea rches for a pattern or relationship among them, search ing for commonal i t ies so a s to s e e 'the big picture'. Th is may be the level that separates mathematical learning from that of other a reas (Pirie, 2001 ; personal communicat ion). At this level, the individual wants to work at the mathemat ics him/herself to see the patterns that exist and the relat ionships among topics. In the development of the Pi r ie-Kieren theory, and in the Mapp ing of individuals' developing understanding of mathemat ical concepts , the upper levels (six, seven and eight), have not been observed and documented as often as the lower levels. It is at these levels of mathemat ical understanding, however, that teachers have to work a s they think about and prepare lessons so that they will be meaningful to students and so that students can develop their own Images and Formal isat ions. Thus , it is important that preserv ice teachers are aware of these levels of thinking s o that they can understand the abstract ions that they are trying to teach their students. 33 2.3.7. Level 7: Structuring W h e n an individual is able to take his/her previous understandings of concepts and establ ish a relationship among them, he /she b e c o m e s more confident about the concepts and is able to verify or justify a p rocess or proof. In working at the Observ ing level an individual may try to re-form his/her own knowledge into an over-arching theorem. Structuring occurs when he /she is able to do so and he/she can s e e a pattern and formulate an encompass ing explanat ion or an abstraction of these Observ ings (Pirie & K ieren, 1992a). A n examp le of a Structuring relationship is that a student, thinking about the linear, quadrat ic and cubic functions studied as separate entit ies real izes that the patterns can be general ized to 'polynomial function'. T h e genera l ized relat ionship then re leases him/her from having to think of separate patterns b e c a u s e he /she is ab le to tie existing knowledge to s o m e other a rea or aspect of knowing. O n c e an individual has abstracted a concept, he /she is able to s e e the genera l relationship and cons ider spec ia l c a s e s under that Structure, thus being freed from the need to classify each example individually. A l though Structuring involves abstract ion, it must be pointed out that it can occur at any age or level of mathemat ica l development. It is the individual's understanding and level of mathemat ica l development that determines his/her level on the mode l , not the abst rac tness of the concept. The Structuring aspect of learning mathemat ics may, aga in , be a level at wh ich the teacher tries to force understanding by present ing it as fact to the student rather than by letting him/her think about the relat ionships involved and 34 determining for him/herself that the Structure exists. In the polynomial function example , for instance, the teacher might give the students the definition without al lowing time for the students to s e e how the previously learned functions qualify. For this study, it was hoped that if preserv ice teacher were aware of this, and if they gave thought to it, they would be ab le to use their own, new understanding of student understanding in order to develop lessons that would al low students to determine the existing Structures for themselves. 2.3.8. Level 8: Inventising Whi le working at the Structuring level, if an individual asks him/herself: "What would happen if I changed a part of this structure?" he/she, on that topic, has reached the Inventising or highest level of mathematical understanding "where, with full structural understanding of a p iece of mathemat ics, he/she can deliberately ask quest ions that break out of this structure" (Pirie & Kieren, 1991b; p. 171). The term, Inventising w a s co ined specif ical ly for this theory, so as not to be confused with the term inventing s ince students may invent at any level of understanding (Kieren, 1992; Pir ie & Kieren, 1991a, 1992). Whi le Inventising, an individual is capab le of "developing, without giving up the previously understood knowledge, a completely new way of looking at and building from phenomena deve loped in the previous structure" (Kieren, 1992; p. 215). He /she is able to create new quest ions and observat ions that have the potential of becoming a new topic or a rea of study. Inventising c a n involve complex concepts (What might be implied by the fifth or sixth d imens ion?) or it 35 can involve quite s imple mathemat ics (What happens if you take 9 away from 7?) depend ing on the age and mathematical development of the individual. The examp les given above for the different levels of the Pi r ie-Kieren theory indicate that at any level of mathematical development, an individual can exper ience different layers of understanding. It is the individual 's knowledge of and manner of working on or with the information that determines his/her level of develop ing understanding with respect to the Pir ie-Kieren model , not his/her age level. 2.4. Features of the Pirie-Kieren theory T h e Pir ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding must be cons idered as a whole interactive, dynamica l unit. The features of it are intrinsically intertwined and yet discrete. S o m e reference has been made to this in previous sect ions. In this sect ion, the focus is on the four features of the Pi r ie-Kieren theory that represent its dynamic nature: Folding Back , Interventions, Don't N e e d Boundar ies , and Complementar i t ies. 2.4.1. Feature 1: Folding Back A n important aspect of the Pi r ie-Kieren theory is its emphas is on the non-linearity of coming to a better understanding. That is, it s t resses the importance of the need to move to-and-fro through the layers of understanding. Whi le working at a particular layer, one may have to work for s o m e time, or one may p a s s through to the next level very quickly. At other t imes, when one faces a cha l lenge that is not immediately so lvable, one may have to move inwards to a prev ious level of understanding in order to reconstruct knowledge or to extend 36 current inadequate understanding (Pirie & Kieren, 1991a). This is referred to as Folding Back. On Folding Back, an individual is "able to extend their [sic] current inadequate and incomplete understanding by reflecting on and then reorganizing his/her earlier constructs for the concept in question, or even to generate and create new images, should their [sic] existing constructs be insufficient to solve the problem" (Pirie, Martin & Kieren, 1996; p. 148). Martin (1999) "explored the act of folding back to examine the actions that a learner engages in when folding back, and to consider the interactions that facilitate, occasion and constitute this process" (p, 74). Highlighting the complexity of the phenomenon as observed in action, he states: Folding back involves more than mere borrowing from a lower level construct on a higher level or to objectify an experience. Instead it is concerned with the process and act of re-visiting, re-constructing and re-shaping these experiences, accepting that one may never have a full understanding and thus growth involves a constant shifting from formal to informal, formulated to unformulated understanding. The notion of folding back recognizes that growing understanding is not defined in terms of a move from the concrete to the abstract, but that one enfolds the other and that the two cannot be split (pp. 57-58) (italics original). Martin (1999) identified three over-arching aspects as discrete but interrelated aspects of folding back. These three aspects are Source, Form and Type (see Table 1). The Source of folding back may be 1. material (manipulatives), 2. an individual (self, peer or teacher), and 3. it can be intentional or non-intentional. Folding Back can take one of four Forms: 1. to collect data from an inner level, 2. to work at an inner level, 3. to move out of a topic and work there, or it can 4. cause a discontinuity (Martin, 1999). Table 1 below summarizes these findings. 3 7 1. "SOURCE" Encompassing four main categories: lnvocative Teacher intervention Invocative Peer Intervention lnvocative Material Intervention Self Invoked Each of these interventions can be divided into two subcategories: Intentional Unintentional An Intentional Intervention can be further divided into two subcategories: Explici t Unfocused 2. "FORM" Encompassing four main categories: Working at an inner layer using existing understanding Collecting at an inner layer Moving out of the topic and working there Causing a discontinuity 3. "OUTCOME" Encompassing four main categories: Uses extended understanding to work on overcoming an obstacle-i.e. effective Cannot use extended understanding to work on solving problem-i.e. ineffective The first of these uses also divided into two further subcategories: Returns to outer layer with external prompt Returns to outer layer without external prompt A special case: *Not taken as invocative' as an outcome Table 1: The framework for describing folding back (adapted from Pirie & Martin, 2000; p. 143) Generally, Folding Back tends to be to a layer that precedes a Don't Need Boundary. At such a level, one is able to re-construct Images, re-consider properties, or re-observe relationships. That is, once a student has formed an Image, he/she proceeds to do mathematics using that Image. If it is a correct Image, then, for the time being the student is likely to be happy with his/her work and will likely 'get the right answers'. If the Image is incorrect, it is likely that this 38 will be ev idenced in that the student will get incorrect answers . If the Image is incomplete, he/she is likely to get contradictory and inconsistently correct and incorrect solut ions. In such c a s e s , he/she would likely be prompted to stop, Fo ld Back , and re-consider the Images held so as to reconstruct them and form new Images that would apply to the situation at hand. The new Images being deve loped would be informed by new understanding and the returned-to level would be 'thicker' with the fold encompass ing or bringing back information which could be used to reconstruct or modify the previous understanding whi le at the s a m e time further extending the outer level of understanding (Pirie & Kieren, 1991a, 1992a). The information e n c o m p a s s e d in the 'fold' would provide the necessa ry data for altering or enlarging the existing Image. Subt le, but signif icant changes can thus enlarge the previous understanding so as to support more formal understanding s o that the new level t ranscends but is compat ib le with the previous level (Kieren & Pir ie, 1991b). However , if a student has D isconnected Understanding, and if pure memory fai ls, then he /she may not be able to Fo ld Back to re-evaluate and re-form the Images that are assoc ia ted with the concept . T h e student will not be able to think back to his/her previous understanding to make the necessary connect ions and will be forced into memor iz ing a new or different method, or may b e c o m e d iscouraged and want to give up. Wh i le it is helpful to identify and classi fy the var ious aspec ts of Folding B a c k s o a s to have a language with which to d i scuss these, it is a lso important to cons ider how these acts affect the teaching and learning of mathemat ics. The act of Fold ing Back does not necessar i ly lead to growth of mathemat ical 39 understanding. Whi le the phenomenon of Folding Back is an integral part of developing mathematical understanding, it is poss ib le that an individual does not return to the outer layer of understanding without help or that he/she may not s e e the usefu lness of the inner layer of working (Martin & Pir ie, 1998). In such a c a s e , Folding Back is cons idered ineffective. Martin (1999) postulates that it is the form of Folding Back and the related Intervention that are signif icant if understanding is to occur. T h e form of Folding Back needs to be appropriate to the needs of the individual, and the individual must be aware of what it is he/she needs (to collect, to move outside the topic, etc.). Martin (1999) indicates that it is teacher Intervention, intentional or unintentional, that plays the greatest role in Invoking students to Fold Back . H e warns, however, that there is real danger if the teacher attempts to give too much gu idance and that the language that the teacher uses is important in determining the ef fect iveness of Folding Back . Explicit teacher Interventions are more directive than unfocussed Interventions and leave little room for explorat ion. Mart in (1999) highlights the similarity between the s e n s e of Discontinuity often created by these explicit Interventions which the student may not take a s Invocative and Jaworsk i ' s (1994) 'didactic tens ion ' created when the teacher 's speci f ic a ims interfere with the student 's own construct ions. H e suggests that this may occur because , when using an explicit Intervention the teacher makes an assumpt ion regarding the student 's understanding. The Intervention then directs the student to the teacher 's perceived need , which d o e s not al low the student t ime for reflection on the limitations of h is/her own thinking. Explicit Interventions 40 may be effective, however, if they follow an unfocused Intervention which has a l ready focused the student on his/her limitation. Genera l ly , intentional, unfocused Interventions appear to be most effective in that, being less directive, they provide 'thinking space ' , al lowing the student to explore his/her own Primitive Knowing or Images more freely (Martin 2000). W h e n a teacher is present, he /she is usually v iewed as the source of authority by a student and his/her Intervention is given c redence over peer Interventions. However , Martin (1999) identified peer Interventions as having the potentiality to play a s great a role in another student 's Fold ing B a c k and he observed that peer Interventions occurred most often when the teacher had made the c lass room a col laborat ive work environment. However , a col laborat ive work envi ronment did not guarantee that the Interventions would be effective s ince the peer intervener is in the p rocess of developing his/her own understanding of the concept s o that any Intervention on his/her part is likely to a lso involve sel f - invoked Fold ing Back to seek ass is tance in his/her own thinking. T o be effective or to deve lop a shared understanding, the Intervention needs to encourage explorat ion and negotiation in light of the intervener's uncertainty, as wel l a s wi l l ingness and the ability on the part of his/her partner(s) to participate in the explorat ion. If the perce ived need by the peer intervener is not seen a s a need by the partner(s), then the Intervention would be ineffective. T h e ability to sel f - invoke Fold ing Back is a useful dynamic in the deve lop ing understanding of a student a s this impl ies a natural tendency to reflective thinking. However , it is unclear whether Folding Back can be taught to 41 students. A s with many pract iced activit ies, Folding B a c k could b e c o m e part of an individual's repertoire of alternative act ions if he /she w a s encouraged to do it often enough through any form of Invocation. A l though most textbooks are linearly formatted, Martin (1999) suggests that they cou ld be written s o as to promote Folding Back as a response to obstac les and that this would promote growth of understanding on the part of the students. For this study, the concept of Fold ing B a c k w a s cons idered to be important in that many preservice teachers do not real ize the s igni f icance of this p rocess . Preserv ice teachers (and many practicing teachers) often do not understand the importance of the p rocess and the need for students to re-create Images as being a major component of the p rocess of developing understanding. M a n y feel that students should be taught correctly and l inearly so there will be no stumbling blocks along the path of developing understanding. However , if this were poss ib le , then the teaching of mathemat ics would be a s imple p rocess , and all students would learn mathemat ics in the s a m e manner . Thus , it w a s hoped, expos ing the preservice teachers to this aspec t of student learning would enab le them to real ize that students have to Fo ld Back to prev ious understandings and Images many t imes and that they have to re-work Images so that they can deve lop a meaningful understanding of a mathemat ica l concept. It w a s hoped that this would help the preservice teachers real ize that student understanding is a lways in a state of development, and that at t imes they deve lop incomplete and incorrect understandings, not as the result of bad teach ing, but as a necessary aspec t of learning. 42 2.4.2. Feature 2: Interventions Interventions are means by which an individual's thinking is st imulated by an act ion, either internal or external, to re-evaluate his/her present working. Pir ie and Kieren (1991b) c lassi f ied Interventions as Provocat ive, Invocative or Val idat ing. A Provocat ive Intervention is an Intervention that prompts a student to cons ider an outer or more sophist icated level of understanding; an Invocative Intervention prompts him/her to see the need to Fold Back ; a Val idat ing Intervention estab l ishes the student within a level, encouraging him/her to s o m e form of express ion (Kieren & Pir ie, 1992). Whi le the teacher often plays an important role in Interventions, in the context of students ' developing understanding it is the student 's response to or action on the Intervention that determines its nature (Pirie & Kieren, 1994b). Thus , al though a teacher may pose a quest ion that is intended to be a Provocat ive Intervention, for example , it may actual ly Invoke the student to Fold Back in order to re-establ ish s o m e concept before being able to elaborate on it (Kieren et al., 1995). Interventions may thus take a form different f rom that intended, and Towers (1998a), with an interest in determining the role of the teacher in support ing and encouraging student learning, "began to investigate the complex web of interactions that interweave to support the growth of understanding in the c lass room" (p. 41). 2.4.2.1. Themes of Interventions Work ing within an enactivist f ramework, Towers (1998a) bel ieved that "understanding can be interpreted a s being dependent on, but not determined by, the act ions of the teacher" (p. 43) and w a s interested in teacher talk and teacher 43 quest ioning strategies, but a lso in the interactions and Interventions between participants in the learning p rocess , specif ical ly in how teacher Interventions occas ion student growth of understanding. Rather than consider ing when a teacher should Intervene in student learning, Towers (1998a, 1999) recognized the need to ask quest ions about how teachers Intervene. Initially applying the three categor ies of Intervention (Invocative, Provocat ive and Val idat ing), and recognizing the need for "a s p a c e for d iscuss ion of understanding and cognit ion which recognizes the in terdependence of all the participants" (Towers, 1999; p. 162), Towers (1998a) found that the three categor ies did not adequately descr ibe the complexity of the observed act ions in the moment. Purposeful ly choos ing active verbs to descr ibe Interventions in act ion, Towers (1998a) identified fifteen Strategies involved in Interventions: 1. Check ing : The teacher determines if students are paying attention and/or conf irms their understanding. 2. Showing and Tel l ing: The teacher presents material, often giving new material and not check ing for understanding. 3. Lead ing: The teacher directs the students, through a ser ies of guided quest ions, toward a speci f ic answer or procedure. 4. Reinforc ing: The teacher repeats for the purpose of confirming the statements of a concept . 5. Inviting: T h e teacher presents open-ended quest ions which encourage alternative interpretations. 6. Clue-g iv ing: T h e teacher purposeful ly directs the student to a line of thinking. 7. Manag ing : T h e teacher e n g a g e s in administrative or behavioral tasks. 8. Enculturat ing: By al luding to the larger mathemat ical community, the teacher introduces the student to convent ional notations and language. 44 9. Block ing: The teacher prevents a student from following a particular path. 10. Model ing: The teacher explicitly mode ls a process or thought pattern. 11. Pra is ing: The teacher commends an individual or the c lass as a whole. 12. Shepherd ing : T h e teacher subtly nudges or coaxes the student on a particular path. 13. Rug-pul l ing: T h e teacher del iberately destabi l izes the student 's thinking. 14. Retreat ing: The teacher deliberately withdraws, leaving the student to think about his/her work. 15. Ant ic ipat ing: Anticipating the student 's thinking, the teacher actively attempts to remove obstac les before they are encountered. 2.4.2.2. Teaching styles Real iz ing that the teacher does not c a u s e learning to happen, but provides occas ions for it to happen, Towers (1998a) cons idered turning points in the student 's understanding - those instances where he/she extended to an outer layer or Fo lded Back to an inner one. Drawing together student Mapp ings of developing understanding with teacher Interventions, she then attended to the teacher Interventions that occas ioned cognit ive shifts, defining occas ion ing as "a situation in which growth of understanding is a l lowed for, not c a u s e d " (Towers, 1998a; p. 229). Us ing these observat ions, she identified three major Teach ing Sty les , later referred to a s Intervention Sty les (Towers, 2001) which differ from Intervention Strategies in that a strategy usual ly involves only a brief interaction whi le a style impl ies a genera l pattern of teaching. T h e identified Teach ing Sty les are: 1. Showing and Tel l ing, 2. Lead ing , and 3. Shepherd ing . A teacher typically 45 displays one dominant Style, while call ing on a repertoire of Strategies. Certa in Intervention Strategies are typical of particular Teach ing Sty les. Tab le 2 below summar izes these classi f icat ions. Consis tent with Towers ' (1998a) observat ions that Showing and Tell ing and Leading are the most common Teach ing Sty les, Truxaw, D e F r a n c o and McGivney-Bure l le (2002) observed that d iscourse within the c lass room tends to be univocal , rather than dialogic, implying these two styles are dominant in the teaching sett ing. Showing and Leading Shepherding Telling Reinforcing Clue-giv ing Clue-giv ing Enculturating Block ing Inviting Model l ing Anticipating Retreat ing Tab le 2: Towers ' intervention styles and related teaching strategies Referr ing to Smith (1996), Towers states: "[TJeaching by telling al lows teachers to build and maintain a s e n s e of eff icacy by defining a manageab le content base and by providing c lear descr ipt ions of how they should present the content" (Towers, 1998b; p. 29). S u c h a Style p laces the teacher-as-authori ty, and breaking away from this may be cause for discomfort. Towers therefore (1998a) descr ibes a "d i lemma of trying not to tell" (p. 225) a s problematic for teachers , and identifies the reform movement a s being partially responsib le as it has encouraged the act ive participation of students in their own development of mathemat ical understanding. In elaborat ing on this, she notes that enact iv ism provides an environment that may make teachers more comfortable in assuming a less directive role. By real izing that the growth of student understanding is, a s 46 stated earlier, dependent on but not determined by the teacher, she bel ieves that teachers will be able to a s s u m e a more Shepherd ing Style of teaching which she feels is more conducive to students developing understanding than are Leading or Showing and Tel l ing. Many preservice teachers, having exper ienced ' teaching by tell ing', a style of teaching which is consistent with Towers ' Showing and Tel l ing and s o m e aspec ts of Lead ing, tend to re-enact this in their own teaching. For this study, it w a s hoped that if preservice teachers were aware of different forms of quest ioning as well as their relationship to the different forms of teacher Interventions it would encourage them to reflect on their own ideas and Sty les of teaching and how these would impact student understanding. 2.4.2.3. Towers' observations on the use of Interventions Towers (1998a) suggests that there exists a kind of cont inuum of Intervention s t reams where the focus of the v iewer shifts f rom teacher to student as Sty le of teaching moves along the cont inuum and that, at the s a m e time, the interactivity of students and teacher a lso shifts. Expectat ion of response to quest ioning shifts from rhetorical, or low level quest ioning where the student 's response involves guess ing what the teacher is thinking, to quest ioning where there is little doubt as to the answer, to quest ions which provide the student with the opportunity to develop ideas through interaction (Towers, 1998b, 2000). In this light, Towers (1998a) noticed that s o m e students exper ienced jumps in understanding. Prior to this, it was bel ieved that a student moved through e a c h layer of understanding as outl ined by the Pi r ie-Kieren theory, al though the length 47 of stay could be very short. Towers noticed that jumps seemed to occur with inviting Interventions for brighter students, and at these t imes they were able to demonstrate a strong understanding of one aspect of understanding while developing a broader knowledge base . Introducing preservice teachers to the different aspec ts of teacher Interventions and the effect they have on student learning, as outl ined by Towers , it w a s hoped would help them in deal ing with their own d i lemmas. Thus , instead of using C l u e Giv ing , Ant ic ipat ing or B lock ing Interventions to e a s e the situation for the student, it w a s hoped that they would real ize that they could leave the student to struggle with the concept so as to develop his/her own understanding. Th is would return "control of the learning to the students and offer them the s p a c e to reach conc lus ions without direct intervention from the teacher" (Towers, 1998a; p. 217-218). To make preservice teachers aware of these aspec ts of teaching, it was felt, would help them in focusing on their own methods and Teach ing Sty les and of other possibi l i t ies that exist. 2.4.3. Feature 3: Don't Need Boundaries A s previously ment ioned, Don't N e e d Boundar ies are indicated on the Pi r ie-K ieren mode l by the darker rings (see Figure 1) which separa te the model into four subsect ions and which represent increasingly abstract modes of thinking. C ross ing over a Don't N e e d Boundary is an important step in a student 's developing understanding in that it f rees him/her from the constraints of less formal and less sophist icated action of concentrat ing on speci f ic c a s e s and al lows for more genera l and abstract thought (Kieren & Pir ie, 1991; Pir ie & 48 Kieren, 1991a). The previous layers are embedded within the new layer and can be a c c e s s e d if and when needed, but the individual working at the new level d o e s not need to constantly reference them. With connected understanding, he /she can , however, Fo ld Back to the previous levels when confronted with a problem where existing pract ices do not provide a solut ion. 2.4.3.1. The importance of the crossover from Property Noticing to Formalising A s ment ioned earlier, Formal is ing is the first level at which the student works at an abstract level of mathematical thinking. M u c h of the original research involving the Pir ie-Kieren theory provides examp les of students working at the inner layers of the model , from Primitive Knowing through to Formal is ing. L e s s has been written about working at the outer layers even though students of all a g e s can work at them. O n e reason that less is known about working at the outer levels may be that teachers often perceive a need to move students beyond the Don't N e e d Boundar ies as quickly as poss ib le so that they are able to work at the Formal is ing level. At this level, students are able to do their mathemat ics ass ignments , often with speed and accuracy. However , if prematurely Forma l i sed , the student may be able to act but not understand (Kieren & Pir ie, 1991). Po lany i (1983) would say this is the difference between konnen (knowing how) and w issen (knowing what) and that understanding is a process , grounded within an individual which co -emerges within an environment. In agreement, Pir ie and K ieren (1994b) indicate that for true understanding to take p lace, a student 's informal understanding must be sufficiently deve loped before an act ion by the 49 teacher can successfu l ly Provoke understanding at the Formal is ing level. The student must be able to apply the abstract meaning to the symbols from within his/her own Image-based sys tem if this understanding is to take p lace. Being dropped into the Formal is ing level by being given the abstract ions and/or formulas before he/she has had t ime to Property Notice affects the individual's mechan ism of growing structural understanding and this in turn may hinder his/her further development . For this study, it w a s cons idered poss ib le that preserv ice teachers may need to work at their own understanding of how students develop an understanding of mathemat ics a s they had already reached or p a s s e d the Formal is ing level of understanding in a learning environment that worked for them - that is, in an educat ion sys tem in which they were able to learn. However , s ince they had been good at mathemat ics the preserv ice teachers may have exper ienced Towers ' j umps in understanding, and may not real ize, un less it is speci f ical ly drawn to their attention, that most students will need to spend time at the Image Mak ing and Property Noticing levels and that they will form incomplete and incorrect understandings of concepts . The tendency for the preserv ice teachers to want to push students into the Formal is ing level may be a strong, albeit an inappropriate act ion for a student struggling with the concept . By expos ing preserv ice teachers to the tenets of the Pi r ie-Kieren theory, it w a s hoped that they would b e c o m e aware of the need for students to deve lop their own understanding in order to make meaning of the mathematical concept being cons idered . 5 0 2.4.4. Feature 4: Complementarities Within each of the six inner layers of the Pi r ie-Kieren Theory for the Growth of Mathemat ical Understanding, there exist two mathemat ical activities (see Figure 3). Referred to as Complementar i t ies, these activities consis t of an appropriate Mathemat ical Act ion and the appropriate Mathemat ica l Express ion which reflects upon and justifies the action (Kieren, 1992). F igure 3: T h e Pi r ie-Kieren model incorporating the Complementar i t ies Data indicates that the express ion of the act ion is an important and necessa ry p rocess in the developing understanding of a student. A n individual 51 can engage in an action such as Image Mak ing, but it is the express ing of that action that al lows the individual to reconsider and consol idate understanding so he/she can progress to the next layer (Kieren, M a s o n , Dav is & Pir ie, 1993). Act ing can be a mental or a physical act, depending on the mathemat ical maturity of the individual. Express ing need not be verbal but it is often the verbal izat ion that facil i tates abstraction or provides the occas ion for an individual to perceive the need to Fo ld Back and correct or enlarge his/her understanding (Kieren & Pir ie, 1991). Thus , Express ing at an inner level is important in the student 's construction of understanding at an outer level a s it provides an orienting action facilitating abstraction (Kieren & Pir ie, 1991) and affects his/her own accep tance of comments made by others as being appropriate to his/her personal understanding (Pirie & Kieren, 1994a). Through Express ing in a more convent ional or appropriate manner, and by refining his/her express ion , the student c o m e s to internalize the meaning of the mathematical object or act ion. Lastly, while it is essent ia l that one engages in Act ing activit ies before Exp ress ing , there is a lso a need to reflect before express ing. Ref lect ing, therefore is a component of the Act ing activity, and cons iders how previous understanding w a s constructed. Express ing articulates what w a s involved in the act ions (Pirie & Kieren, 1994b). Within a level of developing understanding, there is a constant to-and-fro movement between Act ing and Express ing (Kieren et al, 1993; Pir ie & K ie ren , 1994b). Only through s o m e form of Express ing can an individual progress to the next level, and only through s o m e form of external izat ion (visible expressing) can an observer infer what construct ion of 5 2 knowledge has taken p lace. Wh i le for the teacher an Act ion may exist to create an object, for the student the Act ions are often central with the objects created and Express ing of the results indicates his/her understanding of the Act ion. Prospect ive teachers have often not given sufficient thought to how students come to an understanding of a mathemat ical concept and may think that the student understands the concept because he/she can complete ass ignments . Through d iscuss ing with the preserv ice teachers in this study how students come to make meaning of mathemat ics through Act ing and Express ing , it w a s hoped that they would begin to think more about the meaning making involved in learning and that they would think about their own learning and that this would be used to inform their understanding of how their students learn mathemat ics and of how they can des ign lessons that al low for the students to deve lop their own understanding. By reflecting on their own learning, it w a s hoped that they would think about poss ib le incomplete and/or incorrect Images that students might deve lop along the way. 2.5. Extensions and elaborations In a manner similar to the growth of mathemat ical understanding a s descr ibed by the Pi r ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding, affiliated researchers (e.g. Towers , Mart in, Dav is , Covey , etc.) have b e e n extending and elaborat ing the theory, in effect, Folding Back to re-address , re-define and broaden the understandings of the different layers and features within it as well as trying new appl icat ions. S ince 1998 a ser ies of 5 3 publications have dealt with the elaborat ions and extens ions as deve lopments have co-emerged. 