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Drawing as problem-solving : young children’s mathematical reasoning through the act of drawing Saundry, Carole 2006

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DRAWING AS PROBLEM-SOLVING: YOUNG CHILDREN'S MATHEMATICAL REASONING THROUGH THE ACT OF DRAWING by CAROLE SAUNDRY A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Mathematics Education) THE UNIVERSITY OF BRITISH COLUMBIA August 2006 © Carole Saundry, 2006 Abstract This project investigates h o w young chi ldren emp loy m a t h e m a t i c a l reasoning through the act of drawing while solving m a t h e m a t i c a l problems. Included is a n examinat ion of h o w g r a d e 2 chi ldren r e s p o n d e d w h e n presented with two m a t h e m a t i c a l problems to solve, the kinds of pictures or representations they d rew spontaneous ly - a n d the purpose for those drawing - what students were thinking whi le they drew, a n d h o w (or if) the a c t of drawing or the drawings themselves were helping students to reason through the problems. C o n s i d e r e d together , v ideo f o o t a g e of students using drawing as a problem-solv ing strategy, chi ldren's c o m p l e t e d representat ional work a n d student interview d a t a informed the creat ion of a f ramework for examin ing students' d rawing as problem-solv ing strategies a n d behaviours . Four categor ies of student behav iour e m e r g e d from this analysis, namely : the use of virtual manipulat ives, the a p p l i c a t i o n of a system, the use of imagery a n d a n examinat ion of the sophist ication of students' representations. These categor ies form a f ramework that provides a lens for understanding children's m a t h e m a t i c a l reasoning as expressed through a n d der ived from the a c t of d rawing. Questions regard ing p rob lem structure a n d complex i ty a n d their i m p a c t on student reasoning through pictures are also addres sed . Implications of the study are e l a b o r a t e d , a n d involve a re-conceptua l i z i ng of d rawing in m a t h e m a t i c a l problem-solv ing to inc lude a more p e r f o r m a n c e - b a s e d a p p r o a c h , observ ing thinking in the a c t of thinking, reasoning while d rawing. R e c o m m e n d a t i o n s for class room p r a c t i c e are m a d e regard ing h o w teachers c a n support chi ldren in d e v e l o p i n g these a c t i v e a n d metacogn i t i ve skills. Further research directions are suggested in the discussion of the results. Table of Contents Abst ract ii Table of Contents iii List of Tables vi List of Figures vii List of V i d e o Clips ix A c k n o w l e d g e m e n t s x D e d i c a t i o n xi CHAPTER 1 - INTRODUCTION 1 Drawing As Problem-solving: Young Children's Mathematical Reasoning Through the Act of Drawing 1 B a c k g r o u n d a n d rat ionale 1 Conceptua l i z i ng the study: d e v e l o p i n g the research quest ion 7 Re lated studies 8 Methods 10 S ign i f icance of the research p rob lem 11 Structure of this d o c u m e n t 12 CHAPTER 2 - LITERATURE REVIEW 14 Introduction 14 Interpreting the research 15 Imagery a n d m a t h e m a t i c a l problem-solv ing 15 D y n a m i c imagery a n d mathemat ic s a c h i e v e m e n t : C o m p a r a t i v e studies 19 Visualization a n d d i a g r a m m a t i c c a p a c i t y 21 Assessing i m a g e - m a k i n g ; the complex i ty of l a n g u a g e 22 Spatial-visualizing c a p a c i t y a n d number understanding 23 Visualization a n d symbolism in mathemat ics 25 Drawings as d y n a m i c representations 26 Drawings as manipulat ives 27 Explicit instruction in d rawing to problem-solve 28 Reform a n d representat ion: Valu ing d rawing as m a t h e m a t i c a l reasoning 31 Summary 33 Contr ibut ion to a current b o d y of research 35 CHAPTER 3 - METHODS 37 Before the study 37 Who was invo lved 37 The process 38 During the study 42 M y role as observer 42 How the students r e s p o n d e d 43 How my observat ional p r a c t i c e shifted 43 After the study 45 Analysis of the information: How I d e s c r i b e d a n d r e c o r d e d events 45 Sorting a n d chart ing behaviours by group, by task, by behav iour 52 Determining contributors to success 55 Analysis of the interview d a t a 55 CHAPTER 4 - RESULTS 57 Observations by task: How children drew and how it helped them 58 The C o o k i e Problem - G e n e r a l themes 58 O n e chi ld's process ing of the C o o k i e Problem: Orson uses d rawing as a problem-solv ing strategy 61 The Wheels P roblem - G e n e r a l Themes 65 O n e chi ld's process ing of fhe Wheels Problem: Orson uses d rawing as a problem-solv ing strategy 68 Summary - Chi ldren's use of d rawing as problem-solv ing 71 Observations by strategy: Creating the framework 74 Virtual manipulat ives 75 Systems 81 Imagery 87 Sophist ication of representat ion 93 The drawing as problem-solving framework: Summary of the findings 99 Interview data 102 Examples of student's explanat ions of how drawing as problem-solv ing h e l p e d 103 Interview d a t a : Summary of the findings 105 CHAPTER 5 - DISCUSSION 106 Overview 106 Ref lect ing on the problems as illustrative of d rawing as p r o b l e m -solving: A c o m p a r i s o n of the C o o k i e P roblem a n d the Wheels P rob lem. . . , 106 What is drawing as problem-solving? 110 Creat ing a definit ion 110 N e w understandings g iven v i d e o observations From " d r a w i n g the solut ion" to " d r a w i n g thinking" 113 The Drawing as Problem-solving Framework 115 Some questions a n d reflections 115 Virtual manipulat ives 115 From phys ical models to menta l images 116 Sophist ication of the drawing 117 What makes for successful drawing as problem-solving? 123 The a c t i v e use of virtual manipulat ives 123 The flexible use of strategies 126 The p o w e r of a system 130 What a b o u t chi ldren w h o d o n ' t d raw? 135 What a b o u t chi ldren for w h o m drawing is not helpful? 139 Dispositions and mathematical sense-making How they influenced drawing as problem-solving behaviours 140 Asking permission to d r a w 141 But what ' s the division answer go ing to b e ? Class room culture a n d student expectat ions 142 Understanding how drawing helped: Insights from the interview results 144 What the visualizers reported 147 Implications for practice 150 Types of problems 150 Types of learners 151 Address ing a n d support ing m e t a c o g n i t i o n : Questions a n d prompts for c lass room teachers 152 Class room strategies 153 Summary and further research directions 154 References • 157 A p p e n d i x A : Interview d a t a 161 A p p e n d i x B: Consent forms I 6 7 A p p e n d i x C : C o p y of the UBC Research Ethics Board's Cert i f icates of A p p r o v a l 177 List Of Tables Table 1 V i d e o D a t a Record ing Form Excerpt from G r o u p 1A 46 Table 2 Revised V i d e o D a t a Record ing Form Excerpt from G r o u p 4A 49 Table 3 Summary of strategies by group C o o k i e Problem 53 Table 4 Summary of strategies by g roup Wheels Problem 54 Table 5 Summary of type of response against d rawing or not 59 Table 6 Summary of strategies by group for the C o o k i e Problem 72 Table 7 Summary of strategies by g roup for the Wheels Problem 73 Table 8 Summary of Virtual Manipu lat ive use for the C o o k i e a n d Wheels Problems (all groups) 75 Table 9 Summary of Systems use for the C o o k i e a n d Wheels Problems (all groups) 81 Table 10 Summary of Imagery use for the C o o k i e a n d Wheels Problems (all groups) 87 Table 11 Summary of Sophist ication of the Representat ion for the C o o k i e a n d Wheels problems (all groups) 93 Table 12 F requency of Drawing as Problem-solving Behaviour a n d Character ist ics by Cluster of Themes. 100 Table 13 Interview d a t a : Type of support a n d f requency 103 List Of Figures Figure 1 Jason's cook ies 6 Figure 2 O v e r h e a d a n d Handhe ld C a m e r a Views 41 Figure 3 Anthony's lines 59 Figure 4 Melissa's sets 59 Figure 5 Orson works through the C o o k i e Problem 62-63 Figure 6 Lisa's line up 66 Figure 7 Ruby's chart 66 Figure 8 Orson works through the Wheels Problem 68-69 Figure 9 Sara's sets 77 Figure 10 Shauna'ssets 77 Figure 11 David ' s line 78 Figure 12 Anthony distributes 79 Figure 13 Larry distributes 79 Figure 14 Pricilla's counters 80 Figure 15 Jason keeps track with tallies 80 Figure 16 Cathy ' s numbers 82 Figure 17 Susanne's box 83 Figure 18 Susanne's box 2 83 Figure 19 Pricilla's wheels 84 Figure 20 Jessie c h e c k s a n d rechecks 84 Figure 21 Sara eliminates 85 Figure 22 Norman's work , 86 Figure 23 Eyes up 88 Figure 24 Iris's drawing 89 Figure 25 Trent's record ing 91 Figure 26 Lisa's p e o p l e 94 Figure 27 Lisa's p e o p l e 2 94 Figure 28 Martin's vehicles 95 Figure 29 Shauna's wheels 95 Figure 30 Grant ' s tallies 97 Figure 31 Sample interview responses by c a t e g o r y 103 Figure 32 Anthony's d rawing 112 Figure 33 Shauna's d rawing 112 Figure 34 Jason switches 120 Figure 35 Pricilla's lines 121 Figure 36 Virtual manipulat ive use - student samples 124 Figure 37 Sara's flexibility 127-128 Figure 38 Norman's system 130-132 Figure 39 Norman's final work 133 Figure 40 Mark uses imagery 137 Figure 41 John distributes 145 Figure 42 John explains 145 Figure 43 Jessie works it out 146 List Of V ideo Clips V i d e o Cl ip 1 Orson's flexible thinking 64 V i d e o Cl ip 2 Jason keeps track with tallies 80 V i d e o Cl ip 3 Sara el iminates 85 V i d e o Cl ip 4 Grant talks tallies 98 V i d e o Cl ip 5 Sara's flexibility 128 V i d e o Cl ip 6 Norman's system 133 V i d e o Cl ip 7 Mark's use of imagery 137 Acknowledgements Al though only o n e n a m e a p p e a r s on the front of this d o c u m e n t , there were dozens w h o cont r ibuted to its c reat ion . I w o u l d like to then a c k n o w l e d g e the support of the p e o p l e w h o m a d e this possible.. . To Dr. Cynth ia Nicol of UBC, whose friendship a n d g u i d a n c e m a d e this journey a rewarding a n d worthwhile learning exper ience ; to the teachers w h o o p e n e d up their classrooms in the n a m e of professional growth a n d m a t h e m a t i c a l understanding; a n d to my family w h o h e l p e d to m a k e s p a c e for m e to think, p o n d e r a n d c o g i t a t e in while I pursued this invest igation. But mostly, I o w e a d e b t of grat i tude to the chi ldren involved in this study, w h o shared exact ly w h o they are, h o w they think, a n d what ' s important to r e m e m b e r w h e n you d o m a t h . They taught m e wel l . Dedication For Cameron, whose visual approach to mathematical problem-solving inspires me to learn all I can... Chapter 1 - Introduction Drawing As Problem-solving: Young Children's Mathematical Reasoning Through the Act of Drawing Y o u n g chi ldren a p p r o a c h m a t h e m a t i c a l problem-solv ing in a range of c reat ive ways , app ly ing their learning strengths while they work through problems. This study e x a m i n e d h o w a group of g r a d e 2 students a p p r o a c h e d a n d solved m a t h e m a t i c a l tasks using their visual-spatial strengths; specif ical ly, h o w y o u n g chi ldren used drawing as p r o b l e m -solving strategies in the solution of m a t h e m a t i c a l problems. Working in small groups, students so lved 2 problems, a n d the behaviours they exhib i ted while reasoning through the problems were v i d e o t a p e d . These reasoning behaviours were then a n a l y z e d a n d c o l l a t e d into a f ramework descr ib ing drawing as problem-solv ing. From this d a t a a n d the results of interview questions, three b r o a d areas were cons ide red : representat ional complex i ty , the flexible a n d systematic use of drawing as problem-solv ing strategies, a n d the ability to a c c e s s imagery. This study suggests links b e t w e e n these areas a n d problem-solv ing c a p a c i t y in y o u n g chi ldren, a n d proposes instructional strategies for support ing reasoning through the a c t of d rawing . Background and rationale Years of my professional c a r e e r h a v e b e e n spent observing y o u n g chi ldren. In my role as curr iculum coord inator for the R ichmond Schoo l District I h a v e b e e n c h a r g e d with the implementat ion of a distr ict-wide assessment of k indergarten learners, a p l a y - b a s e d p e r f o r m a n c e assessment a i m e d at identifying learner strengths across multiple intel l igences a n d problem-solv ing. The assessment, ent it led DISCOVER: Learner Strengths is a d a p t e d from the work of Dr. June C . Maker of the University of Ar izona. (2001) The results of this assessment on a distr ict-wide level i nd icate that very y o u n g chi ldren enter ing school in R ichmond h a v e strengths in 2 areas predominant ly - l o g i c a l - m a t h e m a t i c a l inte l l igence a n d visual-spatial inte l l igence. From the assessment itself, a judgment of l o g i c a l - m a t h e m a t i c a l c a p a c i t y wou ld b e m a d e on the ex is tence 3 of 4 possible sets of indicators: bas ic m a t h e m a t i c a l c o n c e p t s of more than a n d less than, ability to sort a n d classify using objects a n d l a n g u a g e , the systematic a p p r o a c h to t a n g r a m puzzle problems a n d mak ing constructions that i n c l u d e d notions of measurement a n d order ing. Visual-spatial c a p a c i t y in the assessment was not restricted to a measure of artistic ability in terms of d rawing, but e x t e n d e d to a chi ld's attent ion to pattern a n d design on the p ieces used, the ability to use translations to a d v a n t a g e in the solution of t a n g r a m puzzle problems, a n d the ability to see abst ract objects as something in the envi ronment. The p r e d o m i n a n c e of these strengths in learners of various ethnic, cultural a n d linguistic backgrounds led m e to w o n d e r if there was a c o n n e c t i o n b e t w e e n the ex is tence of visual-spatial a n d l o g i c a l - m a t h e m a t i c a l inte l l igence - if be ing visually a b l e suppor ted m a t h e m a t i c a l c a p a c i t y in some w a y . Likewise, support ing teachers to use these results to inform their p r a c t i c e m e a n t e n c o u r a g i n g them to strengthen these strengths in their y o u n g chi ldren. As a result, I b e g a n to des ign tasks for use in the early primary classroom explicitly c o n n e c t i n g m a t h e m a t i c a l a n d visual aspects . I w o n d e r e d at the ex is tence of a c a u s a l relationship b e t w e e n spatial ability a n d m a t h e m a t i c a l c a p a c i t y ; in part icular, if a chi ld's strength for visualizing, transforming images a n d construct ion c o u l d b e c h a n n e l e d into m a t h e m a t i c a l problem-solv ing ability. In a commun i ty where English as a s e c o n d l a n g u a g e is a major e d u c a t i o n a l c o n c e r n , hav ing options for support ing thinking that inc lude the who le chi ld -e n g a g i n g the who le brain - was a p p e a l i n g , both to m e a n the teachers with w h o m I work. Involvement with the writing t e a m for the B.C. Early N u m e r a c y Project a l l o w e d m e a n opportunity to study some of the early research into the a r e a of imagery a n d m a t h e m a t i c a l problem-solv ing. The t e a m was c h a r g e d with c reat ing a n u m e r a c y assessment for chi ldren in early primary - ult imately focus ing on k indergarten a n d g r a d e o n e students - to assess, identify a n d then support these students at risk in their early years of schoo l . My role in the project was to examine a n d summarize research in spatial sense, imagery a n d visualization, to determine both its s ign i f icance for the project a n d to establish a w a y in wh ich imagery a n d visual thinking might b e built into the assessment a n d instructional c o m p o n e n t s . The work of Grayson Wheat ley , Erma Y a c k e l a n d Paul C o b b (Wheatley & C o b b , 1990; Y a c k e l & Wheat ley , 1990) h a d impl ications for c lassroom p r a c t i c e , in part icular a tangram-l ike task that ult imately b e c a m e part of the Early N u m e r a c y Project assessment. More importantly, though, the process of research h e l p e d to c o n n e c t m e with researchers w h o h a d b e g u n to ask some of the s a m e questions as I h a d regard ing the c o n n e c t i o n b e t w e e n imagery a n d m a t h e m a t i c a l c a p a c i t y . The research I a c c e s s e d (Wheatley & C o b b , 1990; Reynolds & Wheat ley , 1997; Thomas, Mul l igan, a n d Go ld in , 2002; Gray , Pitta, a n d Tall, 2000, Owens , M i tche lmore, Outhred & P e g g , 1996) suggested there was a c o n n e c t i o n b e t w e e n imagery a n d problem-solv ing c a p a c i t y but d id not explicitly state h o w these strengths might transfer, or whether t e a c h i n g a chi ld to visualize wou ld m a k e him or her a better problem-solver. It d id h o w e v e r o p e n my eyes to the signs of visualization in my work in classrooms. As a coord inator I a m invited into classrooms to c o - p l a n , c o -t e a c h a n d co-assess student work. Having a c c e s s to hundreds of students of varying a g e s a l l o w e d m e a w i n d o w through wh ich to observe a n d not ice chi ldren thinking a n d process ing m a t h e m a t i c a l ideas using imagery a n d visualization. Certa in problems in part icular s e e m e d to elicit a n i m a g e - m a k i n g response, caus ing groups of students (and adults in workshop situations) to g a z e upwards, to gesture a n d finger point whi le reasoning through the task. What exact ly were they " look ing " at? A n d h o w d id their i m a g e help them to reason through the p rob lem? I b e g a n to focus in on these students a n d ask them this very quest ion, explain ing that their upwards g l a n c e s m a d e m e w o n d e r what they were seeing in their heads . S o m e c o u l d answer that they were looking at a n ob ject , a picture or i m a g e from the p rob lem; still others reported they were mov ing or manipu lat ing the i m a g e in some w a y . Young chi ldren on the other h a n d , a l though they exhibi ted the s a m e outward signs (up-cast eyes, gesturing) were less a b l e to descr ibe their menta l processes, responding instead that they "just k n e w " the answer. Questions then arose a b o u t l a n g u a g e a n d the ability to a c c e s s metacogn i t ion to descr ibe thinking processes. If young chi ldren were a c c e s s i n g imagery to reason through a p rob lem, they s e e m e d less a b l e to descr ibe it; a n d in our commun i ty of learners of English, c o m m u n i c a t i o n was e v e n more c o m p l i c a t e d . How c o u l d I a c c e s s y o u n g students' thought processes, their i m a g e - m a k i n g activity without requiring l a n g u a g e ? My work with a group of k indergarten chi ldren g a v e m e some possible d i rect ion. As part of a v i d e o project, I set up a n exper iment involving a group of 3 boys in M a y of their k indergarten year . The quest ion I was investigating at that t ime was to determine the extent to w h i c h sharing solutions to a m a t h e m a t i c a l p rob lem i m p a c t e d the behav iour of others working through similar problems. I asked the boys to solve a division p rob lem - 5 cookies shared by 2 chi ldren - a n d g a v e them a c c e s s to w h a t e v e r they n e e d e d to solve the prob lem, including manipulat ives, p a p e r a n d crayons. I h o p e d that sharing their solutions after they h a d worked through the prob lem w o u l d e x p a n d the strategies a v a i l a b l e to a n d used by the chi ldren while they so lved progressively more difficult problems on subsequent days . What h a p p e n e d surprised m e . In the first problem-solv ing interview with this g roup of chi ldren, I p o s e d the p rob lem a l o u d . While two of the boys b e g a n chat t ing a b o u t the p rob lem, o n e ch i ld , Jason, sat quietly staring at the cei l ing, then put his hands together p a l m to p a l m a n d m a d e c h o p p i n g motions 5 times over. He quickly p i c k e d up a p a p e r a n d marker a n d d rew 5 circles, then d rew vert ical lines through e a c h o n e , muttering " c u t in half, cut in half . . . " with e a c h line d r a w n . Next, he d rew 2 p e o p l e under e a c h c o o k i e , n u m b e r e d the p e o p l e a n d exp la ined that everyone wou ld get "5 cook ies . 5 cook ies cut in half." His drawing follows (Figure 1 - Jason's cookies) a n d e v e n as a n art i fact g a t h e r e d after problems solving took p l a c e , it is c lea r that Jason's solution suggests 5 halves b e al lotted to e a c h ch i ld . At the t ime I marve led at his thinking, not ic ing the product more than the process. So too d id the other boys in the group. They left off the manipulat ives - never o n c e using them through the three problems presented - a n d a t t e m p t e d instead to d r a w their solutions like Jason h a d . As the k indergarten problem-solv ing study progressed, two other more c o m p l e x problems were presented. In e a c h c a s e the chi ldren o p t e d only to d raw, using Jason's representations as a m o d e l . Lome bor rowed Jason's structure for the representat ion of the solution to the s e c o n d p rob lem (7+2), a d d i n g a knife to faci l i tate the cutt ing of the p ieces a n d talking a b o u t the act ions suggested by his picture. O n e boy , Char l ie, was less a b l e to c o m m u n i c a t e his solutions with penc i l a n d p a p e r d u e to fine motor issues. In the s e c o n d a n d third problems (7-5-2 a n d 15+2) he d id not p r o d u c e a drawing to show his solution but rather so lved the problems through the a c t of d rawing (creat ing a partial d rawing a n d then stating his answer) or by visualizing the solution without d rawing at all. When I p r o b e d him for more information, he said simply, "I thought it in my bra in . " Figure 1 - Jason's cook ies V \ mil i \ 'Is* Through r e p e a t e d viewings of this f o o t a g e in workshops situations with teachers , I b e g a n to not ice different things - e v i d e n c e of student i m a g e -mak ing, separate from the products they c r e a t e d . E a c h of the boys demons t ra ted visualizing behaviours through upraised eyes a n d gesturing, both whi le they w e r e thinking a b o u t the p rob lem a n d a g a i n afterwards w h e n they were telling h o w they h a d solved the p rob lem. A n d yet afterwards only 2 of the boys h a d p r o d u c e d a p a p e r - b a s e d art i fact for the 3 problems. As a classroom teacher , this situation was a very famil iar o n e . E a c h of the boys h a d solved the prob lem, per fo rmed the cogni t ive work, e a c h h a d a c c e s s e d imagery to support them in reasoning through the problems, but only 2 h a d c r e a t e d a product . M o r e questions e m e r g e d . How h a d drawing h e l p e d e a c h of these chi ldren? To what extent d id it prov ide a scaffo ld for thinking? At what point was drawing as a problem-solv ing strategy no longer helpful? A n d w h e n was visualization e n o u g h ? Conceptualizing the study: developing the research question Several years passed. Jason, Lome a n d Char l ie h a d m o v e d on to g r a d e 2. I r e m a i n e d curious as to whether these drawing as problem-solv ing behaviours persisted in the later grades , a n d h o w these part icular chi ldren, n o w in separate classrooms, might solve c o m p l e x m a t h e m a t i c a l problems if g iven the opportunity to interact with t h e m . Thinking e v e n more broadly , though, I b e g a n to w o n d e r h o w other chi ldren of this a g e group might e n g a g e with m a t h e m a t i c a l problems in this w a y . Ultimately I set up my masters research study to involve the entire g r a d e 2 cohort at o n e school - 33 chi ldren in all - in a n at tempt to address my questions regard ing student imagery a n d h o w it might support students in m a t h e m a t i c a l problem-solv ing. If Jason a n d Lome a n d Char l ie c o u l d a c c e s s imagery a n d drawing as problem-solv ing as early as k indergarten, then w h o else was using these visual strategies? A n d if they were , h o w were these strategies helping? My b a c k g r o u n d a n d e x p e r i e n c e with y o u n g chi ldren a n d the study of k indergarten students a l l o w e d m e to hone in on a more speci f ic set of questions for this study. Imagery a n d the role of representat ion in support ing student thinking or d rawing as problem-solv ing a n d h o w it mediates student reasoning b e c a m e the focus for this invest igation. Specif ical ly, I a r t icu lated the fol lowing research questions: • How d o y o u n g chi ldren a c c e s s visualization a n d imagery in their resolution of m a t h e m a t i c a l tasks? • How does drawing as problem-solving support students in mak ing sense of a n d reasoning through a m a t h e m a t i c a l p rob lem? Related studies Of course, re lated studies h a v e b e e n e n a c t e d . Previous research in m a t h e m a t i c a l representat ion has e x a m i n e d chi ldren's c a p a c i t y to represent their thinking on p a p e r (Diezmann, 2000, 2006; Deizmann a n d English, 2001; Go ld in a n d Kaput, 1996; Owens a n d C l e m e n t , 1998; Smith, 2003; Woleck, 2001). In part icular, this work sought to assess student thinking as shown in the pictures they drew, or the d iagrams they were a b l e to c r e a t e in response to a prob lem situation. Research into representat ion e x a m i n e d student samples to ga in insight into student thinking a n d d e v e l o p m e n t . Smith a n d Woleck, both t e a c h e r researchers, spoke to the rich assessment d a t a ava i lab le to teachers in, as Wo leck terms it, "l istening to their pictures" . Smith a n d Wo leck saw student drawings of m a t h e m a t i c a l thinking as a powerful assessment tool . Research into representat ion, however , attends to a n d seeks to interpret student thinking after the a c t of thinking is c o m p l e t e . That is, student work in the form of d iagrams or pictures was a n a l y z e d after the chi ld h a d finished working on the p rob lem. In read ing this research on representat ion, I h a d to w o n d e r a b o u t the val id m a t h e m a t i c a l thinking that never m a d e it to the p a g e . How was that thinking be ing a c c o u n t e d for? C h a r a c t e r i z e d by Char l ie, w h o in k indergarten just " thought it in [his] b ra in " , my e x p e r i e n c e of working with young mathemat ic ians has b e e n mainly a n oral a n d visual o n e , where scant or no pr inted art i fact is p r o d u c e d . A n d yet these chi ldren are thinking, a n d thinking d e e p l y . By their o w n descr ipt ion, while solving m a t h e m a t i c a l problems, they are c reat ing a n d manipulat ing pictures in their heads . M y research journey then l e a d m e to learn more a b o u t visualization a n d its role in problem-solv ing. I looked to Owens , M i tche lmore, Outh red a n d P e g g (1996), Owens a n d C l e m e n t (1998), Wheat ley (1990), Reynolds (1997), Y a c k e l (1990), Thomas et a l (2002), a n d G r a y et a l (2000) for some clar i f icat ion on h o w students use imagery to solve problems. E a c h of these researchers c r e d i t e d visual imagery or visualization with support ing students in their problem-solv ing efforts. Some e v e n suggested that gi f ted mathemat ic ians use imagery more often a n d more efficiently than other chi ldren (Thomas et a l , 2002; a n d G r a y et a l , 2000). My e x p e r i e n c e working with gi f ted chi ldren in both s e g r e g a t e d a n d regular c lassroom settings certainly a l l o w e d m e to witness this. Still I was not a b l e to satisfy my curiosity a b o u t h o w these visual processes are a c c e s s e d by y o u n g chi ldren a n d a p p l i e d to the solution of m a t h e m a t i c a l problems. In cons ider ing the results of our distr ict-wide DISCOVER: Learner Strengths assessment, m a n y chi ldren in k indergarten t e n d e d to h a v e log ica l -m a t h e m a t i c a l a n d visual-spatial strengths operat ing in a kind of partnership. The literature, however , d id not s e e m to shed light on h o w these processes might b e d e v e l o p i n g concurrent ly in y o u n g chi ldren. In des igning this study, I w a n t e d to set up a situation - a p e r f o r m a n c e task - in w h i c h students' thinking wou ld b e assessed during the a c t of thinking. A c k n o w l e d g i n g the rich assessment d a t a to b e g a i n e d from student representat ion, I w a n t e d to co l lect a n d analyze chi ldren's pictures; but more so, I w a n t e d to observe students wh/7e they were working to represent their thinking processes. I h o p e d to observe a n d descr ibe students' reasoning behaviours while they d rew in response to a m a t h e m a t i c a l p rob lem. Drawing as a problem-solving is the term I use to descr ibe this m a t h e m a t i c a l reasoning through the a c t of d rawing. In add i t ion , for those chi ldren w h o might a c c e s s imagery to support their representat ional activity, I w a n t e d to c r e a t e a research situation that w o u l d p romote student use of a n d talk a b o u t visualization. All this in a n effort to address the main research questions: • How d o young chi ldren a c c e s s visualization a n d imagery in their resolution of m a t h e m a t i c a l tasks? • How does drawing as problem-solving support students in mak ing sense of a n d reasoning through a m a t h e m a t i c a l p rob lem? The structure of the study through wh ich I a t t e m p t e d to answer these questions is d e s c r i b e d be low. Methods The part ic ipants for this study were 33 students from a suburban schoo l district. Two problems were p o s e d of the g r a d e 2 students w h o worked in small groups (3-4 chi ldren). Students were v i d e o - t a p e d while they worked through the tasks, both from a b o v e a n d using a h a n d - h e l d c a m e r a to c a p t u r e both the talk a n d the a c t of drawing as it h a p p e n e d . The problems were both routine a n d non-routine in nature. Students were asked to solve a sharing p rob lem (There are 18 cook ies a n d 12 chi ldren. How m a n y will e a c h chi ld get?) as well as a c o m b i n a t i o n p rob lem (18 wheels , w h a t toys c o u l d there b e ? ) . I r e a d the problems a l o u d to the students, a n d they were presented with a p i e c e of p a p e r a n d a penc i l to use. During the course of the chi ld's problem-solv ing, I p o s e d the fol lowing m e t a c o g n i t i v e questions: " C a n you tell m e what you ' re thinking? What does this part of your drawing m e a n ? " After the problems were c o m p l e t e d , I asked questions a round the a c t of d rawing, inc luding: " W h a t strategy d id you use to solve the p rob lem? How d id it help y o u ? " Students' c o m m e n t s - the questions they asked , their meta-cogn i t i ve talk, a n d their interactions with other chi ldren - were t ranscr ibed verbat im. A further l a n g u a g e sample was r e c o r d e d w h e n chi ldren were asked h o w drawing h e l p e d them solve the p rob lem. These answers h e l p e d to clarify w h a t students were thinking while they d rew a n d the d e g r e e to wh ich drawing was seen as helpful. These l a n g u a g e samples p r o v e d helpful in e laborat ing on students' reasoning through drawing - h o w they used the pictures in their heads a n d the marks on the p a g e to solve problems. Significance of the research problem This study will a t tempt to bring together my b a c k g r o u n d e x p e r i e n c e with the DISCOVER: Learner Strengths assessment, my class room b a s e d observations of visual-spatial thinking across multiple g r a d e levels, my re lated research on imagery for the Early N u m e r a c y Project a n d my informal v i d e o study of problem-solv ing strategies of Kindergarten learners. The project itself g rew out of both structured a n d informal observations of chi ldren; as such, the research has its roots in early primary classrooms. This study is significant in that it attempts to c o n n e c t research with p r a c t i c e , in c reat ing a framework for observing drawing as p r o b l e m -solving behaviours a n d a set of re lated instructional tasks. Significant too is this study's d i v e r g e n c e from previous work. While other researchers h a v e f o c u s e d on students' drawings a n d what c a n b e learned a b o u t thinking after the event , this study will focus on the act of drawing a n d students' reasoning behaviours as they d o o d l e towards a solution. Featuring two p e r f o r m a n c e - b a s e d tasks, this study presents a n opportunity to a c c e s s student thinking as it h a p p e n s ; while v ideo f o o t a g e of students d rawing a n d clips featur ing students explaining their ideas prov ide insight into their reasoning. Students' use of l a n g u a g e - a d e v e l o p m e n t a l cornerstone in early primary - has a cr i t ical role to play in this study, both as a w a y to a c c e s s thinking a n d as a means to expose chi ldren's dispositions. Unlike m a n y other studies featured in the re lated research, this o n e highlights young learners a n d their part icular a p p r o a c h to problem-solv ing - a n a p p r o a c h that c a n b e both mathemat ica l l y structured a n d visually c h a o t i c . A n examinat ion of h o w young chi ldren a c c e s s these seemingly c o m p l e m e n t a r y l o g i c a l - m a t h e m a t i c a l a n d visual-spatial strengths to solve m a t h e m a t i c a l problems m a y shed light on student c a p a c i t y a n d prov ide a w a y to support students w h o n e e d it. Structure of this document This introductory c h a p t e r has out l ined my part icular professional b a c k g r o u n d a n d how the various p ieces of my work h a v e c o m b i n e d to focus in on the a r e a of visualization a n d representat ion as they inf luence a n d support young chi ldren's m a t h e m a t i c a l problem-solv ing. C h a p t e r 2 will e x p a n d on the research re lated to these areas a n d descr ibe the extent to w h i c h my o w n research questions are only partially exp la ined through existing research. Research methods used during the course of this study will b e d e s c r i b e d in C h a p t e r 3 - including h o w I set up the problem-solv ing interviews, h o w I a n a l y z e d a n d c o l l a t e d v ideo f o o t a g e a n d h o w I sorted interview d a t a . C h a p t e r 4 presents a n analysis of this d a t a to determine h o w it is that students reason through the a c t of d rawing, a n d the extent to w h i c h imagery impacts the process. A f ramework descr ib ing character ist ics of these drawing as problem-solv ing activities a n d featur ing student samples is also i n c l u d e d in C h a p t e r 4. In C h a p t e r 5,1 discuss the results, cons ider the range of student strategies, a n d e l a b o r a t e on h o w imagery a n d systematic thinking i m p a c t student success with d rawing as p r o b l e m -solving. Implications for c lassroom p r a c t i c e a n d further research directions are presented in this final c h a p t e r as wel l . Chapter 2 - Literature Review Introduction I turned to the research in cons ider ing h o w young chi ldren a c c e s s visualization a n d imagery in their resolution of m a t h e m a t i c a l tasks, a n d in des igning a research study that w o u l d a c c e s s it. M y research quest ion was largely c o n c e r n e d with visual-spatial imagery; as such I looked to giants in the field - Owens , Thomas, Mul l igan, Y a c k e l a n d Wheat ley - all of w h o m h a v e studied the visualizing efforts of student mathemat ic ians . It b e c a m e c lear that to some d e g r e e , visual thinking was support ing student problem-solv ing. C o m p a r a t i v e studies conf i rmed this. But in addi t ion to imagery, I w a n t e d to determine h o w drawing as problem-solving supports students in making sense of a n d reasoning through a m a t h e m a t i c a l p rob lem. Research in this a r e a p r o v e d harder to f ind. A l though a b r o a d m o v e m e n t in the a r e a of student representat ion h a d y ie lded a large b o d y of work, I struggled to find studies that looked at young chi ldren, or studies that assessed student thinking in the act of thinking. I a t t e n d e d the Psychology of M a t h e m a t i c s Educat ion c o n f e r e n c e in Bergen, Norway in 2004 a n d went to every session I c o u l d find on representat ion, d iagrams, the use of pictures, but found that these studies d id not answer my questions. Instead, they addres sed issues re lated to h o w students might interpret pictures in solving problems, or the notion of explicit instruction to d e v e l o p d i a g r a m m a t i c c a p a c i t y . Largely these studies on representat ion s e e m e d f o c u s e d on assessing m a t h e m a t i c a l understanding after problem-solv ing h a d taken p l a c e . It a p p e a r e d as though I h a d a quest ion worth researching - c o n n e c t e d to previous studies but unique in its p e r f o r m a n c e - b a s e d emphasis a n d in the a g e of its part ic ipants. I w a n t e d to understand what led a chi ld to pick up a penc i l a n d beg in to d r a w in response to a p rob lem, what visual processes took p l a c e before a n d concur rent with d rawing. I w a n t e d to explore h o w visualization was c o n n e c t e d to drawing - the observab le ind icator of i m a g e - m a k i n g - to understand what Jason, the k indergarten student, was seeing in his h e a d w h e n he c h o p p e d the cook ies , what Char l ie thought in his brain. Interpretation of the research My interpretation of these other research projects follows. I h a v e a t t e m p t e d to organize the other research into sets or topics, a d d i n g a h e a d i n g to descr ibe the work where appropr ia te . General ly , the work out l ined b e l o w flows from visualization a n d d y n a m i c imagery a n d their use in m a t h e m a t i c a l problem-solv ing situations, to the analysis a n d assessment of students' m a t h e m a t i c a l understanding through their representations, both d y n a m i c a n d static in nature. Studies highlighting instructional impl ications in representat ion a n d the s t a n c e presented by the Nat ional C o u n c i l of Teachers of M a t h e m a t i c s w r a p up the c h a p t e r . Understanding of my o w n research quest ion g rew from read ing a n d ref lect ing on these works. Other questions that were raised for m e are t h r e a d e d throughout; some were answered in part during the course of my study, still others warrant further invest igation. Imagery and mathematical problem-solving Grayson Wheat ley a n d others (Wheatley & C o b b , 1990; Reynolds & Wheat ley , 1997; Thomas, Mul l igan, a n d Go ld in , 2002; Gray , Pitta, a n d Tall, 2000) h a v e re sea rched spatial sense a n d menta l imagery, a n d h a v e pos i ted a c o n n e c t i o n b e t w e e n a chi ld's c a p a c i t y to m a k e a menta l i m a g e a n d his or her ability to solve m a t h e m a t i c a l problems. Wheat ley asserts that spatial sense (or imagery) is useful in numer ical as wel l as geomet r ic settings, a n d that it is foundat iona l to m a t h e m a t i c a l reasoning (Reynolds & Wheat ley , 1997; Wheat ley , 1990; Wheat ley a n d C o b b , 1990). Wheat ley a n d C o b b (1990) descr ibe mathemat ics as the activity of c reat ing relationships, a n d assert that these relationships h a v e their roots in visual imagery . Their analysis of young chi ldren's spatial constructions outlines the processes chi ldren e m p l o y in the solution of spatial problems, a n d points to d y n a m i c imagery as a powerful tool . Dynamic imagery, the ability to m a k e a n d man ipu late a menta l i m a g e , is seen to b e key in problem-solv ing, lead ing a chi ld to think more flexibly in response to a p rob lem. Wheat ley a n d C o b b state that: M a t h e m a t i c a l p rob lem solving is often a matter of reasoning analyt ical ly , construct ing a n i m a g e , using the i m a g e to support add i t iona l c o n c e p t u a l reasoning. . . a process of bui lding from images to analysis a n d analysis to images [that] m a y cont inue through m a n y cyc les . (1990, p. 161) Wheat ley d e v e l o p e d a series of activities d e s i g n e d to improve menta l imagery , a n d p r o p o s e d that this in turn w o u l d increase student success w h e n a p p r o a c h i n g m a t h e m a t i c a l problem-solv ing tasks (Yackel & Wheat ley , 1990). These tasks, i n c l u d e d in a n instructional p rogram c a l l e d " I m a g e - M a k e r " , involve chi ldren of all a g e s . O n e of a series of s imple line drawings is presented to chi ldren for a few seconds , a n d chi ldren are then asked to " d r a w what they s a w " . The representations students m a k e are then shown a n d the images students c r e a t e "in the mind's e y e " are shared a l o u d . Speci f ic l a n g u a g e re lated to geomet ry a n d transformations e m e r g e . Through this activity, it is h o p e d that students will d e v e l o p the c a p a c i t y to c r e a t e a n d re-present a n i m a g e a n d d e v e l o p the l a n g u a g e to descr ibe menta l images (Yackel & Wheat ley , 1990). I h a v e used these Image-Maker tasks with chi ldren a n d adults al ike, a n d h a v e seen them publ i shed in t e a c h e r resource materials (Buschman, 2003) where they are r e c o m m e n d e d for promot ing l a n g u a g e a n d visual-spat