2.5.1. Elaborating the enactivist nature of the theory In their initial research and Mapp ings , much of the d iscuss ion of Pir ie and Kieren entai led students working at the inner levels of understanding. In 1998 they made a consc ious effort to examine further the more formal layers of developing understanding. Specif ical ly, they cons idered the character of the Formal is ing level and this led to increased d iscuss ion into the enactivist nature of the theory. Understanding a s knowing in act ion (defined as prolonged act ions, which produce a range of responses not predicted or suggested by the teacher) b e c a m e prominent in and central to the d iscuss ion (Kieren & Pir ie, 1998). Observat ion of students in act ion, a lways an integral part of the theory as a m e a n s of understanding understanding, became even more prominent a s a way to gain insights as to how others might Provoke a student to more sophist icated levels of thinking. The simile of set forming and set ordering emerged as a useful one through which to d i scuss observer understanding of learners ' emerging understanding from Images to Formal is ings (Kieren, 2001). Understanding in act ion involves "continual generat ion and col lect ion as well a s working with and express ing" (Kieren, 2001 ; p. 227). Formal is ing requires a method, and a consc ious ability to justify it "supported by act ions which are like one is ordering or has ordered a set in s o m e way and works with that ordering" (Kieren, 2001 : p. 224). Thus , in the p rocess of coming to Formal ised understanding, one must have an Image or 54 Images and then be able to del ineate properties of them to fit criteria in an orderly and sequent ia l fashion, and set forming can be s e e n as a way of deliberately forming a col lect ion that obeys the rules for inclusion into a set. Formal is ings can be ach ieved through pathways other than producing formulas and applying them and can be exhibited without the use of formal ized methods (Kieren & Pir ie, 1998). Within the preserv ice teacher educat ion program, one must cons ider the preserv ice teachers ' Images and possib ly Formal is ings of the meaning of teaching and learning mathemat ics as these will affect the manner in which they v iew teaching and learning the subject. It is poss ib le that through too early developing Formal is ings of the meaning of teaching and learning, the preserv ice teachers will not be flexible enough to reflect upon the p rocess that their students must enact in their developing understanding. Fo r this study, therefore, it was cons idered important to have the prospect ive teachers cons ider these p rocesses and d i scuss their own understanding through the lens of this P i r ie-Kieren theory s o that they would have a foundation on which to build their own p rocesses and understandings. 2.5.2. Alternative forms of Mapping Mapp ing on the Pi r ie-Kieren mode l is often limited to a single set of eight nested circ les (refer back to Figure 1). Th is , however, is a simplist ic v iew of the mode l . Pir ie and Kieren indicate throughout their research that coming to a better understanding is a dynamic p rocess of back and forth activity. In developing understanding, an individual may have to Fo ld Back to collect and/or Fold Back 55 to a different topic and work there. Indicating the complexi ty of this developing understanding, Pir ie & Martin (2000) offer a multi faceted representat ion of the process of developing understanding which involves severa l interconnected models to indicate how a student 's thoughts weave in and out of the necessa ry topics in a dynamic, enact ive p rocess of developing understanding (see Figure 4). Th is Mapp ing clearly indicates the complexity of developing understanding as well as the ability of the theory to capture s o m e of that complexity. Th is is precisely what was needed for my research. A s a "pragmatic move resulting from the difficulty [faced] in an attempt to represent a students understanding over a period of severa l months" (Towers, 1999; p. 164), Towers adapted the Pir ie-Kieren nested circle Mapp ing to a more l inear mode l (see Figure 5). Th is Mapp ing al lowed her to observe patterns, and the connect ion between the student action and the teacher Intervention that may not have been s o evident using the traditional model . Wh i le this appeared problemat ic for s o m e individuals who felt that the integrity of the theory w a s lost in this linearity, a member of the aud ience at a presentat ion (Borgen, 2001) pointed out that the concept of nesting could be maintained by ' fanning' Towers ' mapping. Borgen and Manu (2002) introduced the concept of 'c lustered confus ions ' in their mapping of a student 's misunderstanding of a concept by using a 'broken circle' (see Figure 6) to indicate a grouping of semi-related but confused ideas which the student w a s not able to clarify and which resulted from d isconnected understanding. 5 6 57 5 8 Figure 6: Borgen and Manu 's C lus tered Confus ions Mapp ing 2.5.3. Collective understanding and shared understanding Whi le the Pir ie-Kieren theory al lows for see ing different Images formed by different students and through this to make s e n s e of their del iberat ions and the emerg ing understandings, little research has been done in applying it to group 59 situations. Pir ie and Martin (2000) d i scuss the idea of understanding as shared during interactivity but note that the understanding changes on an individual level. In tracing the growing understanding of two students working together on a problem, Towers and Davis (2002) found that a teacher Intervention could prompt two students to Fo ld Back s imul taneous ly /c reat ing 'coupled activity' or 'overlapping understanding' developing into "a tightly entwined structural dynamic . . . i n which it becomes c lear that e a c h student is complicit in the unfolding of the other" (p. 326). Al though this does not indicate specif ical ly that there is col lect ive understanding, it does indicate that growth of understanding can be a col lect ive act ion and that the col lective action can "simultaneously occas ion differing understanding (which in turn prompts further col lect ive act ion a s they struggle to understand each other and the prompts from the teacher) and converge on shared understanding" (Towers and Davis , 2002; p. 331). Wh i le this is not speci f ical ly detail ing col lect ive understanding, consistent within the enactivist envi ronment on which the Pir ie-Kieren theory is based (an intertwining of the activit ies of teacher, student, and environment), Towers and Dav is (2002) conc lude: "The collective structural dynamic of the two students, the teacher, and the mathemat ical content presents the possibil ity that the mathemat ical understanding is shared" (p. 332). Mart in (2001), drawing on the theory of enact iv ism and Dav is ' (1996) definition of l istening, drew the metaphor of teaching and learning a s listening and shar ing. He states that the Pir ie-Kieren theory "provides a way to cons ider 60 understanding which recognizes the interdependence of all the participants in an environment" (Martin, 2001 ; p. 246). In consider ing understanding not a s a state to be ach ieved but as an ongoing phenomenon whereby individual understanding is entwined with that of the environment and the other participants, he has used this as a bas is to explore the col lect ive understanding of a c lass , using teacher l istening as a means of exploring images and misconcept ions within a group dynamic . 2.5.4. Prospective teachers' growth of mathematical understanding In teaching mathemat ics, a teacher must call upon not only his/her own understanding of mathemat ics, but a lso on his/her understanding of teaching strategies, of the way students learn, and his/her own personal phi losophy of mathemat ics. Berenson , Cavey , Clark, and Sta ley (2000, 2001 , 2002, etc.) have used the Pi r ie-Kieren theory a s a conceptual framework within which to study the growing mathemat ical understanding of prospect ive teachers. Prel iminary results of a longitudinal study indicate that Folding Back is essent ia l in developing this understanding and that conversat ions with others are useful Interventions to occas ion it. A s prospect ive teachers have few teaching strategies deve loped or understood, Cavey , Be renson , Clark and Staley, (2001) suggest that providing them with the opportunity to work together on a lesson plan facilitated their mathemat ical understanding so that they were better able to incorporate different Images of the task at hand and more clearly s e e the connect ions between co l lege mathemat ics and schoo l mathemat ics. 61 After severa l prel iminary studies (Berenson and Cavey , 2000; Berenson , Clark and Staley, 2001 ; Clark, 2001), C a v e y (2002) observed that in planning a lesson , a teacher must Fold Back to his/her own understanding of the topic to be presented, collecting mathematical ly precise information from different Primitive Knowings that are necessary to the lesson. E a c h Folding Back ' thickens' and informs the understanding, affecting how he/she thinks about the topic and how he/she prepares the lesson to present it to students a s effectively a s possib le. C a v e y and Berenson (2005) further demonstrated ' lesson plan study' a s useful in helping preserv ice teachers grow in their understanding of a particular mathemat ical concept. By working together, they noticed that it is often the say ing or verbal izat ion of their Images that Provoked growth in their understanding. "Image say ing is noted as a catalyst for growth because of its potential for creating se l f -awareness in understanding for the learner" (Cavey and Be renson , 2005 ; p. 186). Thus , many aspec ts of the Pi r ie-Kieren theory have recently been deve loped to expand and elaborate its usefu lness, both as a theoretical and a pract ical tool, for studying the growth of mathemat ical understanding, and have provided a languaging by which to d i scuss the concepts and p rocesses deeply e m b e d d e d within the theory. Pir ie & Mart in (2002) indicate that the theory c a n be used to help explain incomprehens ions and sources of problems a s students relate new topics to previously learned material. Mart in 's (1999, 2000, 2001 , etc.) work helps to explain how students can overcome problems by appropriate Interventions and Folding Back . In this light, C a v e y (2002) suggests that further 6 2 use of the Pir ie-Kieren theory in the area of teacher training would be appropriate in helping prospect ive teachers cons ider the teacher- learner situations as they interact with the mathemat ics and the students. By recogniz ing that e a c h activity co -emerges in the environment to become a part of the growing awareness of the individual or group, as teachers they may be able to use concepts from the theory to help inform their teaching and to promote growth of mathematical understanding. 2.6. Modification of the Pirie-Kieren theory to a new context In d iscuss ing the Pi r ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding, it is apparent that it has : evo lved into a theory that can be used by teachers or researchers a s a tool for l istening and observ ing in the context of mathemat ical activity .... It is a theoretical thinking tool for a person who is observ ing mathematical understanding and who might be interacting with students who are engaging in understanding activities (Pir ie and Mart in, 2000; p. 129). Whi le the above presentat ions indicates the usefu lness of the Pir ie-Kieren theory in d iscuss ing the developing understanding of an individual on a speci f ic topic in mathemat ics, for this study, it must be pointed out, that the theory w a s used in two distinct, interconnected ways . Firstly, it w a s presented to the S M I P students as a theory which they could use to cons ider the growth of mathemat ical understanding of their students. They were, in addit ion, encouraged to use the theory a s a theoret ical f ramework to think about their own growth of understanding. Second ly , it w a s used by the researcher, in a modif ied form, to M a p the growth of understanding of teaching and the growth of 63 understanding of learning of the S M I P students. O n e chal lenge of this study, then, w a s to develop definitions that would strictly adhere to the principles of the theory but which could be appl ied to this new context. Be low I present the modif ied definitions that will be appl ied in the Chapter 7 analys is of the growth of understanding of teaching and learning of the preservice teachers . T h e s e definit ions were verified through consultat ion with Dr. S u s a n Pirie as to their integrity. 2.6.1. The modified definitions PRIMITIVE KNOWING (TEACHING & LEARNING): As with the Pi r ie-Kieren Dynamica l Theory for the Growth of Mathemat ical Understanding, Primitive Knowing is the information the individual brings to the situation except that which is related to the topic. It includes his/her background exper iences that affect his/her outlook on life. IMAGE MAKING (TEACHING): A preserv ice teacher is involved in Image Mak ing of what it means to teach mathemat ics when he/she is trying to make meaning out of a lecture, a v ideo or a presentat ion by consider ing what teaching involves. W h e n a preserv ice teacher observes a peer presentat ion and tries to determine if that activity could be useful in teaching, he/she is engaged in Image Mak ing . The preservice teacher observes another's lesson in which a student was asked to explain his incorrect solution and says: "You know, I never would have thought of having the students explain their solution when they did it 64 incorrectly, and I'm thinking, maybe you can teach by taking what they did and building on it." IMAGE MAKING (LEARNING): Image Mak ing regarding what it means to learn mathemat ics occurs when the preserv ice teacher cons iders activities and thinks what students might learn from them. The preservice teacher notices that students MIGHT have different styles of learning. He/she may have observed a presentation by another preservice teacher and realized in the discussion afterwards he/she has made different Images of what the manipulatives were for from those that some others made. He/she says. "Oh, I got something quite different out of the activity than Mary did. I was trying to write it all out in x's and y's, but she just laid out the tiles to make a rectangle and said 'Ta-da!'" IMAGE HAVING (TEACHING): W h e n a preservice teacher, through demonstrat ion, indicates that he/she bel ieves a certain technique is involved in the process of teaching mathemat ics, or when he/she states: "Teaching involves ... ." or "A part of teaching is . . . " , he /she has c rossed the Don't Need Boundary into Image Having of what is the activity of teaching. He /she has determined for him/herself that this particular activity is part of the definition of teaching. The preservice teacher states: "A teacher should provide the background information so that students will know what they have to know." Or he/she indicates: "Teaching involves knowing where your students have made mistakes and fixing that." 65 I M A G E HAVING (LEARNING): Image Having of what it means to learn mathemat ics occurs when the preservice teacher indicates that a particular activity can help students develop understanding. He /she is able to state: "Learn ing involves being able to ..." The preservice teacher states: "I think that getting a good mark on a test indicates that a student has learned this unit." Or he/she indicates: "You can tell that learning is taking place if the student can use the right words." PROPERTY NOTICING (TEACHING): For Property Noticing to occur about what it m e a n s to teach mathemat ics, the preservice teacher will cons ider his/her own Images of teaching mathemat ics and will examine them to try to reconci le them into a meaningful understanding of the process . Th is activity may be quite subtle and he /she may not explicitly state his/her beliefs, but may reason through observed activit ies in trying to rationalize how they can be def ined as teaching. The preservice teacher states that he/she believes that teaching is telling the students what you want them to know, and realizes that he/she also thinks that the teacher should provide opportunities for the student to discover for him/herself. He/she looks at the conflicting Images and tries to determine how they can be accommodated into a meaningful definition of teaching. Or, he/she might say: "You know, it is really interesting all these different ways people present their material, and I never would have thought of some of them. I'll have to think if they will work for me." P R O P E R T Y NOTICING (LEARNING): Property Not ic ing of a develop ing understanding of what it means to learn mathemat ics occurs when the individual 66 real izes that he/she has different ideas of what it means to learn and he/she tries to reconci le these different ideas s o that he/she can expla in what he /she thinks learning means . The preservice teacher says: "Johnny seems to have learned the work because he got a 90% on his test. But, Paul also seems to know what he's doing because he explains it in class - but he didn't do well on the test. Maybe it's not the math that is the problem. Maybe he just can't write tests. " FORMALISING (TEACHING): After a preserv ice teacher has thought about his/her Images (Property Noticing), he /she may come up with a summary statement of the var ious Images. Formal is ing occurs when he/she def ines an activity as teaching mathemat ics. The preservice teacher states: "Teaching is just a matter of leading students to the right conclusion through guided questioning." Or, the preservice teacher presents several examples and each time leads the students to the solution with leading questions. FORMALISING (LEARNING): After consider ing his/her var ious Images of learning (Property Noticing) a preserv ice teacher may reach the conclus ion that a certain activity indicates that the corresponding mathemat ics has been learned and that it has been demonstrated in a particular manner. Formal is ing occurs when he/she offers a definition (visual or verbal) of leaning. A preservice teacher may state: "A student has learned a concept when he can apply a formula correctly in a problem." 6 7 OBSERVING (TEACHING & LEARNING): A t the Observ ing layer of understanding of what it means to teach mathemat ics, there is an interface with what it means to learn mathemat ics. T h e preservice teacher makes a connect ion between teaching and learning and ana lyzes how different teaching methods/approaches affect the learning process of the student. Thus , at this level, the preservice teacher 's understanding of what it m e a n s to teach mathemat ics becomes al igned with what it means to learn mathemat ics. In the p rocess of learning to teach, this is an important step, and after having arrived at this level, it is likely the preserv ice teacher will Fo ld Back to previous understandings and re-examine them. Observ ing is being undertaken when the preserv ice teacher consc ious ly works to al ign his/her definit ions of teaching and learning. It is the corresponding layer to Property Noticing, working on Formal is ings as opposed to Images, and also to Image Having where he/she is working on Primitive Knowings and new ideas come together to form new Images. The preservice teacher has Formalised that teaching is leading the students in the right direction and that learning is demonstrated by getting good marks on a test. He/she states: "I showed the students the best way to do these questions and look, they all did well on the test following my method. If I continue with this method of teaching my students will do well." STRUCTURING (TEACHING & LEARNING): Structuring of the developing understanding of what it m e a n s to teach and learn mathemat ics occurs when the 68 preservice teacher uses his/her Observ ings to make a decis ion about what he/she bel ieves is the essent ia l activity of teaching mathemat ics based on his/her accep tance of what this implies and how it relates to his/her understanding of what it means to learn mathemat ics. He /she has passed through the next Don't N e e d Boundary and has given meaning to the activities that he/she bel ieves descr ibe the process of teaching mathemat ics based on his/her percept ion of the implications this has on the learning of mathemat ics. W h e n an individual answers the quest ion for him/herself: "What does it mean to learn mathemat ics?" and has reconci led this with his/her answer to: "What does it mean to teach mathemat ics?" or answers the quest ion: "How can one teach mathemat ics effectively?" he/she has Structured his/her understanding of teaching and learning. The preservice teacher may conclude: "It is the teacher's job to be sure that a student has had enough practice so that he/she can apply mathematics to any problem" (i.e. uses a Formalisation that learning depends on practice, and teaching is therefore providing practice, Observing the necessary connection between teaching and learning). INVENTISING (TEACHING & LEARNING): Inventising of what it m e a n s to teach and learn mathemat ics involves a preserv ice teacher in thinking 'outside the box'. He /she examines other alternatives to those observed and attempts to determine of a method of teaching mathemat ics whereby the structure creates a different learning environment for the learning of mathemat ics by the students. 69 The preservice teacher, after considering various forms that define teaching (Observing), reaches a conclusion about the basic structure that, in his/her mind unites teaching and learning, then problematizes it and wonders: "What if I didn't tell them if they were right or wrong? What if I just left them to check for themselves? Would that be teaching? Would they be learning?" Within these definit ions, it must be pointed out that, at any level, the preserv ice teacher may have only partial, or even incorrect Images of teaching and/or learning as is the c a s e with the Pi r ie-Kieren theory regarding the learning of mathemat ics and that there is recurs iveness implied in this d iscuss ion . 2.6.2. The modified Pirie-Kieren model The Pir ie-Kieren model for the Dynamica l Theory for the Growth of Mathemat ica l Understanding cons is ts of eight nested circ les to indicate the e m b e d d e d n e s s of the growth of mathemat ical understanding (see Figure 1). In d iscuss ing and analyz ing the growth of the understanding of teaching and learning of mathemat ics of the preserv ice teachers in this study, it w a s found that at the Observ ing level , there w a s a union of the two understandings a s def ined above . That is, when an individual reached the Observ ing level of understanding of teaching and learning he /she real ized that there w a s a connect ion between the p rocess of teaching of mathemat ics and the p rocess of learning of mathemat ics. T h e mode l , therefore, had to be adjusted to take this into considerat ion. T o reflect this, the modif ied model represents the teaching and learning as two separate sets of embedded circ les for the first five levels of developing 70 understanding and the next three levels 'umbrel la ' them. The modif ied model is presented below in Figure 7 and will be used is Chapter 7 to M a p the growth of understanding of teaching and learning mathemat ics for four preservice teachers . Figure 7: Borgen 's Modif ied Dua l Mode l for the Mapp ing of the Growth of Understanding of Teach ing and Learn ing. 71 Chapter 3 Review of Literature - Teacher Education 3.1. A context for discussion of preservice teacher education Over the years, teacher educat ion has undergone a multitude of changes . L icens ing and training/education of teachers in North Amer i ca have a lways involved both a theoretical and a practical component , al though not a lways with the s a m e emphas is (Shu lman, 1998), and have general ly involved a component which provided opportunit ies for c lass room observat ions and pract ice teaching (Fe iman-Nemser , 1990; Shu lman , 1998). A s phi losophies shifted, emphas is shifted from the educat ion of teachers to the training of teachers (Bri tzman, 1991; Shu lman , 2002) with the assumpt ion being that theory learned in isolation would lead to informed pract ice. However , often the theoretical methods learned in isolation were difficult to apply in context (Hiebert, Gal l imore & Stiger, 2002). A n interest in educat ion by cognit ive scient ists, who s a w student exper ience (and thus the student-teacher exper ience) a s a source of knowledge, brought new deve lopments in teacher educat ion. Emphas i s on the importance of prior exper ience led to a redefining of "[t]he recurrent chal lenge of all professional learning [in] negotiating the inescapab le tension between theory and pract ice" (Shu lman, 1998; p. 517). P iaget 's developmenta l s tages created a context for learning through exper ience (Brown, C o o n e y & J o n e s , 1990) and practical exper ience in the c lass room c a m e to be v iewed as an apprent iceship for exper iment ing with different techniques, a s wel l a s a means by which to verify 72 and develop new theory. Emphas i s within educat ion shifted from "preparation of c i t izens, the enhancement of individual abilit ies and talents, [and] the intellectual attainment in academic subjects" (Corr igan & Haberman , 1990; p. 197) to the teaching of bas ic skills as prerequisite to all other goa ls . The effect on teacher educat ion was to "narrow the broad knowledge base formerly required of teachers to emphas ize their preparation with know-how in teaching extremely narrowly def ined basic skil ls" (Corr igan & Habe rman , 1990; p. 197). More recently, teaching has c o m e to be thought of in a more holistic manner (Brown et al, 1990) and as a continual p rocess whereby all those involved in the p rocess learn from each other (Shu lman, 1998). 'Be ing in' is part of the p rocess of 'becoming ' and teaching can be seen a s "an epis temology wherein the knowledge that is teaching is only ev idenced in the pract ice that is teaching" (Roth & Tob in , 2002; p. 21). Th is encourages one to objectify exper iences , reflect upon them and thus engage in activities which al low for the construct ion of self a s ' teacher ' (Br i tzman, 1991). From this perspect ive: [LJearning to teach is not a mere matter of applying decontextua l ized skil ls or of mirroring predetermined images; it is a t ime when one 's past, present and future are set in dynamic tension. Learn ing to teach - like teaching itself - is a lways the p rocess of becoming : a t ime of formation and transformation, of scrutiny into what one is do ing, and who one can b e c o m e (Bri tzman, 1991; p. 8). Thus , in the p rocess of learning to teach it is a s s u m e d that research into teacher educat ion programs has played an important role as it is through the teacher educat ion p rocess that bel iefs and opinions about teaching and learning may be 73 reconfigured from those that are learned through one 's own exper iences in a c lass room as a student. Whi le there have been general changes in the phi losophies surrounding teacher educat ion, there have also been changes speci f ic to subject content a reas . In the area of mathemat ics, specif ical ly, the shifts in phi losophy of educat ion over the past century, possibly beginning with Dewey 's progressive educat ion movement and its emphas is on understanding curr iculum as a connec ted enterprise, have had a profound influence on the phi losophy and activit ies that are cons idered mathemat ics. There was a structural change in the teaching of mathemat ics in the 1960's, manifested through the new mathematics which emphas i zed the ax ioms and structure of the subject. Th is p laced emphas i s on connected understanding as well as on producing more mathematical ly sophist icated students (Brown et al., 1990). Th is resulted in an increased e m p h a s i s on the need for teachers to know more mathemat ics. However , a competency-based teacher education, rooted in behaviour ism, with an emphas is on accountabi l i ty soon fol lowed this movement . L e s s emphas is w a s p laced on the need for subject content knowledge and more on the need to learn and teach the bas ics of the content (Corr igan & Haberman, 1990). Recent ly , schoo ls have been more inclusive of students, and students of all abilit ies and skil ls remain in and are required to complete higher level mathemat ics courses than before. There has been a movement to a problem solving and application approach as "an effort to learn and appropriately invoke heurist ics to dea l with not only what is a l ready known ... but a lso what is unknown" (Brown et al, 1990; p. 641). Prob lem 74 solving and application approaches again create a need for deeper mathematical understanding on the part of the teacher. However , the understanding that is needed may not be the theoretical knowledge base that al lows them to obtain a degree in mathemat ics. R e s e a r c h has indicated that teachers need to better understand the manner in which students learn as many of the students they face will have differing learning styles than their own. It has become important, then, that preservice teacher educat ion programs include some theories of learning in their curr iculum, with theories specif ical ly related to subject content cons idered useful. Thus , during the entire latter part of the twentieth century, with the shift toward mathematics for all, there has been an emphas is on the "concomitant necessi ty to redefine mathemat ical content in terms of the capaci t ies of good workers ... [and] the historically resil ient desi re for bas i c mathemat ical literacy now tends to be paired with the need to deve lop sound reasoning ski l ls" (Davis, 2001 ; p. 18). T h e s e c h a n g e s in v iew a s to the nature of mathemat ics have involved a change from see ing mathemat ics as a set of theorems and postulates to be learned, to see ing mathemat ics as a human endeavor in which communicat ion skil ls are an essent ia l component s o that students will be better ab le to involve themse lves in the technological changes and complexi t ies of the twenty-first century (Cooney & Shea ly , 1997). Ours has become "a culture that is o b s e s s e d with and utterly reliant upon mathemat ics" (Davis, 2001 ; p. 19). Mathemat ics has become an integral part of our l ives, defining the way we look 75 at the world, often being thought of as much a humanity as a sc ience . Th is has created a need for a different form of teacher educat ion. F rom the above d iscuss ion , it is obvious that there have been many changes in the phi losophy of teaching and in the phi losophy of teacher educat ion over the past century. Freiberg and W a x m a n (1990) c la im, however, that there has actually been little change in teacher educat ion. The resistance to change by teachers and administrators, and the popularization of educat ion that has led to an eros ion of the authority by teacher educators in determining the requirements for entrance and exit of programs, they c la im, is partially the cause of this. Whi le there may have been few overal l changes in teacher educat ion at the t ime of Freiberg and W a x m a n ' s writing, there have been s o m e speci f ic changes in the a rea of mathemat ics and mathemat ics teaching as well as mathemat ics teacher educat ion that have resulted from recent changes in v iews of what mathemat ics is and what it means to understand mathemat ics. A l s o , recent studies on the scholarsh ip of educat ion may be having an inf luence on university educat ion, which in turn may be influencing teacher educat ion and thus teaching. T h e s e studies, a imed at consider ing the relationship of teaching as scholar ly activity compared to research as scholar ly activity, have resulted in c h a n g e s in the considerat ion of what is def ined as scholar ly activity and of the teaching done at the university level. Th is may be creating opportunit ies for c h a n g e s in teacher educat ion. 76 3.2. A discussion about the scholarship of teaching and teacher education Teacher educat ion and teaching in schoo ls are obviously interrelated. Recen t research on the scholarship of teaching, with an emphas is on teaching at the university level where, in the past, it was cons idered sufficient that one know one 's subject matter to be able to teach it, now indicates that it is important that teachers at any level have the skil ls to teach. "The p rocess of teaching the subject matter of a discipl ine forces academics to clarify the big picture into which their speci f ic research special izat ion fits" (Marsh & Hattie, 2002; p. 604). Scho larsh ip of teaching emphas i zes that it is important to be aware of the construct ions of knowledge on the part of students. Imparting that knowledge is insufficient, and it is seen as important to deve lop strategies that reward deep , not surface, learning. Exist ing research , however, indicates that there appears to be little relationship between research as scholar ly activity and scholarsh ip of teaching, except to a minor degree, in facult ies of educat ion (Hattie & Ma rsh , 1996; Cunso lo , Elrick, Middleton & Roy , 1996; Sei j ts, Taylor & Latham, 1998; Kreber, 2002). A s a result, much of the preparation of teachers is done by people trained in research methodology rather than by people trained in pedagogy (Cunso lo et al., 1996). Thus , a s students at the university move from their subject a rea special ty into a teacher educat ion program they come with subject content knowledge, but not necessar i ly with pedagog ica l content knowledge. Study of the scholarship of teaching reconf irms the effects of Lort ie's (1975) apprenticeship of observation in which he indicates that, a s students in a c lass , people are actually identifying what teaching m e a n s to them. They are 77 learning to teach in the manner in which they were taught. A s a result, the majority of undergraduates view learning as "acquisit ion of knowledge, memor iz ing, utilization and/or use of knowledge" (Hattie & Marsh , 1996; p. 531) a s this is how they were taught. Students entering teacher educat ion programs often tend to v iew learning a s a reproduction of what is stated as opposed to the more creative "abstraction of meaning, and learning as an interpretive p rocess a imed at understanding reality" (Hattie & Marsh , 1996; p. 531). Scho larsh ip implies integration, application and t ransmiss ion of knowledge, and , it would s e e m , if preserv ice teachers became more aware of the manner in which learning takes p lace, the manner in which students c o m e to understand subject content material, they may approach teaching in a more enl ightened manner. Thus , although scholarship of teaching is usual ly cons idered a university activity promoting an institutional environment support ive of teaching and learning (Kreber, 2002), the concepts may be appl ied to teacher educat ion by providing a language and understanding that permit ways of v iewing pract ice and of promoting critical reflection of pract ice. There is a s e e n need to share teaching a s a discipl ine, going to the literature to learn about the s u c c e s s e s and fai lures of others, grounding one 's own theory in the work of others if one w ishes to improve teaching (Shulman, 2000) s ince, al though the opposi te is not true, scholarship of teaching research has found that inc reased research d o e s improve teaching (Hattie & Marsh , 1996). Th is study, then, which e x p o s e s preservice mathemat ics teachers to theory for understanding - particularly, theory for developing understanding of 78 mathemat ics - is a step which fits into this thinking of understanding teaching a s a scholar ly activity. Therefore, regardless of the substant ive content a rea that teachers are cal led upon to teach, all teachers should have a working knowledge of pedagogica l principles and pract ices which can be brought to bear in the c lass room. 3.3. What is teaching? Viewing teaching and learning from the point of v iew of complex theory, Davis et al, (2000) indicate that these p rocesses do not begin and end in the c lass room but are a dynamic whole with the environment, with learning dependent upon, but not determined by, teaching. T h e process of learning, al though individually exper ienced, is social ly constructed and determined by cultural c i rcumstances (Davis et al., 2000). C larke and Er ickson (in press) def ine teaching as "the professional practice of engaging learners in the construct ion of knowledge directly related to a particular a rea of study" (p. 22). T o be a teacher involves the internalization of the teacher 's role and the teacher 's professional identity. Th is a s s u m e s knowledge and ability, as well as the essent ia l norms, va lues and attitudes of the role (de Ponte & Brunheira, 2001) . Teach ing involves helping the student s e e the relationship between exper ience and the c lass room (Bal l , 1997). In keeping with the more holistic v iew of teaching previously ment ioned, in order to teach effectively, not only must one be famil iar with the subject matter, have an understanding of the learning p rocess , and have knowledge of the students and the culture of teaching, one must a lso have a s e n s e of the complexity of how they fit together (Ball & 7 9 McDiarmid , 1990; Br i tzman, 1991; Davis et al., 2000). Thus , teaching is a complex p rocess in which: T h e teacher 's task is "to ass is t the learners in the cont inuous process of understanding previous exper iences and knowledge in terms of new events and c i rcumstances (Davis et al, 2000; p. 91 (italics added) . The goal of teaching is to ass is t students in the development of intellectual resources that enab le them to participate in, not merely know about, the major domains of human thought and enquiry" (Ball & McDiarmid , 1990; p. 438) (italics added). "The art of teaching is to maintain the difficult ba lance between r ichness of detai l and narrowness of focus ... effective teaching is more a matter of l istening than telling - that is, of attending and responding to the s e n s e that learners are making" (Davis et a l , 2000; p. 10) (italics added) . [TJeaching, fully understood, is an extraordinary p rocess of creating understanding and knowledge" (Shu lman, 2000 ; p.6) and is "a very t ime-consuming activity in that it requires sound knowledge of one 's discipl ine a s well as a good understanding of how to help students grow within, and perhaps beyond, the discipl ine" (Kreber, 2002; p. 9) (italics added) . T h e focus of teaching should therefore not be on how one teaches , but rather on how students learn (Davis et al., 2000). T h e Pi r ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding is a theory about the developing understanding of mathemat ics. Knowledge of this theory, it would s e e m , provides an appropriate p lace to start a d ia logue in the educat ion of prospect ive mathemat ics teachers. Teach ing should involve the possibi l i ty of ' spaces opening in the moment ' and the need and ability to respond and react to them. P lans for teaching must be deve loped s o a s to expect surpr ises and to c h o o s e from different possibi l i t ies 80 in order to help students notice what they know (Davis et al, 2000). Teachers "must be able to work with content for students in its growing, unfinished state, they must be able to do something perverse: work backward from mature and compressed understanding of the content to its constituent e lements" (Ball , 2000; p. 245). Put succinct ly, C larke and Er ickson (in press) indicate that there can be little distinction between teaching and learning: teaching involves cont inuous learning, with l istening and learning on the part of the teacher as an integral part of the dynamic . 