ial thinking in students in the primary grades . They prov ide a powerfu l c o n n e c t i o n point for chi ldren w h o h a v e visual-spatial strengths - s ince be ing a g o o d Image-Maker does not require that a student b e artistic, but visually a d e p t in terms of construct ing menta l images . Wheat ley a n d Reynolds (1997) also suggest that if teachers c a n support students in the use of d y n a m i c imagery, chi ldren will b e more powerful m a t h e m a t i c a l problem-solvers; that cogni t ive structures are e n h a n c e d , a n d the d e v e l o p m e n t of m a t h e m a t i c a l reasoning progresses more fluidly. The Image-Maker p rogram is d e s i g n e d to improve students' visual imagery (Yackel & Wheat ley , 1990), but in their research, a correlat ion b e t w e e n the p r a c t i c e of i m a g e mak ing a n d the d e v e l o p m e n t of m a t h e m a t i c a l reasoning were not m a d e explicit. Understanding h o w this kind of thinking deve lops required a more theoret ical f rame. I turned to the work of Kay Owens for that f ramework. Kay D. Owens has written a n d c o - a u t h o r e d volumes of work on visualization a n d spat io -mathemat ica l p rob lem solving as they relate to p r imary -aged chi ldren. Her research has b e e n extensive, long-term a n d involves large samples of chi ldren; her findings are suppor ted by both qual i tat ive a n d quant i tat ive d a t a . She has d e v e l o p e d a n d ref ined a f ramework for assessing a n d descr ib ing 5 different kinds of imagist ic process ing, a n d offers tasks to assess a chi ld's level of funct ioning. She emphasizes the role of visual imagery in "establ ishing the m e a n i n g of the p rob lem, in channe l ing the problem-solv ing a p p r o a c h e s of the students, a n d in inf luencing the students' cogni t ive construct ions" (Owens & C l e m e n t , 1998, p.216). This art icle was rich with c o m p i l e d work, summarizing a wea l th of research a n d a theoret ical f ramework for assessing a n d understanding students' c a p a c i t y to solve spatial problems using imagist ic process ing. O w e n s ' work with young students has f o c u s e d on students' construct ion of understanding through l a n g u a g e a n d constructivist exper iences . In 1996 she a n d Outhred invest igated h o w older students construct a n understanding of a r e a through unit grids; in her 1998 study of chi ldren's exper iences of tiling areas, she a n d Clements obse rved students cover ing shapes by c o m p a r i n g the lengths of sides a n d different angles by over laying c a r d - c u t o u t representations. In these two studies, Owens makes connect ions b e t w e e n students' m o v e m e n t of objects to find a r e a , a n d tessellation work - the latter requiring more d y n a m i c use of spatial problem-solv ing strategies. In a later art icle, (1999) Owens , Leberne a n d Harrison e l a b o r a t e on this notion of d y n a m i c imagery as a strategy for spatial problem-solv ing. This art icle p r o v e d extremely helpful in beginn ing to def ine a n d c o m p a r e types of imagery - namely static a n d d y n a m i c imagery. Owens et a l , in speak ing of the young student's d e v e l o p m e n t of geomet r ic understandings, defines three important aspects that contr ibute to this d e v e l o p m e n t : or ientation a n d mot ion, part -whole recogni t ion a n d classif ication a n d l a n g u a g e (1999, p. 29). In part icular, the notion of or ientation a n d mot ion were interesting to m e in cons ider ing h o w chi ldren d e v e l o p ef fect ive structures for solving problems using imagery . Owens a n d her co l leagues distinguish b e t w e e n static pictor ial imagery - in w h i c h the pictures in the h e a d d o not c h a n g e - a n d d y n a m i c imagery in w h i c h multiple examples exist a n d are c h a n g e a b l e . The w a y a young chi ld acqui res d y n a m i c imagery, says Owens , Leberne a n d Harrison (1999), is through a c t i v e investigation - the construct ion of understanding through explorat ion. "Through a c t i v e invest igation, imagery is more likely to b e d y n a m i c or representat ive of a pattern, relationship or rule." (1999, p. 27). This s e e m e d to suggest that for y o u n g chi ldren, the w a y to d e v e l o p d y n a m i c imagery that supports pat te rn -based m a t h e m a t i c a l thinking is through a c t i v e explorat ion, rather than through direct t e a c h i n g . D iezmann a n d English (2001) offer a different perspect ive later in this c h a p t e r . Dynamic imagery and mathematics achievement: Comparative studies So what does imagery look like in chi ldren? How does it i m p a c t student p e r f o r m a n c e ? If, a c c o r d i n g to Owens et al (1999) d y n a m i c imagery is more flexible a n d c h a n g e a b l e than static imagery, a n d students w h o possess d y n a m i c imagery t e n d to c r e a t e images that are more representat ive of a rule or pat te rn -based understanding, d o students w h o use d y n a m i c imagery out-perform those with static pictor ial imagery? The fol lowing studies descr ibe c o m p a r a t i v e projects to this ef fect . Thomas, Mul l igan, a n d Go ld in (2002) e x a m i n e d the d e v e l o p m e n t a l s e q u e n c e through wh ich e l e m e n t a r y - a g e d chi ldren progress in the internalization of the number system (the count ing s e q u e n c e from 1-100) a n d pos i ted that this d e v e l o p m e n t c o u l d b e witnessed in y o u n g chi ldren's external representations of the number system. The authors sought to invest igate the speci f ic role of imagery in representations a n d the construct ion of relat ional understanding in mathemat ics . High ability chi ldren (classified gifted) were interv iewed a n d c o m p a r e d to those of a v e r a g e ability; distinctions were m a d e b e t w e e n the types of imagery (the internal representations of number) used by both groups. High ability chi ldren t e n d e d to use d y n a m i c , fluid a n d c h a n g i n g imagery to descr ibe number , demonst rat ing flexibility in their understanding. A v e r a g e ability students, by contrast, d i sp layed more static visualizations of number - their images were c o n c r e t e , a n d less mobi le . The authors state that var ied exper iences of the number system a n d a n array of contexts for imag ing a n d visualizing number relationships significantly d e e p e n number c o n c e p t s a n d a d d " m o v e m e n t " to the images c r e a t e d by chi ldren (p.129); that powerful problem-solvers utilize d y n a m i c imagery in solution f inding, by manipu lat ing internal representations of the b a s e ten system in flexible ways . Gray , Pitta, a n d Tall also c o m p a r e d two groups of students - low a n d high achievers - in their study (2000). Chi ldren a g e d 8-12 were g iven a series of menta l ar i thmetic problems to solve a n d were asked to descr ibe " w h a t c a m e to m i n d " as they worked through the problems, descr ib ing their strategies a n d images . In solving the ar ithmetic problems, low achievers cons t ructed images that were story-based, ep i sod ic a n d rich in detai l , e v e n colour . Chi ldren in this group car r ied out "p rocedures in the m i n d " a n d d e s c r i b e d the numbers as " sp inn ing" a n d "in a jumble " (p.14). By contrast, high achievers reported images that were more abst ract , a n d featu red relat ional understandings (p. 12). They d e s c r i b e d symbol ic images a n d numbers " f lash ing" in their heads - as though they were be ing presented with the answer. For these chi ldren, images of number a n d number relationships were generat ive a n d meaningfu l . Gray , Pitta, a n d Tall (2000) a n d Thomas, Mul l igan, a n d Go ld in (2002) a g r e e that spatial patterns a n d visual imagery p lay a n important role in establishing relationships b e t w e e n numbers, a n d help to solidify part-who le thinking. They c o n c u r with other researchers that imagery c a n b e static or d y n a m i c in nature (Gray, Pitta, & Tall, 2000; Owens & C l e m e n t , 1998; Owens , M i tche lmore, Outhred, & P e g g , 1996; Owens , 1999; Wheat ley & C o b b , 1990). G r a y et al (2000) a n d Thomas et al (2002) m o v e b e y o n d O w e n s ' 1999 study to suggest that imagery - a chi ld's abil ity to bui ld, re-present a n d transform images - c a n b e a p p l i e d to other problems, e v e n those that are not geomet r ic in nature. Higher a c h i e v i n g m a t h e m a t i c a l p rob lem solvers, say G r a y a n d Thomas, tend to use d y n a m i c a n d flexible imagery in their solution of problems, whi le those w h o use static or u n c h a n g i n g images tend to rely on a n d over -app ly p r o c e d u r a l understandings (Gray, Pitta & Tall, 2000; Thomas, Mul l igan & Go ld in , 2002). Despite the d i f fe rence in the a g e s of the students involved in these studies (Gray, Pitta & Tall, 2000; Thomas, Mul l igan & Go ld in , 2002; Owens , Leberne & Harrison, 1999), there was cons is tency seen in the results. These researchers seem to a g r e e within the realm of their research projects, that d y n a m i c imagery is a powerful m a t h e m a t i c a l thinking tool . This m a d e m e recons ider my thinking a b o u t c o n c e p t u a l a n d p rocedura l k n o w l e d g e in mathemat ics . Perhaps c o n c e p t u a l understanding is der ived from d y n a m i c , c h a n g i n g images , while p rocedura l k n o w l e d g e c o m e s from static a n d inflexible ones. I b e g a n to w o n d e r then if there was a w a y to a d d m o v e m e n t to these otherwise static images - to m a k e d y n a m i c that w h i c h was u n c h a n g i n g a n d support students w h o struggle. D iezmann brings up the issue of struggling students in her art icle, " M a k i n g sense with d iagrams: Students' difficulties with feature-similar problems"(2000). Visualization and diagrammatic capacity Diezmann (2000) makes connect ions b e t w e e n d i a g r a m m a t i c representations a n d spatial a n d number sense. She proposes that students w h o struggle to represent their thinking d iag rammat ica l l y m a y also struggle with either (or both) spatial sense or number sense; that the inability to visualize a m a t h e m a t i c a l situation affects a student's ability to represent that situation in a d i a g r a m . D iezmann writes: Students' difficulties a n d errors in generat ing a c c u r a t e a n d ef fect ive d iagrams are general ly a s soc ia ted with students' lack of expertise in d i a g r a m m a t i c representat ion. However the results of this cross-study compar i son suggests that ef fect ive d i a g r a m m a t i c representat ion also d e p e n d s on a sound m a t h e m a t i c a l k n o w l e d g e b a s e , w h i c h includes sense-making in m a t h e m a t i c a l situations, (p. 233) If not all students c a n a c c u r a t e l y g e n e r a t e ef fect ive d iagrams, as suggested in Diezmann's study, what are the impl ications for d rawing as problem-solv ing c a p a c i t y ? Are there students for w h o m drawing as problem-solv ing is too difficult or not access ib le by virtue of a shaky m a t h e m a t i c a l foundat ion? Assessing image-making; the complexity of language Diezmann (2000) states that representations funct ion as both a student response as wel l as a n indicator of emerg ing m a t h e m a t i c a l understandings; that is, a ch i ld c a n represent a n answer through a d i a g r a m (a representat ion or picture) but also that the chi ld's m a t h e m a t i c a l understandings c a n b e inferred from these artifacts. In contrast, this study proposes that the a c t of d rawing itself presents a rich opportunity to observe a n d assess student thinking in the a c t of thinking. While research into representat ion highlights the many ways students c a n show what they know a n d c a n d o , the study of d rawing as p r o b l e m -solving provides p e r f o r m a n c e assessment d a t a - a w i n d o w into the m a t h e m a t i c a l activity in progress. Measur ing menta l imagery is difficult, s ince it relies on a student's ability to use l a n g u a g e to descr ibe the i m a g e a n d process be ing under taken mental ly. These " internal representat ional c a p a c i t i e s " must b e inferred (Goldin, 1996), as they p r o d u c e no formal p roduct to b e e x a m i n e d a n d d e s c r i b e d . Rather, it is the talk a b o u t these images a n d the internal processes that is the focus of study. A young chi ld's c a p a c i t y to descr ibe the pictures in his or her h e a d - i n d e e d to b e metacogn i t i ve at all - poses a part icular c h a l l e n g e for researchers looking to examine menta l imagery as a factor in m a t h e m a t i c a l thinking, particularly in very y o u n g learners. Without a formal p roduct or l a n g u a g e to descr ibe the process of thinking, e v i d e n c e of m a t h e m a t i c a l reasoning through imagery must b e g e n e r a t e d through carefu l , informed observat ion on the part of researchers. A n d without l a n g u a g e to descr ibe, explain a n d reflect on thinking, does student learning c o a l e s c e ? Sue Gifford (2005) discusses the relationships that exist b e t w e e n l a n g u a g e , visual i m a g e - m a k i n g a n d n u m e r a c y a n d makes r e c o m m e n d a t i o n s for students w h o struggle to mathemat ica l l y . Spatial-visualizing capacity and number understanding L a n g u a g e is featu red in the work of Sue Gifford (2005). In her review of research re lated to dysca lcu l ia (the impai red ability to solve m a t h e m a t i c a l problems) in y o u n g chi ldren, Gifford speaks to the inter-relatedness of l a n g u a g e , visualizing a n d the d e v e l o p m e n t of number sense. However , number representat ion also d e p e n d s on l a n g u a g e in a c o m p l e x w a y , a n d on the d e v e l o p m e n t of symbolis ing. ... A c o m p l e x picture emerges , w h e r e b y visualising, f inger use, l a n g u a g e a n d symbol ic representations of numbers d e v e l o p interdependent ly , suggest ing that various areas of the brain work together to d e v e l o p number understanding, (p. 38) She highlights the in te rdependent processes of the d e v e l o p i n g brain, a n d proposes a mult i -modal intervention strategy including support with visual reasoning for students w h o are d i a g n o s e d with dysca lcu l ia . Specif ical ly , she r e c o m m e n d s c reat ing networks of images a n d vivid associations to support memory a n d mean ing-mak ing (p. 50). She proposes a focus on t e a c h i n g to students' strengths, in part icular visual c a p a c i t y , wh ich emerges early in young chi ldren: "It seems that young chi ldren first solve number problems by visual images , a n d so a spatial visualising c a p a c i t y is i n d i c a t e d . " (p. 37) I a p p r e c i a t e d two things a b o u t Gifford's research. First, she promotes t e a c h i n g to a student's a r e a of strength. This is consistent with the phi losophy beh ind the DISCOVER: Learner Strengths assessment a n d its instructional c o m p o n e n t , in w h i c h address ing areas of n e e d is a c c o m p l i s h e d through a chi ld's a r e a of greatest strength. In our district, for e x a m p l e , l a n g u a g e d e v e l o p m e n t is p r o m o t e d by e n g a g i n g y o u n g learners in visual-spatial, log ica l m a t h e m a t i c a l tasks - areas of relative strengths for t h e m . Gifford suggests a similar route for support ing chi ldren w h o struggle, by a c c e s s i n g visual c a p a c i t y to address m a t h e m a t i c a l difficulty. S e c o n d , Gifford speaks to the creat ion of networks of images a n d vivid associations to support m e a n i n g - m a k i n g . Her means of support ing chi ldren w h o struggle takes the form of a b r o a d - b a s e d , mult i -modal a p p r o a c h in w h i c h visual images a n d l a n g u a g e are key. This rings of O w e n s ' work (Owens, Leberne a n d Hamilton, 1999), in w h i c h d y n a m i c imagery is d e f i n e d as flexible, mov ing a n d c o n c e r n e d with relationships a n d patterns. It seems as though Gifford's a p p r o a c h to support ing students with difficulty a d v o c a t e s for the same flexible, sense-making network as Owens suggests exists for those students w h o d e v e l o p d y n a m i c imagery . So h o w d o these images - a n d more importantly, these systems of images - d e v e l o p a n d link to c o n c e p t s ? Dav id Tall outlines o n e theory. Visualization and Symbolism in Mathematics In his 1994 address to the Commission Internationale pour I'Etude et I'Amelioration de I'Ensignement des Mathematiques in Toulouse, Dav id Tall e x t e n d e d Bruner's theory of representat ion to cons ider students visualization a n d conceptua l i z ing of symbols. Tall found that chi ldren use number (the symbol for the numeral) to represent the process of count ing a n d also to symbolize the c o n c e p t of that number; that operat ional ly chi ldren over lap c o n c e p t a n d process within symbol ic representat ion. Tall a n d G r a y c o i n e d the term " p r o c e p t " to descr ibe the students' encapsu la t ion of the c o n c e p t a n d process into o n e symbol ic representat ion. A l though he speaks here a b o u t formal symbols (numerals, signs for operat ions, a l g e b r a i c symbols a n d Euc l idean formulae), I w o n d e r e d at the appl icabi l i ty of Tall a n d Gray ' s notion of p r o c e p t to drawing as problem-solv ing representations. The symbol c a n b e spoken, h e a r d , seen a n d r e a d a n d the c o m b i n a t i o n of these sensory percept ions a n d act ions gives it a cogni t ive ex is tence as a m a t h e m a t i c a l ob ject . But it is more powerful than that - the process c a n b e used to d o mathemat ics a n d the ob ject c a n b e used to think a b o u t it. (p. 2) In terms of drawing as problem-solv ing, the representat ion itself - the tally, the st ick-man, the c o o k i e - stands for both the ob ject a n d the a c t i o n be ing per fo rmed on it. The duality of the representat ion as both a w a y to do the math a n d a w a y to think a b o u t the m a t h poses a n interesting c o n n e c t i o n to my masters research project, in w h i c h students' visual images are c o n v e r t e d to representations a n d then o p e r a t e d on by w a y of reasoning through the problems presented. Drawings as dynamic representations Operat ing on representations to reason through a p rob lem suggests that drawings - the symbol - are a w a y to d o a n d think a b o u t the m a t h , a n d not simply a w a y to record a response. When chi ldren d r a w in response to a m a t h e m a t i c a l p rob lem, there is more go ing on than simply representing a solution; Tall maintains that a chi ld's interact ion with the symbol on the p a g e is far more a c t i v e , more d y n a m i c than that. The c reat ion of these representations on the p a g e a p p e a r s to h a v e a d y n a m i c , cognit ively involved c o m p o n e n t . Wo leck (2001) a n d Smith (2003) present a similar perspect ive on drawing in math class. They maintain that drawing supports chi ldren in model ing a p rob lem a n d therefore in arriving at a solution for it (Smith, 2003, Woleck, 2001). Kristine R e e d Wo leck writes from a c lass room-teacher/ researcher perspect ive in her art icle "Listen to their Pictures: A n investigation of Chi ldren's M a t h e m a t i c a l Drawings" (2001, p. 215-227). She descr ibes her g r a d e o n e students' drawings in response to math problems a n d explains that these d y n a m i c representations signify a n essential cogn i t ive br idge b e t w e e n the c o n c r e t e a n d the abst ract in mathemat ics . As chi ldren strip a w a y the unnecessary features of a p rob lem, removing all but the essential e lements of the problem's structure, they b e c o m e genera l i zab le to more than the single context in wh ich they are be ing used. Rather than drawing p e o p l e , c l o t h e d a n d s e a t e d in bus seats to represent h o w m a n y buses are n e e d e d to a c c o m m o d a t e the class for a field trip, a ch i ld using a more abst ract representation m a y opt for stick figures or e n c i r c l e d tally marks to ind icate groups of chi ldren a n d bus seats. These latter forms of representat ion a l low for more flexibility of use - the essential feature of divisive thinking is ma in ta ined, a n d less attent ion is p a i d to the context of the p rob lem. I w o n d e r though, what is it that leads a chi ld to strip a w a y the unnecessary features of a p rob lem over t ime, a n d represent it in a less idiosyncrat ic w a y ? Or c a n pictor ial , idiosyncrat ic representations support learning b e y o n d early c h i l d h o o d ? A l though Woleck a c k n o w l e d g e s that chi ldren m a y use drawings as scaf fo ld for thinking, a n d mentions specif ical ly the use of pictures as a p l a c e - h o l d e r or pictorial manipulat ives to b e c o u n t e d during p rob lem-solving, the focus of her act ion- research is on the use of drawings to c o m m u n i c a t e m a t h e m a t i c a l ideas after problem-solv ing. My research quest ion is more c o n c e r n e d with Tail's notion (1994) of doing a n d thinking a b o u t the math in the generat ive sense. That is, h o w d o chi ldren create symbol ic representations that a l low them to act ive ly e n g a g e with the images in the mind's e y e as they a re be ing translated to the p a g e ? What does this process look like, a n d h o w does it d e v e l o p ? Drawings as manipulatives Woleck (2001) a n d Smith (2003) d id prov ide some insight into h o w y o u n g chi ldren interpret a n d use their representations in a m a t h e m a t i c a l sense. In S tephen Smith's study of third graders ' response to " the c a n d y bar p r o b l e m " , he n o t e d that e a c h of the chi ldren involved in the study d rew -both to man ipu late the objects in the p rob lem (drawing as p rob lem-solving) a n d to represent their thinking afterwards (drawing of p rob lem-solving). He n o t e d that the chi ldren's use of drawings dif fered in terms of their idiosyncratic nature - two of the chi ldren used a n d rel ied on the context of the p rob lem in their c o m m u n i c a t i o n of both their m a t h e m a t i c a l processes a n d the final p rob lem product , whi le two others t e n d e d to strip a w a y context a n d represent their thinking in less idiosyncrat ic ways . Smith emphasizes that the artifacts chi ldren p r o d u c e in solving m a t h e m a t i c a l problems - including l a n g u a g e , drawings a n d constructions - c a n n o t b e cons ide red separate from the students' m a t h e m a t i c a l reasoning. Here, a n d in Woleck ' s 2001 study, these researchers suggest that y o u n g chi ldren use drawings in the s a m e w a y as a manipulat ive - that the drawing serves as a perfect ly des igned m a t h e m a t i c a l tool to represent w h a t e v e r is be ing m a n i p u l a t e d in the p rob lem. Unlike tradit ional manipulat ives (like Cuisinaire Rods) that h a v e b e e n cons t ructed by those w h o h a v e a firm grasp of a c o n c e p t to represent that c o n c e p t , a d rawing is uniquely m a t c h e d to the p rob lem a n d the p rob lem context . The d e g r e e of abst ract ion from the a c t u a l ob ject is quite low, a n d as such, these drawings serve the purpose of taking chi ldren from c o n c r e t e representations to more abst ract ones, such as structured d iagrams or equat ions . In terms of my study, the notion of drawings as a manipulat ive p r o v e d to b e a helpful w a y of cons ider ing young chi ldren's mean ing-mak ing efforts, a purpose for their interact ion with the representat ion they c r e a t e d . Virtual manipulat ives, a term c o i n e d later in my project, was b a s e d on Wo leck (2001) a n d Smith's (2003) work, with a more d y n a m i c spin sugges ted in the qualif ier "v i r tual" , cal l ing to mind the d y n a m i c imagery a n d mean ing-mak ing work of Owens (Owens a n d Clements,1998; Owens , Leberne a n d Hamilton, 1999), Wheat ley (1990) a n d Tall (1994). Explicit instruction in drawing to problem-solve For Smith (2003) a n d Wo leck (2001), pictures are seen as o n e c o m p o n e n t of m a t h e m a t i c a l representat ion in genera l , wh ich includes oral l a n g u a g e , written words a n d numbers. These drawings are g e n e r a t e d by chi ldren, represent understandings a n d are purposeful . Other researchers cons ider that idiosyncrat ic d rawing serves little purpose, c louds the structure of a prob lem a n d detracts from the big m a t h e m a t i c a l ideas . Like manipulat ives, they suggest that these drawings b e d i s c a r d e d as soon as possible a n d a n efficient a n d s tandard ized structure for representing a n d organiz ing information b e a d o p t e d . In Cypress, for e x a m p l e , this is precisely w h a t h a p p e n s at 4 t h g r a d e , w h e n chi ldren are taught structured d iagrams to m a t c h the operat ions a n d are d i s c o u r a g e d from using a n y other form of representat ion (Pantziara, Gagats i s , & Pitta-Pantazi, 2004). Current Cypr ian research in the use of visual images to support m a t h e m a t i c a l understanding focuses on the read ing of pictures - the interpretation of a n i m a g e a n d the d e g r e e to wh ich the i m a g e supports or det racts from a chi ld's c a p a c i t y to problem-solve (Elia & Phi l ippou, 2004). Focused largely on intermediate aged-ch i ld ren ' s interpretations of images , the research task does not ask chi ldren to solve m a t h e m a t i c a l problems by drawing, but rather by read ing information presented pictorially. In terms of i m a g e product ion , the research repor ted that i n t e r m e d i a t e - a g e d chi ldren were taught to use very speci f ic s c h e m a for representing m a t h e m a t i c a l operat ions; however , the research does not address a chi ld's spontaneous a n d cons t ructed representations or the d e v e l o p m e n t of these images over t ime a n d with exper ience . v a n G a r d e r e n a n d M o n t a g u e ' s study of g r a d e 6 students of varying abilities (students with learning disabilities, a v e r a g e achievers a n d gi f ted learners) descr ibes students' visual-spatial representations during m a t h e m a t i c a l problem-solv ing (2003). The researchers found that chi ldren with learning disabilities t e n d e d to d r a w primarily pictor ial representations that illustrated persons, p laces , or things d e s c r i b e d in the p rob lem, whi le gi f ted students t e n d e d to c r e a t e representations that illustrated relationships b e t w e e n elements of the p rob lem. O n e wonders whether this d e p e n d e n c e on pictorial representations is b a s e d on students' lack of sense-making of the m a t h e m a t i c a l situation (Diezmann, 2000), or whether this strategy, like count ing all for very young chi ldren, was the only o n e ava i lab le to a n d a c c e s s e d by these students. Like count ing all, the use of pictor ial representations - a l though less eff icient a n d with greater potent ia l for i n a c c u r a c y - c a n still g e n e r a t e a solution. It is this sort of result, however , that leads researchers to seek to support students in mov ing b e y o n d their cons t ructed idiosyncratic or pictor ial drawings to more structured a n d formal representations. Novick, Hurley a n d Francis (1999) h a v e def ined 4 genera l -purpose d iagrams that c a n b e used to m a t c h the structure of any g iven p rob lem; D iezmann a n d English (2001) a n d Diezmann (2006) descr ibe ways that teachers c a n support their students in understanding the relationship b e t w e e n a p rob lem a n d the d i a g r a m n e e d e d (2001, p. 86-87, 2006, p. 2-439). D iezmann a n d English p romote d i a g r a m l iteracy - " the ability to understand a n d use a n d to think a n d learn in terms of i m a g e s " in their study of 10 year o ld mathemat ic s students (p. 77). The authors' desire to c o n n e c t i m a g e -making a n d m a t h e m a t i c a l understanding is consistent with the work of Wheat ley a n d C o b b (1990), Y a c k e l a n d Wheat ley (1990) a n d Owens a n d Clements (1998) a m o n g others. A n important distinction is the app l icat ion of Diezmann's research findings. While D iezmann a n d English highlight the i m p o r t a n c e of talk a n d model ing w h e n working with d iagrams, they also r e c o m m e n d that students b e explicitly taught these four speci f ic d iagrams a n d w h e n to a p p l y them, a n d should p r a c t i c e them within structured p rob lem sets. They maintain that students should b e presented with sets of problems that c a n b e represented with a part icular d i a g r a m a n d that the relationship b e t w e e n a p rob lem a n d its cor responding d i a g r a m should b e explicitly taught . I be l ieve that Owens , Leberne a n d Harrison (1999) w o u l d d i sagree with this a p p r o a c h . Their research supports the d e v e l o p m e n t of d y n a m i c imagery, flexible networks of spat ial understanding through explorat ion a n d investigation - particularly for young students. A l though the part icipants in the Diezmann study (Diezmann a n d English, 2001) are 10 years o ld a n d a p p r o a c h i n g a d e v e l o p m e n t a l s tage at w h i c h these relationships c o u l d b e in t roduced a n d discussed, caut ion must b e exerc i sed w h e n a c c e l e r a t i n g this d e v e l o p m e n t a l process for younger students. The desire to support d i a g r a m l iteracy (Nickerson, 1994) in e lementary s c h o o l - a g e d students should not s u p e r c e d e the natural d e v e l o p m e n t a n d cons t ructed mean ing-mak ing for young chi ldren p r o m o t e d by the NCTM a n d researchers like Woleck (2001) a n d Smith (2003). Smith (2003) a c k n o w l e d g e s that chi ldren n e e d to br idge from idiosyncrat ic to m a t h e m a t i c a l representations in order to understand a n d c o m m u n i c a t e mathemat ics (p.273) a n d makes the c a s e for sharing, discussing a n d analyz ing peers ' solutions as a deve lopmenta l l y appropr ia te w a y to br idge from o n e to the other. Reform and representation: Valuing drawing as mathematical reasoning A c k n o w l e d g i n g the n e e d to d e v e l o p l a n g u a g e to descr ibe m a t h e m a t i c a l processes a n d thinking, proponents of the reform m o v e m e n t in mathemat ics e d u c a t i o n h a v e highl ighted the s ign i f icance of representat ion a n d c o m m u n i c a t i o n in mathemat ics . The NCTM Principles and Standards (2000) m a k e this explicit. Teachers are d i r e c t e d to support chi ldren's use of representations to m o d e l a n d solve problems, a n d to p romote the use of representations to organize, record a n d c o m m u n i c a t e m a t h e m a t i c a l ideas . The c o n c e r n here is this: a l though the NCTM e n c o u r a g e s teachers to h a v e their chi ldren represent their thinking with manipulat ives a n d with drawings, these drawings m a y b e seen in tradit ional math classrooms to b e superfluous p ieces that det ract from the purpose of problem-solv ing rather than support ing it. Like the use of manipulat ives, pictures c a n b e interpreted as a temporary c ru tch , to b e d i s c a r d e d as quickly as possible in favour of more s tandard representations a n d structures. In my e x p e r i e n c e as Curr iculum Coord inator , more tradit ional math classrooms s e e m to emphas i ze the use of abst ract a n d genera l representations over the idiosyncrat ic ones e m p l o y e d by young chi ldren. The N C T M Principles and Standards g o on to say that a constructivist s t a n c e should b e a d o p t e d w h e n cons ider ing students' drawings in math class, s ince a young chi ld's c a p a c i t y to represent m a t h e m a t i c a l ideas deve lops over t ime. With e x p e r i e n c e a n d a n e d u c a t i o n a l p rogram that includes sharing a n d talk, a chi ld's representations of m a t h e m a t i c a l ideas will d e v e l o p a n d b e c o m e less idiosyncratic a n d more genera l i zab le (Smith, 2003). Most e d u c a t i o n a l researchers e m b r a c e this d e v e l o p m e n t a l not ion; h o w it is interpreted a n d i m p l e m e n t e d , however , varies. Summary My research journey a l l o w e d m e to refine my study, both in terms of its intent a n d the nature of the questions addres sed within it. Wonder ing h o w young children access visualization and imagery in their resolution of mathematical tasks l e a d m e to the work of Owens a n d Outh red (1996), Owens a n d Clements (1998) a n d Owens , Leberne a n d Hamilton (1999), Wheat ley a n d C o b b (1990), Reynolds & Wheat ley (1997) a n d Thomas, Mul l igan a n d Go ld in , (2002). Their studies conf i rmed that imagery is a c c e s s e d in the solution of m a t h e m a t i c a l problems - both those that are g e o m e t r i c a n d numer ical in nature. In fact , d y n a m i c imagery contr ibutes to the d e v e l o p m e n t of number sense, in al lowing for flexibility in thinking a b o u t constructions of number (Thomas, Mul l igan a n d Go ld in , 2002). Flexibility a n d m o v e m e n t in imagery are key to success with p r o b l e m -solving, as obse rved in the p e r f o r m a n c e of students of various abilities (Gray, Pitta & Tall, 2000; Thomas, Mul l igan & Go ld in , 2002). Higher a c h i e v i n g m a t h e m a t i c a l p rob lem solvers tend to use d y n a m i c a n d flexible imagery in their solution of problems, while those w h o use static or u n c h a n g i n g images t e n d also to rely on p rocedura l k n o w l e d g e (Gray, Pitta & Tall, 2000; Thomas, Mul l igan & Go ld in , 2002). Gifford's work (2005) p rov ided a summary of ways to address the needs of students with dysca lcu l ia , r e c o m m e n d i n g , like G r a y et al (2000), Thomas et al (2002), that support with visual thinking a n d the d e v e l o p m e n t of d y n a m i c imagery w o u l d inf luence m a t h e m a t i c a l undeers tanding. This p rov ided conf i rmat ion for the di rect ion of my study; if I c o u l d observe a n d recogn i ze d y n a m i c imagery in a c t i o n - a c k n o w l e d g e d as a powerful contr ibutor to m a t h e m a t i c a l reasoning a n d number s e n s e - t h e n perhaps I c o u l d beg in to answer the quest ion of how y o u n g chi ldren a c c e s s visualization a n d imagery in their resolution of m a t h e m a t i c a l tasks. My s e c o n d research quest ion asked : " H o w does drawing as problem-solving support students in mak ing sense of a n d reasoning through a m a t h e m a t i c a l p r o b l e m ? " . For support in d e v e l o p i n g my o w n understanding a n d in des igning a study to beg in to answer this quest ion, I looked to research on representat ion. M u c h of this research concerns itself with the assessment of student thinking after problem-solv ing h a d taken p l a c e a n d so the notion of reasoning through a p rob lem was not clar i f ied for m e through these works. Nonetheless, I d id see connect ions through the notion of drawings as manipulat ives (Smith, 2003; Woleck , 2001) a n d the fascinat ing c o n c e p t of the p r o c e p t (the encapsu la t ion of the symbol as a w a y to both d o a n d think a b o u t the math) suggested by Tall a n d G r a y (1994). These three studies presented a w a y to cons ider representations as more a c t i v e - more d y n a m i c in their use in p r o b l e m -solving. Further in my survey of the literature on representat ion, I e n c o u n t e r e d a w a y to descr ibe a n d distinguish b e t w e e n the types of students' representations. Students' representat ion show different stages: idiosyncrat ic or pictor ial , in w h i c h the level of abst ract ion b e t w e e n the picture on the p a g e a n d the context of the prob lem is quite low, a n d abst ract representations that are more genera l i zable. (Woleck, 2001; Smith, 2003; v a n G a r d e r e n a n d M o n t a g u e , 2003). In my explorat ion, I felt c o n c e r n regard ing the instructional directions suggested by certa in researchers (Diezmann a n d English, 2001; Novick, Hurley, Francis, 1999; Nickerson, 1994). In a n effort to support chi ldren in d e v e l o p i n g the most eff icient representat ional strategies possible, some suggest di rect instruction a n d p r a c t i c e , while others pose a more constructivist a p p r o a c h (Gifford, 2005; Owens , Leberne a n d Hamilton, 1999, NCTM, 2000). M o r e than the abst ract ion of the marks on the p a g e , attent ion needs to b e p a i d to h o w students are using their representations, doing a n d thinking a b o u t the m a t h . A l though research into representat ion provides some clues into h o w students solve problems a n d c o m m u n i c a t e m a t h e m a t i c a l understanding, this study proposes pay ing attent ion to the processes that p r e c e d e , or take p l a c e concur rent with the product ion of these representations. Contribution to a current body of research Research in the a r e a of young chi ldren's d rawing as p rob lem solving is l imited. While there is a c k n o w l e d g e m e n t that drawings assist chi ldren in representing their m a t h e m a t i c a l thinking a n d that this form of representat ion has merit (Smith, 2003), there is little research f o c u s e d on h o w representat ional c a p a c i t y deve lops in young chi ldren, or what imagist ic thinking transpires while d rawing. In my project, I h o p e to link research on imagery - particularly d y n a m i c imagery - a n d research on representat ion, to cons ider h o w students' visual images are c o n v e r t e d to representations a n d then (or concurrently) o p e r a t e d on by w a y of reasoning through the problems presented. The h o p e is that this study might contr ibute to a larger b o d y of research re lated to the representat ion of m a t h e m a t i c a l thinking, a n d to fill a g a p in cons ider ing h o w the process of d rawing - not simply its p roduct - has a cent ra l role to p lay in fostering chi ldren's m a t h e m a t i c a l reasoning. Kyriakides (2006) concurs . In his art icle entit led " M o d e l i n g Fractions with A r e a : the Sa l ience of Vert ical Part it ioning", he argues that w e as mathemat ic s researchers a n d educator s should " . . . d r a w our attent ion not so m u c h on the images (either menta l or concrete) but through the  i m a g e s " in order to understond the regsoning chi ldren d o in the o c t of p rob lem solving. Quot ing John M e s o n , he offirms thgt mgthemgt ics images 'a re not mere marks on p a p e r but ind icate or speak to entities that are almost p a l p a b l e , almost substantial ' (p. 4-17). By address ing key questions re lated to this top ic a n d by identifying observab le indicators of m a t h e m a t i c a l reasoning whi le chi ldren are e n g a g e d in d rawing, I aspire to contr ibute to the field of imagery a n d m a t h e m a t i c a l representat ion, a n d in turn support c lassroom teachers in implement ing strategies for strengthening student learning; helping them to focus instructional decis ion mak ing not on the images but through t h e m . Chapter 3 - Methods BEFORE THE STUDY Who was involved This study involved 33 chi ldren in s e c o n d g r a d e , 16 boys a n d 17 girls. The chi ldren were all from the s a m e school , but d rawn from 4 different classrooms. Chi ldren were invited to par t ic ipate; parents g a v e consent for part ic ipat ion in the study a n d v i d e o t a p i n g of students whi le they worked (see A p p e n d i x B - Consent forms). Problem-solving sessions took p l a c e in groups of 3 or 4, with students from the s a m e classes g r o u p e d together . E a c h problem-solv ing session lasted approx imate ly 45 minutes, a l lowing for the c o m p l e t i o n of 2 problems. Problem-solving sessions took p l a c e in late M a y 2005. D a t a from the chi ldren's responses to the problems was c o l l e c t e d a n d a n a l y z e d to determine a basel ine or f ramework for descr ib ing drawing-as-problem-solv ing themes, including student behaviours a n d character ist ics of the drawings they m a d e . The school in quest ion is a suburban school with approx imate ly 350 students in grades K-7. This is a multi-cultural schoo l , in wh ich more than 22 different nationalities are represented. English as a s e c o n d l a n g u a g e is a n a r e a of o n g o i n g support at this schoo l . Select ing chi ldren from the s a m e schoo l a n d from the s a m e g r a d e level was d o n e intentionally. In c reat ing a f ramework, the h o p e was to work with students w h o h a d h a d a similar b a c k g r o u n d a n d early school expe r ience . The process The problem-solv ing tasks The p rob lem solving session was in tended to a c c e s s students' m a t h e m a t i c a l reasoning through drawing as wel l as to prov ide a context for observing drawing as problem-solv ing behaviours. As such, quest ion structures were c h o s e n a n d the complex i ty of those questions cons t ructed to elicit as m u c h as possible from the chi ldren. Having worked with some of these chi ldren w h e n they were in k indergarten, I h a d a sense of their c a p a c i t y a n d areas of strength. In k indergarten, I h a