3.3.1. T h e role of theory in teach ing A s ment ioned in Sect ion 1.3, there occurs in teacher educat ion, and thus in teaching, a perpetual struggle between mediat ion of theory and pract ice. Learn ing environment research indicates that e a c h theoretical f ramework and/or theory involves a set of constructs and provides only one window through which to v iew the learning and the environment. Activity theory indicates that w e are co -creators of our learning and have the power to act based on subject, object and tools avai lable in a communi ty guided by rules and regulations (Roth & Tob in , 2002) . T h u s , awareness of different educat ional theor ies and research may help deve lop different forms of praxis that apply to different teaching situations. The "s igni f icance [is] not just understanding that much of teaching occurs a s habitus that genera tes pract ices but a l so of understanding the critical need to connect theor ies to the exper ience of teach ing" (Roth & Tobin , 2002; p. 296). Whi le theory cannot capture the pract ice/enactment of the situation, "it is in the 81 explication of why particular roles might be productive that theorizing can be benef ic ial" (Roth & Tobin , 2002; p. 168). Having a theoretical framework with which to d i scuss their understanding of teaching and learning, therefore, would be useful to preserv ice teachers . The Pir ie-Kieren theory g ives preservice teachers, not only a theory upon which to build their understanding of student understanding, but a lso a language with which to d iscuss and cons ider student, as well as their own, learning. A teacher must continually quest ion what he/she teaches and how he/she teaches it. It is essent ia l that teachers understand the va lue of theor ies a s models of understanding which can then be adapted to speci f ic content. "Us ing theory to inform pract ice then cannot, and should not, occur in the form of a direct appl icat ion of a recipe to a given problem. It rather impl ies a dec is ion-making p rocess on the part of the teacher" (Kerber, 2002 ; p. 11). T h e Pi r ie-Kieren theory provides a forum that can Invoke preserv ice teachers to quest ion their own understanding of mathemat ics and the manner in which they bel ieve students learn mathemat ics. Whi le indicating that research may not be the best foundation on which to build an essent ia l professional knowledge base , K ieren (1997) notes that research should be used for its exploratory power and that ideas drawn from theor ies will not tell one how students will use material but rather provides a f ramework for consider ing different approaches students might take. By listening to the student, the teacher is ab le to inform his/her pract ice through developing a deeper understanding of the student 's understanding. Theory may help in this 82 explorat ion. " B e c a u s e teaching is a highly cognit ive task, a c c e s s requires shar ing otherwise covert thought p rocesses by reading the works of others, actively seek ing feedback, and d iscuss ing teaching dec is ions with other" (Seijts et al, 1998; p. 160). Teach ing itself thus b e c o m e s a means of theorizing, and separat ion of theory and pract ice is impossib le (Shulman, 2000). 3.3.2. The role of reflection in teaching Reflect ion on action provides an important method by which teachers can monitor their knowledge in act ion, using teaching acts as objects of reflection, whi le applying theory (Roth & Tob in , 2002). Mature reflection on teaching shares many of the s a m e features a s the scientif ic method in that it involves quest ioning, ev idence, interpretation based on that ev idence, critical appra isa l , and conc lus ions drawn from this. Through thinking about teaching and d iscuss ing it with co l leagues, teachers can improve learning by forcing the considerat ion of strategies that make the content knowledge more access ib le to the learner (Cunso lo , et al, 1996). C la rke and Er ickson (in press) argue that teaching b e c o m e s routine without enquiry and reflection and that practice b e c o m e s less professional when teachers are not inquisitive about how students learn. Thus , a long with S c h o n (1988) they agree that how students learn - their act ions and words - must be the cornerstone on which to build activities to help students m a k e s e n s e of the world. Sei j ts, Taylor and Laytham (1998) suggest that research indicates that observat ion of others teaching and reviewing v ideo- tapes of onesel f teaching could help in identifying concerns and create opportunit ies to critically ana lyze alternative mode ls of teaching. 83 In this study, preservice teachers observed themse lves and other preserv ice teachers during presentat ions and were encouraged to think about the teaching and learning that took p lace. A s wel l , they observed v ideos of teachers in regular c lassrooms and d i scussed these as they p roceeded through their teacher educat ion c lasses . Whi le they were in situ during their pract icum and had a chance to observe students developing understanding, and when they returned to their university c l a s s e s , they d i scussed these activit ies in terms of the teaching and learning that took p lace. In this manner, they were encouraged to d i scuss their understanding of teaching and learning and the Pi r ie-Kieren theory provided them with a format and a vocabulary with which to do so meaningfully. 3.3.3. The role of knowledge in teaching Teach ing invokes many kinds of knowledge such a s knowledge of the subject matter, knowledge of pedagogy, pedagog ica l content knowledge, and knowledge of how students learn. A s wel l , there is a practical knowledge involved in teaching, s o m e of which may only be learned through exper ience, observat ion and/or through discussion/ref lect ion. 3.3.3.1. The role of subject content knowledge in teaching For instruction to be meaningful to a student, it must be based on what he /she knows and the teacher must b a s e instruction on the cognit ive connect ions he /she a s s u m e s exist and can be further deve loped (Cooney & Shea ly , 1997). B a s e d on the tenets of the Pi r ie-Kieren theory, this impl ies that the teacher must try to a c c e s s the student 's Primit ive Knowing and his/her Images. T o do this, the importance of subject content knowledge on the part of the teacher cannot be 84 downplayed (Ball , 2000 ; Dav is et al, 2000 ; Thompson , 1992; Bal l & McDiarmid , 1990). "Subject matter, or content knowledge includes knowledge of facts, concepts , and procedures that define a given field and understanding of how those facts fit together. It a lso includes knowledge about knowledge - where it c o m e s from, how it grows, how truth is estab l ished" (Fe iman-Nemser , 1990; p. 221). A s well , subject content knowledge includes understanding about the subject, "the relative validity and centrality of different ideas or perspect ives, the major d isagreements within the field ... how c la ims are justified and val idated, and what doing and engaging in the d iscourse of the field entai ls" (Ball & McDia rmid , 1990; p. 440) . A wel l -developed understanding of subject matter on the part of the teacher, and a deep conceptua l understanding of a topic al low the teacher to respond to all aspects of the pupils' needs more readily so that he/she can teach the content for understanding (Prestage and Perks , 2001 ; Davis et al, 2000; T h o m p s o n , 1992). "Knowing content is a lso crucial to being inventive in creat ing worthwhile opportunit ies for learning that take learners ' exper iences, interests and needs into account" (Bal l , 2000 ; p. 242) s ince learners learn in different ways . A teacher with inaccurate and/or narrowly conce ived understandings will not be able to respond to the s e n s e the students are making of the content, and this lack of understanding will be p a s s e d on to his/her students through failure to cha l lenge their misconcept ions which will perpetuate a state of not knowing (Carter, 1990). Thus , the teacher must know the subject content and be able to approach it from different perspect ives in order to make it meaningful to the 85 student. He /she must be capab le of see ing him/herself as an authority in evaluat ing materials and practice while being flexible enough to change those beliefs in the face of conflicting ev idence (Cooney & Shea ly , 1997). The teacher, at the very least, must know the subject content wel l enough to be able to determine if the student 's express ion of it indicates understanding. "Both the ability to create flexible tasks and the capaci ty to fol low student leads clearly depends on fairly broad and flexible understanding of the task at hand" (Davis et al, 2000; p. 143). Thus , the ability to respond to the s e n s e that the learner is making requires subject content knowledge. S ince "meaningfulness of all knowledge, regardless of its level of abstract ion, der ives from one's own exper ience with the world" (Davis et al., 2000 ; p. 26), knowledge of the students here and now is important in knowing how to approach the teaching of a topic to make it understandable to students. Accord ing to the Pir ie-Kieren theory, this impl ies a need to a c c e s s the students' Primitive Knowing. 3.3.3.2. The role of mathematical content knowledge in the teaching of mathematics Discipl inary knowledge impacts the organizat ion and instruction of a subject a s well as the teacher 's ability to re-present it to students (Carter, 1990). Greater knowledge of mathemat ical content leads to more conceptual teaching strategies in mathemat ics, the ability to relate it to outside activities and the ability to engage students in problem solving activit ies (Steinburg, Haymore & Marks (1985) cited in Car ter (1990)). A n important component of mathematical content knowledge is understanding poss ib le representat ional sys tems that students may 86 have for a topic as well as the potential misconcept ions they may have (Shu lman, 1986). S imple subject content knowledge is likely to be insufficient to be an effective teacher of mathemat ics (Ball , 2000). Alternative images of that knowledge are necessary so that one is able to understand the mathemat ical concepts from the different perspect ives the students might have, as wel l as the ability to be able to relate it to activities that are involving for students. B a s e d on the Pi r ie-Kieren theory, this implies that the teacher must have a vast repertoire of Images on which to draw if he /she is to understand student representat ions. Wha t is important, then, in teachers ' mathemat ical knowledge, is not just what they know, but a lso how they know it and how they represent it. 3.3.3.3. T h e role of p e d a g o g i c a l content k n o w l e d g e A central tenet of teaching is "an understanding of how to recognize, evaluate, and implement activities with pupi ls ' learning in mind" (Carter, 1990; p. 295). However, s ince people tend to become teachers in subject or content a reas in which they did wel l , not all teachers know the content in a way that they can use it to help students. Be ing an effective teacher involves having the "ability to transform subject content through a range of techniques, suitable examp les , analog ies, etc. to make the content more a s s e s s a b l e and meaningful to the learner" (Smith, 2001 ; p. 117) or to be able to translate one 's own knowledge into c lass room curricular events that make that knowledge access ib le to students, using their interests and motivations to learn a particular topic (Carter, 1990). Th is is pedagogica l content knowledge, knowledge which goes beyond knowing 87 subject content knowledge, to knowing ways of presenting it to make it comprehensible to others (Shulman, 1986). Teachers draw on their own wits, observations, intuitions, and articulations of what it is that they do and how they do it to guide their practice. Often idiosyncratic, rarely documented, and always complicated, this knowledge is the essence of their teaching and bears the imprint of an authentic rendering of the complexities associated with a highly social and inherently situated practice (Clarke & Erickson, in press; p. 27). When teachers develop expertise, they not only mediate theoretical knowledge about education with their knowledge derived from personal teaching experience, they also develop increasingly better ways of helping students understand the subject matter. When expertise in the discipline is effectively combined with knowledge of how to teach, the latter being derived from both educational theory as well as experience, we witness the construction of pedagogical content knowledge (Kerber, 2002; p. 15). Pedagogical content knowledge thus implies the blending of content knowledge and knowledge of pedagogy and involves knowing and applying "useful ways to conceptualize and represent commonly taught topics in a given subject, plus an understanding of what makes learning those topics difficult or easy for students of different ages and backgrounds" (Feman-Nemsor, 1990; p. 221-222). Within the context of the Pirie-Kieren theory, pedagogical content knowledge implies being aware of the different Images students might hold, and the different levels of understanding at which they might be working. Pedagogical content knowledge thus involves formal and informal knowledge constructed about teaching by combining knowledge of the discipline with knowledge of teaching, declarative knowledge (which can be gained through reading books and articles about teaching and theory), along with procedural knowledge about teaching (which can be gained through experience) and implicit 88 knowledge of self-regulation (Shulman, 1986). It can be cons idered as knowing both the how and the why of the content or subject matter - knowing the structure of and knowing the potential sources of error within the topics. Th is can only be ach ieved through being able to recognize the knowledge in its bas ic , tactic d imens ions (Shulman, 1986), through being able to understand the content from different perspect ives and being able to d iscuss it in different manners and at different levels. 3.3.3.4. The role of pedagogical content knowledge in mathematics teaching Pres tage and Perks (2001) refer to the distinction between "learner-knowledge in mathemat ics [and] teacher-knowledge in mathemat ics, the former is the knowledge needed to pass examinat ions, to find the solut ions to mathemat ical problems; the latter is the knowledge needed to plan for others to c o m e to learn the mathemat ics" (p. 102) (italics original). In order to des ign appropriate c lass room activit ies, analys is of topics through reflection "to recons ider their own personal understandings of mathemat ics" (Prestage & Perks , 2001 ; p. 108) is necessary for an individual to develop the teacher-knowledge with which to inform teaching that will cha l lenge and meet the needs of the students. Thus , teaching mathemat ics for understanding requires knowledge of mathemat ics, knowledge of student learning and knowledge of mathemat ics pedagogy. Th is implies the need to have both conceptua l knowledge, the "rich relat ionship, linking new ideas to ideas that are al ready understood" (Stump, 2001 , p. 210) and procedural knowledge, the "formal 89 language and symbol sys tems, as well as a lgebraic algorithms and rules" (Stump, 2001 ; p. 210) of the content. "Pedagog ica l content knowledge highlights the interplay of mathemat ics and pedagogy in teaching. Rooted in content knowledge, it compr ises more than understanding the content o n e s e l f (Ball , 2000; p. 245-246). It is important that teachers know something about how students come to know the mathemat ical content, what representation and Images students might have, a s wel l a s being ab le to identify the teacher 's role in developing this. The Pi r ie-Kieren theory, being a theory for the learning of mathemat ics provides a good base upon which to build an understanding of poss ib le Images and representat ions that students might have a s wel l a s consider ing where they might exper ience difficulties or obstac les in their developing understanding. 3.3.3.5. The role of practical knowledge in teaching Whi le Pres tage and Pe rks (2001) agree with Shu lman and others that pedagogica l content knowledge is important, they indicate that even it is insufficient. Exper ience, persona l theor ies and opinions, and ways of interacting are a lso important aspec ts to teaching. T h e s e affect one 's act ions in a situation and one 's ability to reflect-in-action. On ly when combined with practical knowledge, "the knowledge teachers have of c lass room situations and the practical d i lemmas they face in carrying out purposeful action in these sett ings" (Carter, 1990; p. 299) or with "the kinds of knowledge that practit ioners generate through act ive participation and reflection on their own pract ices" (Heibert et al, 2002; p. 4) does pedagog ica l content knowledge help inform the teacher a s to 90 how students learn. Informed by one 's practical knowledge, knowledge of self, and one 's va lues and intentions can enhance teaching possibi l i t ies. Th is knowledge, it s e e m s would best be deve loped through reflection on one 's own i learning, on the process of developing understanding, and on one 's beliefs about the subject matter. 3.3.4. The role of attitudes and beliefs in teaching It is commonly bel ieved that the manner in which a curr iculum is implemented is dependent on one 's knowledge and beliefs about the nature of the subject, about teaching and about how students learn (Pajares, 1992; Thompson , 1992; C o o n e y & Shea ly , 1997). "What teachers and students bel ieve about [the subject matter] is influential to how they understand themselves , others, knowledge, and future possibi l i t ies" (Davis et al, 2000 ; p. 234). Pa jares (1992) suggests that not only do beliefs inf luence how tasks are def ined and organized but a lso that they predict behaviour to a greater extent than does knowledge. Hattie and Marsh (1996) point out that there are "two teaching concept ions: teaching that involves preparing concepts so that the student can learn without too much interpretation, and teaching that a ims to involve the students in interpretive and structuring work" (p. 530). T h e s e , they say , co inc ide with two configurations of teaching; the instructional method and the student-centered method. If knowledge is bel ieved to be objective and, a s involving creat ion or d iscovery, "it would s e e m consistent to think that it requires the t ransmiss ion and absorpt ion through a separate ly conceptua l ized teaching p rocess" (Marsh & Hattie, 2002; p. 629). But if knowledge is v iewed as a product 91 of communicat ion and negotiation, then "the relationship between, teaching and learning becomes an intimate one" (Marsh & Hattie, 2002; p. 629). However , beliefs are difficult to determine. Wha t one says one bel ieves is often more a reflection of what one thinks one should bel ieve than of one 's true beliefs (Davis et al, 2000). A l s o , "[t]here is no reason to bel ieve that one 's beliefs are consistent and logically ar ranged, as if cons iderable thought had gone into the networking of beliefs. S o m e beliefs are held in isolation from one ano the r . . . [others] can exist in separate clusters, segregated and protected by one another" (Brown et al, 1990; p. 652). If bel iefs are held peripheral ly or in isolation, they may contradict other beliefs held in the s a m e manner, leading to apparently contradictory incorporation of activities (Cooney & Shea ly , 1997). Ref lect ion on exper ience is cons idered paramount to learning, particularly in determining what one truly bel ieves about teaching and learning. Th is reflection can occur only when the individual is able to "step out of the st ream of direct exper ience, to re-present a chunk of it, and to look at it a s though it were direct exper ience, while remaining aware of the fact that it is not" (von G lasers fe ld , 1991, p. 47 , cited in C o o n e y and Shea ly , 1997; p. 100) (italics original). Th is reflection can be used to help one determine what one truly be l ieves about the nature of the subject and the nature of student learning and thus impacts the manner in which one implements the curr iculum. 92 3.3.4.1. T h e importance of c h a n g i n g attitudes and bel ief about the nature of mathemat ics Recen t changes in the way we view and enact mathemat ical activity has led to changes in what we consider mathematical proof and mathematical understanding (Cooney & Shea ly , 1997). The naive idea that mathematics is hierarchical , that mathemat ical learning involves the learning of a priori knowledge in an axiomatic, deduct ive sys tem with irrefutable solut ions and that per formance indicates that learning has taken p lace has been chal lenged (Davis & Hersh , 1986; Brown, et al, 1990; C o o n e y & Shea ly , 1997). T h e s e changes in perspect ive have required more than s imple changes in quest ioning and ways of problem posing in mathemat ics, but necessi tate changes in one 's beliefs of the nature of mathematics and to chal lenging one 's phi losophic v iews of the nature of the subject and its assessmen t . Th is shift in perspect ive in mathemat ics has led to a widening v iew of the nature of mathemat ics and has created uncertainty in a field previously cons idered certain, and to a quest ioning of bas ic beliefs and epis temological perspect ives (Franke, F e n n e m a & Carpenter , 1997). Test ing for understanding, formerly cons idered e a s y in mathemat ics because of the structure and organizat ion of the subject, has become more complex and problemat ic (Cooney & Shea ly , 1997; Brown ef al, 1990). T h e paradigmat ic shift in thinking about the nature of mathematics forces o n e to bel ieve that students have mathemat ical understanding which they can further develop without direct instruction. It fo rces the tenet that the teacher can learn from listening to the students and that this listening can and should be used 93 to inform instructional dec is ions. T h e teacher, with the students, deve lops an understanding of mathemat ics negotiated through interaction in a cultural setting (Cobb, Jaworsk i & P r e s m e g , 1996). However, the inevitable resistance to change by s o m e teachers who hold absolutist v iews about mathemat ics and mathemat ics teaching m e a n s that they are unlikely to participate in such activity (Cooney & Shea ly , 1997; Hart, 2002). Thus , while there are fundamental changes in the phi losophy of what mathemat ics js, and this affects the educat ion of students and teachers, there are pockets of res istance in the field. In order to actuate change in the teaching of mathemat ics, "mathemat ics teachers must change the way they learn before they c a n change the way they i t each" (Pereira, 2000; pg. 205). Th is will need to be a requisite of teacher educat ion programs, and it can possib ly be approached through present ing preserv ice teachers with alternative v iews about the nature and mean ing of mathemat ics, present ing them with ways of thinking about and present ing material that make them quest ion their own understandings, and by al lowing them to pract ice the new app roaches in a safe environment. 3.4. W h a t is effective mathemat i cs t e a c h i n g ? "Effective mathemat ics teaching requires understanding what students know and need to learn and then chal lenging them and support ing them to learn it wel l ' ( N C T M , 2000; p. 16). For mathemat ics to be meaningful and understandable to the student, the teacher must be able to merge these var ious aspec t s of teaching and learning together into a meaningful whole that c reates a more understandable unit for both. Effective mathemat ics teaching goes wel l 94 beyond knowing the subject matter to being able to draw on a broad background of mathematical knowledge, having s o m e understanding about how students learn it, and having the ability to reflect upon what mathemat ics m e a n s to the student as well as to onesel f ( N C T M , 2000). It involves knowing different representat ions that students may hold and being able to s e e the relationship between these representat ions and the students ' understanding of them. In many c a s e s preservice teachers must learn these representat ions and relat ionships during their teacher educat ion programs as their previous learning, based on their own ability to do mathemat ics with relative e a s e , may not have involved the concept of different understandings. 3.5. W h a t is learning to t e a c h ? Learning to teach may be thought of as "a p rocess of learning to understand, deve lop, and use onesel f effectively. T h e teacher 's own personal deve lopment is a central part of teacher preparat ion" (Fe iman-Nemsor , 1990; p. 225) and forms the bas is for his/her personal deve lopment as a teacher. Students entering professional programs other than teaching must def ine their posit ion in that profess ion, but students entering the field of educat ion are enter ing, albeit at a different level, the s a m e s c e n e (the c lassroom) that they have previously exper ienced (Pajares, 1992). They have al ready formed an image of teaching based on their own exper iences in the c lass room and those exper iences and Images will serve a s the filter through which their teacher educat ion is p rocessed (Carter, 1990). Thus , grounded in one 's personal exper iences , one 's own student exper iences and fash ioned through one 's 95 university exper ience, the transformation from student to teacher will involve knowledge of self, knowledge of how students learn, knowledge of subject matter, and knowledge of the system into which they fit (Davis et al, 2000). However, learning is a soc ia l interaction and learning to teach must therefore a lso be thought of as a socia l act ion, intertwined in a socio-cul tural setting in which individuals work together to create shared meanings. Exper ience is an important aspect in learning to teach (Smith, 2002). During the p rocess of learning to teach, there is a need for observat ion of others to b e c o m e famil iar with the task as well as the structure and nature of the learning p rocess (Roth & Tob in , 2002). Th is observat ion of the c lassroom is from a different perspect ive than that which the individual has previously exper ienced, and it is through the p rocess of see ing and exper iencing theories in pract ice, theories that had been presented in abstract, that they will become internalized (Smith, 2002). It is important that preserv ice teachers, in the p rocess of learning to teach, b e c o m e act ive inquirers into their pract ices (Clarke & Er ickson, in press) so that they will deve lop into inquiring teachers who are capab le of support ing their students ' develop ing understanding. 3.5.1. The effect of prior schooling on beliefs and Images of teaching "Learning to teach, like teaching itself, is a t ime when des i res are rehearsed , refashioned and refused. The construct ion of the real , the necessary , and the imaginary are constantly shifting a s student teachers set about to accentuate the identities of their teaching se lves in contexts that are overpopulated with the identities and discurs ive pract ices of others" (Br i tzman, 9 6 1991; p. 220). The ability to experiment and to try something new, however, is affected by the fact that in the process of learning to teach an individual is constrained by various forces including the political context in which he/she works (Britzman, 1991), the textbook used (Stump, 2001), the school climate (Roth & Tobin, 2002), one's own ideas and experiences (Lortie, 1975; Britzman, 1991) as well as one's conflicting position as both a student and a teacher (Roth & Tobin, 2002). As mentioned, Lortie (1975) refers to the years spent as a student in school as the apprenticeship of observation and suggests that these years may be more influential than any teacher-training program. Feiman-Nemser (1990) concurs: "Many people believe that teacher education is a weak intervention incapable of overcoming the powerful influence of teachers' own personal schooling or the impact of experience on the job" ( p. 229). As students, prospective teachers "draw from their subjective experiences constructed from actually being there" (Britzman, 1991; p. 3) in developing their understanding of what teaching means. The habitus that they have developed through their own experience - the practices, perceptions and expectations that they have - form the catapult for what and how they perceive the experience of teaching (Roth & Tobin, 2002). Thus, in developing an Image of what it means to teach or to learn, prospective teachers are greatly influenced by their own experiences in the classroom (Lortie, 1975; Britzman, 1991; Pajares, 1992; Smith 2001; Roth & Tobin, 2002). However, these Images are often superficial and may impose stereotypes on what teaching and learning mean (Britzman, 1991). This 97 may impact the view that the individual has of the content knowledge which, in turn, will affect his/her future practice (Pajares, 1992). Many preservice teachers have a naive idealism in which they assume that they can be taught the right way to teach while others believe that they already know it, being unrealistically optimistically biased in favor of the belief that they themselves hold the attributes most important for successful teaching (Shealy, 1993; Cooney & Shealy, 1997). Still others hold views that are incomplete and possibly dysfunctional. For example, they may perceive the teacher as 'friend' or 'guide' to students (Carter, 1990). Other prospective teachers enter the teacher education program with a view of teaching as a replication of the learner knowledge they hold ( Prestage and Perks, 2001 ) but with the conviction that they will avoid the teaching styles that they found unhelpful or inappropriate when they were students, not realizing that for different learners, those styles may be helpful. As Britzman's (1991) Jack August stated: "[There is a] desire to do something creative, something other than the traditional lecture model so endemic both to the high school classes observed recently and to his own educational biography" (p. 128). These preservice teachers are convinced that they will teach differently than they were taught (Britzman, 1991; Davis et al, 2000; Hart, 2002). Thus, one's past experiences affect one's views, and it is essential that teacher education programs address these issues. 