d e n g a g e d a small g roup of chi ldren in solving a series of division by sharing problems with var ied results in terms of strategies, ways of dea l ing with remainders a n d complex i ty a n d range of representations. A l though I h a d p l a c e d manipulat ives on the tab le at that point, these very y o u n g chi ldren all p i c k e d up markers a n d p a p e r to work through the prob lem. I was surprised to see them choos ing this m o d e of problem-solv ing. M o r e so, I was intr igued a b o u t the i m a g e - m a k i n g taking p l a c e for these chi ldren. I b e c a m e increasingly curious a b o u t the i m p a c t of visualization on problem-solv ing; in part icular, how young chi ldren's c a p a c i t y to m a k e a picture in their heads supports them in their m a t h e m a t i c a l reasoning. For these chi ldren, visualization p l a y e d a n important role in f inding the answers - eyes were raised, hands " c h o p p e d " a n d chi ldren spoke a l o u d a b o u t the a c t i o n they were orchestrat ing in their minds e y e . For this current study, I h o p e d to interact with these s a m e chi ldren in researching my n e w quest ion: How does the act of drawing support children in reasoning through a mathematical problem? The Cookie Problem In order to address this research quest ion, I n e e d e d to des ign a p rob lem that was sufficiently c o m p l e x that it c o u l d not b e solved without menta l effort, but still prov ide chi ldren with a c c e s s to the math no matter w h a t their c a p a c i t y . The division as sharing context was the starting point, a n d the numbers careful ly c h o s e n . The first quest ion I asked of chi ldren was as follows: There are 18 cook ies on a p late a n d 12 chi ldren w h o want to share t h e m . How c a n the chi ldren share the cook ies? How m u c h will everyone get? This quest ion required chi ldren to div ide a who le number by another who le number a n d d e a l with a remainder. A l though this kind of division p rob lem is not part of the g r a d e 2 curr iculum, the real-world context a n d the range of cor rect answers a c c e p t e d (1 a n d a half, 3 halves, o n e a n d some left over) m a d e this quest ion both sufficiently cha l leng ing a n d acces s ib le . During the study itself, I d id a d a p t this quest ion for chi ldren w h o answered it effortlessly (asking: "If you h a d 9 cook ies a n d 6 chi ldren w h o w a n t e d to share, h o w many cook ies wou ld everyone g e t ? " ) ; or for those chi ldren a b o u t w h o m I sought more information in terms of m a t h e m a t i c a l reasoning through drawing ("Answer this similar quest ion -but without drawing this t ime.") . The Wheels Problem The wheels p rob lem presented a different kind of c h a l l e n g e . This was a part -part -whole p rob lem in wh ich the parts were var iab le . This p rob lem featu red more than o n e right answer a n d more than o n e m e t h o d for solving it. In fact , I h a d used this quest ion with another g roup of g r a d e 1 a n d 2 chi ldren in another school district with interesting results. The o p e n -e n d e d structure of this p rob lem was intentional - my h o p e was that chi ldren w o u l d use drawing as a problem-solv ing strategy in order to solve it. Likewise, the p rob lem was too difficult to b e so lved mental ly, a n d so the thought was that drawing w o u l d b e e m p l o y e d as a system for record ing a n d m a n a g i n g the elements of the p rob lem. The p rob lem r e a d : C a m got 18 wheels for Christmas. What toys c o u l d he h a v e got ten? V i d e o t a p i n g Two v i d e o c o m e r a s were used for this study. The first was m o u n t e d on a tr ipod a n d r e c o r d e d o v e r h e a d f o o t a g e of the group as they w o r k e d . This c a m c o r d e r was d i r e c t e d at the entire g roup in a n effort to c a p t u r e interactions b e t w e e n the chi ldren (looking at o n e another ' s work, talking, c o m p a r i n g ideas), the point at wh ich the chi ldren put penc i l to p a p e r (to monitor pers istence in terms of t ime on task) a n d the f requency of visualization activity as observed through upraised eyes. The h a n d h e l d c a m c o r d e r was used to c a p t u r e close-ups of the chi ldren's hands, their penci ls in a c t i o n a n d a n y c o m m e n t s or self-talk they of fe red. In focus ing in closely on the chi ld while working on the task, I h o p e d to film the " a c t " of d rawing as problem-solv ing, including a n y process ing, self-cor rect ion or the app l icat ion of n e w strategies while problem-solv ing. The two c a m e r a s in c o n c e r t p rov ided 2 important v a n t a g e points a n d a means of cross-referencing the v ideo d a t a ; for those moments w h e n I was f o c u s e d in on o n e part icular ch i ld , the o v e r h e a d c a m e r a was a b l e to c a p t u r e v i d e o d a t a from the other 2-3 chi ldren. Figure 2 - O v e r h e a d a n d H a n d h e l d C a m e r a Views O v e r h e a d C a m e r a Handhe ld C a m e r a * • O n e group works on the cookies p rob lem, as f i lmed o n the o v e r h e a d c a m c o r d e r . 1 a m gett ing a c lose up of Orson... ...using the h a n d h e l d c a m c o r d e r , while he works on the p rob lem. DURING THE STUDY My role as observer Chi ldren were taken from their classrooms to work in small groups of 3 or 4 for a brief 45-minute problem-solv ing session. During this 45-minute b lock, both problems were presented a n d the interview quest ion was asked of the chi ldren. Working in groups with classmates was i n t e n d e d to prov ide chi ldren with a familiar, schoo l like situation. W e worked in a small resource room a round a round tab le . O n the tab le were pencils a n d erasers a n d nothing else. I b e g a n by setting the context with the chi ldren. I to ld them that I was interested in knowing more a b o u t h o w chi ldren think w h e n they solve math problems, a n d that I was go ing to b e filming them while they w o r k e d on a series of problems. (A letter to this ef fect h a d b e e n r e a d to all chi ldren before they e l e c t e d to par t ic ipate in the study.) I turned on the v i d e o c a m e r a s a n d h a d them int roduce themselves on film, then I r e a d the c o o k i e p rob lem out loud to the chi ldren, a n d answered a n y questions that arose to b e sure everyone unders tood the p rob lem. I turned on the h a n d h e l d c a m e r a a n d b e g a n to film chi ldren as they w o r k e d , zooming in on their fingers a n d their emerg ing drawings. While chi ldren worked , I tried to b e non-invasive with my quest ioning. W h e n I d id want some more information, I was invitational in my prompts, asking questions a b o u t chi ldren's thinking: • What are you thinking a b o u t ? • What ' s your i d e a ? • What are you do ing? I also asked a b o u t the chi ld's d rawing or parts of it, to determine the purpose: • What d o these l ines.mean? • What does this part of your drawing m e a n ? • How does that help you? How the students responded The problems were p o s e d of the chi ldren orally, a n d a blank p a p e r with only the p rob lem written on it was presented to the students. The initial quest ion asked h o w 12 chi ldren c o u l d share 18 cook ies . There were no manipulat ives on the tab le , a n d no directives g iven by m e for solving the p rob lem. All of the chi ldren a t t e m p t e d the problems, a n d e a c h chi ld tr ied, w h e n asked, to descr ibe his or her thinking. All students worked on the p rob lem for the al lotted 20 minutes without prompt ing from m e , ind icat ing their pe r seve rance a n d e n g a g e m e n t with the problems. Working in small groups with their c lassmates p rov ided the chi ldren with a famil iar learning situation, a n d a l l o w e d them to interact with o n e another . Occas iona l l y chi ldren w o u l d e n c o u r a g e o n e another to stick with the task or e v e n g a v e e a c h other support with problem-solv ing strategies ("Just d r a w f a c e s - just d o it like mine. . . " ) . Some chi ldren spontaneous ly c o m p a r e d their results at the e n d of the problem-solv ing session (eg. Jason a n d Grant , Norman a n d Lome), the content of w h i c h is discussed later in this p a p e r . How my observational practice shifted What I a t t e n d e d to While working with the s e c o n d group of 3 chi ldren, I b e g a n to w a t c h for repetitions of the first group's strategies. I w a t c h e d for the creat ion of sets, the distribution of items, a n d p a i d attent ion to the m o v e m e n t of the chi ld's hands while process ing the p rob lem. I b e g a n to a t t e n d to chi ldren's st rategic, systematic a p p r o a c h . Systems like el imination (crossing off items) a n d keep ing track behaviours (count ing a n d recount ing, d rawing boxes, etc) were observed careful ly. The sophist ication of the drawing, a l though interesting, w o u l d remain after the chi ldren h a d finished working; I r e c o g n i z e d that the p a p e r art i fact c o u l d b e a n a l y z e d afterwards. I m o v e d my attent ion elsewhere, focus ing instead on fingers a n d penci l tips, on postures a n d upraised eyes - the p e r f o r m a n c e - b a s e d indicators that wou ld not remain after the chi ldren h a d left, e x c e p t in the form of v ideo f o o t a g e . A d d i n g in the Interview quest ion Still, the chi ldren's work wos curious to m e . Their behoviours were conforming to o pgttern somewhot , g n d yet b e c c u s e the chi ldren t e n d e d to work independent ly , there wos not o lot of folk to e a v e s d r o p o n . Likewise, I was hesitant to ask guestions while chi ldren worked for feor of distracting them or redirect ing them inodvertently. Clegr ly the get of drawing (or the drawings themselves) were help ing, but I c o u l d not b e sure how. I d e c i d e d to osk g speci f ic quest ion b e t w e e n the two tosks to try to elicit some thinking from the chi ldren. So, b e t w e e n the C o o k i e Problem g n d the Wheels Problem, I osked the fol lowing quest ion: Whot strategy d id you use to solve the prob lem? How d id it help you? I cons t ructed this quest ion coreful ly, not wont ing to suggest to chi ldren thot d rawing would he lp, so osked instegd whgt h g d h e l p e d them, a n d then inquired as to how. The chi ldren answered that d rawing h a d h e l p e d t h e m ; their descriptions of how var ied. Of those a b l e to ort iculgte gn gnswer, gng logous thinking e m e r g e d ("Drawing is like..." "D rawing is..."). Student thinking is clossif ied g n d discussed later in this p a p e r . During the course of the study itself however , students' descr ipt ion of both their drawings a n d h o w the a c t of drawing h a d h e l p e d a l l o w e d m e to observe their drawing-as-problem-solv ing in a different w a y , a n d to ant ic ipate act ions I h a d not up to that point. AFTER THE STUDY Analysis of the information - How I described and recorded events After the study I g a t h e r e d student work a n d m a d e several cop ies , classifying some a c c o r d i n g to strategy used or sophist ication. I soon real ized however that the task of assessing the students' p roduct was not go ing to prov ide m e with information on drawing-as-problem-solv ing behaviours. In order to assess those behaviours I h a d to w a t c h a n d ana lyze the videos. I w a t c h e d e a c h of the h a n d h e l d c a m c o r d e r v ideos a n d t imed a n d a n n o t a t e d events by ch i ld . E a c h event (shift in d rawing, a n e w phase of p rob lem solving) was d e s c r i b e d with a running c o m m e n t a r y using a n e c d o t a l l a n g u a g e a c c o r d i n g to character ist ics. Student c o m m e n t a r y was r e c o r d e d verbat im a n d key clips were c r o p p e d a n d s a v e d for later v iewing. As I w a t c h e d more v ideo I establ ished a c o d e a n d a more sophis t icated system for record ing the behaviours that resur faced. My o w n c o m m e n t s or questions were a d d e d in a separate c o l u m n as a w a y of keep ing track of emerg ing questions or themes. Below is a sample from early v ideo analysis notes from G r o u p 1 A. It shows d a t a from the C o o k i e Problem a n d h o w the 4 chi ldren r e s p o n d e d within a part icular t ime per iod. Student quotes a n d what they d id in response to the task are n o t e d ; my questions or reflections are c o n t a i n e d within the final c o l u m n . Note the t h e m e c o l u m n where I a t t e m p t e d to assign a labe l to the obse rved behaviours. Table 1 - V i d e o D a t a Record ing Form - Excerpt from Group 1A The Cookie Problem: What students did, what they said Time Observations Behaviour Questions, comments CLIP2 2:30 Mark - eyes raised, finger count ing - "Trying to count by like stuff - 3's a n d 2's a n d ones" Did a n y of those things help? "Not really." SPATIAL IMAGERY More spatially or iented than others? Drawing d id not help him, but was using a system in his h e a d CLIP2 3:28 J o a n n a - d rew two sets - o n e group of cookies , o n e group of chi ldren c i rc led groups of cook ies in 2's, split some. Asked " w h a t a g e are the ch i ld ren?" SET CREATION Looking for proport ional re lat ionsh ip-smal l kids, less cookies Fairness a n issue CLIP2 1:47 Melissa: "This is go ing to take a whi le. " What? "Drawing these pictures!" PICTORIAL USE OF DRAWING Drawing not simplified - still pictorial CLIP2 4:00 Sara drawing faces on p e o p l e PICTORIAL USE OF DRAWING How m u c h was her solution a f f e c t e d by other chi ldren? Eyes on Melissa, J o a n n a , but then a b a n d o n e d their solution ideas for her own CLIP2 4:06 Melissa - n u m b e r e d chi ldren 1-12. Then number ing cookies on front of p a g e to match/represent distribution. "They all get o n e c o o k i e but then there's extras. 1 haven ' t thinked a b o u t that yet. I'll think a b o u t that after 1 d o that. " DISTRIBUTION KEEPING TRACK The descriptors a n d their origins The p r e c e d i n g formot wgs Igbour intensive. It split up the chi ldren's responses by t ime o n d m g d e ref lecting on o single chi ld's behaviours cha l leng ing . As behaviours su r faced multiple times, the list of "behav iou r s " b e c a m e overwhelming a n d distracting. In all, I h a d obse rved a n d identif ied 14 different behaviours, listing them in the order of their o c c u r r e n c e within the first group's session. They were : C reat ing sets Distribution Keep ing track Elimination C h e c k a n d r e c h e c k Counters Solution representat ion Spatial imagery Pictorial use of drawing Set c reat ion D y n a m i c imagery Visualization Eyes up Iconic use of pictures This list of behaviours g rew out of the chi ldren's p e r f o r m a n c e in response to the problems. Like assessing any p e r f o r m a n c e task, I c o u l d not descr ibe student p e r f o r m a n c e until I h a d seen it. A n d , as e a c h n e w behav iour was obse rved, it was a d d e d to the growing list a b o v e . A n at tempt was m a d e to organize the list a n d to establish commonal i t ies within it. Upon ref lection, there arose 4 categor ies or themes under wh ich e a c h of these behaviours fell. Dup l icate behaviours were d e l e t e d a n d a m a l g a m a t e d under o n e t h e m e h e a d i n g . Theme and behaviour Virtual manips Imagery 1. Creat ing sets 1. Dynamic imagery 2. Distribution 2. Visualization 3. Counters 3. Eyes up Systems Sophistication of representation 1. Keep ing track 1. Pictorial use of d rawing 2. Elimination 2. Iconic use of pictures 3. C h e c k a n d r e c h e c k 3. Abst ract thinking Represented The tab le on the fol lowing p a g e (Table 2 - Revised V i d e o D a t a Record ing Form - Excerpt from G r o u p 4A) shows a n n o t a t e d v ideo d a t a from G r o u p 4A using these n e w themes a n d behaviours. It outlines the C o o k i e Problem, Trent's response to it a n d my questions a b o u t his response. Note the numbers in e a c h t h e m e c o l u m n , wh ich cor respond to the behav iour o b s e r v e d . Table 2 - Revised V i d e o D a t a Record ing Form - Excerpt from Group 4A The Cookie Problem - What students did, what they said SAVED AS Time Behaviour noted and Comment JAL MANIPS CO .. ... • | I :. IMAGERY IISTICATION Questions, comments CLIP: VIRTl SYSTE IMAGERY SOPH Trent's image Trent - very faint finger count ing movements , looking u p . . . quiet . . . "1 d o n ' t know what to write." Aud io only - c a m e r a po in ted at the floor... Trent took m e off c a m e r a to say: 1, 2 ,3 Not putting anyth ing o n paper . . . risk taking? He motions m e a w a y from the group to ask if his i d e a works - does NOT w a n t to b e on c a m e r a a n d Trent explains and re-thinks CLIP2 0:45 "1 c o u n t e d up to 12 in my h e a d a n d then 1 c o u n t e d up to 18 - 13, 14, 15, 16, 17, 18..." (showing m e on his fingers h o w he counted) " then 1 c o u n t e d the cookies split in 2 - 2, 4, 6, 8, 10, 12." That works, doesn't it? "whisper ing. . . - " W h a t should 1 wr i te??" Can you just write what you said? He reads his solution a l o u d . . . wrong. . . Clearly Trent has so lved the problem BEFORE he begins to write it d o w n . This solution is very sophist icated a n d complete ly visualized. Dynamic imagery. Inf luence of other research Two of these sets of descriptors were der ived from other research. The terms icon ic a n d pictor ial refer to the type of drawing that chi ldren c r e a t e d in response to the p rob lem situation. These terms speak to the artistic detai l a n d sophist ication of the drawing. Pictorial as a descr iptor was bo r rowed from the work of D iezmann (2000), Smith (2003) a n d Wo leck (2001). Pictorial representations are deta i led a n d idiosyncrat ic; that is, they are closely if not specif ical ly m a t c h e d to the context of the p rob lem. In contrast, icon ic representations are more abst ract , genera l i zab le , a n d feature few context - speci f ic details. This terminology a n d definit ion was d rawn from the research of Tall (1999), Gray , Pitta a n d Tall (2000) a n d Thomas, Mull l igan a n d Gold in (2002). Imagery, including d y n a m i c imagery a n d visualization, references the work of Owens et a l (1996,1999). O w e n descr ibes imagery as a menta l process that includes " . . . the recognit ion of shapes a n d parts of shapes, the transformations of images in the mind, the visual analysis of shapes a n d mental ly v iewing shapes from other perspect ives " (Owens, M i tche lmore , Outhred & P e g g , 1996). Normally a s soc ia ted with g e o m e t r i c understandings, imagery has a role to p lay in the resolution of m a t h e m a t i c a l problems. Wheat ley a n d C o b b state that: M a t h e m a t i c a l p rob lem solving is often a matter of reasoning ana ly t ica l l y , . . . a process of bui lding from images to analysis a n d analysis to images [that] m a y cont inue through many cyc les . (1990, p. 161). In descr ib ing this state of c h a n g e , of cyc l ing through p rob lem solving, Wheat ley a n d C o b b refer to the d y n a m i c nature of i m a g e - m a k i n g a n d its potent ia l for support ing reasoning through m a t h e m a t i c a l problems. Mark's p e r f o r m a n c e was a g o o d e x a m p l e of what Wheat ley calls d y n a m i c imagery. His upraised eyes a n d finger count ing , as wel l as his oral descr ipt ion of the process he used all support the label of d y n a m i c imagery . While o n e c a n n o t assume that raised eyes is proof of visualization, it o c c u r r e d often e n o u g h to warrant not ing, e v e n while looking solely through the h a n d h e l d c a m e r a . The o v e r h e a d c a m e r a a l l o w e d for more observat ion of this behav iour whi le Mark a n d other students so lved the problems. Another behav iour worth noting in students' response to these problems -was their use of gestures. Their gesturing - h e a d b o b b i n g , f inger count ing , point ing, a n d c h o p p i n g - likewise p rov ided e v i d e n c e of the internal construct ion a n d manipulat ion of menta l images . In Presmeg's earliest research on teacher s ' use of gestures to transmit visual information to their pupils, (1985, q u o t e d in Presmeg, 2006) she s tated that " t e a c h e r s ' use of gesture was o n e of the surest indicators that they h a d a menta l i m a g e that they were intentionally or inadvertently c o n v e y i n g to their students". I w o u l d suggest that in this study, chi ldren's use of gestures (whether consc ious or unconsciously performed) was a n indicat ion that students were operat ing on a menta l i m a g e in a d y n a m i c w a y . Presmeg descr ibes a gesture as a sign veh ic le ind icat ing a n ob ject in s o m e o n e ' s cogn i t ion . (2006) The inter-relation b e t w e e n gestures a n d menta l objects - a n d the c o m p l e m e n t a r y w a y in w h i c h gestures support visualization a n d m a t h e m a t i c a l reasoning - m a d e student gestures worth not ic ing a n d annotat ing in the course of this study. Questions g e n e r a t e d through analysis While examin ing student work a n d record ing the type a n d f requency of their responses to the task, I b e g a n to reflect on my o w n thinking, my p r e c o n c e p t i o n s a n d what I h a d e x p e c t e d to see in the students' p e r f o r m a n c e . In part icular, I b e g a n to w o n d e r a b o u t the appropr iateness of the task for certa in chi ldren, those w h o were not a b l e to use d raw ing-as-problem-solv ing b e c a u s e they h a d not c r e a t e d a stable i m a g e of the p rob lem. I w o n d e r e d as wel l a b o u t those visual spatial students w h o d id NOT d r a w a n d the menta l processes these students were using. This resur faced my initial inquiry a n d r e n e w e d my curiosity a b o u t visual spat ial thinkers a n d their c a p a c i t y in terms of m a t h e m a t i c a l problem-solv ing. Two further questions arose from this. How c a n chi ldren b e suppor ted in their efforts to problem-so lve? A n d h o w c a n teachers b e suppor ted in recogniz ing drawing-as-problem-solv ing in all its forms? These issues will b e addres sed in the discussion c h a p t e r . Sorting and charting behaviours by group, by task, by behaviour O n c e v ideo analysis was c o m p l e t e for all 11 groups of chi ldren, d a t a was c o l l a t e d into o n e larger d o c u m e n t in order to ga in a sense of the f requency of the behaviours within a n d b e t w e e n groups. This d o c u m e n t resulted in a c o n s o l i d a t e d list of the number of behaviours by type, by task a n d by group. E a c h of these was to ta led to get a sense of w h i c h behaviours su r faced the most of ten, by wh ich group(s) a n d h o w they i m p a c t e d the relative success of the chi ldren. Some groups were more visual spatially or iented (Group 4B); some groups e m p l o y e d systems more readily (Group 3A); some groups d rew pictorially (Group 3B). Examining these behaviours by task also highl ighted h o w the nature of the task (the c o o k i e p rob lem as o p p o s e d to the wheels problem) invited different strategies. See Tables 3 a n d 4, Summary of Strategies by G r o u p - C o o k i e Problem a n d Summary of Strategies by G r o u p - Wheels P roblem. The results of these tables will b e discussed in C h a p t e r 4. Note: The numeral in e a c h cel l refers to the total number of o c c u r r e n c e s of the behav iour or strategy used by a n y m e m b e r of the g roup. CO ro ro CO tn J* CO ro CREATING SETS VIRTUAL MANIPS to o o to CO CO Mi ro CO DISTRIBUTION VIRTUAL MANIPS ro o> ro CO CO CO ro o 4^  COUNTERS VIRTUAL MANIPS CO tn o - CO ro CO KEEPING TRACK SYSTEMS o o CO ELIMINATION SYSTEMS ro J*. CO CO CHECK & RECHECK SYSTEMS O o o o o o DYNAMIC IMAGERY IMAGERY o o ro ro to mm* o VISUALIZATION IMAGERY ro o ro o — • o o ro ro EYES UP IMAGERY - CO o ro - ro ro PICTORIAL SOPHISTICATION CO o - ro ro ro -u o ICONIC SOPHISTICATION o o - o MM* o ro ABSRTACT THINKING REPRESENTED SOPHISTICATION Stephanie, Shauna, Jorge, Iris Hannah, Zarah, Susie, Jessie Trent, David, Charlie Ruby, Mona, Rebecca, Karl Norman, Lome, Cathy Anthony, Avril, Orson Pricilla, Martin, Larry, John Jason, Lisa, Grant, Susanne Mark, Joanna, Melissa, Sara Students' names Note : The numeral in each cell refers to the total number of occurrences of the behaviour or strategy used by any member of the group. - CO O O O o O IO O CREATING SETS VIRTUAL MANIPS o o o o O IO to o DISTRIBUTION VIRTUAL MANIPS CO IO IO IO CO O COUNTERS VIRTUAL MANIPS IO CO CO CO o CO CO KEEPING TRACK SYSTEMS © o O o o o IO IO ELIMINATION SYSTEMS CO IO CO o CO o IO CHECKS, RECHECK SYSTEMS o o o o o o o o o DYNAMIC IMAGERY IMAGERY o o co —t co IO to —J VISUALIZATION IMAGERY CO co CO CO o o o EYES UP IMAGERY K> o IO o to PICTORIAL SOPHISTICATION o O —ft O o CO ICONIC SOPHISTICATION o o o o o o o o ABSTRACT THINKING REPRESENTED SOPHISTICATION Stephanie, Shauna, Jorge, Iris Hannah, Zarah, Susie, Jessie Trent, David, Charlie Ruby, Mona, Rebecca, Karl Norman, Lome, Cathy Anthony, Avril, Orson Pricilla, Martin, Larry, John Jason, Lisa, Grant, Susanne Mark, Joanna, Melissa, Sara Students' names Q C£ CD I co c 3 3 Q -5 O CD CQ CD GO D" -< CQ O C "O I CD GO o CT CD Determining contributors to success From these more organ ized charts it was possible to d o a somewhat more quant i tat ive analysis. A n assessment of the successful c o m p l e t i o n of the questions by task a n d by group fo l lowed, later c o m p a r e d to the type a n d sophist ication of the strategies used. This was d o n e in a n at tempt to cor re late the type of drawing as problem-solv ing strategy a n d the sophist ication of the drawings with a chi ld's c a p a c i t y to arrive at a solution that a p p r o a c h e d 1 a n d a half for the C o o k i e Problem. Success was then more general ly de f ined to inc lude distributing all the cook ies in some w a y (one e a c h now, give the remaining 6 a w a y ) . " Succes s " c o u l d not b e m a t c h e d directly to a single strategy used or e v e n the sophist ication of the d rawing, however . Rather, success s e e m e d a f f e c t e d by a variety of interact ing factors. The predisposit ion to use visualization a n d i m a g e - m a k i n g in c o n c e r t with d rawing, a n d the a p p l i c a t i o n of a system to the solution of the p rob lem h a d a significant ef fect on students' c a p a c i t y to resolve the p rob lem. This will b e exp lored in more detai l in the discussion c h a p t e r . Analysis of Interview data A g a i n , the students' v i d e o t a p e d responses to the interview quest ion ("What strategy d id you use? How d id it help you?" ) were a n n o t a t e d , t ranscr ibed a n d a n a l y z e d . There e m e r g e d 2 sets of interview d a t a , descr ib ing h o w m u c h a n d why drawing was helpful to students. The first c a t e g o r i z e d students' statements regard ing the d e g r e e to w h i c h drawing h a d h e l p e d them solve the c o o k i e p rob lem. The s e c o n d looked more closely at students' statements to classify the w a y in w h i c h drawing h a d h e l p e d . Themes e m e r g e d in both sets of d a t a ; cross-referencing these two d a t a sets h e l p e d to clarify h o w students w h o d id not find drawing helpful we re using their drawings as o p p o s e d to those w h o were a i d e d by their drawings or the a c t of d rawing itself. Chapter 4 - Results Overview Students c o m p l e t e d 2 problems during the small g roup interviews, the C o o k i e Problem a n d the Wheels P roblem. E a c h of the 11 groups of students a n d their responses to the problems are d e s c r i b e d be low, with trends in student a p p r o a c h to the individual tasks. Following the "Observat ions by Task" sect ion for e a c h of the C o o k i e a n d Wheels Problems is a n overv iew of the drawing as problem-solv ing behaviours n o t e d during all of the problem-solv ing interviews. These behaviours h a v e b e e n c o l l a t e d a n d g r o u p e d into a drawing as problem-solv ing f ramework, with e l a b o r a t e d descriptions of e a c h cluster of behaviours a n d exemplars to illustrate e a c h o n e . This s e c o n d sect ion is entit led "Observat ions by Strategy" as it focuses on the part icular d rawing as problem-solv ing strategies used by the students across both of the problem-solv ing tasks. Col lat ion a n d analysis of this d a t a helps to e l a b o r a t e a response to the main research quest ion: How do young children access visualization and imagery in their resolution of mathematical tasks? How does drawing as problem-solving support students in making sense of and reason through a mathematical problem? OBSERVATIONS BY TASK How children drew and how it helped them The Cookie Problem - General Themes Trends in the solution (product) The first quest ion p o s e d of the chi ldren asked them to share 18 cook ies a m o n g 12 chi ldren. Chi ldren e m p l o y e d several different methods to solve the p rob lem, a n d twenty of the 33 chi ldren were a b l e to arrive at " o n e a n d a half cook ies e a c h " as a n answer. Of the thirteen w h o d id not, div iding the 6 remaining who le cookies into f ract ional parts p r o v e d cha l leng ing . Rather than giving up, students p r o p o s e d a range of a l ternate solutions for the p rob lem. Some a t t e m p t e d to determine a who le number solution to the prob lem to a v o i d break ing up the cook ies . Karl* thought that every s e c o n d chi ld should h a v e 2 cook ies , a n d J o a n n a asked wh ich chi ldren were older so she c o u l d distribute more to the " b i g g e r kids". Other solutions dea l t with the remainder by exc lud ing or subtract ing it from the who le . Martin suggested giving a w a y the remaining 6 cook ies to " s o m e other ch i ld ren" ; while Jessie d e c i d e d that the 6 leftover cook ies c o u l d b e "for tomor row" . Three chi ldren d id not demonst rate a n understanding of the c o n c e p t of division as sharing, a n d as a result, were not successful in arriving at a solution that m a d e sense. Their responses were very different from those d e s c r i b e d a b o v e as "a l te rnate solutions" to the p rob lem. M o n a , for ins tance, d rew two sets of pictures on her sheet of p a p e r - 18 cookies a n d 12 chi ldren - a n d then a d d e d the two sets together to arrive at 30 as a response. When asked , " W h a t does 30 m e a n ? " M o n a r e s p o n d e d , "It's w h a t the answer is for the who le th ing. " Implications for these three chi ldren will b e addres sed in the discussion c h a p t e r . * All names in this document have been changed. Table 5 - Summary of type of response against d rawing or not Response type Drew to solve Did not draw 1.5 cook ies e a c h 16 4 A l ternate response 10 0 Mistaken response 3 0 Trends in the d rawing (process) Twenty of the 33 chi ldren interv iewed arr ived at the answer, most of t h e m through drawing or some kind of manipulat ion of a n i m a g e o n the p a g e . For those w h o d r e w to solve the prob lem, the a p p r o a c h e s used fo l lowed part icular patterns; most b e g a n by drawing sets of cook ies a n d / o r chi ldren a n d then operat ing on these images in some w a y . Figure 3 - Anthony ' s lines Figure 4 - Melissa's sets V/1M f ' ¥ ft a A X t J -••M'ffs» L^/H'-YX \ " ^—^C^' / Some students d r e w cook ies a n d chi ldren then c o n n e c t e d o n e to the other with lines (eg. Anthony - Figure 3); some chi ldren d r e w circles, c o u n t e d out, l a b e l e d or part i t ioned off 12 of t h e m , then split the remain ing cook ies in 2 parts (eg. Melissa - Figure 4), whi le still others d r e w 18 cook ies , cut them all in half a n d distributed 3 halves to e a c h " p e r s o n " (eg. John). John explains his thinking be low: (Note: My c o m m e n t s to the students are in italics.) You break all the cook ies in half a n d then you h a v e 36 cook ies . Then you c o u l d g ive 36 to the kids. You give 36 to the kids? Tell me about that part. C u z you break all the cook ies in half in the midd le a n d it gives you 36 halts. Then you give the 36 halts to the chi ldren until there's no more. ...they get a who le a n d a half. Chi ldren's successful attempts at solving the c o o k i e p rob lem i n d i c a t e d that they unders tood the notion of sharing, a n d that they were c o n c e r n e d both with using up all the cookies as well as mak ing sure e a c h of the chi ldren got a fair share. All but four (Mark, Trent, S tephanie a n d Charl ie) used drawing in process ing the prob lem in some w a y , to represent the sets of chi ldren a n d cookies , to separate the d e d i c a t e d who le cook ies from the remaining six a n d in some cases to c a n c e l out those cook ies that h a d b e e n " e a t e n " . E a c h of these drawing as p r o b l e m -solving strategies is d e s c r i b e d in more detai l in the "After Problem-solv ing" sect ion of this c h a p t e r ; o n e chi ld's use of several of these strategies is out l ined in the next sect ion. One child's processing of the Cookie Problem Orson uses drawing as a problem-solving strategy Orson is a chi ld w h o m a d e several attempts at this p rob lem. His p r o b l e m -solving attempts were fairly typ ica l of chi ldren in the cohort of 33, a n d so his p e r f o r m a n c e on both the C o o k i e Problem a n d the Wheels Problem is d e s c r i b e d be low. Following Orson through his part icular process of thinking helps to e l a b o r a t e a number of strategies a n d m a k e the framework more c lear . Field notes from v ideo observat ion are i n c l u d e d b e l o w to highlight key elements a n d to t r a c e his process ing of the prob lem. Orson b e g a n to solve the c o o k i e p rob lem by drawing 18 cook ies in a ci rc le at the b o t t o m corner of his p a g e . " I 'm drawing them on a p l a t e , " he told his fr iend. He c o u n t e d out the cookies a n d put a small mark on e a c h o n e , doubl ing up on three of the 18 cookies . " A r e they like taking 12 a w a y ? " he asked m e , explain ing, "It says 12 there. " (Figure 5a) At this point, Orson s t o p p e d a n d re- read the quest ion. "This is hard . . . " he c o m m e n t e d , (Figure 5b) then asked for a n e w sheet of p a p e r , " to write lines." Figure 5 - Orson works through the C o o k i e Problem Figure 5a Figure 5b A | I Orson's first a t tempt seems to ind icate he thinks this is a subtract ion p rob lem. Orson has shifted his thinking somehow. Drawing lines is a w a y to distribute cook ies - to assign them to a n owner . What c l i c k e d for him in re- reading the p rob lem?  Figure 5c Figure 5d Orson's strategy n o w involves count ing , c reat ing sets of 4 a n d distributing these sets by c o n n e c t i n g lines to " c h i l d r e n " . Orson's use of lines. Orson realizes he has run out of cook ies to cont inue with this solution. Note the lines dividing cook ies into sets of 4, a n d a line joining these sets to a ch i ld .  O n his n e w sheet of p a p e r , Orson d rew 12 chi ldren (stick men) a n d ske tched 18 cook ies underneath . Beginning on the right s ide of the p a g e , he c o u n t e d a n d part i t ioned off 4 cookies with a line. He d r e w a line c o n n e c t i n g this set of 4 cookies to a n individual ch i ld . (Figure 5c) He persisted in this w a y , c reat ing 4 sets of 4 cook ies a n d then he s t o p p e d , realizing he h a d run out of cook ies . (Figure 5d) Embarrassed, Orson f l ipped his p a p e r over. I a sked him to expla in his thinking. I d r a w e d 12 kids a n d 18 cookies a n d I m a d e lines of 4 a n d then m a d e a line to o n e person a n d then to the other persons. Why did you choose 4? B e c a u s e I thought it w o u l d m a y b e b e 4. How are you going to change your idea now? M a k e it m a y b e like 2. Figure 5e Figure 5f Orson uses circles to enc lose sets of o n e chi ld a n d 2 cook ies , starting with 2 cook ies e a c h , as he i n d i c a t e d he w o u l d try next. Orson's self corrects a n d erases. "On ly 9 kids c a n h a v e cook ies , so, I'll just erase o n e e a c h [erasing] ...that d idn ' t work either!" This t ime, Orson tried 2 cookies per chi ld, but d id not use the partit ioning system he h a d e m p l o y e d before. Instead, he d r e w circles to enc lose sets of cook ies a n d chi ldren. (Figure 5e) He used self-talk as a running c o m m e n t a r y to gu ide his thinking: "On ly 9 kids c a n h a v e cook ies , so, I'll just erase o n e e a c h , [erasing]...that d idn ' t work either!" (see Figure 5f) In the e n d , Orson settled on o n e c o o k i e for e a c h chi ld a n d 4 left over for the next d a y as his attempts to erase extra cook ies left him with only 16 cook ies rather than 18, a n d , it a p p e a r s , 14 chi ldren as wel l . To v iew Orson working through the C o o k i e Problem, p lease w a t c h V i d e o Cl ip 1, ent i t led: "Orson's flexible thinking" In e a c h o n e of Orson's attempts to solve the p rob lem, he used drawing as a problem-solv ing strategy to m a k e sense of the p rob lem; that is, Orson p rocessed a n d tested out possible groupings (4's then 2's then 1 's) by drawing a n d redrawing his emerg ing ideas. It was through his drawings that Orson m a n i p u l a t e d the elements of the p rob lem. He c r e a t e d a n d then c o u n t e d the items in e a c h set, he m a d e estimates, he distr ibuted sets of cook ies to chi ldren, he se l f -cor rected a n d adjusted his thinking. In the e n d , Orson persisted for more than 15 minutes on this p rob lem, d rawing a n d thinking a l o u d , a n d presented three different d rawing as problem-solv ing strategies. OBSERVATIONS BY TASK How children drew and how it helped them The Wheels Problem - General Themes Trends in the solution (product) The wheels p rob lem asked chi ldren to determine w h i c h vehicles c o u l d b e represented by 18 wheels . O n c e a g a i n , the chi ldren solved the p rob lem in a variety of ways . Drawing was used by most of the chi ldren to support them in keep ing track of sets of wheels , a n d h o w m a n y wheels they h a d a c c o u n t e d for in solving the prob lem. Chi ldren w h o used drawing as a problem-solv ing strategy for this task h a d to cons ider the multiple parts of the who le of 18 in assigning wheels to different vehicles. Chi ldren p r o d u c e d d e t a i l e d drawings that closely resembled the a c t u a l ob ject in solving this p rob lem (ex. drawings of scooters, rol lerblades, toy cars a n d trains). Trends in the drawing (process) Students used drawing as a means of keep ing track of the number of wheels that h a d a l ready b e e n ass igned to vehicles. Dav id , for e x a m p l e , d rew 2 cars, p a u s e d , then m a d e a truck with 10 wheels to c o m p l e t e his solution. For Dav id , d rawing p rov ided him with a w a y to record a cumulat ive total . This was conf i rmed w h e n he r e c o r d e d numbers at the top of the p a g e to c o m p l e m e n t his pictures: 10, 14, 18. Lisa d rew 18 wheels in a long line, then c i rc led wheels a n d a d d e d vehicles a b o v e t h e m , s topping w h e n the 18 wheels were used up. (Figure 6 - Lisa's line up) Figure 6 - Lisa's line up Cam got 18 wheels for Christmas. What toys could he have gotten? Ruby pa i red d rawing a n d strategies incorporated from other disciplines to solve the p rob lem. She wrote the word for the veh ic le she w a n t e d to use, s ke tched the a p p r o p r i a t e number of wheels bes ide it a n d then r e c o r d e d a running total of the wheels used so far. (Figure 7 - Ruby's chart) I a s k e d her why there were no vehicles on her p a g e . She repl ied, "I d r e w wheels . Just the wheels . " Did this help? How? " C u z it was a shortcut. If I d r a w e d the p l a n e it w o u l d take too long, if I d r a w e d the m o t o r c y c l e , the truck a n d everything it w o u l d take too long for m e to d o it." Figure 7 - Ruby's Char t The Wheels Problem Cam got !8 wheels for Christmas. What toys could he have gotten? Ruby's solution s h o w e d sophist ication in thinking - a n d a n a c k n o w l e d g e m e n t that sets of circles c a n " s tand for" a p l a n e or c a r . Her systematic a p p r o a c h a n d log ica l thinking in a p p r o a c h i n g the p rob lem were ev ident here. One child's processing of the Wheels Problem Orson uses drawing as a problem-solving strategy Orson b e g a n to solve this n e w prob lem by asking: " C a n w e m a k e a list? C a n I m a k e a w e b ? " He g e n e r a t e d a brainstormed list of possible vehicles (Figure 8a) a n d then chose careful ly from the pr inted list, c o m m e n t i n g : "I n e e d 4 things with 4 wheels.. . 4+4+4+4 is 18." When I asked him to " p r o v e it", Orson suggested 4 vehicles from his list a n d c o u n t e d a l o u d using his fingers from 8 to 16 - "4, 8, 9, 10, 11, 12, 13, 14, 15, 16..." He p a u s e d , looked up a n d e x c l a i m e d , "It's 4 plus 4 plus 4 plus 4 plus 4 plus 2! A scooter ! " then a d d e d the word " s c o o t e r " to his list. (Figure 8b) Figure 8 - Orson works through the Wheels Problem Figure 8 a Figure 8b Orson's list, in w h i c h he brings in his "It's 4 plus 4 plus 4 plus 4 plus 4 plus l a n g u a g e - b a s e d strength to 2! A scooter ! " support him in problem-solv ing. Orson visualizes the wheels on e a c h veh ic le a n d counts a l o u d : "I n e e d 4 things with 4 wheels. . . 4+4+4+4 is 18."  