9 8 3.5.2. E f fects of p r e v i o u s mathemat ica l e x p e r i e n c e s o n bel iefs a n d Images of teach ing mathemat ics In the areas of mathematics and mathematics education, the NCTM Principles and Standards (2000) indicate that students' understanding of mathematics and their confidence and willingness to do it are reflections of their own classroom experiences. Those experiences are reflections of their teachers' attitudes and beliefs about teaching and learning and "many teachers' traditional experiences with and orientations to mathematical pedagogy hinder their ability to conceive and enact a kind of practice that centers on mathematical understanding and reasoning and that situates skill in context" (Ball, 1993; p. 162 cited in Stump, 2001; p. 208). In considering the education of prospective high school mathematics teachers, this impacts the experiences and practice of their preservice observations and student teaching experiences. Thus, in mathematics education, with the changing perspective on the nature of mathematics and mathematical understanding, there is a great need to challenge pre-established absolutist beliefs and practices early in the education program so that preservice teachers will be more likely to accept the uncertainties of the subject, uncertainties which they possibly did not experience in their own learning. The willingness to accept uncertainty must be made strong enough so as to be held, even when confronted by others who hold absolutist views on the nature of mathematics (Cooney & Shealy, 1997). In effect, then, presenting preservice mathematics teachers, at the beginning of their teacher education program, with a theory such as the Pirie-Kieren theory which 99 e n c o m p a s s e s the changing understandings of the nature of mathemat ics and the manner in which students grow in understanding may have an impact on the preserv ice teachers ' understanding of teaching and learning mathemat ics. 3.5.3. T h e effect of teacher e d u c a t i o n p r o g r a m s o n bel iefs a n d Images of t e a c h i n g Chang ing beliefs about teaching may be a difficult task s ince, a s ment ioned previously, prospect ive teachers are entering a s c e n e that they have al ready exper ienced and, "(f)or insiders, changing concept ions is taxing and potentially threatening. T h e s e students have commitments to prior bel iefs, and efforts to accommodate new information and adjust existing beliefs can be nearly imposs ib le" (Pajares, 1992; p. 323). Thus , preserv ice teachers often exper ience internal conflicts between how they have been taught and how they think teaching should be enacted (Bri tzman, 1991), between their own bel iefs and va lues and the norms of the traditional setting in which they find themse lves (Hart, 2002), as well as between what they are presented with at the university and the reality of the c lass room in which they are expected to enact the curr iculum in its entirety (Smith, 2002). Even if beliefs are cha l lenged and changed during a preserv ice program, and prospect ive teachers indicate a change of perspect ive on teaching, the strength of their prior bel iefs and the p ressures of the situation may b e such that these beliefs are not enac ted or they may d isso lve when the individual is confronted with a problematic situation in pract ice. They may take much longer to move from intellectual belief to pract ical act ion. Thus , the quest ion remains a s to whether teacher educat ion programs 100 can help prospective teachers develop a philosophy or vision of teacher and teaching and of learner and learning that is different from that which they hold when they enter the program and which is strong enough for them to maintain when they begin teaching (Cooney & Shealy, 1997). The teacher education program must therefore address this issue, as well as the issue of changing perspectives. It must provide the preservice teacher with a view of teaching and learning that is strong enough so that he/she can see the benefits of it and put it into practice. To do so, he/she would have to be able to, not only learn theory, but see it in practice, and have an opportunity to experiment with new perspectives him/herself so as to develop and understanding of the working of the theory. 3.5.4. T h e effects of the p r a c t i c u m exper ience o n beliefs a n d Images of teach ing Da Ponte and Brunheira (2001) consider the student teaching experience as a highly valued activity in the process of learning to teach. They believe that diversity of observations or of objects of observation during the student teaching experience provides a means whereby preservice teachers can be helped to become aware of the complexity that is teaching. Field experience should provide prospective teachers with the opportunity to put their theories into practice (Hart, 2002) and while many schools seem to expect them to want to try new ideas (Smith, 1996) other school organizations may resist their attempts at pedagogical experimentation (Britzman, 1991). Thus, preservice teachers are learning to teach by working within the given constraints and a political situation 101 in which interests, values and contexts are used to maintain the status quo, and the orientation toward power and identity may make it difficult for them to develop a critical voice in order to participate in the negotiation of change. The school, its culture and its climate, have a great influence on the experience of and views expressed by preservice teachers (Roth & Tobin, 2002). During preservice field experience, while sponsor teachers often encourage preservice teachers to try a variety of activities, they do not generally suggest any alternative approaches, but rather focus on ""the Ex's", i.e. explanation, examples and exercises" (Smith, 2001; p.27). Often the sponsor teacher will advise the use of the same method that he/she would use. The student part of the student-teacher necessarily responds to the authoritative/supervisoratory aspects of the program, limiting his/her attempts at trying new teaching techniques. In order to cover the content matter and to meet the expectation and approval of the supervisory teacher, the preservice teacher often finds him/herself slipping back into the manner in which he/she was taught (Britzman, 1991; Davis et al, 2000) or the manner that the sponsor teacher advises (Smith, 2001) as the expectations of the in-school sponsor teacher may have a greater influence on the preservice teacher's enactment of the curriculum than the theories and practices learned in the university environment of the teacher education program. Generally, then, teacher education programs make it difficult for preservice teachers to change in that, in the process of teacher education, there is a pattern of observing and being observed, of evaluating and being evaluated 102 (Roth & Tobin , 2002), a structure which dictates that a prospect ive teacher is part student and part teacher (Bri tzman, 1991). The authoritative/evaluative posit ion of the inservice teacher or university superv isor during preserv ice practice creates a tension between the theoretical view held by, and the practice enacted by the preserv ice teacher. A s a result, he /she may struggle between real izing his/her own ideologies and subjectivit ies and the demands p laced on him/her by practicing teachers in the exist ing institutional sett ings. Caugh t between the official expectat ions (spoken and unspoken), the expectat ions of supervisory personnel , and his/her own des i res , interests, expectat ions and phi losophical v iews on teaching and learning, "[t]he student teacher is confronted with [the di lemma] between tradition and change - ... confronted not only with the traditions assoc ia ted with those of past teachers and those of past and present c lass room lives, but with the personal desire to carve out one 's own style, and make a dif ference in the educat ion of students" (Br i tzman, 1991; p. 19). 3.6. What can teacher education programs do? T h e above d iscuss ion concern ing what is meant by teaching and by learning to teach and what factors impact these, leads one, of necessi ty , to cons ider what teacher educat ion programs can actually do to support and promote changing attitudes, bel iefs and pract ices regarding these. S o m e researchers bel ieve that early field exper ience can positively inf luence the teaching attitude and per formance of preserv ice teacher whereas others bel ieve that it "can foster bad habits and narrow v is ion" (Freiberg and Waxman ,1990 ; p. 626). Thus , one must cons ider the role of the preserv ice educat ion program and 103 how it should be applied if it is to be effective in preparing teachers to teach in the present day classroom. Prior to field experience, preservice teachers need to be adequately prepared in both content and method (da Ponte & Brunheira, 2001). However, there is, at present, a perceived gap between what is taught at the university in teacher education courses and what is required as a teacher (Prestage & Perks, 2001; Ball, 2000; Smith, 1996). Smith (2001) states specifically: "There is not a close match between the teaching required of the student teachers and the activities that had been suggested in college" (p. 37) and Roth and Tobin (2002) indicate: "What they hear in university courses is generally declarative and procedural knowledge about teaching that has a timeless character ... . In their teaching, on the other hand, they have to cope with situations without having time out to reflect on the next move" (p. 5). Thus, the preservice teacher, in his/her practicum, faces two conflicts. The one being that there is a disconnection between the theory he/she learns at the university and the enactment of curriculum that is expected in the school setting, and the other being the disconnection between theory about teaching practice and actions and the reality of the time commitments and response times in the actual classroom. Teacher education courses should help preservice teachers develop pedagogical content knowledge as well as help them develop alternative teaching approaches (Ball, 2000). To do so, the program should help them know about the pupils (cultural influences, economic influences, etc.) and should help them become aware of key educational influences and school expectations. This 104 can be achieved by focusing on both conceptual and procedural knowledge, which allows prospective teachers to think about students' knowledge in different ways and to expand their own views of a topic while considering areas of possible difficulty and misunderstanding (Stump, 2001). "The overarching problem across many examples is that the prevalent conceptualization and organization of teachers' learning tends to fragment practice and leave to individual teachers the challenge of integrating subject matter knowledge and pedagogy in the context of their work" (Ball, 2000; p. 243). This issue must be addressed through re-organization of content and pedagogical knowledge, combining the two so that preservice teachers learn the how as well as the what to teach within their subject specialty. Unfortunately, teacher education often aids in the conditioning and socialization of the past when methods of instruction are taught without the context of subject matter (Britzman, 1991). The perceived gap between university preparation courses and classroom experience could be decreased if there was a re-examining of what content knowledge is needed by the teacher (Ball, 2000). In order to bring out and develop both a deep understanding of the pedagogical issues and the subject matter understanding to help understand ways that students come to understand, Roth and Tobin (2002) point out the importance of what they refer to as 'cogenerative dialogue', dialogue which uses current understanding to describe, identify and articulate pedagogical instances and/or problems students might have been experiencing, as a means of framing options for change which reflect on occurrences and problems students might 105 have been exper iencing with another. They suggest that through dialoguing with others in the field, knowledge increases in both subject matter and pedagogica l matters. Cogenerat ive dialogue, they feel, should be an integral part of preserv ice teacher programs because of the deeper structural understanding of the subject and of teaching that is thus generated. Attention in teacher educat ion programs must be given to how knowledge is constructed as well as to focused-ref lect ion-on/self-regulated-learning-about teaching. "The act of reflecting on bel iefs and behaviors al lows teachers to make connect ions between thoughts and act ions and to recognize, expose and confront contradictions and inconsistencies" (Hart, 2002; p. 6). "Transforming pract ice, then, is hinged to the exerc ise of uncover ing core assumpt ions and webs of belief about what knowledge is (an object? an act ion?), what learning is (acquisi t ion? transformation?), what schoo ls do (inform? enculturate?)" (Davis et al, 2000; p. 41). A requisite of teacher educat ion programs should be to cha l lenge pre-existing beliefs and ideas and to deve lop practit ioners who learn about their students and their teaching through reflective practice involving col lect ive dia logue. T h e S M I P program was des igned s o a s to accompl ish this task. 3.7. Mathematics teacher education programs Most secondary preserv ice mathemat ics teachers choose the area b e c a u s e they were 'good ' at mathemat ics. However , much of the content knowledge that is taught in secondary mathemat ics is not revisited at the university level (Bal l , 2000). Th is m e a n s that secondary mathemat ics preserv ice 106 teachers must call upon their own high school exper iences when deciding how to teach the content. Lerman (1999) suggests that preserv ice courses do not provoke students to confront their preconceived notions of teaching mathemat ics. A s a result, many teachers of mathematics enter the profession applying loose fragments of knowledge g leaned from their own learning exper iences a s to what the art of teaching and the meaning of mathemat ics are (Bal l & McDiarmid , 1990) and do not have at hand even the requisite subject content knowledge to be able to predict student behavior (Ball , 2000). In many c a s e s ... educat ion students' understanding of schoo l mathemat ics lacks the depth, vas tness , and thoroughness required to teach wel l . Understanding coursework, taken as a ser ies of isolated independent courses does little to help students s e e the connec tedness of mathemat ics within the discipl ine or outs ide of it. Students ' mathemat ical knowledge for an undergraduate degree in mathemat ics is not the s a m e a s the mathematical knowledge needed for teaching mathemat ics for understanding (Nicol, 2002; p. 30). Mathemat ics teacher educat ion must provide preserv ice teachers with the opportunity to deve lop and pract ice different representat ional forms and expose them to nontraditional situations to broaden their percept ions on mathemat ical understanding (Stump, 2001) if we expect them to pract ice enl ightened teaching methods of the subject matter. Lack of exposure to a rich meaningful knowledge of how students so lve mathemat ics problems may lead to an inability to make coherent connect ions among different aspects of pedagog ica l knowledge (Carter, 1990). Mathemat ics teacher educat ion must help guide prospect ive teachers in f inding a method whereby they can learn/discover alternative teaching approaches and m e a n s of understanding student understanding. 107 Using student work a s a means of analyz ing and interpreting what the student knows and is learning can help preservice teachers work on learning the content so that they can present it f lexibly and meaningful ly to students (Ball , 2000). C h a n g e in practice is limited for preserv ice teachers who learn mathematical content differently than they learn their mathematical teaching methods (Hart, 2002). Us ing content knowledge as the bas is for learning pedagogica l methods is another way in which teacher educat ion courses are able to support preservice teachers in their understanding of how students learn a s well as to help them develop new methodologies (Hart, 2002). Therefore, an integrated approach to learning to teach , an approach which e n c o m p a s s e s theory and methodology and which helps develop pedagogica l content knowledge, would be a useful approach to teacher educat ion. Thus , in order for the teacher educat ion program to be meaningful for prospect ive mathemat ics teachers, "just as teaching mathemat ics needs fluid and connected knowledge of mathemat ics ( teacher-knowledge) s o too mathemat ics educators need an art iculated, fluid and connected understanding of teaching mathemat ics educat ion - the teacher knowledge of mathemat ics educat ion" (Prestage and Perks , 2001). That is, in order to help prospect ive teachers of mathemat ics deve lop in their role of becoming effective mathemat ics teachers , the teacher-educator must be an effective teacher of teaching how to teach mathemat ics, being cognizant of subject content knowledge, pedagog ica l content knowledge and 'pedagogica l teaching content knowledge' , that knowledge which enab les him/her to help the prospect ive teacher become aware 108 of the need for, and to develop the ability to develop pedagogical content knowledge. Only then will the divide between theory and practice be balanced. 3.8. C o n c l u d i n g remarks o n teacher educat ion The meaning of teaching and learning has undergone many changes over the years. The constructivist and enactivist perspectives of recent times have greatly influenced our understanding of how students come to a greater understanding of subject content matter. This is particularly true in the area of mathematics learning which, once thought to be linear and cumulative, is now perceived as a dynamic growth of building upon previous knowledge which involves the need to revisit and rebuild structures to make them more meaningful to the present situation. This has led to a reform movement in mathematics teaching and a need to re-evaluate teacher education programs. Many of the present preservice teachers have been exposed to this more enlightened/reform perspective, but many have not. If, as Lortie (1975) indicates, beliefs about teaching are developed through years of apprenticeship of observation that occur as a student in a classroom, then we may not be on the road to a more enlightened teaching force. Also, as the most recent mathematical experiences of most preservice teachers has been the lecture methods of their university mathematics courses, this may have had an effect on their ideas of teaching and learning so that they will have to be re-initiated into the more progressive methods. Thus, there is a need in the mathematics teacher education program to consider applying new concepts and theories and to present preservice mathematics teachers with theories and applications that 109 address this issue of re-view of the nature of mathemat ics and of learning mathemat ics. Preserv ice teachers need to be exposed to and exper ience the dynamic of the two changed percept ions if they are to enact them in their future c lass rooms. 110 C h a p t e r 4 R e v i e w of the literature: V i d e o data a n d Portraiture The purpose of this study was to determine whether using the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding as a model enhanced the development of preservice teachers' understanding of the meaning of teaching and learning mathematics. Video was used as the main source of data collection for the first phase of the study. This data was used to paint the portraits of four preservice teachers, Sophia, Lance, Ellie and Wayne, based on their actions and discussions as they progressed through their preservice teacher education program. The portraits, and the data from which they were drawn, were then used to analyze the growth of understanding of teaching and learning mathematics of the four preservice teachers using the modified version of the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding which was specifically modified for this purpose. In this chapter, I present a review of the literature related to the method of data collection, video, and for the initial stage of analysis, the drawing of the portraits. 4.1. A context for v i d e o a s a s o u r c e of data co l lec t ion Originally used in ethnographic studies in anthropology, video recording as a means of data collection has been used in quantitative studies for the purpose of counting incidents that were difficult to enumerate in live observation (Bottoroff, 1994). Its use in qualitative studies has increased greatly in the past I l l few decades , particularly in the sphere of educat ional research, as an aid to understanding c lassroom culture, interactions and the implication of these (Pirie, 1996b). V ideo data has a lso been used in educat ional research in demonstrat ing speci f ic teaching methods, as a means of observing and reflecting on one 's own pract ices to improve teaching, and in analyzing student behaviour a s a m e a n s of determining understanding (Wood, C o b b &Yacke l , 1991; Maher , Mart ino & Dav is , 1994). 4.1.1. Collecting video data Whi le the p resence of a v ideo camera might s e e m intrusive in a c lass room situation, establ ishing a good rapport with the participants and explaining the reason for which the data is being col lected, with assu rances of the confidentiality of the data will help al leviate problems of behaviour distortion (Mar land, 1984). Having the camera present for a period of t ime before the actual data col lect ion begins may a lso help the participants forget about its p resence or at least be less distracted by it (Bottoroff, 1994; Cudmore & Pir ie, 1996). A s wel l , peop le have become more used to being v ideoed (Er ickson, 1992) and thus tend to be only temporari ly affected by the p resence of the camera (Mercer, 1995) with the intrusive effect decreas ing the longer the v ideo camera is present (Towers, 1998). Whi le v ideo is a rich source of data col lect ion, there are obvious limitations p laced on the quality of the data col lected based on the limitations of the recording dev ices , the lighting, or location of the camera (Mar land, 1984). T h e s e can obscure the sound and/or the features of the individuals being v ideoed and 112 in a group situation, it is often difficult to p lace the camera so that one is able to s e e the faces of all those involved. The number of and p lacement of the camera(s) during data col lection as well as the quality of the audio must therefore be cons idered. W o o d , C o b b and Y a c k e l (1991) and Maher , Mart ino and Davis (1996) are among those who have used severa l cameras in their v ideo data col lect ion. Combin ing this data with field notes, students' work, and meet ings with teachers, they felt, a l lowed them to make better select ions of particular incidents for further study. Cudmore & Pir ie (1996) preferred to use a f ixed camera at the back of the room a s they felt that this w a s less intrusive to the c lass room act ion. Us ing one or two fixed cameras on a tripod Pir ie and Kieren (1989, 1991, etc.), in developing their theory of developing mathemat ical understanding used in this study, focused on smal l groups of students working on a speci f ic problem for a short t ime, occas ional ly drawing these students out of c lass and recording them while they worked on a speci f ic mathemat ical task, whi le at other t imes recording them in c lass . In extending the theory deve loped by Pir ie and Kieren, Towers (1998) a lso focused on a smal l group of students, v ideoing them during their regular mathemat ics c l asses for the duration of a unit of work and , to obtain more speci f ic data, v ideoing them working on speci f ic mathemat ical tasks outside the regular c lass room. G o l d m a n - S e g a l (1998) chose an interactive method of v ideoing, tucking her camera under her arm and mingling with the students involved in her 1 1 3 research. She and her camera thus became an integral part of the classroom during the year in which she obtained her video data. Other researchers have found that the zoom capabilities of video data collection allowed for closer observation of individuals without infringing on the individual's personal space (Bottoroff, 1994). This method has been useful in classroom observations in that it allows for focusing on different groups of students at different times without disrupting the class by moving the camera. In considering the quality of the audio aspect of videoing, simply letting the camera pick up the sounds of the classroom in process is a suitable means by which to view the class in action. An external microphone can be used if one wishes to record the conversations of a small, selected group of students. Cudmore and Pirie (1996) chose small clip-on microphones in order to minimize the background noise. Since these microphones were fairly inconspicuous, students tended to forget that they were there, thus lessening their impact on the collection of data. Towers (1998) used a more conspicuous microphone in her study on teacher interventions. This allowed her to record the 'teacher talk' and the general classroom questions while minimizing the distraction of other students' talk, thus allowing her to focus on the students being studied. It did, however, have a negative effect on one of her subjects who withdrew from the study because of self-consciousness. Video recorded data is selective in that a conscious decision is made by someone (usually the videographer) with regards to when, what and on whom to focus, and it is limited by what can be picked up through the camera lens 114 (Bottoroff, 1994; Pir ie, 1996b) and the poss ib le b ias of the interpreter. "Who we are, where we p lace the cameras , even the type of microphone we use governs which data we will gather" (Pirie, 1996; p. 553). E v e n the angle of a shot and the point at which one begins and ends a taping affect the data col lected (Go ldman-S e g a l , 1993). V ideo data is thus data col lected in the eye of the beholder, the videographer, and this p laces limits on the act ions that are recorded (Bottoroff, 1994) creating a b ias, albeit usual ly an unconsc ious one, by the subjective nature of this cho ice of focus (Pirie, 1996b). During a v ideo sess ion s o m e members of the group may move out of range of the camera , limiting the interpretive aspec t of facial express ions and other physical s ign that are important to the data. A l s o , if an individual who is not part of the research enters the field of focus, un less the researcher can obtain his/her permission to use the data, this portion of the v ideo will have to be el iminated. Other limitations are implied in v ideo research when the camera operator is not the primary researcher . If the v ideographer is not the primary researcher, he/she may not shift the emphas is of focus to the speci f ic aspec ts of the situation that are most significant to the research quest ion and this can affect the r ichness of the data col lected (Go ldman-Sega l , 1993). Data col lected by v ideo recording, therefore, al though offering more insights into occur rences than do many other forms of data col lect ion is still incomplete and not without problems. 115 4.1.2. A d v a n t a g e s of v ideo data One of the main advantages of using video data in qualitative research is that it allows for re-viewing the data and re-examining actions in light of past and future actions (Mercer, 1995; Pirie, 1996) which reduces the tendency to make premature interpretations of events (Erickson, 1992). As well, video allows one to both hear and see the individuals being studied, thus providing more detailed information than field notes and anecdotal responses. Many of the signs that we respond to are subtle, and not obvious to us in situ. Body language and facial expressions often reveal information that cannot be captured by means other than video, the physical actions revealing more than the spoken word, often indicating that the spoken word should not be taken 'at face value' (Pirie, 1996b; Bottoroff, 1994). Video is a means by which one can help determine the 'do/say problem': What a person does may in contradiction to what he/she says (Ruhleder and Jordan, 1997). That is, when asked specifically about a problem or a situation, an individual will often state what he/she thinks is the expected or desired response but actions may reveal something quite different (Davis, Sumara & Luce-Kapler, 2000). With video, one is able to re-view the actions to determine if they agree with the statements and to compare the two at a later date. Video also frees one from "the limits of sequential occurrence of events in real time" (Erickson, 1992; p. 209). It allows for re-observations of interactions, and enables the viewer to see actions that he/she did not originally observe. Instantaneous and brief reactions of a teacher to a student, missed in a 'live' 116 observat ion, might be picked up in the re-viewing of a v ideo. With v ideo, one is able to focus on a speci f ic hand movement that, in pass ing , may not have s e e m e d important but on c loser scrutiny may reveal the individual's attitude or feel ings about the situation. "What v ideo tapes can do is give us the facility through which to re-visit the aspect of the c lass room recorded, granting us greater leisure to reflect on c lass room events and pursue the answers we seek" (Pirie, 1996b; p. 553). In the re-view, one may s e e and hear things that were not originally seen or heard but which may have occas ioned an action or reaction on the part of an individual. The re-view al lows for the separat ion of the 'smal ler picture' from the 'bigger picture' and v ise ve rsa , and for concentrat ing on those aspec ts that are important to the research at the t ime. With v ideo, one is ab le to consc ious ly pay c lose attention to speci f ic occur rences or activities in order to make s e n s e of the situation based on both the initial understanding as wel l as what has been g leaned from further, c loser , observat ion. By consider ing an incident from different perspect ives and re-examining the original interpretation, the complexi ty of the interactions may become more apparent and help in understanding earl ier act ions (Bottoroff, 1994), putting them into a new context, and providing the opportunity to obtain confirming or alternative interpretations of the results drawn from the original data (Bottoroff, 1994; G o l d m a n - S e g a l , 1993). V ideo al lows for the placing of the incident into context both visually and orally (Pir ie, 1996b). Re-examin ing v ideo data al lows one to observe the timing and sequence of events and behaviours in order to deve lop rich descr ipt ions and accurate 117 records of the situation without having to prematurely exclude behaviours or make interpretive judgments. "[N]o longer constrained by the sequential occurrences of events in real time, instances of similar recorded events at different points in time can be compared and contrasted easily" (Bottoroff, 1994; p. 246). By comparing later actions to earlier ones, one can determine if individuals are responding significantly differently in similar situations or if the response is simply a different manifestation of the same behaviour, brought on by different stimuli but not showing a change in attitude. "[Bjecause the behavior is analyzed in context, the likelihood of being able to identify possible antecedents and consequences of behavioral or interactive patterns is enhanced" (Bottoroff, 1994; p. 258). This is essential to any research looking at change. With video data, this refinement of interpretation is based on the actions that take place, not just the recollection of the action that took place (Ruhleder & Jordon, 1997). Together with the verbal, the visual can provide a more complete, albeit, a more complex picture of the situation or individual being studied than either can provide individually (Pirie & Kieren 1991). Thus, a major advantage of video data over field notes or audio data is that it allows one to actively listen to and view the data after the fact any number of times (Pirie & Kieren, 1991). Observing video data frees one from the tendency to specifically notice frequently occurring events and allows one to observe for less common, but possibly revealing incidents (Erickson, 1992). 118 4.1.3. A p r o c e s s for ana lyz ing v ideo data Video data is an exceptionally rich source of data and there are techniques which allow for the analysis of the data in an unbiased, orderly manner. The process of analyzing video data is likely to involve viewing the data several times, doing a 'trace' every three to five minutes. A' trace' is a statement of occurrences made, without bias, for that period of time. By doing a trace, the researcher becomes familiar with the data and may be able to temporarily eliminate some that do not appear to relate to the research question and he/she can also flag incidents that appear significant (Pirie, 1996). Major concerns in the interpretation of video data are that the amount of information collected can distract from the research question (Mercer, 1991) and that the plethora of raw data collected renders the analysis of video data time consuming and difficult (Goldman-Segal, 1993). Simply stated, video data cannot be scanned quickly as can field notes and/or transcripts. The researcher may be required to watch hours of video that is of little or no relevance to the study (Pirie, 1996). He/she is faced with the difficult task of determining what to look for in the data. This, naturally, is guided by the research question and is facilitated by the fact that one can re-view the data in its original form (Cudmore & Pirie, 1996). However, in the beginning, anything may be relevant (Pirie, 1996b). In non-video research, due to an immediate decision that has to be made by the researcher in situ, much information is omitted from the data because the researcher has to make an immediate decision with regards to its relevance. This is not the case with video data. The videographer could possibly suggest the 119 elimination of some of the v ideoed data through the use of field notes, indicating its i rrelevance at the t ime, but this contravenes the purpose of using v ideo as a means of data col lect ion. Interpretation of v ideo data thus involves an interactive ana lys is with multiple v iewings and layer ings of data to obtain nuances and subtlet ies of the situation, which may not be observed at the time of v ideoing. El iminating parts of the data before re-viewing it could el iminate important information (Go ldman-Sega l , 1993). Interpreting v ideo data is much more difficult than interpreting text (Pirie, 1996b; Go ldman -Sega l , 1993) and a major concern is that it p roduces an overabundance of data (Hammers ley & Atk inson, 1983). Due to the profusion of data that becomes avai lable, organizing it requires a systemat ic approach to analys is . Wh i le var ious computer programs such as V P r i s m , MacSHAPA®, VideoNoter®, NVivo , Nud*ist and T ransana are avai lable that can ass is t in analys is , and s o m e of them can interface with v ideo and audio editing programs s u c h a s App le ' s F ina l C u t P ro and S o n i c Foundry 's Ac i d link (Thorn, 2002) this does not el iminate the hours of observat ion required. V ideo -based interaction analys is (Ruhleder & Jo rdon , 1997) is a p rocess which al lows for in-depth multidisciplinary ana lys is of v ideo data. In v ideo-based interaction analys is , the "analysts al low the categor ies to emerge out of a deepen ing understanding of the taped part icipants' interactions ... [and] emerg ing patterns of interactions are checked against other s e q u e n c e s of tape, and against other forms of ethnographic observat ions including field notes, interviews, transcripts, documentary materials, etc." (p. 6) to neutral ize b ias whi le 120 generat ing a set of categor ies for exploration. Fol lowing a p rocess similar to F lanagan 's (1954) critical incident technique which al lows one to look at "observable human activity that is sufficiently complete in itself to permit inferences and predict ions to be made" (p. 327), and G lase r and S t rauss 's (1967) constant comparat ive method which al lows for "generating and plausibly suggest ing (but not provisionally testing) many categor ies, propert ies and hypotheses" (p. 104), v ideo-based interaction analys is provides o n e m e a n s to study v ideo data. 4.1.4. Val idity in v i d e o research Eisner (1991) suggests that when one is descr ib ing or interpreting complex acts such as those that occur in the p rocess of teaching and learning in a c lass room, one is not actually trying to present things as they are, but is seek ing to interpret them as s e e n . S u c h subjectivity leads many to quest ion the validity of qualitative research. S ince "[vjideotape records as a constant reference point, a point of departure, permit grounded, well cons idered conc lus ions during interpretation of original field data" (Schaeffer, 1995; p. 256), the subject ive viewing and interpretation of the data by different persons may help estab l ish validity of the data by confirming the original interpretation. O n the other hand, a second observer might v iew the data differently, repudiating the initial interpretation. Alternat ive concerns c a n be d i scussed and initial interpretations can be repudiated or a deeper , thicker interpretation can result. "[Hjaving to articulate and argue for my percept ion helps to crystal l ize them into either useful or inappropriate descr ipt ions" (Pirie, 1996b; p. 556). 121 Whi le thick descript ions (Geertz,1973) can be used "to address the validity in ethnographic descr ipt ions of cultures [whereby it is] the role of the ethnographer to build thick enough descr ipt ions of events, such as the closing of the eyel ids, so that the reader of the descript ion can come c lose to understanding what the gesture might have meant for the person whose act ion is being descr ibed" (Go ldman-Sega l , 1993; p. 262) interpreting video data is affected by and determined by the interpreter "in much the s a m e way that a writer of fiction determines the telling of the story" (Go ldman-Sega l , 1993; p. 262). A n individual's descript ion of occur rences and movements is subjective and the descr ipt ions he/she provides may not instill the s a m e image to another person. Act ions, descr ibed in a transcript, lose tone and demeanor and once an interpretation of an action has been written, the act ion is less likely to be cons idered differently. 4.2. C o n s i d e r i n g c a s e s t u d y a n d portraiture 4.2.1. C a s e s t u d y C a s e studies are a useful m e a n s of developing knowledge of action and reflection on p rocess and provide a particularly effective method for studying educat ional programs (Stake, 1995). Accord ing to S n o w and Anderson (1991), c a s e studies are relative holistic ana lyses of sys tems of act ions that are bounded social ly, spatially, and temporal ly; they are mult i-perspectival, and polyphonic; they tend toward tr iangulation; they al low for the observat ion of behavior over t ime and thus facilitate the processua l analysis of soc ia l life; and they have an open-ended, emergent quality (p. 152). 122 A c a s e study can transform an individual exper ience into a group d iscuss ion so that pract ice in action becomes a focus for reflective pract ice, helping focus attention on speci f ic professional act ions (Shulman, 1998). Th is can provide a language and a procedure by which to general ize from a particular instance for the development of possib le act ions in other situations (Roth & Tob in , 2002). A c a s e study involves circumscribing units of analys is , contextual izing the observat ions, and adding rich details where necessary . It is in "the study of the particularity and complexity of a single c a s e , coming to understand its activity within important c i rcumstances ... [emphasizing] ep i sodes of nuance, the sequential i ty of happenings in context, the who leness of the individual" (Stake, 1995; xi-xii) that g ives strength to c a s e study. A c a s e is c h o s e n , then, not s o much for its generalizabil i ty, but "to maximize what we learn" (Stake, 1995; p. 4). In terms of generalizabil i ty it might be better to think in psycholog ica l terms rather than mathematical probability (Donmoyer, 1990) and to use the results to "expand and enr ich the repertoire of socia l construct ions avai lable to the practit ioners and others" (p. 182). Thus , while a c a s e may be c h o s e n for its typicality, it may a lso be chosen because it appears intrinsically interesting. 4.2.2. Portraiture Portraiture is a method of qualitative research which combines art and sc ience s o as to blur the boundar ies of aesthet ics and empir ic ism in an effort to capture the complexi ty, dynamics , and subtlety of human exper ience and organizat ional life. Portrait ists seek to record and interpret the 123 perspect ives and the exper ience of the people they are studying, document ing their vo ices and their v is ions. ... T h e relat ionship between the two is rich with meaning and resonance and b e c o m e s the arena for navigating the empir ical , aesthet ic, and ethical d imens ions of authentic and compel l ing narrative (Lawrence-Lightfoot and Hoffman Davis , 1997; p. xv). Portraiture is used in an attempt to capture the e s s e n c e of the situation, creating what Lawrence-Lightfoot and Hoffmann Davis (1997) refer to as "life drawings" (P- 4). Resea rch involving portraiture has been conducted in teacher educat ion programs (Broyles, Ku lawiec & Fryl ing, 1988; Koz lesk i , 1999) and within this genre has combined the impress ions of the researcher with the hard, scientif ic approach, attempting to emphas ize the goodness of the situation rather than its def ic iencies (Lawrence-Lightfoot and Hoffmann Davis , 1997). "With its focus on narrative, with its use of metaphor and symbol , [portraiture] s e e k s to seduce the readers into thinking more deeply about i ssues" (Lawrence-Lightfoot & Hoffmann Dav is , 1997; p 10). 4.2.3. Choosing case study and portraiture In qualitative research , the un iqueness of the individual is important (Stake, 1995). The a im of qualitative research is to "d iscover e s s e n c e s and then to reveal those e s s e n c e s with sufficient context, yet not become too mired trying to include everything" (Wolcott, 1990; p. 35). In writing c a s e studies, it is important to emphas i ze the person within the context of t ime and p lace. C a s e study is a suitable approach to take to observe and d i scuss ongoing situations. 124 Resea rche rs have classi f ied portraiture as a c a s e study approach (Merr iam, 1988; Y i n , 1999) and have indicated that, by use of this methodology, there is a blurring of the line between the researcher and the researched. T h e subject matter is more about rare events than commonly occurr ing events (van Mannen , 1988) in that the tale is the tale of an individual 's journey rather than a repl icable f inding. Engl ish (2000) argues that the portraitist cannot find the truth of the situation, and that he/she uses his/her posit ion to control the information that is inc luded. Marb le (1997), however, indicates that portraiture al lows for multiple ways of understanding the data and Lawrence-Lightfoot and Hoffmann Dav is (1997) suggest that the portraitist's approach is an attempt to expla in the situation from an insider point of view. Stor ies and narratives are useful means of providing v icar ious exper iences (Stake, 1995). A s the anthropologist, Clifford Geer tz (1973) stated, a fundamental aspec t of any cultural descript ion is the researcher 's imaginat ion, and the depict ion will bring us to better understand the un-exper ienced situation. The depict ion must pay c lose attention to the reality of the soc ia l and human exper iences , paying attention to all aspec ts of interaction. In this type of research, then, the pictures drawn of the individuals represent interpretations based on observat ion. By providing a thick descr ipt ion as outl ined by Geer tz (1977), of the individuals in act ion, a portrait b e c o m e s embedded into context, with the interpretation of act ions based on understandings g leaned from observat ions over t ime and in a number of situations s o that the portrait painted is a detai led and layered understanding of the portrayed. 125 4.3. Writing of a portrait Kozlesk i (1999) relates portraiture to photography. Accordingly , when using portraiture, like when taking a photograph, the research would result in a snapshot of their work within a speci f ic t imeframe [much like a photograph]. The aperture of the lens, the angle that the photographer shoots from, the time of day, and the avai lable natural lighting are s o m e of the var iables that shape the final image. In the end , any photograph likely omits far more than it reveals but it remains an artifact of what w a s (p. 4). Portraiture, like collecting v ideo data, is thus restricted to the viewing and interpretation that takes p lace and the results can be cons idered as that which has occurred under the research eye of the observer. Individuals' subject ive realities vary accord ing to the context and the events of their exper iences (Lincoln & G u b a , 1985) and thus, in order to obtain a larger v iew of the landscape, one might cons ider using a number of portraits drawn from the s a m e surrounding. "The portraitist c o m e s to the field with an intellectual f ramework and set of guiding quest ions. The framework is usual ly the result of a review of relevant literature, prior exper ience in similar sett ings, and a genera l knowledge of the field of inquiry." (Lawrence-Lightfoot and Hoffmann Davis , 1997; p. 185). Accord ing to the anthropologist Clifford Geer tz (1973), the author will, of necessi ty , present his/her s e n s e of the situation which means that a fundamental aspec t of the descript ion is the researcher 's imagination. Thus , in order to present an authentic portrait, the portraitist must l isten for a story. T h e author must evaluate and juggle different k inds of data to produce coherence through 126 themes which emerge from the data so as to represent the order that exists in what, to the outsider appears as disarray. Interpretation is an important feature of art. The artist is the one who draws attention to particular aspec ts of the portrayed, but it is the v iewer who d o e s the interpretation. S o it is with portraiture. The portraitist is involved in a p rocess of evaluat ing and judging the data in order to determine which are the signif icant features of the themes that emerge and the ones to which he /she wants to draws attention. However , it is the reader who interprets them and puts them into a context that he /she understands. A n intention of portraiture, then, is to capture, from an outsider's view, the insider's understanding of the situation, to capture the e s s e n c e of the situation, so that it makes meaning to the outsider but a l lows him/her to draw his/her own conc lus ions about the veracity of the work. Speci f ical ly, portraiture "makes the researcher 's b iases and exper iences explicit, in e s s e n c e becoming a lens through which the researcher p rocesses and ana lyzes data col lected throughout the study" (Hackmann, 2002, p. 52) keeping in mind the five essent ia l features of vo ice, relationship, context, emergent themes and aesthet ic whole (Lawrence-Lightfoot and Hoffmann Davis , 1997). V o i c e is that personal input through which act ions are interpreted and descr ibed through the eye of the portraitist. Relat ionships of trust must be deve loped so that the portrayed al lows the portraitist to s e e his/her inner feel ings and context must be cons idered s o that the resulting themes can paint a picture which results in genuine representat ion of the portrayed. 127 4.3.1. Rigor in portraiture Tradit ion dictates that researcher b ias should be el iminated or at least limited in research. More recently, especia l ly in qualitative research, it has been acknowledged that the researcher will impact the interpretation of data (Van M a a n e n , 1988; Marshal l and R o s s m a n , 1999) and specif ical ly that portraiture "makes the researcher 's b iases and exper iences explicit, in e s s e n c e becoming a lens through which the researcher p rocesses and ana lyzes data col lected throughout the study" (Hackmann, 2002, p. 52). The argument that r igorous data analys is is not a part of portraiture is countered by Lawrence-Lightfoot (1986) who indicates that in portraiture the researcher must use a number of different p ieces of data collection and, using these, must try to find points of convergence, a form of triangulation. T h e m e s may be revealed through the rituals and normal activities of individuals (Lawrence-Lightfoot and Hoffmann Davis , 1997). 4.3.2. Trust in portraiture A s is necessary when col lect ing v ideo data, in portraiture there is a need for a strong bond of trust and relationship between the researcher and the researched, and for the researcher 's p resence to be accepted a s non-interfering and non-judgmental (Lawrence-Lightfoot & Hoffmann Davis , 1997). T h e "[pjortraitist must a lways be mindful of the se r iousness of their [sic] work on site and the e a s e with which their [sic] act ions can unintentionally be injurious" (p. 167-168) whi le maintaining "the attitude of respectful learners, insiders with the opportunity to perceive themse lves a s experts and teachers" (Lawrence-Lightfoot 128 and Hoffmann Davis, 1997; p.167). It is imperative, then, that in research involving portraiture, rapport and trust exist between the researcher and the portrayed. 4.3.3. Authenticat ion in portraiture "It is the actors who are most vulnerable to whatever distortion they may find in the image reflected in the portraitist's mirror" (Lawrence-Lightfoot and Hof fmann Davis , 1997; p. 172). In writing, the portraitist "must guard the relat ionships that are establ ished throughout the writing of the final portraits" (Lawrence-Lightfoot and Hoffmann Davis , 1997; p. 173). T h e rapport between portraitist and portrayed "is imprinted on the portrayal, embod ied in the carefu lness with which descr ipt ions of individuals are shaped and presented" (Lawrence-Lightfoot and Hoffmann Davis , 1997; p. 176). S i nce the portrait drawn must reflect the fact of the situation, the complexity of the situation must be mainta ined, a s well as the authenticity of the individual 's exper iences (Marshal l & R o s s m a n , 1989). Thus , in organizing data, complexi ty must be retained and each recorded incident must embody the e s s e n c e of the individual. The portrayal must be authentic. Th is authentif ication, it is sugges ted , can be ach ieved through al lowing the portrayed an opportunity for reciprocity, al lowing him/her the opportunity to review his/her portrait and to comment on its integrity, putting constraints around factual error (Lawrence-Lightfoot and Hoffmann Davis , 1997). 129 4.4. Concluding remarks on video data collection, case study and portraiture Portraiture as a means of presenting data offers many advantages when one w ishes to depict situations that are open to interpretation. Just as art depicts the situation as seen by the artist but is interpreted by the beholder, portraits depict the individual as seen by the portraitist, but it is the reader who interprets them. In this study, the development of understanding of teaching and learning of the four preserv ice teachers, Soph ia , Lance , Ell ie and W a y n e is presented through the lens of the researcher 's eyes and this development is thus open to the interpretation of the reader. Chapter 5 Methodology - Outline of Study 130 5.1. The setting A s indicated in Chapter 1, Wubbe ls , Kor thagan and Broekman (1997) bel ieve that prospective teachers need to be exposed to innovative teaching sty les throughout their teacher educat ion program if change in their pract ice is to occur . Smith (1999) states that the use of imagery in teacher educat ion programs may be useful in helping preservice teachers redefine their images and attitudes about what it means to teach. A n d S c h o n (1987) bel ieves that images are important in creating knowledge that can inform new pract ice. P i ecemea l modif icat ions to existing programs are not seen to bring about the necessary change in prospect ive teachers ' beliefs and there is a need to improve the context in which they are p laced if change is to occur (Ross , 1990). In order for a program to be success fu l , a more inclusive approach is needed . In their university c lass room, preserv ice teachers need to be presented with an Image of teaching and learning, an Image that is appl icable to the act ions and activities of that c lass room, and which incorporates their peer d iscuss ions and activities s o that they can s e e how that approach to teaching can be manifested in day-to-day high schoo l c lassroom activities. T h e teacher educat ion program at the University of British Co lumb ia provided an opportunity to test these ideas and the Pi r ie-Kieren Dynamica l 131 Theory for the Growth of Mathemat ica l Understanding provided a suitable model to present to the preserv ice secondary mathematics teachers involved. Students in the Secondary Mathemat ics Integrated Program (SMIP) were enrol led in an intensive one-year program in which their Methods, Pr incip les of Teach ing and Communica t ions courses were integrated and taught by one instructor. Th is provided the opportunity to present and practice with them, an Image of the manner in which students learn mathemat ics, an Image that incorporates the present understanding of the teaching and learning of the subject a s "a whole, dynamic , leveled but non-l inear, t ranscendental ly recursive p rocess" (Pirie & Kieren, 1991a; p.1). Th is Image of learning mathemat ics is implicit in the Pi r ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding (the Pi r ie-Kieren theory) and it was thus cons idered a good Image to present to prospect ive teachers regarding the teaching and learning of mathemat ics. A s this situation provided an excel lent opportunity to study preserv ice teachers ' develop ing understanding of teaching and learning mathemat ics, a Reques t for Ethical Rev iew was completed and permiss ion to carry out the study w a s granted (See Append ix A ) . At the beginning of their teacher educat ion program, prospect ive secondary mathemat ics teachers were presented with the Pi r ie-Kieren theory a s a way to cons ider the manner in which schoo l students come to understand mathemat ics in order to determine if knowledge of and model ing of the theory would e n h a n c e their understanding of the p rocess of teaching and learning mathemat ics. The theory w a s revisi ted, specif ical ly, twice during the preserv ice 132 in-c lass sess ions during the fall term of the teacher educat ion program as well a s being used throughout the term as a lens through which to d i scuss the teaching and learning that was taking p lace by the prospect ive secondary mathemat ics teachers, by schoo l students observed in v ideos and by the high schoo l students in the preservice teachers ' short pract icum exper ience. The purpose of the study was to examine the growth of understanding of teaching and learning of preservice secondary mathemat ics teachers , consider ing this growth through the lens of the modif ied Pi r ie-Kieren theory, a s they progressed through their intensive one-year teacher educat ion program. The speci f ic quest ions that it addressed were stated in Chapter 1. 5.2. Framework under which the study was be carried out The Pir ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding is a theory deve loped by Dr. T h o m a s Kieren and Dr. S u s a n Pir ie to d i scuss the manner in which students develop in their understanding of a mathemat ical concept. A s d i scussed in Chapter 2, the theory is a theory for the growth of mathematical understanding, not a theory of mathemat ical understanding. A s such , it is not a theory that can be used in assessmen t , but rather, it can be used to d iscuss the deve lopment of mean ing making by an individual student about an individual concept . Fol lowing is a descript ion of the manner in which the theory w a s used in the S M I P program. The proposed course outline is provided in Append ix B. 133 5.2.1. Presentation of the Pirie-Kieren theory For the purposes of this study, the Pir ie-Kieren theory w a s presented as a mode l to descr ibe and d iscuss the manner in which students develop mathematical understanding. A s such , it provided a language with which preserv ice teachers could d iscuss their own understanding of the development of mathematical understanding as well as providing them with a way of thinking about the development of mathematical understanding. Specif ical ly, the theory w a s used in the following manner: 1. Prospect ive teachers enter a teacher educat ion program with bel iefs and Images of what it means to teach and of the manner in which students learn. During the first two days of the S M I P program, the prospect ive secondary mathemat ics teachers were presented, by Dr. S u s a n Pir ie, with the Pir ie-Kieren theory as a poss ib le alternative method to think about and d i scuss the manner in which students come to a greater understanding of mathemat ics. During this presentat ion, the students were provided with the model of the theory, with examp les of work depict ing the different levels, with v ideos of students working s o that they could d i scuss their working with reference to the theory, as wel l a s being given opportunit ies to work at mathemat ical activities and d iscuss their own understanding of the working with reference to the theory. A s s u c h , they were given a short, intensive course on the theory. 2. C h a n g e in behaviour is not random, but condit ions can be des igned s o a s to ass is t obtaining the desi red result. Model ing of desi red behaviours is 134 one method that appears to have a posit ive effect (Bandura, 1997; Hart, 2002). However, what is being modeled is often not apparent to those to whom it is being demonstrated and it is often necessary to specif ical ly address the model ing s o that the benefits can be more not iceable (Hanney, Smel tzer Erb & R o s s , 2001). Thus , the Pir ie-Kieren theory was modeled in the teaching of the S M I P program s o that the preservice teachers would be better ab le to s e e how it could be used as a means of consider ing student understanding. Preserv ice teachers were asked to cons ider their own mathemat ical learning exper iences and to d i scuss their impact on their learning opportunit ies. Regu lar references to and re-visitations to aspec ts of the theory in c lass room d iscuss ions were made. For example , the preserv ice teachers observed v ideos of students working on mathemat ical tasks and were then given the opportunity to d iscuss the understanding that they observed . Twice during the term, guest speakers were invited to d iscuss and reinforce the tenets of the theory. Dr. J o Towers presented s o m e of her f indings to the c lass in mid-October, and Dr. S u s a n Pir ie returned in la te-November to further d i scuss aspec ts of the theory. T h e s e presentat ions were used to reinforce the preservice teachers ' understanding of the embedded ideas. In order to teach a concept effectively, one must cons ider what background knowledge is required and what knowledge it is a s s u m e d that the students have about the topic. (Schon , 1987; Wubbe ls et al, 1997). Through the lens of the Pi r ie-Kieren theory, one could express this a s 135 specif ical ly access ing and assess ing the Primitive Knowing of the students, and in so doing, possibly Folding Back to one 's own learning exper ience. During the program, preservice teachers were therefore encouraged to Fold Back to their own learning exper iences and understandings and to reflect upon their assumpt ions about student understanding while planning for and d iscuss ing their l essons for presentation in the S M I P c lass and during their pract icum. That is, they were asked to cons ider the learning of mathemat ics as a recursive p rocess by returning to inner layers of their own understanding to inform their understanding of outer layers to help in planning meaningful lessons. Speci f ical ly, they were given mathemat ical tasks to work on and were asked to d i scuss their own s tages or levels of understanding with reference to the theory. Thus , as pract ice is cons idered important in learning new behaviours (Wubbels et al, 1997), the preserv ice teachers were encouraged to use the Pi r ie-Kieren theory and the language which it provided in their university c lass room thinking and d iscuss ions and during their preserv ice pract icum. T h e Pi r ie-Kieren Dynamica l Theory for the Growth of Mathemat ica l Understanding thus provided both a theoretical model for the prospect ive teachers to use in thinking about and d iscuss ing the teaching and learning of mathemat ics a s wel l as a practical model around which to plan lessons and to d i scuss the p rocesses . 136 5.2.2. Data collection Video w a s the main form of data for this study, but the data a lso included quest ionnaires, field notes, e-mail m e s s a g e s and c lass ass ignments . Data was col lected throughout the fall term of the S M I P program as well as throughout the pract icum exper ience of the four students who were se lected for further study. O n the first day of their teacher educat ion program, prospect ive candidates for this study received an explanat ion of the purpose of the study. It w a s expla ined that those who did not wish to participate would not be v ideoed and that their data would not be used in the study. Demograph ic information w a s obta ined, and all part icipants, including the instructor were given the chance to choose a pseudonym. Other than Dr. S u s a n Pir ie and Dr. J o Towers , people who appear in the d iscuss ion and who did not provide pseudonyms were given fictitious names . T h e p lacement schoo ls of the four preserv ice teachers fol lowed in the pract icum have a lso been given fictitious names . 5.2.3. Initial questionnaire Preserv ice teachers ' past exper iences form the bas ic knowledge upon which they draw in thinking about teaching and learning (Pajares, 1992; Kreber, 2002; Roth & Tob in , 2002). In the language of the Pi r ie-Kieren theory, this forms their Primitive Knowing, their Images, and even , possibly, s o m e Formal is ings of what it means to teach and to learn mathemat ics. A s the instructor of the S M I P program wanted to obtain an initial v iew of the prospect ive secondary mathemat ics teachers ' attitudes and beliefs about teaching and learning the subject, and a s it w a s important that I have an 137 unbiased pre-view of their beliefs and attitudes, the S M I P students were asked to complete an initial quest ionnaire des igned to obtain this information. T h e first portion was a survey (See Appendix C : Images Quest ionnaire) consist ing of thirty statements which were rated on a four-point Likkert-l ike sca le to indicate how strongly the prospect ive teachers agree or d isagree with the statements. Mode led on a survey used by Hart (2002) to determine preserv ice elementary teachers ' attitudes and beliefs about mathemat ics teaching and learning, the quest ions were modif ied to suit the purposes of this study. T h e S M I P students were asked to complete this quest ionnaire as quickly a s poss ib le so as to obtain their 'gut' react ions or unbiased v iews with regard to the statements. They were then given ample time to complete and elaborate on eight sen tences (See Append ix C : V iews Quest ionnaire) concerning their v iews about teaching in genera l and how teaching other subjects is the s a m e as or different from teaching mathemat ics. 5.2.4. Outline of video data collection Whi le the quest ionnaires were des igned to obtain an initial overv iew of the stated images and beliefs held by the prospect ive teachers , there w a s need to obtain their lived beliefs and understandings. Wha t is important to people will be revealed in their day-to-day conversat ions and interactions and this can be determined by watching and listening to them with as few pre-concept ions a s poss ib le (Richer, 1975). Thus , to best capture the honest react ions of the S M I P students in their natural environment and to determine their true bel iefs about teaching and learning mathemat ics, v ideoing their day-to-day d i scuss ions and 138 interactions as unobtrusively as poss ib le w a s an appropriate means of data col lect ion. V ideo data al lows one to put the stated and enacted understandings of the people into the context of present, past and future act ions and statements and enab les one to review any relevant incidents any number of t imes in as authentic of a setting as possib le, confirming or altering an interpretation in light of later data (Pirie & Kieren, 1991a). For the above reasons, v ideo was chosen as a means of data col lect ion for this study. A single, semi-f ixed v ideo camera was p laced on a tripod at the back of the room. It was smal l and light enough s o that it could be moved around the periphery of the room to get a better angle when videoing smal l groups. In this manner, I w a s able to v ideo the c lass as a whole, an individual presenter or a se lected smal l group. The microphone on the camera w a s of high quality and w a s used a s audio for large groups. For smal l group d iscuss ions a smal l directed microphone which cut out background noise w a s used . Of the twenty-two students in the S M I P c lass , nineteen initially agreed to participate by signing the Informed Concep t Fo rms (See Append ix E) . Of the three remaining students, one indicated that she did not think that she 'fit' the criterion as she was from another country and w a s recertifying a s a teacher s o that she could teach in C a n a d a . S h e b e c a m e a willing participant and s igned the form when I expla ined that I thought she might add an interesting perspect ive to the d iscuss ions as she al ready had c lass room exper ience. O n e week into the study, one of the other two indicated that the camera did not affect him as much a s he had thought it would , and therefore he would like to be an integral part of 139 the study. W h e n he real ized that he was the only one not being v ideoed, the remaining student approached me, indicating that he was a 'nervous' person but that he did not mind being videotaped providing he was not one of the people chosen for in-depth study. He s igned a form to this effect (See Append ix F). I was therefore able to v ideo all of the students in the S M I P program in any combinat ion, an important issue as , at the beginning of the research, I did not know who would be fol lowed during the pract icum situation. Thus , while it may be argued that the p resence of a v ideo camera is likely to have an effect on individual's react ions (Cudmore & Pir ie, 1996), in this c a s e , the p resence of the camera over t ime minimized its effect on the c lass (Mar land, 1984). A l so , the intrusion of the v ideo camera w a s also dec reased in that, a s v ideographer, I became an observant participant (Towers, 1998a), capturing not only the S M I P students, but mysel f on v ideo as I w a s interacting with them and the instructor, participating in activities and helping with presentat ions. A s one of the students indicated: "It's just like having another person here, watching but not talking." 5.2.5. General data collection A s ment ioned, Dr. S u s a n Pir ie presented and d i scussed the tenets of the Pi r ie-Kieren Dynamica l Theory for the Growth of Mathemat ical Understanding dur ing the first two days of the S M I P program. Fol lowing the initial presentat ion of the Pi r ie-Kieren theory, the next three w e e k s of c l asses involved the preserv ice teachers in d iscuss ions of their understandings and beliefs about mathemat ics as a subject, about teaching in genera l , and specif ical ly about the teaching and 140 learning of mathemat ics. Al l d iscuss ions were v ideotaped. The camera was at t imes focused on the c lass as a whole, on smal l groups, or on individuals, which ever s e e m e d appropriate at the t ime for the purposes of obtaining the best data. A s it was not known at this t ime which students would be fol lowed during their pract icum, an effort was made to obtain data on as many, and as equal ly as possib le, all S M I P students and to have them working and d iscuss ing in different combinat ions s o a s to obtain a comprehens ive overv iew of the act ions and beliefs of all of them and of their interactions with different groupings. Th is w a s made poss ib le s ince , as ment ioned, all the preserv ice teachers had agreed to be v ideoed A second set of v ideo data w a s obtained during the second and third weeks of c l asses when the S M I P students were v ideoed in sel f -selected groups of two to three outside c l ass t ime. T h e s e v ideo sess ions were arranged at a t ime convenient for the students, and during the sess ions they d i scussed their understanding of the nature of the teaching and learning of mathemat ics, and specif ical ly their understanding of the Pi r ie-Kieren theory as it was presented to them in this context. The purpose of these d iscuss ions w a s to obtain, in a more relaxed atmosphere, the preserv ice teachers ' stated beliefs and understandings and their opinions and understandings of the Pir ie-Kieren theory and to determine if there w a s a connect ion between the two. Throughout the Fal l term, there were many informal meet ings with the S M I P students - s o m e s imple conversat ions in the hal lways walk ing to c lass , s o m e organ ized over coffee, and others at p lanned socia l events. Whi le these 141 were not v ideotaped, any informative statements made by students were recorded as soon after the event as possib le in the form of field notes. 