To this point, Orson h a d b e e n using visualization as a problem-solv ing strategy ( imagining the wheels a t t a c h e d to the vehicles), pair ing his menta l i m a g e with his list. Orson's list a l lowed him to k e e p t rack of the possible vehicles so that he c o u l d then visualize the wheels a n d record his solution in a systematic w a y . Next Orson d rew the solution to the prob lem, mak ing a picture of e a c h of the vehicles, a n d count ing the 4 wheels on e a c h veh ic le as he a d d e d e a c h o n e . (Figure 8c) Lastly he a d d e d numbers a n d "plus" signs to the d i a g r a m . (Figure 8d) Figure 8 c Figure 8d P * ^ . i f • T^^ ^ Cam got 18 wheels for Christmas. What toys could he have gotten? <^\r~~7~i Orson draws pictures to prove his thinking. His drawings a l low him to count to c h e c k ; his list supports him in be ing systematic. Orson's c o m p l e t e d p a g e - note the addit ion statements. Orson's solution o n c e a g a i n demonstrates a r a n g e of d rawing as problem-solv ing strategies a n d flexibility in the a p p l i c a t i o n of those strategies. Within this context , Orson uses strategies that were different from the ones he e m p l o y e d in the c o o k i e p rob lem: in part icular, his use of visualization pa i red with the scaf fo ld of the list a n d then pictures used to represent the solution rather than a c t i n g as the veh ic le for thinking through the p rob lem. Summary - Children's use of drawing as problem-solving As I obse rved the chi ldren while they so lved these problems, I n o t i c e d several commonal i t ies both in terms of the products they c r e a t e d a n d in the processes they used to solve the problems. A l though the problems h a d different structures, chi ldren used drawing as a p rob lem solving strategy in both cases to construct m e a n i n g for the task, man ipu late the elements of the problems a n d arrive at a solution. The next sect ion of this c h a p t e r outlines these d rawing as problem-solvng strategies a n d behaviours a n d the commonal i t ies n o t e d across both the cook ies a n d wheels problems. A tab le summarizing the o c c u r r e n c e of these strategies a n d the f requency of their o c c u r r e n c e follows on the next 2 p a g e s . ZL Note : The numeral in e a c h cel l refers to the total n u m b e r of o c c u r r e n c e s of the behav iour or strategy used by a n y m e m b e r of the g roup. CO to ro CO tn CO ro CREATING SETS VIRTUAL MANIPS ro O o ro CO CO ro CO DISTRIBUTION VIRTUAL MANIPS ro CO CO CO ro o COUNTERS VIRTUAL MANIPS in o -£» CO ro CO KEEPING TRACK SYSTEMS o o CO ELIMINATION SYSTEMS ro CO CO CHECK & RECHECK SYSTEMS o © o o —J o o DYNAMIC IMAGERY 5 > o m 70 •< o o ro —* ro ro o VISUALIZATION 10 o ro o - o o ro ro EYES UP CO o (O - ro ro PICTORIAL SOPHISTICATION CO o mmm* ro ro ro o ICONIC SOPHISTICATION — 1 o O o o to - ABSRTACT THINKING REPRESENTED SOPHISTICATION Stephanie, Shauna, Jorge, Iris Hannah, Zarah, Susie, Jessie Trent, David, Charlie Ruby, Mona, Rebecca, Karl Norman, Lome, Cathy Anthony, Avril, Orson Pricilla, Melissa, Larry, John Jason, Lisa, Grant, Susanne Mark, Joanna, Melissa, Sara Students' names ez Note : The numeral in e a c h cel l refers to the total n u m b e r of o c c u r r e n c e s of the behav iour or strategy used by a n y m e m b e r of the g roup. CO o o o o O to o CREATING SETS VIRTUAL MANIPS o o o o o to to O DISTRIBUTION VIRTUAL MANIPS CO to IO IO CO O - COUNTERS VIRTUAL MANIPS to co co CO o CO CO KEEPING TRACK SYSTEMS o o o o o o IO IO - ELIMINATION SYSTEMS CO IO co o CO o to CHECK & RECHECK SYSTEMS o o o o o o o o o DYNAMIC IMAGERY IMAGERY o o CO CO IO to VISUALIZATION IMAGERY CO CO co - CO o o o EYES UP IMAGERY - to o to o to PICTORIAL SOPHISTICATION o o o o CO - ICONIC SOPHISTICATION o o o o o o o o ABSTRACT THINKING REPRESENTED SOPHISTICATION Stephanie, Shauna, Jorge, Iris Hannah, Zarah, Susie, Jessie Trent, David, Charlie Ruby, Mona, Rebecca, Karl Norman, Lome, Cathy Anthony, Avril, Orson Pricilla, Martin, Larry, John Jason, Lisa, Grant, Susanne Mark, Joanna, Melissa, Sara Students' names OBSERVATIONS BY STRATEGY Creating the framework Chi ldren's d rawing as problem-solv ing strategies a n d their f requency will b e e x a m i n e d in some detai l be low. Through examinat ion of the v ideo d a t a , 4 clusters of drawing as problem-solv ing themes including student behaviours a n d drawing character ist ics were d e f i n e d : virtual manipulat ives, systems, imagery a n d sophist ication. What follows is a descr ipt ion of the cluster of indicators by t h e m e , a n d then examples -print a n d v ideo samples - of e a c h type of strategy within the cluster. Within e a c h sect ion is a chart ind icat ing the cluster of indicators by prob lem, m a t c h e d to the groups of chi ldren w h o demons t ra ted these behaviours . Together, these 4 clusters m a k e up a framework of d rawing as problem-solv ing behaviours a n d character ist ics. Virtual manipulatives Within this set of behaviours, the most c o m m o n l y occur r ing cluster of behaviours, chi ldren used their pictures, sketches or representations as manipulat ives - like counters to b e m o v e d or a c t e d u p o n . Table 8 -Summary of Virtual Manipu lat ive use for the C o o k i e a n d Wheels problems (all groups) - provides a summary of these act ions for the groups, o rgan i zed by quest ion. Note that the numeral in e a c h cel l refers to the total number of o c c u r r e n c e s of the behav iour or strategy used by a n y m e m b e r of the group. For e x a m p l e , within the first g roup, only o n e of the chi ldren used their pictures like counters to solve the Wheels P roblem, while in the C o o k i e Problem, there were 4 o c c u r r e n c e s obse rved within the g roup. Table 8 - Summary of Virtual Manipu lat ive use for the C o o k i e a n d Wheels problems (all groups) VIRTUAL MANIPULATIVES COOKIES PROBLEM WHEELS PROBLEM CREATING SETS DISTRIBUTION COUNTERS >«/> o z < LU OS U o i— 3 CO on t— CO a OH ' LU 1— Z-O u Students' names / f 2 3 4 0 0 1 Mark, Joanna, Melissa, Sara 3 2 0 2 2 o Jason, Lisa, Grant, Susanne 1 1 2 2 Pricilla, Martin, Larry, John 4 3 3 0 2 Anthony, Avril, Orson 5 3 3 0 Q X : Norman, Lome, Cathy 3 2 3 0 i •: Ruby, Mona, Rebecca, Karl 2 0 2 0 1 2 Trent, David, Charlie 2 0 6 3 0 2 Hannah, Zarah, Susie, Jessie 3 2 2 1 0 3 Stephanie, Shauna, Jorge, Iris Through observat ion during p rob lem solving I establ ished that chi ldren were using their drawings like phys ical objects . The notion of a c t i o n is key here; chi ldren's solution-finding a n d reasoning was suppor ted by mode l ing the a c t i o n in the p rob lem. As a virtual manipulat ive, pictures, tallies, circles a n d cook ies c o u l d all b e t ransformed. Chi ldren w h o demons t ra ted these behaviours d rew both in i c o n o g r a p h i c a n d pictor ial style - that is, the sophist ication of their drawings d id not p r e c l u d e their use of virtual manipu lat ive strategies. Lisa, for e x a m p l e , d rew very e l a b o r a t e f a c e s for the c o o k i e p rob lem a n d used those pictor ial representations as counters in her solution-finding, whereas John d r e w simple circles to represent both cookies a n d chi ldren a n d " m o v e d " them with lines. Both processes involved using pictures like manipulat ives, but the level of abst ract ion of those pictures di f fered. Creat ing sets Chi ldren w h o c r e a t e d sets o rgan i zed their representations into distinct groups. Sometimes these groups were c i rc led or b o x e d ; occas iona l l y the sets were o rgan i zed in rows or columns, or d rawn on o n e side of d i v ided p a g e or another . The point of c reat ing sets was to organize the elements of the p rob lem in a visual w a y so that a n act ion c o u l d take p l a c e on t h e m . Circles or boxes h e l p e d chi ldren k e e p track or separate the elements of the p rob lem. The fol lowing examples show clear ly the elements of the p rob lem in 2 separate groupings; o n e represents a d e v e l o p e d solution (Figure 9 - Sara's sets) the other shows Shauna's groups a n d final solution (Figure 10 - Shauna's sets). Figure 9 - Sara's sets Figure 10 - Shauna' s sets Another form of c reat ing or del ineat ing sets is shown in Dav id ' s work. Dav id d r e w 18 cook ies in a n initial set, then c o u n t e d out 12 of t h e m , marking e a c h o n e with a dot, a n d then d rew a div iding line to c r e a t e a set of e a t e n a n d a set of u n e a t e n cookies . When asked to tell a b o u t this last mark on the p a g e , he r e s p o n d e d : "The line is s topping it there. " He then wrote numbers to the left a n d the right of the line (12 a n d 6) to show h o w m a n y who le cook ies were on e a c h side. (Figure 11 - Dav id ' s Line) Drawing was important here b e c a u s e it a l l o w e d Dav id to de l i neate b e t w e e n the parts of his solution; in this c a s e to mark h o w m a n y of the who le cook ies h a d b e e n al lotted to chi ldren a n d wh ich ones w o u l d h a v e to b e t reated differently. This is a n e x a m p l e of d raw ing-as -p rob lem-solving; it shows the purposeful use of a line, w h i c h Dav id himself descr ibes as " s topp ing it". "It" might b e interpreted as the number of chi ldren in possession of w h o l e cook ies . Dav id later conf i rmed that the strategy that h e l p e d him the most in solving the prob lem was: " W h e n I d r e w the l ine." Figure 11 - David ' s Line David ' s thinking Cookies and Kids There are 18 cookies pn a plate and 12 children who want to share them. How can the children share the cookies? How much will everyone get? BOQO Occoeo exqw| \ $ & Dav id draws a set of 18 cook ies - t h e 12 chi ldren are unders tood. The vert ical line, says Dav id , divides the who le cook ies from the half ones: "The line is s topping it there. " Distribution Distribution is a n a c t i o n , characte r i zed by the " m o v i n g " of the e lements of a p r o b l e m . Sometimes the m o v e m e n t was a c c o m p l i s h e d by t rac ing lines from o n e set of objects to the items in another set; thereby distributing or sharing out e a c h of the items. Distribution a p p l i e d to both w h o l e a n d part ial cook ies as a w a y of a l locat ing the set. In order to establish the response to the p rob lem, e a c h of the lines c o n n e c t i n g the port ion a m o u n t to the receiver then h a d to b e c o u n t e d , a n d the shared portion (the quotient) verif ied for fairness (see Figure 12- Anthony distributes). This drawing as problem-solv ing strategy lent itself well to division problems, where the a c t i o n of sharing was rep l icated by the phys ical distribution of items. Larry's work gives a c lear indicat ion of this. (Figure 13 - Larry distributes) During the process of solving this p rob lem, Larry lost t rack of the cook ies he h a d distr ibuted a n d used finger count ing to verify his i d e a . "This is tricky," he c o m m e n t e d . " O o p s ! I forgot to join this." Pricilla's thinking Figure 14 - Pricilla's counters It's lines, so 1 c a n he lp. So 1 c a n think, (count ing a n d then boxing groups of lines) What are the lines for? So 1 c a n count t h e m . So when you put those lines there, Pricilla, what are you counting? ^ , , " How m a n y kids will h a v e cookies . So each of those lines is a cookie? Y a . C o o k i e s a n d K i d s V i d e o of Jason shows a range of drawing as problem-solv ing strategies, including the use of drawing as counters. In the cl ip i n d i c a t e d b e l o w , Jason counts concurrent ly with his finger a n d his penc i l , m a t c h i n g cook ies to chi ldren (Figure 15 - Jason keeps track with tallies). The marks on the p a g e (simple lines for peop le) are not part of his solution but rather are the tools for arriving at that solution. To w a t c h Jason as he keeps track, p lease v iew V i d e o Cl ip 2, ent it led: " Jason keeps track with tall ies" Figure 15 - Jason keeps track with tallies Systems The fol lowing strategies descr ibe ways in wh ich chi ldren a p p r o a c h e d the task systematical ly. E a c h of these strategies was d e v e l o p e d a n d a p p l i e d spontaneous ly by the ch i ld . The drawings e m p l o y e d in e a c h c a s e were either the ob ject of the system (they were " o p e r a t e d o n " in a systematic way) or the drawings or symbols were the system itself (tallies, e tc) . Even chi ldren w h o struggled with the problems initially h a d more success o n c e they a d o p t e d a systematic a p p r o a c h than those w h o a t t e m p t e d other strategies. Table 9 outlines the different systems used by the chi ldren a n d their f requency . As before, the numeral in e a c h c o l u m n indicates the number times e a c h strategy was used by members of the group. For e x a m p l e , o n e of the members of the first group used a n el imination strategy in the resolution of the Cook ies Problem, but no o n e in the s e c o n d group used the c h e c k a n d r e c h e c k strategy to solve the Wheels P roblem. Table 9 - Summary of Systems use for the C o o k i e a n d Wheels problems (all groups) SYSTEMS COOKIES PROBLEM ^WHEELS PROBLEM KEEPING TRACK ELIMINATION CHECK&RECHECK . u • bZ LU LU O z S , 15 LU 1 o LU Students' names > ^ - | * ^ "y 3 1 3 1 1 ] 2 Mark, Joanna, Melissa, Sara 2 1 4 3 2 0 Jason, Lisa, Grant, Susanne 1 4 4 f^2*-A 3 Pricilla, Martin, Larry, John 3 4 1 0 1 ' Anthony, Avril, Orson 4 3 1 1 o 0 Norman, Lome, Cathy 1 0 3 i " 3 ;-/*<> ^ ; / 3 Ruby, Mona, Rebecca, Karl 0 0 1 3 0 2 Trent, David, Charlie 5 1 4 3 0 1 Hannah, Zarah, Susie, Jessie 3 1 2 2 o 3 Stephanie, Shauna, Jorge, Iris Keep ing track Chi ldren o p e r a t e d on their drawings in this c a s e . Keep ing t rack behaviours o c c u r r e d w h e n chi ldren l a b e l e d or n u m b e r e d their drawings, to k e e p track of w h i c h ones h a d b e e n o p e r a t e d o n , distr ibuted or used up. Sometimes these labels or numbers were a d d e d initially a l o n g with the pictures. Other times, they were a d d e d after the initial set was d rawn as a w a y of verifying or record ing a partially c o m p l e t e d menta l strategy. Figure 16 - Cathy ' s numbers Cookies and Kids There are 13 cookies on a ptofe a n d 12 children who want to share them, how can the children share the cookies? How much will everyone get? Here, C a t h y has n u m b e r e d her cook ies in wholes a n d parts as a w a y of keep ing track of w h i c h ch i ld is a l lot ted wh ich c o o k i e or port ion of it. In her solution, the numbers serve 2 purposes - o n e as a strategy for k e e p i n g t rack a n d the other as a w a y of label ing the chi ldren, with the numbers from 1 to 12. In e a c h c a s e , the number cor responds to the chi ld w h o will e a t the cook ies . Circl ing or boxing objects was another indicat ion that chi ldren were using drawing to k e e p track. The drawing as problem-solv ing o b s e r v e d in this c a s e was fairly abst ract (ie - in the form of the box or the ci rc le) , but it served the purpose of separat ing those objects that h a d b e e n o p e r a t e d on from others. (Figure 14 - Pricilla's counters, a b o v e ) Below, Susanne's d rawing (Figure 17 - Susanne's box) shows h o w she thought a b o u t the p rob lem a n d used drawing to k e e p track of cook ies - a n d to show the cook ies she has " t a k e n a w a y " to the right of the b o x e d cook ies . Later, her box serves to separate who le cookies (a single c o o k i e a l lot ted to a single child) from part ial ones (split in half a n d distr ibuted to chi ldren by a n identifying number) . (Figure 1 8 - S u s a n n e ' s box 2) Figure 17 - Susanne's box Susanne's thinking Figure 18 - Susanne's box 2 "I d r e w 18 cook ies a n d I c o u n t e d t h e m - to 12 - a n d then I took these a w a y . . . " Susanne's box indicates a set of cook ies (12) g iven as wholes to the 12 chi ldren - not p ic tu red, but sugges ted by the d raw ing . In the s e c o n d part of her solution, Susanne explains: "I cut t h e m in half, like count ing by 2's, -2,4,6,8,10,12 ...12 of the half cook ie s . " Still more abst ract was the use of tallies to k e e p track of objects m o v e d , o p e r a t e d on , or ass igned to a p l a c e . For some, these marks w e r e cent ra l to the solution, or represented the solution-finding process in its entirety (Figure 29 - Grant ' s tallies, be low) . Other times the use of a tally was a supp lementa l system, used to support the al lotment of items, as in Pricilla's work. (Figure 19 - Pricilla's wheels) C h e c k a n d r e c h e c k C h e c k a n d r e c h e c k as a system required chi ldren to return to their original set of drawings over a n d over, to count a n d re-count the items to verify or disprove a partial solution. This system was akin to "guess a n d c h e c k " in d rawing form, but a l l o w e d chi ldren the scaf fo ld of go ing b a c k over their thinking during problem-solv ing. In some cases , c h e c k a n d r e c h e c k behaviours a l l o w e d chi ldren to self-correct. C h e c k a n d r e c h e c k as a drawing as problem-solv ing strategy supported chi ldren in nav igat ing through their o w n process ing of the information in the p r o b l e m . Here, (Figure 20) Jessie circles pairs of cookies to distribute to the chi ldren. He c h e c k s his work par tway through, then stops. Figure 20 - Jessie c h e c k s a n d rechecks Jessie's thinking "I'm letting eve ryone get 2, but it's actua l l y not work ing. " Elimination As a drawing as problem-solv ing strategy, el imination was a fairly sophis t icated system. Elimination required the use of 2 s e p a r a t e but concur rent act ions - the m o v e m e n t of o n e ob ject a n d the c a n c e l i n g of another . Chi ldren d e v e l o p e d a range of strategies for t rack ing w h i c h items h a d b e e n " u s e d u p " , including erasing, crossing out, stroking off or colour ing in objects . To observe Sara using her system of el imination, p lease v iew V i d e o Cl ip 3, ent it led, "Sara el iminates" . Figure 21 - Sara el iminates Sara's thinking Tell me about these x's here. That m e a n s p e o p l e h a v e those cook ies . Of all the systems used, this o n e was the most support ive of chi ldren's solution-finding efforts; that is, w h e n chi ldren used the el iminat ion system they were more likely to arrive at a cor rect answer, e v e n after several false starts. Norman' s work (Figure 22) is be low. He b e g a n by d rawing sets of cookies a n d then chi ldren. He worked d o g g e d l y for 11 minutes a n d e m p l o y e d a r a n g e of strategies, but only so lved the p rob lem o n c e he used el imination to help him (colouring in e a c h of the 6 remain ing cook ies while re-writing them as a fraction a b o v e the receiv ing chi ld's h e a d ) . Figure 22 - Norman's work Norman's thinking 1 d r a w 18 cook ies a n d 1 a c c i d e n t a l l y d r a w 22 so 1 crossed it out, a n d 1 d r a w 12 kids a n d 1 put every number in the cook ies a n d then 1 put the s a m e a m o u n t of cook ies a n d w h e n 1 got to 12, ... a n d then 1 c r a c k two a n d 1 got o n e half... a n d 1 s h a d e it in like this... How did the shading in help you? It doesn ' t m a k e m e 'lost' c o u n t . . . " Imagery Imagery or visualization behaviours were i n d i c a t e d through the chi ldren's oral descriptions of what they were thinking (I c a n see it..., you just break the cook ies . . . , you put these ones a n d those ones together . . . ) , in student h a n d signals or gestures ( c h o p p i n g , sl icing, separat ing groups, mot ioning " o v e r there") , or wordlessly through upraised eyes, or murmuring a n d count ing on fingers. Table 10 outlines student use of these obse rvab le indicators of i m a g e - m a k i n g a n d instances of visualization a n d d y n a m i c imagery use as repor ted by the chi ldren. Table 10 - Summary of Imagery use for the C o o k i e a n d Wheels problems (all groups) IMAGERY COOKIES PROBLEM WHEELS PROBLEM ION ION > ,••/'. ' s ^  . ' VISUALIZAT DYNAMIC IMAGERY EYES UP VISUALIZAT l l i p i i i y >->< o ? 2, CO LU . >-Students' names 4 1 2 " 0 Mark, Joanna, Melissa, Sara 0 0 2 2 0 0 Jason, Lisa, Grant, Susanne 1 0 0 • 4^ o 0 Pricilla, Martin, Larry, John 2 1 0 2 0 Anthony, Avril, Orson 2 0 1 3 >°< v: /3 Norman, Lome, Cathy 1 0 0 1 0 1 Ruby, Mona, Rebecca, Karl 2 1 2 • 3 V . 