5.2.6. Practicum data collection "A student cannot at first understand what he [sic] needs to learn, and can learn it only by beginning to do what he [sic] does not yet understand" (Schon, 1987; p. 93). For the purpose of this study, it was felt that it w a s appropriate, if not essent ia l , to follow students during their pract icum a s it is in doing that one often reveals one 's true understanding and beliefs. By observ ing the students in in situ, a more realistic or honest v iew of their actual beliefs about teaching and learning mathemat ics would be obtained. It was physical ly impossib le to fol low all twenty-one of the S M I P students who agreed to be a part of the study during the pract icum. A sample of four students w a s thus selected for more in-depth study. The select ion, al though somewhat random, w a s a lso based on conven ience . They would e a c h be observed severa l t imes during the pract icum, thus students who were p laced outs ide the Lower Main land were el iminated. B a s e d on their initial quest ionnaires and the observat ions during the fall term, students with somewhat different profi les and stated beliefs were then se lec ted. That is, in consider ing poss ib le se lect ions, I looked at their stated beliefs and exper iences and se lected s o m e students who indicated that they held quite traditional v iews of the teaching and learning of mathemat ics and s o m e who s e e m e d more progressive. I a lso cons idered their background exper iences and chose s o m e who had 'outside' exper ience and others who had fol lowed through the standard educat ional 142 sys tem. A s a result, it w a s possib le to follow two male and two female S M I P students who were p laced in different soc ioeconomic a reas of the city. The gender ba lance and different p lacement a reas , it w a s felt, would al low for a comprehens ive view of the S M I P students' pract icum exper ience. Necessa ry permiss ion from the Schoo l District, the principals of the schoo ls involved, as wel l a s the sponsor teachers was obtained (See Append i ces G , H, and I respectively). A min imum of four c l asses taught by e a c h of the four se lec ted S M I P preservice teachers was v ideoed. 5.2.7. Reflection videos Since "[t]he act of reflecting on beliefs and behaviors al lows teachers to make connect ions between their thoughts and act ions and to recognize, expose , and confront contradict ions and inconsistencies" (Hart, 2002 ; p. 6), it w a s important to have the preserv ice teachers reflect on their own teaching. They were therefore involved in two further v ideo sess ions . The first of these took p lace as soon as poss ib le after the c lass that w a s v ideoed. The sponsor teacher w a s cons idered important in these d iscuss ions because he/she has a signif icant impact on the attitudes and teaching behaviours of the student teachers and how they enact the curr iculum (Koster et al, 1998). Therefore, the sponsor teacher w a s invited to attend these sess ions and be part of the d iscuss ion . The second post -c lassroom video sess ion w a s a retrospective viewing which occurred after the pract icum exper ience w a s comple ted. A retrospective viewing involved the S M I P student observ ing and comment ing on one of the v ideos of him/herself teaching. Retrospective viewings differed from v ideo-143 stimulated recal l s e s s i o n s (Bloom, 1953; Mar land, 1984) a s the intention w a s not to try to determine what the individual w a s thinking at the time, but rather, what he/she observed in retrospect, and to d iscuss this in light of other exper iences. In this case , the S M I P students were to d iscuss their observat ions through the lens of the Pi r ie-Kieren theory. T h e s e sess ions al lowed the preservice teacher to cons ider his/her interactions with the students in the c lass and further consider how he/she enacted the teaching process and to d i scuss if these were in agreement with his/her stated beliefs about mathemat ics teaching and learning. 5.2.8. Debriefing A s w a s p lanned at the beginning of the program, a sess ion was arranged in which the students in the S M I P program could meet as a whole after their pract icum. A s such an opportunity was not provided in the teacher educat ion program, it was arranged outside c lass time, one week after their pract icum ended . Twenty of the twenty-two S M I P students were able to attend. Th is s e s s i o n w a s v ideoed, and while they did share their exper iences , it was difficult to obtain speci f ic information on any one student. T h e S M I P students tended to use it more as a soc ia l activity, building on and maintaining their s e n s e of unity and support. 5.2.9. Field notes and class assignments A s ment ioned earlier, v ideo was the major source of data col lect ion. However , field notes were a lso used and c lass ass ignments were col lected throughout the study. Reca l l that, as v ideographer, I w a s interacting with the c lass , and therefore, the field notes were not detai led, but were used to draw 144 attention to speci f ic act ions or activities that I thought might be important and which took p lace out of the focus of the camera . I a lso used field notes to record interactions that occurred out of c lass and which I thought might have impact on the data col lected during the study. For example, after the socia l even ings and coffee sess ions that I at tended, I would often make brief notes concern ing students ' statements. Individual preservice teachers were contacted when I felt that further clarification was needed and emai l m e s s a g e s within the S M I P list-serve through which we communicated on a regular bas is , especia l ly during the pract icum were a lso used . Al l notes were dated and identified as to where and how the information w a s obtained so that it could be fitted into the correct p lace in data analys is . 5.2.10. Discussion of the collection of the data A s outl ined above, the col lection of data for this study provided a comprehens ive col lect ion of the preserv ice teachers ' activities and as such revealed much about their changing understanding about teaching and learning mathemat ics. By col lect ing data in this manner, it w a s poss ib le to capture the exper iences of the prospect ive teachers in var ious forms and to be able to isolate those activities which depicted their Images, attitudes and beliefs and their develop ing understandings of the teaching and learning of mathemat ics. Th is provided the necessary information to answer the research quest ions. 5.3. An outline of the data analysis Data analys is involving v ideotapes is a t ime-consuming p rocess , and this study w a s no except ion. Whi le , during the col lection of the data, I 145 reviewed v ideos on a random basis , I left the detai led reviews until after all data was col lected. I chose to do this as I would then have an overview of the entire development and would be viewing the data with greater knowledge of the act ions in context. By the time I had col lected all the data, I had more than two hundred forty hours of v ideo. I began by viewing the tapes chronological ly to re-famil iarize myself with the initial data. S ince at this point, I knew which preserv ice teachers I had fol lowed through the pract icum, a s I v iewed the v ideos I kept that in mind. After viewing approximately forty hours of v ideotapes, I randomly se lected other v ideotapes from the fall term and v iewed approximately forty more hours of v ideo. At that point, I felt that I w a s ready to proceed with the formal analys is as I w a s immersed in the data. B e c a u s e of the data I had obtained and the information I hoped to extract from it, I chose not to use any of the avai lable programs des igned to assist in v ideo analys is as I felt that they constr icted my intentions and purposes. Instead, I set up my computer bes ide my v ideo monitor and engaged in post-field-note-taking. A s in a 'trace', a s descr ibed earl ier, I noted the t ime and activity at regular intervals. However , my post-field-notes were more directed to the interactions that took p lace, in that I typed notes related to my observat ion, making spec ia l note of when any of the four individuals chosen for further observat ion were involved. Th is viewing w a s done chronological ly and I w a s able to identify a number of v ideos that s e e m e d unrelated to my purpose as wel l as identify many that s e e m e d specif ical ly signif icant. A l though I use the term 'chronological ly ' 146 here, I do not mean that I did not go back and re-view previous v ideos in light of new information. That is, if, for example , on v ideo #93 I w i tnessed an act ion that reminded me of a previous incident, I would go through my notes, locate the v ideo that it reminded me of, and re-view that v ideo to determine if there w a s in fact a connect ion or a contradiction to the previous observat ion. In this manner, I was constantly Folding Back to my previous interpretation, refining and redefining them as necessary . During the viewing and post-field-note taking, I would rewind and review v ideos s o that I could obtain verbat im, statements of preserv ice teachers wh ich s e e m e d relevant at the t ime as well a s making note of physical react ions and to re-observe act ions that s e e m e d to be instances of understanding of teaching and learning. S i nce the validity or trustworthiness of v ideo data can be conf i rmed and/or den ied through shar ing interpretations with others, a samp le of my v ideo data w a s shared with Dr. S u s a n Pir ie for d iscuss ion and confirmation or repudiation of interpretation. After having v iewed and re-viewed all the in-c lass sess ions , I v iewed and re-viewed the v ideos of the four preserv ice teachers in their pract icum sett ings. T h e s e I v iewed chronological ly, one student at a t ime, again noting verbat im, the interactions that appeared significant and making notes as to their interactions with the c lass and sponsor teacher. By the t ime I began to write my portraits, I had approximately three hundred and twenty-five pages of notes. Ana lys i s thus began with the initial impress ions made during the observat ions in c lass , and cont inued with the v iewings and re-viewings of the 147 videos, and with making of post-field-notes. Since I had used video as the primary means of data collection for the first part of the study, I had been able to observe the case in two ways - first during the' actual proceedings, and then in hindsight, observing the videos later. This enabled me to put actions in the context of past and future occurrences. Throughout the study, I was neither teacher nor learner, but a participant in both, as well as being an outsider, observing the situation from afar so that as I collected data and analyzed it I was able to live through the experiences of the preservice teachers both in reality and vicariously. In the viewing of the video data, I had to observe for themes that developed in the understanding of teaching and learning mathematics for the preservice teachers and to observe how they portrayed this understanding in their presentations to students in class, both at the university and in their practicum. As I wrote my observations of the videos and chose quotes to use to portray the individual's understanding, I had to review previous data to determine more explicitly what had transpired and the context in which the quotes were made. Was this a 'one-timer' or did it truly capture the individual's understanding? 5.4. The drawing of the portraits A portrait must provide readers with "good raw material for their own generalizing" (Stake, 1995, p. 102). In this case, this meant that I had to portray each of four individuals, Sophia, Lance, Ellie and Wayne, as they developed their understanding of the meaning of teaching and learning mathematics, as I 148 perceived it through the lens of the modified Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding. By observing the preservice teachers in a number of different situations, keeping the modified theory in mind, but leaving the interpretation of actions open and ongoing, I was able to obtain an extensive overview of each individual under consideration and his/her reactions to different situations. The video data and resulting notes along with all other data gathered through the fall term formed the 'paint' with which I would portray each of the four preservice teachers. A portrait can lead one to assume the sitter to be of calm and intelligent disposition, although this is not stated in any one feature of the picture, but is surmised from the whole portrait. As such, my intention was not to simply describe the preservice teachers as I observed them, but to truly 'paint a portrait' of each. The portrait would, of course, be my interpretation of the individual, just as was, for example is Whistler's portrait of his mother. And, just as a viewer interprets Whistler's portrait to form his/her own image of the lady, the reader would interpret my portraits to form their own images of the preservice teachers. Thus, as the portraitist in this case, the portraits would be my interpretation of each individual after having listened to them and empathized with them, and their experiences would be interpreted through my own experiences and understandings as a teacher and researcher. As Marshall and Rossman (1989) suggest, this part of the analysis was challenging in that I had to try to maintain the authenticity of each individual's experiences and 149 understandings while I was interpreting them through my own. Not only was it important to organize and classify the incidents that I was selecting, it was essential that the complexity of each be maintained and that each incident presented in the portrait embodied the essence of the individual's understanding. In construction of the portrait, I had to search for patterns that developed in each person's profile and choose quotes that authentically represented their development. At the same time I had to watch for "the deviant voice", incidents that seem to contradict the norm (Lawrence-Lightfoot & Hofmann Davis, 1997) for that individual and determine how they fit into the whole. Even the manner in which the preservice teachers expressed themselves when describing their classes was significant in determining their understanding of teaching and learning and the behaviour they expected of their students as these provided their metaphors for understanding and learning. By finding threads of connection between themes, I was able to present each person as an individual who reacted to the environment in a specific manner and who interpreted outcomes in a particular manner. Having had experience as a preservice teacher (albeit many years previous), as a teacher, as a sponsor teacher and as an instructor of preservice teachers, and having done an extensive review of the relevant literature of teacher education and teachers' attitudes and beliefs, I was able to consider the data collected in this study from the perspective of the relevant literature, my personal experiences and general knowledge of the area, all of which were part of my "anticipatory schema" (Lawrence-Lightfoot and Hoffmann Davis, 1997; p. 1 5 0 186). From the perspective of the Pirie-Kieren theory which was used in this study, these formed part of my Primitive Knowing. Thus, I entered the field of research with a framework and plan, while expecting that both my intellectual agenda and my methods of obtaining data may have to be adapted to the situation. Also, as I chose to portray more than one preservice teacher, I would be allowing my interpretation of their actions to be seen through different lenses which, in essence, would help authenticate the portraits. In painting the portrait of each individual, consideration had to be given to his/her positioning in the whole. The depiction of the situation had to pay close attention to the reality of the social and human experiences, and to all aspects of interaction. Thus, by initially choosing to video not only the individuals who would be followed through during their practicum, but rather by videoing the entire SMIP class, I was considering not only "the voices and experiences of the range of actors of focal concern but also the perspectives and actions of other relevant groups of actors and the interaction among them" (Snow & Anderson, 1991; p. 154). By observing the preservice teachers in the larger class, and not pin-pointing specific students from the beginning, I was able to gain a more thorough understanding of the individuals' attitudes and beliefs without interference of guidelines that limited their expression. In portraiture, an important aspect is trust. In this study, there were several layers of trust demonstrated, and evidence of willingness to share experiences. Through the building of relationships with the subjects, it was possible to learn more about their conceptual and emotional understandings of students' ways of 151 learning - empathy, trust - s o a s to understand the other (the preserv ice teacher) and to s e e him/her in light of my own exper iences. The first ev idence of trust was that all S M I P students al lowed me to v ideo them and were not restricted in their act ions in c lass while being v ideoed. At severa l points, they even 'p layed the . part' of the instructor and myself, knowing that all would be taken in good s tead. The instructor and I were a lso invited to the many socia l gatherings organized by the S M I P students. For those who would be fol lowed into the student teaching c lass room, the bond of trust needed to be strong enough for the them to feel at e a s e in al lowing me to v ideo them during their teaching sess ions , a t ime when they would be practicing newly learned techniques and when errors might occur and mis takes might be made. It w a s thus important that they not perceive me as judgmental a s this would have diminished the l ikelihood of shar ing of negative exper iences (Lawrence-Lightfoot and Hoffmann Davis , 1997). T h e trust and wi l l ingness of the four individuals to have me in their c l asses w a s demonstrated by the fact that they took the initiative to ask their sponsor teachers to al low me to v ideo their c l asses and that e a c h one arranged for me to meet with the sponsor teachers and the principals of the schoo l in which they were p laced to obtain the necessary approval . E a c h of the preservice teachers indicated a wi l l ingness to share negative exper iences with me. For example , one of the preserv ice teachers, W a y n e , on one occas ion , indicated that his students "were apprehens ive" and that it might be best that I not be there the next day. H e sa id that his interactions with them at that precise time were not as "healthy" a s 152 they should be and he needed time for their relationship to become smooth again. It was essent ia l that I honor his interpretation of the situation so as not to lose his trust or to have my p resence interfere with the natural behaviour of the students, even though the observat ion might have provided s o m e useful data to determine what it was that made W a y n e feel this way. Similarly, when Lance indicated that his c lass was uneasy due to the fact that one girl "injured herse l f , it w a s important that I not ask too many quest ions about the situation and that I wait until he felt that the incident w a s under control and he wanted to share the exper ience with me, which he did later in conf idence. Ell ie d i scussed her concerns over her sponsor teacher thinking her presentat ions were too theoret ical and Soph ia chatted about how p leased she w a s with her own approach to teaching the solving of equat ions and her unease about the teaching that she had wi tnessed, knowing that I would not reveal her statements to her sponsor teachers. T h e s e examp les indicate that there w a s a bond of trust deve loped in which we could share exper iences - the posit ive, the negative and those of concern. After consider ing the data col lected in this study and the e s s e n c e of portraiture, I wrote my portraits. E a c h portrait begins with a statement made by the individual, a statement which depicts his/her understanding of the process of teaching and learning at a particular s tage in his/her development. The individual 's personal history, his/her own Primitive Knowing, if you like, is provided for background s o as to create a hol ism in each portrait which al lows for the analys is of each within the context of the c a s e , taking into considerat ion the 153 viewer's own observations and experiences, that is, my own Primitive Knowings and Images. This is in agreement with recent acknowledgements that, especially in qualitative research, the researcher's bias can, and likely should impact the interpretation of data (van Maanen, 1988; Marshall and Rossman, 1999). Portraitists often present individuals as omnipotent and as if they can see things that others cannot. The portraits presented here, however, are written using a collage of the individual's own words extracted from class discussions or written assignments. They therefore capture the essence of the individual in a style that represents his/her mode of expression in real, day-to-day activities. Since the portrait is written in the words of the portrayed, he/she should appear to the reader as a real person, not as an individual who is 'bigger than life' or as one who has been 'airbrushed'. He/she should appear as an actual person who is having day-to-day experiences. In drawing the portraits of the four preservice teachers, I drew on my interpretation of the situation, but I wanted to ascertain that the portrayed felt that I had captured their images. To verify this, I gave each participant an opportunity for reciprocity. After the initial study, while writing up the data, I kept in e-mail contact with and met with the participants on several occasions, both individually and in groups. We discussed, not only the study, but also our present lives and we shared information and ideas on teaching and learning. After the writing of the portraits, I gave each of the portrayed the opportunity to review his/her own portrait and to comment on its integrity, thus allowing them the opportunity to authenticate my Image and to note if there were factual errors. Putting 154 constraints around factual errors gave them the impression that the portrait was indeed a finished product, even if at times there were aspects of the portrait that they would have preferred not be made public (Lawrence-Lightfoot and Hofmann Davis, 1997). I wanted them to be able to say "Yes, that is me." even if it was not how they would like to portray themselves. All four accepted the portrait as an authentic Image, and as a result, no changes were necessary. The final portraits of Sophia, Lance, Ellie and Wayne are presented in Chapter 6 as data for the next stage of analysis, the discussion of their growth of understanding of teaching and learning mathematics, which is presented in Chapter 7. The analysis of the growth of understanding of teaching and learning mathematics of each individual is presented in the same order as the portraits are presented so that the reader is easily able to follow one individual through, or may read the portraits as a unit so as to feel the interaction of the individuals and obtain a sense of the SMIP class. 5.5. Analysis of the portraits Analysis of the portraits, like analysis of the video data, began long before sitting down to do the analysis. It began before the construction of the portraits or the collection of the video data, activities during which I was actively involved in Image Making. Since the Pirie-Kieren theory was being used in this study in several ways, a thorough understanding of it was essential, and thus, developing this understanding was the beginning of the analysis. After having immersed myself in the theory, the next stage, involving it, was to re-define the terms of the Pirie-Kieren Dynamical Theory for the Growth of Mathematical Understanding to 155 fit the new context of growth of understanding of teaching and learning mathemat ics. In order to do this, I had to consider the meaning of mathemat ical understanding and its similarit ies and/or di f ferences to the meaning of understanding understanding. That is, I had to think about the relationship between what it means to learn something onesel f and what it m e a n s to understand the p rocess of developing understanding. Th is , then, had to be bifurcated to the two separate a reas of developing understanding of the meaning of teaching and developing understanding of the meaning of learning. Beginning at the beginning, s o to speak, it w a s fairly e a s y to def ine Primit ive Knowing a s this is the information one brings with them, except that information directly related to the topic at hand. However , s ince everyone has had exper iences with learning, it was important to think in terms of developing an understanding of the learning p rocess , or developing an understanding of how one learns, and to keep this distinct from learning (or simply develop ing understanding). The p rocess of developing further definitions w a s somewhat more complex. Develop ing understanding of mathemat ics is quite different from develop ing an understanding of how one learns mathemat ics, and quite different from developing an understanding of how one teaches mathemat ics. Thus , in develop ing the definit ions for the different levels of understanding, it w a s first necessary to determine if indeed these levels did exist and if so , to think of exemplars for them. A s a practicing teacher, I had to Fold Back to my own learning of how to teach and my own learning of how I had deve loped a n 156 understanding of how students learn. Using this and the Pirie-Kieren theory as my lens, the process began with examining my own understanding of the original definitions and re-formatting them to fit the new contexts. Indeed, it appeared, in so doing that the different levels existed. However, after several trial definitions and thinking of examples that fit, I realized that at the outer levels of developing an understanding of teaching and learning, the two became amalgamated so that at the Observing level, Structuring level and Inventising level one had to be able to rationalize the relationship between teaching and learning. In order to accommodate this, a dual model following the tenets of the Pirie-Kieren model was designed (See Figure 7 in Chapter 2) and which can be used to map the growth of understanding of both teaching and learning mathematics. During the process of developing definitions and thinking about exemplars that fit the new context of developing understanding, I was in regular contact with Dr. Susan Pirie to ensure that the integrity of the original definitions was maintained. Once I thought that the definitions were completed and exemplars determined, it was time to analyze the portraits. It became apparent that I had over-simplified some of the definitions and that some activities that I would have defined as Formalisings were really Property Noticings, Recording. The Formalising levels for both growth of understanding of teaching and learning were redefined, as was the Observing level. New exemplars were determined for the Inventising level. Thus, throughout the process of developing definitions, my thinking about them was following a path of the to-and-fro movement of a 157 mapping on the Pirie-Kieren model. I was using my own understanding to develop Images of what I thought was involved at that level. I then used these definitions to analyze the developing understanding of teaching and learning of the SMIP students, and when I met with an obstacle, I had to Fold Back and re-examine my Images, reconciling them (Property Noticing) into a more meaningful understanding. This, in my mind, re-confirmed that the model could be used to discuss the growth of understanding in the new context. Once the definitions were complete, the final analysis of the portraits was begun. Again, the process was not linear. I first had to determine which statements and actions of the preservice teachers seemed to indicate a particular level of understanding according to my new definitions. Classifying the incidents onto the appropriate level took great concentration, and it soon became apparent that I wanted, too quickly, to assume that the preservice teacher had developed beyond the level that he/she had. I wanted him/her to show development since the aim of the study had been to determine growth of understanding, and the purpose of the program had been to give them a foundation upon which to build understanding. However, I was trying to force onto the individual's profile, an understanding that was not there. Thus, often if he/she was at the Reviewing level of Image Making, or if an individual made an emphatic statement or repeated a statement, I would classify it as Formalising. I then had to go back and re-classify when I realized that he/she had not had engaged in Property Noticing. I had to be reminded that I was tracking the growth of what actually happened, not what I hoped had happened: "Tracking his growth is about what 1 5 8 actually happened. If I plant a seed in the ground and take infinite care to water it, warm it, protect it, etc. and a mole c o m e s along and digs it up (as they do here), then however good my intentions, however little it was my fault, I cannot say the seed grew to be a plant" (Susan Pir ie, personal communicat ion, 2006). I a lso had to be reminded that, at this stage of their development, most preserv ice teachers would likely be involved, as much , if not more, at the Image Mak ing level than at the Property Noticing level and that this was good because it meant they were open to new ideas. I therefore had to 'back off' and be less emotional ly involved. T h e distinction had to be made between making a statement (Property Noticing - Recording) and Formal is ing, when the individual actually enacted and d i scussed the understanding. Thus , I had to reconsider and re-classify severa l incidents, realizing that enactment was necessary before the preserv ice teacher could be sa id to have c rossed a Don't N e e d Boundary. Another d i lemma in classi f icat ion occurred when an individual stated that he /she bel ieved one thing but enacted another. A t face value, the statement would be taken as an Image, but the later act ion enacted a different Image. Thus , the statement was likely an Image Mak ing - Rev iewing statement and had not been e n c o m p a s s e d into the individual's repertoire of Images. In these situations, it w a s only after consider ing the later Images and act ings that I w a s able to properly c lassi fy the incident. A s a result, in the p rocess of identifying levels on the theory it w a s necessary to identify incidents, c lassi fy them at the time, and then, possibly, re-classify once they were p laced into context. 159 The final dec is ion with respect to the ana lys is w a s on how to best d isplay the developing understanding as a form of Mapping s o that the movement between levels was apparent. The modif ied, duel model would be used as a summary of the results, but it could not be used to indicate all the connect ions of approximately ninety incidents for each preservice teacher. After consider ing severa l different models and trying a number of methods, I chose a vertical d isplay of the incidents, charting the individual 's growth through the levels of the modif ied Pir ie-Kieren theory using numbers provided in the text of the analys is . In this way, it was possib le to v isual ize the growth of understanding of teaching and learning at the s a m e time, and to see the connect ions between them. This form of d isplay provided a good visual representat ion of the developing understanding of the individual. The results of this analys is are presented in Chapter 7. 160 C h a p t e r 6 Presentat ion of the Portraits 6.1. Introduction In this chapter I present the portraits of the four preservice teachers, Soph ia , Lance , Ell ie and W a y n e , whom I fol lowed through their pract icum to observe their growth of understanding of teaching and learning mathemat ics. T h e purpose of these detai led portraits is to give the reader a 'picture' of the four students as they passed through the S M I P course. It is hoped that by forming s u c h a picture, the reader will be in a better posit ion to understand the data and follow and agree with the analys is performed in Chapter 7. For the analys is , it is essent ia l that the students be seen as individuals, as real people in a real situation and I felt that merely quoting snippets from the v ideo data would not have suppl ied enough information and background for the reader to be conv inced of the analys is . The analys is of the individuals will be presented in the s a m e order as the portraits. Thus , the reader can follow one individual by reading a sect ion of Chapter 6 fol lowed by the corresponding sect ion of Chapter 7. However , reading the four portraits first will enhance the understanding of the group dynamics of the S M I P program and will better situate each individual in the group. A s ment ioned, the portraits have been written as much as poss ib le in the words of the individual. In each portrait, italics print was used to give background information or when I interpreted the individual's reaction. B o l d print was used when the individual emphas i zed a word or phrase. 161 6.2. A portrait of S o p h i a It is the first day back in class after the short practicum. Preservice teachers are excitedly talking about their experiences: The experience was enjoyable. But, it was also found to be a real 'eye-opener". They would have to work on questioning, pacing, being authoritative. Sponsor teachers were great, students were great, things were happening. And suddenly: Sophia: It was horrible. I attended six classes and was bo red out of my sku l l in every one. Lome (instructor): What do you mean? Sophia: It was really - REALLY - boring - nothing - teacher was trying to - mayhem - nothing - like nothing happened. Trinity (student): What do you mean, nothing happened? Sophia: Nothing! Nobody learned a thing! Val: How could nothing happen? No instruction? Like turn to chapter 2 ? Sophia: "You're grade 8 - doing fractions since grade 4 — so you know how to do this." So he gave them an assignment - kids don't work - some a m a z i n g behaviour so he just walked away - not one spot of math for the whole time -Originally from a small city in the interior of the province, immediately after high school, Sophia entered the Faculty of Science to study engineering. After only two years, she realized that this was not what she wanted to do so went to Ghana for six months on a cultural studies program. Returning to university, she switched focus and finished a degree in mathematics. After that, she spent two years in Peru as a volunteer working in a rural community development project where she had a chance to teach English and practice her favorite activity, mountaineering. "I'm an idealistic girl and I want to make the world a better place 162 through educat ion. To tell you the truth, I'd rather be teaching outdoor educat ion than mathemat ics". However, the enthusiasm she displays for mathematics seems to belie this statement. ******************** "I don't think a teacher can understand the confusion or difficulties in a student 's head by evaluat ing written work - or heaven forbid a multiple cho ice e x a m - one needs to s e e a student work through the problem, hear their 'umm's ' to know where they are having logic problems etc. A l so by encouraging understanding of problem solv ing, and not just procedures. There are a few students for whom mathemat ics is fabulous — just c o m e s naturally and placing a textbook in their hand would be more than sufficient. But for others — probably the majority — they need to be taught more clearly — more precisely — and it's the teacher 's responsibi l i ty to open up their hor izons with her pass ion and energy." ******************** A sense of comradery develops in the SMIP class and Sophia is quickly identified as someone who not only has good ideas, but also as someone who is not afraid to present provocative ideas. Discussions in which Sophia is involved are always lively and she freely expresses her thoughts while, at the same time, being willing to listen to others. To her, human nature is complex, and so is learning mathematics, so when Evan, a fellow student, suggests that learning mathematics is a linear process, she jumps in: 163 "Mathemat ics has a natural learning progression. — But it's from tangent to tangent, not book chapter to book chapter. Sure , there is stability in mathemat ics. Truths you learn today will be true tomorrow, but your understanding is a lways expanding and changing and going back. A s e n s e of continuity is important, and math can't be presented as a bunch of little units outl ined in the textbook. I think it's really important that a teacher understands where the students are coming from - like their Primit ive Knowing , wasn' t it? -and that she l istens to the kids in order to understand what they are say ing. E a c h learns differently and — you can't just teach them all the s a m e . I really want to learn how to reach all k inds of learners — to find ways to chal lenge the top of the c lass without leaving the bottom behind — as wel l as v ice versa . Teach ing , I think, is like a guided tour - fabulous — like mountaineer ing. The guide knows genera l direct ions and knows the terrain and overal l picture s o — s o — wel l , she can detour depending on the needs of the tourists and then the tour is different e a c h time depending on the tourists' interests and abilit ies to dea l with the terrain. Y e a h , I think that that's what teaching is l ike. I like that - mountaineer ing. That 's a good simile. Fabu lous . A n d anyway, learning — I think it has to be a cont inuous process and the teacher must be a cont inual learner of - wel l , of both mathemat ics and of how to teach it. I mean — you even have to look at it differently depending where you're teaching. I remember that when I w a s teaching in Pe ru , the students had lots of problems with the concepts of how to enquire about quantity in Eng l ish . It w a s completely foreign to them to use different express ions for, I mean like t ime — how long — or quantity — how 164 many/how much — or maybe distance — how far — 'cuz in Span i sh all would be expressed using the word 'Cuanto. ' A n d so , if I were trying to define the nature of mathemat ics in Span i sh , I would simply say that Mathemat ics is what happens whenever you try to answer the quest ion 'Cuan to? ' ... To me this is the bas ic nature of mathemat ics, and many of the other ways we def ine mathemat ics are simply descr iptors of this bas ic nature. I would compare it to my bas ic nature as a human being. There are many ways to descr ibe me - tall or short, old or young, hairy or bald, introverted or extroverted, etc. - but these are only descr iptors of my quali t ies and do not themse lves sum up my nature of being human." ******************** Sophia is impressed by a video the class observed. Alwyn, the teacher of a low-level mathematics class introduces algebra using boxes rather than x's or some other variable. He begins his lesson with a fairly complex question, not the usual kind where the student can just look at it and tell the answer. " Y o u know, that makes s e n s e — to do it that way — you know, what the teacher did there. The boxes really show that something goes there. It w a s fabulous the way the kids just dug in and got going on the problems. Understanding is important and maybe — juuuust maybe — those boxes might help students make s e n s e out of it - they can maybe s e e that something goes Into that box — the answer — it's not just s o m e abstract entity — it actually fits in there. Fabu lous ! I think I might — yeah, I will definitely try that in my teaching. I think it w a s good. " 165 When the preservice teachers are given a problem to work on, they are asked to work in groups and to think about and discuss how they are solving the problem. "Hey, I just Formal ised there! W o w ! I never thought of that before. Fabu lous ! I didn't real ize that even we went through these s teps in the theory when we worked on a problem. I think it is neat, just fabulous, to d i scuss the levels that we are working at - use the terminology and d iscuss . — see ing how w e go through problem solving and d iscuss ing it with peers — will --- will — wel l -- will help understanding what our k ids will go through and how to push them to get beyond the level at which they are working. ... I think that labeling a lways serves a purpose and what I like about the Pi r ie-Kieren theory is that you can look at a problem and you can say that I went to Structuring or whatever and you could s e e what level you progressed to whereas you just sa id that w a s a good problem - that's good , but it doesn' t define the limitations of the problem or the benefits of the problem s o I think that if we d i scuss the problems in terms of this and how we can push the students to conceptual ize further or less or what level will most of the students get to and it puts a nice exac tness to the problem that you can then put on other problems. O h — this is how I can make my teaching better - it's more of a retro-scale of your teaching the problem of how your teaching goes — that w a s a nice exper ience, a good problem but you don't really — but if you don't use the label, def ine it more discretely — I think it is more nebulous about how good your teaching is - how good their learning is". 166 In discussing homework, many of Sophia's classmates indicate that they think students are given too much. Students should be able to decide how much they need to do, but Sophia does not totally agree. "Students don't learn just by being shown how to do a quest ion or two. They do have to practice. They don't have to do like 20 quest ions, maybe, but they — wel l , to understand, they — practice g ives kids the chance to break through a barrier so that the work can become — more automatic — at a higher level. They might be able to do the quest ions by following an examp le - but being able to do is only a part of understanding. Somet imes , maybe the doing can lead to the understanding — Like going through the knowledge boundary. By doing, you can maybe suddenly c o m e to understand — like move up a level in math, that's supposed to be like — its like spel l ing — not supposed to have to think about every step — you should — you have to just do as a functional p rocess ... if you have to think of each one as a cognit ive p rocess there is just too much and they get los t " . ******************** Each student in the SMIP program is required to make several presentations to the class. Sophia's lessons are always interactive and involve social contexts to which high school students can identify and with which they can engage: linears are introduced using mountaineering, a class to which she brings her rope and compass, and for which she fabricates an enticing story; in a lesson on areas and perimeters, Harry Potter is 'chained to a building'; and, manipulatives are a standard part of her presentations. 167 "I think it is really important that students feel engaged with their work. I think they have to have fun with it — and they have to understand. Somet imes it's hard for them, but the teacher has to encourage — tell them they are doing fabulously — she has to have them make connect ions s o m e way or other - and it's not really too hard for the kids to get the connect ions if they come up with the connect ions. The teacher should be helping them maneuver through math. Like when Trinity gave her problem on probability — often not a really interesting thing — but she made it about A I D S with real numbers and got us to guess at the answer before we started. Boy, was I off — but then she — wel l , it w a s just fabulous the way she made us do all the work - and figure it out — ourselves. We' l l remember that. Fabu lous ! " ******************** A positive attitude and the use of encouragement will go a long way in helping students learn, from Sophia's perspective. "Not that it will make everything perfect — it's okay to make mistakes. Y o u can learn from them if you don't get all g loomy — But, the teacher must give students s o m e chal lenging work to do. W h e n I w a s in schoo l , well — M y own grade 7 teacher as really strong - encouraged us to learn more. But, in grade 8 and 9, it w a s boring - so boring because those teachers didn't push . O r maybe it w a s the curr iculum, I don't know, but I remember — Finally — in grade 10 we learned something new — s o m e trigonometry — and it was 'L ike — Wahoo! Fabulous! Finally we learned something new. Pra ise be to G o d ! ' 168 "Al l students should be given the opportunity to do the best they can . A n d , I think they all can do s o m e things well . They just need to be encouraged -develop their own skil ls - and think about what they a l ready know. A major goa l of mathemat ics educat ion should be to get students to be independent thinkers and problem solvers. More hands on — Manipulat ives — give the phys ica l and work with them until they can s e e where it c o m e s from — the standard, r igorous form - maybe all don't need it — the rigorous Formal isat ions — un less they are going into sc ience or math, but I think you are doing a great d isserv ice to your kids if you don't prepare them to the level that they can get out there and that they are able — I mean , they'll be good problem solvers and they will l ikely be able to so lve the problem — but I think you have to give them the tools to be able to read the language that everybody e lse is writing in and to be able to exp ress their thoughts in those ways . Not all need to be able to do the abstract stuff — the typical standard form. For them, it is the brain impulses that are important — but give the weaker student a chance to physically f igure things out while — make it less boring as kids who are strong — they might be ab le to pick up other ideas in the hands-on approach. Y o u know, sort of like we did when w e worked with the Alget i le - we figured out s o m e stuff we hadn't thought of before --- maybe the better kids would too. "Real ly , you shouldn' t judge students by their mathemat ical ability, but rather — wel l , you should a s s u m e that they can do it - and let them know that it is their responsibil i ty to do so . " 169 Sophia is not so naive as to think that all students will reach the same level, and admits that not all students will love and understand mathematics. "There are s o m e who just want to learn the process and get out of there -but even these should be encouraged to understand and maybe they will c o m e to a better feeling about the subject. A l l students should be given the opportunity to deve lop their own way of thinking. The teacher should be one with the students - see ing how they are learning it — and she really has to encourage them." With this positive attitude and 'fabulous' outlook, Sophia entered her short practicum. ******************** Sophia's experience during her short practicum is encapsulated at the beginning of her portrait. ******************** "The attitude of the schoo l st inks! Nobody should be made to feel that they can't do good — the teacher needs — the students are fabulous and they need to know that. H ow can they learn if the teacher - if they're not expected to? It sucks ! It was horrible the way students got p laced in s o m e c lasses . They were put in an Essent ia ls c lass just because there w a s no room in the Pr inciples c lass there w a s this one girl - in Essent ia ls - and she was a fabulous little thing -from G u a t e m a l a - talked to her in Span i sh and she really knew all the math — 170 could do all sorts of things — her parents, engineers — don't speak much Engl ish and don't know - scared to go and push. It w a s terrible!" Sophia was not impressed with the laissez faire attitude of many of the teachers she met, nor with the school's policy of assigning second language students to lower level mathematics classes based on their lack of English skills or simply because the other classes were full. The experience made Sophia think about how she will teach while on her "long practicum" at John's High. "My Father once said - after he retired and had time to think — that if schoo ls focused on teaching kids how to think, boy, educat ion would be amaz ing ! That fairly s u m s up the academic s ide of my goa ls as a teacher. But — one really has to think about the soc ia l aspect a lso. I hope, well — that my c lass room will represent — a mic rocosm of the world I'd like to live in — where everyone is respected and va lued. Schoo l is a p lace to learn about the wor ld. Peop le of all sorts are a part of society, and so students with except ional i t ies shou ld be included in c l asses . I d isagree with you, W a y n e — Peop le have to live in society with everyone and so they may a s well learn it in the c lass room — soc ia l benefits outweigh the negative - potential negative implications on learning. Y e s , you have to cons ider the majority, and yes , people do spend the majority of their t ime with people of similar interests — but you have to cons ider the minority, not just the majority, at all t imes, and respect people for their d i f ferences. Just because it — well benefi ts the majority doesn' t mean you can ignore the minority - for example , people with fetal a lcohol syndrome — they aren't go ing to find a group of people with similar disabil i t ies to interact with — 171 they have to be part of the society — we have to augment them in the regular c lass room. But, then, a lso, somet imes we don't address the problems of the gifted. The educat ion sys tem, however, often focuses too much on the lower end and doesn' t do enough for the gifted because they will m a n a g e anyway. I think we have to think about this — equity — chal lenging — spend t ime on making quest ions that are d iverse enough s o that people in the lower end won't feel right out of it but people at the upper end are chal lenged but if you could theoretically c o m e up with extension quest ions where people at the upper end could be cha l lenged, there is no reason why you couldn't have people of completely var ied abilit ies in one c lass room. Equity - and a part of that is technology. — It is a fundamental for the information and communicat ion age we live in and there are so many fabulous ways it can be used in a math c lass . Jus t imagine how — if used properly — it can help enhance and deepen the students' understanding of certain mathematical concepts . For many, just giving them exper ience of using technology with e a s e is useful — helps create 'a zone of equality' — help limit the inequality between the haves and the have nots". ******************** After various presentations in class where the SMIP students have been presenting on different topics, they talk about the methods they used and ones that they think might be used in the classroom to help students understand. " P e e r people, peers expla in ing, a s long a s the peer understands is so much more effective than the teacher explaining s o that I would probably do it in groups and then just like — c a n you s e e a faster way to do it? - just ask 172 quest ions — and somebody is going to come up with a more efficient way — even if its not your way ... like I'll bet there are four or f ive people who c o m e up with a more efficient way and then get them to teach the people nearest them and then you'l l have everyone progressing at a c loser to uniform rate than if you let everyone work on their own and you have someone progressing and helping everyone e lse get done." After some work with algetiles, Sophia is concerned the switching from one representation to another might be confusing for students. "Whatever method the students are using, they need t ime to develop. They need to notice this, this and this, and they notice in s o many different ways , ' cuz it's all new to them and s o the teacher needs to be aware of all sorts of different possib le connect ions. S h e shouldn't outline everything too much — you can't show the — you can't lead them through everything because then the students will not be able to develop the understanding that will be necessary when new topics are introduced. They' l l just have the teacher 's understanding and when they try — wel l , they have to cal l upon their own previous understanding." ******************** Looking at the world as a whole, and taking a pragmatic perspective, Sophia thinks that tests are useful but the design of them needs to be considered: "Tests are a part of life and kids will have to take tests of different forms throughout their l ives. Not all will be paper and penci l — answer this quest ion 173 and that — but they will be tested — one way or another — so why not prepare them when they are working in a — well — safe envi ronment? But — most tests given in schoo ls don't really evaluate a student 's understanding. Open-ended quest ions — ones with no right answer. — then you have to rely on how well the student can argue and you can better determine the p rocess they go through — that's what 's important. A n d , during assessment , it is important to cons ider the k inds of mistakes students make: S o m e are just care less , and these should be pointed out — but if they don't affect the understanding, students shou ld not be penal ized too greatly for them. But, if something is something — completely wrong — wel l , the teacher needs to be concerned and try to determine what epistemological obstacle is there and help them fix it. Multiple cho ice tests. We l l , they shouldn't be completely d iscarded — but to be useful , you really have to know the kinds of mis takes the kids will make and why and go over all this with them otherwise they will be use less . Don't want them to do what L a n c e did — just substitute back in." ******************** John's High is not designated as an inner-city school in which case it would receive extra funding, but it has many of the characteristics of such a school. There is a large number of Special Needs students, some in special classes, but integrated into regular classrooms for some of their courses. There are also many students who are not designated as special needs, but who need extra help. Many are first or second generation immigrants and poverty is a problem. In some families the children have to won\ to help make ends meet. 174 However, there are also families in which there is a stay-at-home mom, there are relatively few families on welfare, and there is a relatively low proportion of single parent families. Because of cultural traditions and financial necessities, parental involvement in school activities is low. While academic performance at the school is poor, the school prides itself on its athletics. This is a concern for some teachers and the school has recently introduced a mini-school program for the more able students. There is, however, a significantly large and vocal group of teachers who express the opinion: "With what we've got to work with, what can you expect?" A few days into her long practicum, one of Sophia's sponsor teachers, the one she had previously complained about, has put her in charge of the class. Although no longer meeting regularly at the university, the preservice teachers keep in email contact with each other. Sophia expresses dismay at the situation, again, but delights with the way her class was responding. Wel l , i thought instead of being the last ones to write this t ime, I'd start it off. S o hope everyone 's pract ica are going spendidly Hey - not fabulously!). I've had the eventful t ime a s a lways, one of my sponsor teachers had to rush back to Eng land for a family emergency, and s o there's been a different T O C every day now. and the other, wel l . . . . Let me just say that i w a s once again ranting to someone about the teaching i'm wi tnessing. ... but my own lessons today went great. L o m e (FA) c a m e to my second one. I did that Cubey -Doob ie -Doo thingy we learned, (looking for patterns in the cube structure, translating it into equat ions) and the kids W E R E E N G A G E D ! Al though lots of them gave up quickly, but s o m e were really problem solv ing, good emot ion in their eyes , and hardly anyone threw the b locks at e a c h other, and lots of them built guns and tanks ... got to love it. ... it w a s lovely, (email communique) 175 Sophia is pleased that she will be able to introduce algebra to her class. She models the unit on the lesson that she observed in the Methods class as she thought she would. She doesn't copy Alwyn's lesson, but rather adapts it to suit her personality and her students - a relatively small class of 17 boys and 8 girls which has been described as 'rowdy'. She begins the first class of the unit by writing on the chalkboard: • + D 5 = n + n + n + i 2 . "What number do you think goes in the box?" The students don't know. " G u e s s - guess any number." Someone guesses 52. "Fabu lous , now, plug it in and work it out. ... D o e s that work? ... N o ? ... How do you know? ... Okay , then what do we need a bigger or a smal ler number? ... Smal le r? How do you know? Okay . Wha t would you choose now? ... W h y ? " Students are not afraid to answer. They seem to feel comfortable giving their opinion because even if it is wrong, Sophia doesn't say it's wrong. She lets them work with their number until they see that it 'doesn't work'. Sophia gives them another question on the chalkboard and they work on it in groups, discussing the process and their results. She hands out a worksheet. Within minutes, the classroom buzzes with sound. Students are working in small groups. Hands are shooting up faster than Sophia can get around to them. Her identification badge, hooked on the waistband of her slacks, she makes no attempt to hurry, acknowledging by eye contact or a little wave that she will come as soon as she can. Meanwhile: 176 "Could you try working it out? Ge t help from s o m e o n e ? " It is hard to believe that Sophia has taught this class for only three days. After a few examples, Sophia tries to introduce variables, but many students resist. She does not force them to switch. She lets them work with the boxes because they feel comfortable with them and they seem to understand what they are doing. The next day she hands out another worksheet. "It was too hard to do all those boxes with the computer, so I just used the letter B. But, if you want - when you write it in your book, if you want a box, you can use it". Over the next few days, most students do change from using the boxes to using variables in their equations, many using the 'B' that Sophia used as her first example. They continue to refer to it as 'the box' and talk about the answer as 'what fits in it'. " Y o u know, learning mathemat ics is a p rocess of thinking — not a step by step p rocess — you can't just fol low rules, you have to think — and verify and interpret — always check ing and justifying. B e sure to substitute your answer back in to s e e if it works," Sophia says to the class. ******************** There is respect in Sophia's classroom - respect and laughter. "Today you can choose who you want to work with. But, think. Don't necessar i ly choose your best buddy. Th is is for marks and you will want to have s o m e o n e in the group who knows how to do the work." The class laughs. In the after class discussion, Sophia states: 177 "Students need to be reminded that everything they do is important and marks are a lways important to students, so this is a good way of reminding them - you know, when I told then it was for marks — that everything they do is important and will affect their future. They — they will a lways be marked — its not graded - like outside schoo l , I mean — so I think they should know it is a lways important to do your best." Sophia expresses her respect for her students in many ways. She spends very little time explaining and doing questions - she trusts their ability to learn. At the front of the class, she listens as students explain their methods. They come and do the work on the board and then discuss it. "It doesn' t matter if it isn't done correctly" she tells the class. "Errors s imply provide an opportunity for learning to take p lace - a chance to deve lop new understandings and create better/more accurate images. Y o u can't learn without making mis takes. No one is perfect". Even when she is reprimanding someone for inappropriate behaviour, Sophia shows respect for the student. Often it is just silence - the reprimand -silence with a smile - a little lop-sided - accompanied by a slight tilt of the head and a raised eyebrow. Or sometimes, it is just a reminder: Soph ia : C h a n , did you just throw that e raser? C h a n : Y e s , ma 'am. Soph ia : Do we throw erasers in this c l a s s ? C h a n : No, m a ' - wel l , yes , ma 'am I did but I shoudn' t ' a . Sorry. However, at times, discretions must be brought to the fore: " J a s o n , you're busted. You ' re s o busted " Sophia retains her ever-present smile, but her sparkling eyes speak volumes as she walks over to Jason and puts out her hand 178 so that he will give her his Discman. There aren't many rules in Sophia's class, but Jason has been caught breaking one of them and he is called on it. Since the students helped in drawing up the few rules that exist, Jason knows he is in the wrong and he does not feel demeaned or put down. Nor do any of the other students make fun of him. Not saying anything more about the indiscretion, Sophia's eyes and tone of voice tell Jason that she expects better from him as she reminds him that he will have a detention, a fairly strict form of discipline for her to give. But, even on this Sophia puts a positive spin: "You can get extra help" she says, "and if anyone e lse wants help, wel l , I'll be here visiting with J a s o n and you are we lcome to join us." ******************** 'Received wisdom' in North America often regards teaching as a 'stand and deliver1 process whereby each class begins with a five minute review followed by ten minutes to go over students' problems with their homework. Ten to twenty minutes are then spent on presenting new work, interspersed with time for questioning and for students to try progressively harder questions. On the surface, then, it seems that Sophia seldom teaches. She may put a question or two on the chalkboard. She then walks about the class, answering questions with questions. "The students can do the work. They are capab le . N o doubt they will have prob lems, but that is when you have to ask them what their problem is and in explaining it, they somet imes figure it out themselves. Or, they should talk to their 179 fr iends and d iscuss . They just have to think about what they know and they learn s o much from each other - P e e r teaching is fabulous. It s a v e s my voice!" ******************** Sophia continues to move about the class, returning to a group of four boys at the back for the fourth time. Two of them are quite 'good' at mathematics, she has indicated, but the other two have 'low skills'. This time they have stopped working. Soph ia : Wha t are you do ing? C h a n : Nothing. W e can't do this quest ion. Soph ia : Try using different numbers - eas ier ones . A l i : O h , yeah ! Learning was taking place! ******************** " E a c h c lass is a surpr ise" says Sophia. "It never goes as expected and that is what keeps me going - the exci tement of developing deeper understanding, of quest ioning and probing for more knowledge." ******************** After viewing herself on videotape, Sophia reflects on her teaching and recalls various incidents through the practicum. "They really s e e m to have — really s e e m to have gotten the equat ions stuff. I l iked the conceptual izat ion I s a w when the kids were doing it. Instead of x being s o m e unknown weight — It w a s fabulous. We l l , they s e e m e d to s e e the box as s o m e number and they got the clue that it had to be the s a m e e a c h time. I thought the box thing was great to have them work with numbers because their other teacher had them use calculators s o much I found that their mental numeric 180 abilities were so - so very poor — so I thought with this - with lots of calculat ions as wel l , it helps them. There 's a lot — and just the concept that this box is not a weight — it's a number and it's always the same and I think it's fabulous the way — it really took off — like it was a puzz le and something to f ind. Peop le understand something hidden - puzz les - but with x it's a lgebra and a lot of kids have this mental thing even before they start ' cuz they've heard about it. S o the boxes — they didn't even real ize, wel l s o m e of them did — sort of like your situation analys is you told us about instead of cal l ing it problem solv ing. Anyway , when we made the transit ion, they — even the weak ones — they will have a good mental v isual izat ion of what x m e a n s and that sort of thing — build their Images, sort of and they won't just forget it or try to fol low rules. I think it was a fabulous way to try the - to teach equat ions. "There are s o m e — wel l , they just don' t want to be there, s o you have to make it like they want to c o m e — J a d e , wel l , he sort of knows what 's going on , but S teven — wel l — s o a lot of the stuff is trying to let them have fun in a socia l context and hopefully there is s o m e math go ing on at the s a m e t ime - if they work individually its just chaos — at least this way - group work — they may pick up something from their neighbors and get the concept but really a lot of it is their own Image in this sett ing. P e e r teaching is great and they can learn so much. I remember at one t ime, there w a s a hysterical d iscuss ion with a group of girls — the ones at the front — remember them? Anyway , about '2x' when V was '6 ' . W a s that two, 2 t imes 6 or 6 t imes 6 or someth ing e l s e ? — like they got 60 out of it s o I guess they thought it w a s 6 t imes 10 — and there w a s one student going 181 'It's this way. ' and another student saying 'No it isn't'. There w a s someth ing about the — the energy they were putting in to defend their argument — that w a s fabulous math going on right there. Then there was Serb who kept explaining things to R o n who is very low level math — just like — I don't know — the care that I s a w - just the good mathematical learning. That was good — just with a few quest ions a lot of teaching goes on without my having to lose my voice. I don't have to talk all the time. I think they really learn it this way and I don't have to lose my voice. I really s a w s o m e great teaching going on between the kids and I think maybe it worked best when I let them pick their own groups b e c a u s e of the increased — fraternity that was in the groups. I thought that went really wel l " About punishment: "Wel l , I learned long ago as a c a m p counsel lor , that somet imes it is best to get angry before you really are angry. Pun ishment — has to have different levels for different students. I don't cons ider mysel f a discipl inar ian, but I expect the kids — students — to behave. A few rules are good , but I never really had a problem. Somet imes , I think it is just keeping the kids occup ied and engaged . If they have something interesting to do and they can do it, I think they are all just fabulous. They' l l do it." ******************** T h e next year, Soph ia took a job in a remote northern area of the province. 182 6.3. A portrait of Lance It is the end of the first class Lance taught for Mr. Forsyth. The class went fairly smoothly but Lance is concerned that he really doesn't know how difficult he should be making the assignments. Lance : Someth ing tells me if I just do what is in the book, I'm not trying very hard. I have to look for other things to do — Mr. Forsyth: First of all, THIS is the curr iculum. (Picking up the textbook in two hands he lays it firmly down on Lance's worksheet.) Textbook writer wrote the textbook based on the curr iculum. They spent a lot of t ime — Everything you need is here. If you go beyond that — first of all you could be out of your riding — research has been done into this (tapping the textbook with his right pointer finger) and this is what you teach. Lance : O h . Okay . ******************** Lance is a quiet, sincere young man who has lived in the city all his life. He is the youngest of three sons of a University professor and a stay-at-home mom. As was expected of him, he went to University immediately after high school, and, after almost "failing out", finished a degree in Commerce, studying part-time and trying his hand at a few jobs - delivery boy, bartender, magazine distribution manager. None of these proved satisfying, except, possibly when he had to train people for new responsibilities, so he decided to follow in his father's footsteps and become a teacher. He did not find that his own teachers had been an inspiration for him, and he was hoping that there was a way to make mathematics classes a little more interesting than the ones he had had. ******************** "I've g iven it a lot of thought and I feel this is a good dec is ion, and the right one for me. I have to say that I a m beginning the program a bit concerned about 183 my own understanding of high school mathemat ics, but s ince mathemat ics is ordered and logical, I guess I will be okay. Su re — the ideas are still there, but I'm — uh — just not sure if I remember all the ... umm — you know, the words. S e e , I bel ieve that learning mathemat ics requires higher- level thinking than many other subjects. For example , socia l studies you can memor ize it. But in mathemat ics everything is either right or wrong, and can be proven true or fa lse through a ser ies of s teps. Learning mathemat ics requires practice, focus and pat ience — and I think it is the teacher 's job to guide and educate my students, providing - well — thoughtful, precise, you know — detai led lessons. Except for the incredibly gifted who'l l understand e a c h concept immediately, mathemat ics may s e e m strange and foreign to kids until there is a sudden insight. A s a teacher of mathemat ics, I'll have to put mysel f in their — the students — s h o e s and try to remember how I learned the concepts . It's much eas ier to teach a concept if I remember what it w a s like for mysel f to learn the s a m e concepts . " ******************** At the end of the second day of the SMIP program, the preservice teachers were left with a proof to do, based on a hand- on activity they had done. Although Lance was hesitant about his own understanding of high school mathematics, he was the only student who came to class the next day ready to demonstrate his 'proof of the theorem. True to his admitted weakness of the language of mathematics, a weakness that persists throughout the course, Lance talks about "six-gons" and "squished squares" instead of hexagons and trapezoids. He is not nervous in front of the class, and although he makes a few 184 mistakes which are corrected by his fellow classmates, this does not bother him and he discusses his presentation. "You know now I think about it, I reckon my lesson w a s O K — a bit jumpy and not very organized maybe 'cuz I hadn't really thought about how the others would follow my presentat ion and I wonder if, — B e c a u s e — you know, mathemat ics c o m e s e a s y to us students in S M I P , if we will be able to anticipate the problems students will have in schoo l . I don't know, I'd say I was a learn by doing kind of guy who likes to figure things out by myself. W h e n I had to train s o m e people at work, I p lanned my, you know — planned my lessons on my understanding and the manner in which I learned to do the work. Learn ing mathemat ics is a bit like construct ion of a building — I read that somewhere and I agree. It sa id that a building must have a strong foundat ion, but if you don't fill it, it's use less - and if the foundation is not strong, it will co l lapse. S o that's like math, and the teacher should direct the student 's learning — create a strong structure so ... you know ... more learning can take p lace." ******************** Lance's honesty and sincerity make him a person who is easy to talk to, and everyone in the class seems to like him. His female classmates tease him that all the high school girls will fall in love with him because of his classic good looks and his athletic body. He self-consciously smiles this aside. "That 's not s o important. Y o u have to cons ider what people do. Think about Armie . Here she is, all the way from Iran where she taught, and here she has to re-certify and she ' s doing it. That takes real strength, courage. I'm 185 impressed by that and I hope — well , I know I can learn a lot from her. Y o u know, she ' s from a different country, but you know, mathemat ics is everywhere - the multiplication of rabbits is a ser ies , shel ls are spi ra ls - you can' t talk about anything without mathemat ics. Even linguistics is structured mathematical ly and they don't have the math to do the structuring - it just happened naturally -maybe the math we do is just trying to make s e n s e of what we s e e . But how do you explain that to s tudents? Y o u know, they should know that it is important, that they really have to use it - s o maybe word problems are good , but maybe what that teacher in the video — you know, the one L o m e showed us where he kept telling them that they must be smart if they could learn this stuff ' cuz it w a s hard. Maybe telling them that they are smart is a substitute for re levance. — I'll have to think about that 'cuz, everything w e know and do in mathemat ics is based on what we al ready know. Y o u can't go ahead unless you al ready have a p lace to start and maybe if you tell the kids - let them know you think they can do it, that that will, you know, make them think they can . — A n d then that makes me wonder — Is math created as a product of our environment, as a way to relate? If we grew up on the moon , with no gravity, would that affect it? If an al ien race c a m e to earth, would they have math and , you know, would it be the s a m e as ou rs? Cou ld we have gotten to this point without math?" 1 ******************** Lance watches carefully as other students present their lessons and when they are given videos to watch of teachers in classrooms. 186 "Actually, talking in these groups in S M I P about what everybody did in schoo l and see ing all these different teaching techniques that are effective, I real ize how much I don't know - because - I keep analyz ing - but I don't think I ever had a teacher in high schoo l that ever, you know — that used a quest ioning style or that — wel l , I rarely remember working in groups as a project - they actually d iscouraged it because they s a w that as a way for kids to cause trouble -- just do your work and they made sure the desks were far enough apart — s o those are a lot of the things I'm learning - keeping kids involved, getting them to learn without being told - working together, hearing information from all s ides -you just naturally absorb. But, you know, somet imes I can't f igure out what it is that you're trying to teach - W a s it factoring or multiplying that that you were teaching, Trinity? It really wasn' t very c lear in my mind. "Real ly , when I entered the program I thought I would have liked it if I had just been given a teaching certificate and went out to teach so ' s I could learn on the job. ' C u z I pictured the program being more a telling kind of thing and I never really l iked that. Sti l l , in this first month of the teacher ed program, I can s e e how much I'm actually — you know — learning. I mean learning techniques and tricks of how to get through to students, things I'd never previously even thought about. T h e teaching I got was the lecture-type and — but that's the way it — wel l - it's amaz ing how many different teaching styles there really are, you know — and how much better they s e e m to work than what I suffered! I had to try to figure it out mysel f and we are learning to help students figure it out for themse lves s o they can remember - c o m e s from your own head rather than someone telling 187 you and that's what I a lways wanted but never — and it is good to s e e that happen so much because I didn't know how to do it — I 'm sure if I'd gone out to teach I'd have been just like my teachers. " ******************** For one of his presentations to the SMIP class, Lance begins the lesson with a game/puzzle: "P ick a number. ... Doub le it. ... A d d 9. ... A d d the original number. ... Divide by 3. ... A d d 4. Okay , subtract the original number. ... What ' s your answer , Er in? N o ! Let m e guess . " His hand goes up to his head, he thinks for a few seconds, then: "The answer is 7." The preservice teachers are intrigued. Lance goes on to explain how/why he knew the answer would be 7. He had made this question up himself and he demonstrates the algebraic reasoning to the class, purposefully making errors. His classmates correct him. "Just making sure you are paying attention. I think that this might be a good idea in teaching — m a k e mistakes to s e e if the kids are paying attention. Someth ing I never thought of before. Wha t do you guys th ink?" As is often the case, Lance has again started a discussion. Later, when discussing the presentation: " Y o u know, I just told you guys how it worked - and I know this is not the best way for students to learn. W e talked about quest ioning and all that. Y o u know, I guess I'm just used to — I g u e s s I need to work on this - not tell ing, but trying to get the c lass to f igure it out for themselves. It's really hard to break habits. Th is is what I'm used to, but it's not the best." 188 "I suppose I'd a lso descr ibe mysel f as a v isual learner. I really enjoyed geometry. I could look at a geometry quest ion and s e e a ten-step proof, but a lgebra was more abstract and I had to start it - write things down you know — before the steps would fall into p lace. That 's the way my mind works. That 's probably why I'm learning a lot about teaching by watching you guys present -you know, the min i - lessons L o m e has us do — where we have to try someth ing new — those fifteen-minute lessons on something and we talk about it. I think I'll probably try to incorporate s o m e of those activities in my lessons . Y o u know, like I'm eager to use what Peter - that arithmetic ser ies - I'm a v isual - and I never thought to relate a reas to addition of sequences . Th is will be good to teach with. I like getting things others have done. T h e s e ideas - I can use them - s o they're good . There were a lot of good ideas in those demos. " Lance seems able to find something good to say about almost anything he has to critique. After presenting a teach-by-telling lesson in which Armie simply presented material to students in a lecture format, she criticizes herself, but Lance cuts in with: "You ' re teaching in a "second language, but you are so knowledgeable . Don't be so hard on yourse l f . And when another student, Mer presented a class in which he faced the board for the entire explanation, and during which it was hard to hear what he was saying, Lance commented: " Y o u were s o ca lm. I could learn from that - not putting me under pressure." ******************** Lance gives some thought to student behaviour, and as is often the case, his thoughts jump from topic to topic. " Y o u know, maybe we have to think about 189 inappropriate behaviour by the kids. Y o u know, it might be caused by something outside the c lass - like the student might have personal problems or learning problems. How will a kid feel if the teacher cuts him off or did a quest ion differently from the way that the kid sugges ted? Y o u know, E v a n fol lowed the way we suggested that he do the quest ion and later told us that that wasn' t what he intended. I felt really actuated when you did that. It m a d e me feel , you know, like you cared how I thought of the quest ion and how I would do it. But, what if he hadn' t? How would I have felt, and how would a kid who thought he knew what he was doing didn't have the teacher, you know, accept it? Wou ld he think he w a s wrong or what? A n d then, what about a s s e s s m e n t ? Do you a s s e s s where the kids are or how far they have c o m e ? It may not be the student 's fault that he /she doesn' t know everything - it may be the result of previous, poor teaching - or even my own." ******************** Lance confided: "I really felt sorry for W a y n e that he w a s thinking of leaving the program - you know, when he m issed c lass the other day. H e sa id he didn't really know why, but I think it's, you know, ' cuz he really hasn't had any work exper ience. I had a few jobs before - you know to make money - but that w a s all they were. They didn't mean much , but - and I hope this doesn' t sound too corny - now I feel commit ted. Y e a h , I think teaching takes commitment. A n d , it is more beneficial to society than my other jobs. That d o e s sound corny, but that's how I feel and if — wel l , hopefully W a y n e will remain. For mysel f - even if I don't stay, I've learned a lot. T e a m dynamics , relating to people, understanding 190 where they c o m e from. - you know, even these are enough for W a y n e to remain if he learns these. That 's what I told him, anyway. "In schoo l , you know, maybe it's the focus on grades that's the problem. I know I was , you know — stifled by being "told" what to learn and by worrying about marks. I worried about marks and I wasn' t really learning -1 could maybe do the work, but I wasn' t understanding. I think its good the way this course , you know, is a pass-fai l course 'cuz I'm not focused on getting that percentage — not like — quest ions like this will be on the test and I maybe focus on the wrong thing and don't s e e the big picture. W h e n I was studying math, it was , you know, learn how to do it and it didn't really matter. But here, because of the structure - not focused on that score I'm learning a lot more. Its important for kids to "look at this and this" and try to make the connect ions - try to digest and so to go through a p rocess ... I g u e s s that it makes sense , I guess , I don't know, but maybe 'cuz that's the way I learned - not memoriz ing like I w a s supposed to. That 's the way I really learned, and maybe that's the way it should happen - not focusing on marks in schoo l , but on d iscuss ion , getting kids to think about what they do." ******************** "Teach ing involves a lot of organizat ion and you have to be able to exp ress ideas in different ways . I don't want to dominate and control my c lass but to teach I will, you know, have to express my knowledge. I want to find out from all you guys the best way to do things 'cuz I don't want my students to get the wrong idea. If you learn something wrong, it will be hard to get back on track. Y o u know — I taught mysel f how to subtract in e lementary schoo l by adding on . 191 Y o u know, 10 subtract 7 is 8, 9, 10, I would think, so — three. But then I had to learn real subtraction 'cuz that wasn' t the right way and now I just draw the numbers in my head — the line and all, you know — and do real subtraction. Learning takes p lace in the head and if you figure it out yourself it will make s e n s e and you will remember it — but if your learned it wrong, you'l l have to re-learn it and that's hard — I don't want my kids to have to re-learn things 'cuz they learned them wrong." Worry may be a way to describe Lance, worry and concern over fairness. Lance is always a part of every conversation, often through posing questions that stimulate the discussion. "What does it mean to know? - does it mean knowing more or knowing better. A r e they trying to put too much into the high schoo l curr iculum? What is the best configuration of d e s k s in a c l a s s ? If students are seated together, won't they talk too much and not do their mathemat ics? Y o u know, the best teaching may be done by giving a naturally occurring example . " ******************** "My short pract icum went pretty good . I watched a lot of c l asses that were, you know, right out of my league — band — texti les — wow — but they had to have different techniques and that was neat to s e e . Then , there was a great examp le in a mathemat ics c lass where the teacher used graphics calculators to demonstrate s lope and y-intercept. Th is really worked great, and you know, I'll probably use this with my own c lasses . " 192 Lance gives thought to what it means to teach mathematics after his short practicum. "I read, you know — in elementary schoo l there are specia l is ts for F rench and physical educat ion, s o why not mathemat ics? It's important, but by grade 3, 7 0 % of the boys say they hate mathemat ics, and — wel l , I don't want to stereotype, but most elementary teachers fear mathemat ics. It isn't giving the students a fair shake - specia l is ts for P E and French but not math." Lome interjects, and says that he doesn't think he understands mathematics well enough to teach elementary school. Other students concur. This surprises Lance. "That really sca res me, but, you know, as E l i sa says , that may true for most of us. W e are beyond that. W e do all that math stuff that they do but automatical ly. Hey, d o e s that mean we 've c rossed a Don't N e e d Boundary? Neat ! — but — doesn' t that mean we should be able to - what w a s the word? Y o u know, go back and retrieve it? I'm not sure if I could now that you mention it." ******************** As time goes on Lance continues to use incorrect terminology - perfect polygon, for example, instead of regular polygon - but he is not afraid to speak up, and his understanding of the topics is developing as he discusses them with the class. He remains open to learning other methods, but he is comfortable with his own. "I l iked working with a lgebra ti les the second t ime. I w a s able to think more about how they worked. I thought about the problems the students might 193 have by making up the problems for myself: Y o u know, like I made up one where there weren't enough x 's to take away and I tried to figure out how I would do it. Then , when E l isa demonstrated her method I saw that it works, but I understood mine. Then I real ized with factoring using a lgebra ti les, I saw that if it doesn' t make a rectangle, you can't factor it. Y o u know, these are good , but I wonder if you would show them to the students first and then teach decomposi t ion or if I found them interesting because I a l ready knew decomposi t ion. Wha t do you think about this - which way does the learning g o ? Don't you think it will take too much t ime if the — the kids all have to figure it out by themse lves? I don't want to just give rules, but there is a lot of work to cover. There must be s o m e way to work this out — you know, s o m e way so they can learn it without you telling them everything, but not so ' s they have to do it all t hemse lves?" ******************** Lance has been developing different approaches to teaching during the SMIP program. At the beginning of the year, when he did his presentation on the proof of the theorem about the shapes, he was serious and matter of fact. Now he and Sophia dress the part of Lome and myself and act out the question for their class presentation as Frank reads it. Lance goes to the chalkboard and draws a representational diagram. He practices various questioning techniques to elicit information from his fellow classmates. "Do you think the d iagram should be more real? Like I made the banks c rooked, not straight l ines and then, you know, I wasn' t very accurate with my 194 angle. In a real c lass do you think it should be sort of accura te? Y o u know, looking more like 40 degrees. I think that that is what I would really do so ' s not to lead them astray." ******************** As the fall term nears its end, and thus, also, the close contact with his SMIP cohorts, Lance wants to find out as much as he can. When a visiting professor, Dr. Jo Towers, presents to the class, he is one of the few students who remains behind and talks to her. He is impressed with how openly she criticizes her own teaching. He wonders if she would teach it differently, now. "I'm often not sure what to do. Y o u know, even Dr. Towers made mistakes. S h e didn't s e e that the one girl had the right answer at the beginning. S o , w a s this a good learning exper ience for that girl? A n d then people talk about testing as a good learning tool. Do they mean the actual test or the preparing for it? O n mult iple-choice tests — like I sa id earlier, I used to just play with the answers , plugging them in, to s e e which fit the quest ion. Th is w a s not a learning exper ience! S o , what would make a good learning activity? A good activity should make the students want to play with it and they sure don't, you know — want to play with tests. Doing something with the c lass like Soph ia and Frank and I did would be fun if you could get another teacher to help and they could then do the quest ion. Or, you know, remember Soph ia ' s quest ion about the — I can' t remember, but it had something to do with Harry Potter — But, you know, for a lot of activities — I — it would be hard to make the leap from the activity to the abstract. I'm not sure how I could get the kids to do that. They 'd have the 195 picture — uh — Image — but I don't know if they could switch. A n d , you know — what — how much time do you need? What if your lesson takes too much t ime? Not enough t ime? How can you make sure those open-ended quest ions will get to what you want them to — you know, will the kids — will they learn what you want them to? How do you grade them? C a n you ever give anyone a 5 out of 5? Isn't there a lways room for improvement? A n d who is to say that this student 's approach is better than that one ' s? It might just be that the one said it — I guess looked at it more like you did and so you might give a better grade. Is that fair? A n d , then, if a kid — you know, student — does switch from physical manipulat ions to drawing, have they gone over s o m e Boundary - or are they just Property Not ic ing? How do you know if the student made a lucky guess or if he really understood the p rocess? T h e s e things really bother me." During the last week of classes, before the long practicum, Lance was unusually quiet. ******************** Eric's High is in a well-to-do area of the city. It is known to be relatively traditional in its attitude, and staff and students alike are proud of their academic performance. About 80% of the students go on to post-secondary education. Mathematics is considered to be an important subject, and the mathematics office is adorned with awards and trophies - tribute to the good performance by the students on the mathematics contests from the University of Waterloo. Eric's High is also proud of its performance in other areas, and have many athletic and fine arts awards displayed in the front hall to prove this. Thus, at this school, 196 there seems to be emphasis on a well-rounded education. Students are expected to do their best in whatever they are doing. Teachers are friendly and cooperative and seem happy with the administration. Because of its reputation Eric's High is a sought after school by teachers to transfer to. ******************** Mr. Forsyth announces to the class that Lance, who had been observing for several days, will be teaching them for the next 2 1A months. He moves to the side of the room and Lance walks to the front of the class. Although this grade 8 class had obviously not been previously told that Lance would be teaching them, they accepted it without question as they did Lance's announcement that I would be videoing the class - him, not them. There are 30 students in the class, 15 boys and 15 girls, mostly of Asian descent. The lack of reaction by the class to Lance's taking over and to the videoing is almost eerie, but Lance is relaxed as he talks about rules and about learning mathematics. "Respec t - Mutual respect is what I expect . I expect your attention when I a m talking and w h e n s o m e o n e e l se is talking you shou ld pay attention to them. T o learn new mathemat ics, you a lways have to rely on what you a l ready know so you shouldn' t be afraid to ask quest ions. To let you know about myself, I've worked in other a reas and I have found that it is a lways important to ask if you don't understand. But after working at other p laces , I found I don't want to, you know, sit at a desk. I want to teach, s o maybe we should start." Lance's lesson begins with a review, section by section, of the formulas from the work covered by Mr. Forsyth the previous day. He asks the students if 197 there were any problems, and having a student read the first question, he uses a diagram and some algebraic notation, writing the information on the chalkboard. He acknowledges that this is a very difficult question (It is a question that asks about maximizing perimeter of a rectangle with a fixed area.) and has students try to identify the important information given. Many indicate that they do not really know where to begin. "Okay, what do you know about a reas? — Right they are a lways square units. Now, if this is a rectangle, what do you know about it? ... That 's right, there are all right ang les . ... Wha t ' s that? ... Y e s , the s ides can be the s a m e length. Wha t do you cal l a rectangle where the s ides are the s a m e length? ... Right - a square . ... S o , suppose the a rea w a s 10,000 square meters. Wha t would the d imens ions be, knowing it is a squa re? ... Now, what might they be if it isn't a squa re? " Lance makes good use of diagrams as he has different students describe 'their" rectangle. One girl, Grace, talks about making a 'less square rectangle', and Lance, knowing what she means (after all, this is the way he talks about them), follows through, drawing first a rectangle where the length is about twice the width, then makes other rectangles that are 'skinnier and skinnier", as Grace describes them. Lance is unconcerned about the terminology. Grace seems to know what she is talking about. The students conclude: 'The less square the rectangle is, the greater its perimeter will be.' " G o o d go ing! Right! Right on. The skinnier the rectangle, the greater the perimeter. N o w this is a problem that takes s o m e time. I only expected that you 198 could think about it and maybe come up with the idea that the square has the least perimeter. But, I guess — when it works out — maybe you can think that if you maximize either the base or the height — that you will get — you know — the perimeter will be biggest. G o o d work. S o , let's go on to the next sect ion. Ci rc les. Wha t can you tell me about c i rc les?" The class reviews the terms associated with circles: radius, circumference, pi. "P i is the ratio of c i rcumference to diameter. At the end of the period I will be giving you an activity that will help you understand this. But we have to get more work done today, s o we'l l take a look at the next sect ion. Compos i te f igures. D o e s anybody know what I mean by a composi te f igure? ... Right, G r a c e . They are made up of different shapes . Now, they can be divided into distinct regions — do you know what I mean by distinct reg ions? — Y e s ? — Okay , and the a reas of these regions can be added together to find the total a rea . Let 's do one together." Lance draws an L-shaped figure on the board, divides it into two rectangles, finds their areas and adds them together. The class, not really engaged with the question, but listening attentively and copying the work in their notebooks, agrees that this would be the area of the composite figure. Lance assigns a few questions, and, working individually, the students begin the assignment. By raising his hand to get their attention, Lance 're-groups' the class to discuss their work. They discover that they have not all divided the figures up in 199 the same manner. Using this as a lead, Lance chooses one question that he has noticed has been done three different ways. Drawing the diagram on the board, he asks three students to come up, one at a time to describe their different methods of finding the area. "Now, what do you notice about these answers? A r e they different? No. — You ' re right. It's the s a m e figure s o it doesn' t matter how you divide it up. It has the s a m e area irregardless. Now, turn to page 228. S e e the activity there. That 's what you're going to do . Here 's a photocopy - one for e a c h person - I want you to do this ' cuz it's really important to know what you are doing — I want it to make s e n s e . Y o u can't do more math un less you understand — know what you're doing. Th is activity will teach you about pi." For the last fifteen minutes of class, students wori< individually or in pairs on the activity and Lance walks about the class. The activity, the students find, is simple enough, and there is little questioning. After class, Lance expresses concern to Mr. Forsyth over the depth to which he should go in teaching and if he should be bringing in outside activities. It is clear (see the opening statement) what Mr. Forsyth believes. ******************** Lance's next lesson is based on the next two sections in the textbook. " B e sure to draw d iagrams, and remember, you can work together. Y o u don't have to work a lone." But, it seems that the students are used to working alone, and most continue to do so. "Remember , you c a n do more than I ass ign if you want to. It never hurts to do extra work. John — I can s e e that you aren't 200 working — and don't throw that!" John is surprised that Lance saw his movements as he appears to be facing the other direction. Lance seems to have 'eyes in the back of his head' and the students are now aware that they will likely be caught if they try to act up. ******************** Lance phoned me a few nights later. Concern in his voice, he says: "I don't think it would be a good idea if you c a m e tomorrow. I don't want to talk too much about it now, but there has been a problem with a student - a personal problem, and she did harm to herself. I'm afraid that if you're there with the camera the c lass might react - and you know — when she c o m e s back, we should probably wait a day or two to s e e if she 's okay." Lance's concern over his student is sincere. The privacy of the girl involved and the welfare of all the class are foremost in his mind. Over the next while, he does keep in touch with me via the telephone, and with the whole SMIP group by taking an active part in the SMIP email list-serve. He lets down his natural reserve, and reveals a sense of humor that had not originally been obvious. For s o m e reason, they [the students] don't share my enthus iasm in solving quadrat ic equat ions s o I've gotten into the routine of telling silly, innocent, one line jokes: Wha t did the tide say when it rolled in? — Nothing, just waved . I'm starting to run out of ones that aren't too r isque. Anyone have s o m e to sha re? ... A n d , no, Soph ia - none of the girls has asked me to marry them — yet. (email communique) And later: Whi le teaching grade 8's composi te a reas , it is not recommended to use the example of a cyl inder with a semi-c i rc le at tached to it, especia l ly if you are no great artist and your cyl inder is slightly curved ... G igg les spread 201 ac ross the c lass and I had no idea why until I looked back ... It's hard not to laugh yourself, (email communique) ******************** Lance lets me know that he thinks it would be okay if I retuned to his class. When I do, he is teaching the 'stem and leaf plot' and the 'box and whisker plot' as outlined in the text. After doing a few questions involving stem and leaf plots, he gives the class a worksheet on which he has outlined the method for drawing a box and whisker plot. After class, he discusses his reasoning and reactions. "Rather than teaching it directly, I like to give them instructions on paper. They still like to work by themselves , but — you know, for this activity I wanted them to work together. But they got confused. They couldn't even s e e m to follow the instructions. I tried to let Z a o explain ' cuz she 's pretty good , but they were noisy. I had to raise my hand to quiet them down severa l t imes. I ended up having to explain it myself. A s I d id, I had to use my arms to demonstrate the different proport ions, holding them out to represent the whole amount, bringing them in to indicate the center half, etcetera. But it just gets frustrating with the pace I have to go in getting through the work. T h e s e kids don't really have a very good understanding of what 's going on . They really just learned m e a n , median and mode, you know - only yesterday! — and now they have to use it here to do this stuff. It s e e m s that they don't ever have t ime to think and absorb what they need to learn. — But they just need t ime to make s e n s e of the whole thing." 202 After Lance observed the video of himself teaching, he spoke freely about how he felt about the experience and his reaction to the teaching he had observed at Eric's High. And, even though he had displayed a presence in the class that let the students know that he was the authority, (a presence that kept him distanced just enough from his students so that, as his fellow SMIPPERs had feared would happen, none of the students 'fell head over heals in love with him') he indicated that he was not totally sure of his position as a teacher. "I encouraged the students to ask quest ions and I'm pretty sure I w a s able to answer them, but I don't think I really got to do a lot that w a s differentish. I think I'm better at using the right words, now, but I — you know — I w a s often really tired. I was working a night job during the pract icum, and s o m e nights I didn't get enough s leep. Somet imes I just didn't connect with what the kids sa id , and I just went on . Like I really m issed it when - who was it - oh wel l , it doesn' t matter who it was , but s h e asked about s tem and leaf plots that were in the hundreds. I just passed over it. I guess , partly ' cuz it wasn' t in the book and I hadn't thought of it - and partly ' cuz Mr. Forsyth told me that everything I needed w a s in the book - and ' cuz he said I shouldn' t teach anything e lse 'cuz I might — can't remember his word, but you know - somebody might compla in . I felt that working that way really limited me. Y o u know, put all sort of constraints on me. "I think I got down the quest ioning a bit and didn't a lways just tell them what to do, but I had trouble - couldn't really get them working on activities that wou ld , you know, get them to figure it out themse lves . I didn't s e e any of my sponsor teachers do anything differentish. They just sort of stuck to the book. But 203 when I s e e people do good things — different — and I like it, I can use it. I need to see somebody do these different things when they are teaching s o I can s e e how they work --- like we did in S M I P . But it, you know, didn't really happen at Er ic 's High. "Teaching is really different from other jobs. You ' re just thrown in the fire and away you go - not like taking training and then being watched or working with someone . Y o u know — it's really — once you're there, you're there. I'd like to watch more dif ferences and then try them out - different teaching styles, that is - I guess I learned that being a teacher — knowing how to teach is a big thing. Y o u have to — wel l , it's more than knowing your subject matter — even more than understanding it — there's a dif ference between a mathemat ic ian and a math teacher. The mathemat ic ian loves math but the math teacher first loves teaching - am I being corny? But math is - teaching math, wel l , you have to know the math and know what the kids will think like in the P - K theory. It m a k e s s e n s e . Y o u can't just jump in at any old point. Y o u know, you have to have all the knowledge — and be able to Fold Back - can't Property Not ice un less you know s o m e properties and - Most teachers I watched just showed , you know — here is how it works now do it — forcing them through the Property Noticing to Formal ised . "You know - I watched one c lass where they were working on standard deviat ion and the teacher just — you know, said: ' You don't really have to figure it out. Y o u have a computer that will work it out.' That was s o frustrating and I — 204 well , that's what happened and — wel l , you know I wasn' t impressed. But, what does teaching m e a n ? "I think maybe it should involve s o m e leading of students in the right direction, but a lso, l istening. Wha t do the kids s a y ? I know there was one girl who sa id something and I sort of ignored her when she asked . I guess that happens somet imes, but if it isn't really related, how much time — if it's a tangent - how much t ime do you g ive? I a lways felt that I couldn't spend t ime 'cuz Mr. Forsyth sa id to teach what was in the book. But what if the kids were interested in someth ing e l se? A n d if they have a wrong idea, I think you need to d i scuss that s o they can understand what is wrong — not just tell them: 'This is the right way. ' I know I somet imes just droned in front of the c lass , but I really — wel l , somet imes I wasn' t really engaged , but I think I would like to get more activities that would involve the c lass . A n d there were students who shouldn't be in that c l ass — didn't have the prior knowledge they needed - two of them - you know, when we were doing the circle - knew circle, but not radius or pi or d iameter -and couldn't do much for them." ******************** T h e following year, Lance took a job teaching mathemat ics in a relatively new, smal l private schoo l . 205 6.4. A portrait of Ellie It was the beginning of the third week of the fall term in the Teacher Education program. The preservice teachers had been given some student work to grade. The mathematics question was open-ended, asking students to explain and justify their choice of which player they would choose for a bowling team given only the scores of two boys from six games. In the discussion that followed: Ell ie: I never would have thought of giving a quest ion like that! — but if I — It's still bothering m e - but a good quest ion but I'm not sure if I'm right - I would have serious reservations giving this — what is expec ted? — looking for c lear justifica