0 Trent, David, Charlie 0 0 0 0 >;;()••<); 3 Hannah, Zarah, Susie, Jessie 0 0 2 3 Stephanie, Shauna, Jorge, Iris Chi ldren w h o used these strategies d id so to support solution f inding. The timing of the use of imagery var ied - some chi ldren visualized initially to m a k e sense of the p rob lem (Jason, Shauna), some s t o p p e d to visualize w h e n they b e c a m e s tumped, (Iris, Grant) others used only visualization to solve the p rob lem. This last group of chi ldren d id not d r a w anyth ing o n their p a p e r to help them - no tally marks, no sets of counters - instead they r e c o r d e d only their solution to the p rob lem. (Charl ie, Trent, Mark, Stephanie) Eyes up Any t ime a chi ld was seen to b e "th inking", with eyes raised upwards a n d to the right, this was c o u n t e d as a n o c c u r r e n c e of "eyes u p " behav iour . "Eyes u p " was the observab le indicator of i m a g e - m a k i n g or visualization. (Figure 23 - Eyes up) A l though many chi ldren raised their eyes to "think" whi le problem-solv ing, students were not a lways a b l e to desc r ibe what they were do ing or w h a t they were " looking a t " w h e n raising their eyes (Iris, C a t h y , Lome). Figure 23 - Eyes up — I •*> » Grant Lome Jason Visualization This behav iour was obse rved w h e n chi ldren d e s c r i b e d " s e e i n g " sets of objects in their h e a d s a n d then imagining a n a c t i o n , or a n operat ion be ing per fo rmed on them. Chi ldren w h o used visualization as a p r o b l e m -solving strategy d id not always d r a w the i m a g e they c r e a t e d a n d used mental ly. Rather, they c o u l d " see the answer" or the process to fol low in their heads . S o m e chi ldren used menta l strategies in c o n c e r t with d raw ing to sort out h o w m a n y cook ies e a c h chi ld should get (Iris, Jason). A l though these chi ldren struggled to explain their process, their intensity, their raised eyes, gestures a n d h e a d - b o b b i n g i n d i c a t e d they were working hard to man ipu la te the e lements of the prob lem. Iris used her picture a n d visualizing strategies to solve the prob lem (Figure 24) a n d then ta lked a b o u t w h a t she d i d : Iris's Thinking Figure 24 - Iris's d rawing There are 12 kids so I k inda count 12 a n d the other ones I l e a v e d them there a n d . . . there's 6 left... a n d 6 plus 6 equals 12... so if you cut them in half there's 2 of e a c h o n e so I m a d e e a c h o n e h a v e half of it so all of them w o u l d h a v e o n e a n d a half. i Four chi ldren were successful in their a t tempt to solve the c o o k i e p rob lem, but d id not d r a w pictures or representations to support their thinking. Rather, these chi ldren so lved the prob lem visually using menta l images a n d then r e c o r d e d their answers afterwards. (Mark, Trent, Char l ie , S tephanie) . W h e n I obse rved Mark with his eyes raised to the cei l ing, count ing on his fingers, I a sked what he was thinking. He r e s p o n d e d : "I 'm trying to count by like stuff — 3's a n d 2's a n d ones. . . " He returned to the task, gaz ing upwards , count ing on the fingers of his other h a n d . S tephanie too spent t ime looking up at the cei l ing, o n c e she h a d d rawn 18 cook ies on her sheet. I asked her what she saw in her h e a d w h e n solving this p rob lem. "Just see like cookies a n d kids a n d like a n d see like, h o w c a n you share them in your h e a d . " Trent's response to the c o o k i e p rob lem is a g o o d e x a m p l e of " see ing things in your h e a d " . When asked to solve the p rob lem, Trent g rew quiet, a n d looked up at the cei l ing. He wrote nothing on his p a g e for a few minutes a n d then c a l l e d m e over to him - he mot ioned for m e to fol low him a w a y from the tab le a n d a w a y from the o v e r h e a d c a m e r a . I kept the h a n d h e l d c a m e r a rolling to record his v o i c e , but po in ted it at the floor. Whispering, Trent told m e that everyone wou ld get 1 a n d a half cook ies - a n d then exp la ined h o w he h a d solved the p rob lem: Trent's thinking: Figure 25 - Trent's record ing 1 c o u n t e d up to 12 in my h e a d a n d then 1 c o u n t e d up to 18 -13,14, 15, 16, 17, 18... (showing m e on his fingers h o w he counted) Then 1 c o u n t e d the cook ies split in two - 2, 4, 6, 8, 10, 12 1 I lo V» .\^ v CXCVA \ , <^ Ux-W ^ gov ^ ^ 1 : t ; ; ^ Trent's record ing of the solution (Figure 25) is a pr inted version of the process he fo l lowed in his h e a d to solve the p rob lem, a n d ind icates his use of visualization as a problem-solving strategy. D y n a m i c imagery When students e m p l o y e d d y n a m i c imagery in the solution of these problems, m o v e m e n t or c h a n g e in the elements of the p rob lem were key features. However , unlike the distribution behav iour (as d e s c r i b e d in the virtual manipulat ives sect ion), d y n a m i c imagery was a menta l process that d id not result in a d rawing. The manipulat ion a n d division of the cook ies , the des ign of the vehicles was d o n e through pictures in the students' heads - pictures some chi ldren d e s c r i b e d as " m o v i e s " . The students' hands a n d fingers m o v e d to mirror the a c t i o n taking p l a c e in their heads . W h e n a s k e d , students c o u l d descr ibe the m o v e m e n t of the objects a n d what was h a p p e n i n g to them. Mark arr ived at solutions to both problems through d y n a m i c imagery . For the c o o k i e p rob lem he sat without writing or drawing for over 11 minutes, a l though his fingers m o v e d as d id his lips while he murmured. He exp la ined that e a c h chi ld wou ld get 3 halves. If you cut six cook ies in half everyone gets half a c o o k i e , but if you cut six cook ies three times - cuz 6 plus 6 plus 6 is 18 - so you cut six cook ies in half a n d you cut 6 cookies in half a n d you cut 6 cook ies in half so e v e r y b o d y gets 3 halts of a c o o k i e . Where did you get that idea from? It just c a m e in my h e a d . For Mark, the i m a g e d id more than just " c o m e into his h e a d " . The i m a g e of the cook ies a n d chi ldren was p ictured in his visual working memory , where he m a n i p u l a t e d , cut a n d distr ibuted the cook ies in 3 sets of 6 a n d al lot ted e a c h chi ld 3/2 cook ies . A n in-depth look at Mark's p r o b l e m -solving process a n d links to v ideo e v i d e n c e follows in the discussion c h a p t e r . Sophistication of representation Indicators g r o u p e d under this h e a d i n g speak to the d e g r e e of complex i ty of the drawing - the detai l i n c l u d e d a n d the sophist ication of the thinking invo lved. In some cases , drawings were little more than marks o n the p a g e (tallies, lines, circles) whi le in other cases the representations were very realistic, full of detai l a n d des ign. In this sect ion, artistic drawings a re cont ras ted with more abst ract representations. Table 11 summarizes these character ist ics of student representations. Table 11 - Summary of Sophist ication of the Representat ion for the C o o k i e a n d Wheels problems (all groups) SOPHISTICATION OF REPRESENTATION COOKIES PROBLEM *WHEELS PROBLEM PICTORIAL ICONIC ABSTRACT THINKING REPRESENTED "/> ^  v £ -on ->>£» » ' u - ' a-J; y y ABSTRACT THINKING REPRESENTED ; ^ ' Students' names 2 0 1 1 s; > ; Mark, Joanna, Melissa, Sara 2 4 2 2 i 1 Jason, Lisa, Grant, Susanne 1 2 0 ^ 3 -7 0 Priciila, Martin, Larry, John 1 1 1 1 0 * 0 Anthony, Avril, Orson 4 2 0 2 0 \, 0 Norman, Lome, Cathy 2 2 1 ^ 1 0 Ruby, Mona, Rebecca, Karl 0 1 0 0 Trent, David, Charlie 3 0 0 4 0 , o Hannah, Zarah, Susie, Jessie 1 3 1 V i ; , . 0 Stephanie, Shauna, Jorge, Iris Pictorial use of d rawing Pictorial drawings closely resembled reality a n d c o n t a i n e d e l a b o r a t e deta i l . Chi ldren's work was "pretty" - a n d i n c l u d e d artistic renderings or cartoon- l ike pictures, (see Figures 26 a n d 27) Contextual ly , these drawings were very closely l inked, a n d idiosyncrat ic in their p roduct ion . Figure 26 - Lisa's p e o p l e Figure 27 - Lisa's p e o p l e 2 Pictorial drawings took more t ime a n d effort to d r a w than more icon ic representations - for Lisa, drawing her e l a b o r a t e p e o p l e took more than 7 minutes, a n d in her final drawing (Figure 27), w e see that Lisa miscalculates, div iding the 18 t h c o o k i e into 6 p ieces . Melissa n o t e d herself that pictor ial representations were t ime consuming: "This is go ing to take a whi le, ... d rawing these pictures!" For some, the p roduct ion of these pictor ial representations was a w a y of drawing the solution rather than process ing the information in the prob lem; that is, the i m a g e o n the p a g e was i n t e n d e d by the chi ld to b e the product of the problem-solv ing event rather than process- re lated. For others, there was a greater d e g r e e of process ing, thinking through the images p r o d u c e d - a l though the product ion of these images d id i n d e e d take more t ime. Iconic representations Iconic representations were less deta i led than pictor ial ones. In this kind of d rawing, images were more abst ract a n d genera l i zab le b e y o n d this p rob lem - p e o p l e were represented as circles or sticks or numbers, vehicles were shown by lines or circles. (See Figure 28 - Martin's vehicles, a n d Figure 29 - Shauna' s wheels) Iconic representations are simple, but they demonst rate sophist ication in their a p p l i c a t i o n ; these simple images stand for another more c o m p l e x o n e . The icon ic representations used by the chi ldren were short-term memory aids, a temporary p l a c e holder or marker to support solution f inding. Their simplicity a n d lack of detai l a l l o w e d chi ldren to process information without b e c o m i n g d i s t racted by deta i l . Figure 28 - Martin's vehicles Figure 29 - Shauna' s wheels H p <p & • ••• iMiiiiiiimiiiiiiiiiin «m Martin has p r o d u c e d only wheels a n d distributes them into "veh ic les " . Next, he names the vehicles. Shauna begins with words to descr ibe her vehicles, a n d , like Ruby, draws wheels as p l a c e -holders for her to count . Abst ract thinking represented Sophis t icated m a t h e m a t i c a l thinking was ev ident in the students' process ing of their task, at times displaying a n u n e x p e c t e d level of abst ract ion. For e x a m p l e , there were chi ldren w h o represented the solution to the c o o k i e p rob lem as a single unit; that is, chi ldren s h o w e d the answer to the p rob lem as a single thing - a p late with o n e a n d a half cook ies on it. (Sara, Jorge) Other chi ldren used a short cut to simplify the a m o u n t of record ing necessary or to k e e p track of multiple objects with a single i m a g e or set of character s . Like a n "x" in a n a l g e b r a i c expression, this i m a g e stood for multiple other sets of objects . Chi ldren w h o used a short cut demons t ra ted a level of sophist ication in terms of their m a t h e m a t i c a l thinking; their d rawing was not necessari ly sophis t icated, but their thinking certainly was. In the C o o k i e Problem, Grant asked questions a b o u t the quality of the drawings. " D o the pictures h a v e to b e like real n ice a n d all t h a t ? " a n d then e n c o u r a g e d his peers to d r a w simply: "Just d r a w h a p p y f a c e s , " he told Lisa. In the wheels task, Grant d e v e l o p e d a system for representing his vehicles that d id not involve drawing - a system that actua l ly m a t c h e d his thinking a b o u t the p rob lem. (Figure 30 - Grant ' s tallies) Grant asked if he c o u l d "use tallies for car s " . He wrote the word cars on his p a p e r a n d then r e c o r d e d tallies, count ing 4, 8, 12, 16 as he d rew e a c h o n e . He d rew a fifth tally stroke through his first four, then p a u s e d a n d e rased the line, realizing he h a d m a d e too m a n y a n d n e e d e d something with just 2 wheels . Instead, he wrote "dirt bikes" a n d a single tally to c o m p l e t e his solution. I a sked him what he h a d d o n e . Figure 30 - Grant ' s tallies The Wheels Problem C o m got 18 wheels for Christmas. What toys could he havjigotten? A I wrote "4 wheels for car s " so I d rew 4 tallies. It ends up with 16 wheels . Every one of those tallies means what? Four, like instead of drawing 4 wheels.. . So if I was going to read this... (Grant, touch ing tallies) So that's eight - these two - a n d that's 12, that's 16. A n d a dirt bike has 2 wheels so after 16 plus 2 is 18. And this little line here? It's a tally. It's a tally for just one dirt bike? Y a . Grant ' s process shows a different type of thinking; more abst ract than that of his peers w h o d r e w e a c h veh ic le a n d all the wheels on t h e m . Grant ' s tallies represent a single veh ic le a n d its wheels. His solution is mult ipl icative in its representat ion as o p p o s e d to addi t ive like David ' s cumu lat i ve total or subtract ive like Lisa's line of wheels ; he counts by 4's a n d then a d d s 2 to find the solution. Grant ' s thinking a b o u t the p rob lem d e m o n s t r a t e d level of sophist ication both in its process a n d in the abst ract ion of his drawings. To listen to Grant ' s exp lanat ion of his thinking, p lease v iew V i d e o Cl ip 4, entit led " G r a n t talks tall ies". The Drawing as Problem-solving Framework Summary of the findings There were four clusters of drawing-as-problem-solv ing behaviours obse rved in the chi ldren's work that cont r ibuted to the creat ion of the framework. While the sophist ication of representat ion cluster speaks more specif ical ly to the abst ract nature of the p roduct , the virtual manipulat ives a n d systems clusters descr ibe h o w chi ldren used their drawings a n d the a c t of d rawing to process the elements of the problems. Imagery d e s c r i b e d the visual thinking that students e n g a g e d in while solving these m a t h e m a t i c a l problems. A discussion of these categor ies a n d impl ications for c lassroom p r a c t i c e fol low in the final c h a p t e r . These four categor ies or clusters of drawing-as-problem-solv ing behaviours a n d character ist ics were obse rved across both the C o o k i e Problem a n d the Wheels P roblem. In both cases , the behaviours or character ist ics of the drawings assisted chi ldren in process ing the information in the problems. The f requency of their app l icat ion di f fered, however . In Table 12 - F requency of Drawing as Problem-solving Behaviour a n d Character is t ics by Cluster of Themes - these trends were n o t e d . Numerals in the columns are a n indicat ion of how m a n y times chi ldren in the cohort of 33 were obse rved performing these behaviors (eg. Chi ldren were obse rved 9 times with eyes upraised in the Cook ies Problem). Table 12- F requency of Drawing as Problem-solving Behaviour a n d Character is f ics by Cluster of Themes CLUSTER OF THEMES FREQUENCY COOKIES WHEELS VIRTUAL MANIPULATIVES 66 26 E • Creat ing sets 25 6 or ed on probli • Distribution 16 5 D) or ed on probli • Counters 25 15 c > lages, focus ng as SYSTEMS 56 39 o > lages, focus ng as • Keep ing t rack 20 19 c t of in" itions; drawi • Elimination 13 5 T5 0) t of in" itions; drawi • C h e c k a n d r e c h e c k 23 15 o n o r o IMAGERY 24 30 i c 0 u ovem prese ocess living • Dynamic Imagery 3 0 ovem prese ocess living • Visualizing 12 16 < E 2 a « • Eyes Up 9 14 c SOPHISTICATION 37 21 o r oduct ol r awing a: r oblem-)lving • Pictorial Representat ion 16 13 r oduct ol r awing a: r oblem-)lving • Iconic Representat ion 15 7 D u o r oduct ol r awing a: r oblem-)lving • Abst ract Thinking 6 1 \j Q.-0 a V, Represented The C o o k i e Problem, the more c o m p l e x p rob lem of the two, requi red significantly more use of pictures as virtual manipulat ives for c reat ing sets, count ing a n d distribution. Both problems requi red the use of systems; however in the Wheels Problem these systems were largely m a d e up of keep ing track a n d c h e c k a n d r e c h e c k behaviours (34/39 examples) . In the C o o k i e Problem, chi ldren a c c e s s e d el imination as a strategy 8 times more of ten. The type of p rob lem i m p a c t e d the f requency of both visualization a n d pictor ial representations in the c a s e of the Wheels P rob lem. That is, chi ldren visualized the vehicles a n d drew them in detai l for the wheels p rob lem, whereas for the C o o k i e Problem they w e r e equal ly as likely to use icon ic or pictorial representations in their work. Abst ract thinking was more often seen with the more c o m p l e x C o o k i e Problem than in the Wheels Problem that required a more straightforward response. Visualization took the p l a c e of virtual manipulat ives in the Wheels Problem situation, whi le in the C o o k i e Problem students a p p l i e d both virtual manipu lat ive strategies a n d visualization in c o n c e r t to solve the p rob lem. The complex i ty of.the p rob lem a n d the nature of the p rob lem structure in f luenced the ways in wh ich chi ldren in te racted with it. M o r e distribution was seen in the division as sharing p rob lem, whi le more "eyes u p " visualizing behaviours a n d pictorial representations were n o t e d in the o p e n - e n d e d Wheels P roblem. Interview data In the interview d a t a , there were four themes ev ident in students' responses. The first c a t e g o r y i n c l u d e d chi ldren w h o d id not answer the quest ion " H o w d id drawing h e l p ? " These chi ldren d e s c r i b e d their process for solving the p rob lem rather than talking a b o u t d rawing as a p r o b l e m -solving strategy; they were not a b l e to reflect metacogni t ive ly a b o u t the quest ion. The s e c o n d a n d third groups i n c l u d e d chi ldren w h o i n d i c a t e d that d rawing h e l p e d them but c o u l d not say how, a n d then students w h o said that d rawing as a problem-solv ing strategy h a d h e l p e d them to k e e p track of the parts of the p rob lem a n d man ipu late the p ieces . A fourth g roup of students r e s p o n d e d that drawing h a d not h e l p e d them at all as they h a d used " the pictures in their h e a d s " to solve the p rob lem ins tead. Within chi ldren's responses, there were likewise patterns in the reasons they c i t e d for the helpfulness of drawing as a problem-solv ing strategy. Students d e s c r i b e d the ways in w h i c h they h a d used their drawings like manipulat ives, to count , m o v e , distribute or otherwise a c t out the p rob lem. Chi ldren spoke a b o u t h o w drawing h e l p e d them to k e e p track, to test out a n i d e a or h o w chi ldren were a b l e to a p p l y a system to their solution-finding through drawing. A summary of students' responses a n d the reasons for their responses follows. The tab le , (Table 13 - Interview d a t a : Type of support a n d f requency) gives a numer ical overv iew of the interview d a t a for chi ldren w h o r e s p o n d e d that drawing h a d h e l p e d them. This is fo l lowed by examples of e a c h type of support, or h o w drawing h a d h e l p e d , as d e s c r i b e d by the chi ldren (Figure 30 - Sample interview responses by c a t e g o r y ) . A c o m p l e t e list of chi ldren's interview statements is c o n t a i n e d within A p p e n d i x 1 - Interview Data. Table 13 - Interview d a t a : Type of support a n d f requency TYPE OF SUPPORT (HOW DRAWING HELPED) NUMBER OF REFERENCES TO EACH TYPE OF SUPPORT BY STUDENT Subject of a n a c t i o n 7 Distribution 5 M e m o r y a i d 7 Elimination system 3 Pictures as icon ic representations 1 Examples of student's explanations of how drawing as problem-solving helped Figure 31 - Sample interview responses by c a t e g o r y Drawings were the subject of a n a c t i o n ; d rawing is a m a t h e m a t i c a l a c t Ruby says: I d r a w e d o n e c o o k i e for e a c h chi ldren then I saw the 6 leftover a n d I knew that I h a d to cut them into p ieces. . . Grant says: C u z you c a n d r a w f a c e s a n d d r a w cookies to the faces . . . instead of just keep ing it in your h e a d . Anthony says: [Drawing is] ...like a c t i n g what ' s h a p p e n i n g . Drawing's like a c t i n g b e c a u s e . . . the kids take cook ies . Orson says: It's like a c t i n g b e c a u s e it's like what p e o p l e d o . D id you have any acting in yours, Orson? Yes, like taking a w a y , like giving p e o p l e . Jessie says: ...the drawing is sort of like do ing the m a t h . Like drawing the m a t h . What part of the math were you drawing, Jessie? It's like w e ' r e do ing take-aways . Like I d id o n e , there's 1, 2, 3 a n d 1, 2, 3, a n d it makes six. A n d it makes 12. Drawing as a n Elimination System Norman says: . . .and I s h a d e it in like this. How did the shading in help you? It doesn ' t m a k e m e lost count . . . Drawing a l l o w e d for trial a n d error Shauna says: It helps b e c a u s e I c a n try different things, like 3's or ones if that works. C a t h y says: It helps m e d r a w so like, uh, in c a s e I d o like four in half a n d it doesn ' t work then like sometimes it c a n help m e , so if I so if I h a v e to cut it in half or just leave it... Drawing as a M e m o r y a i d : Zarah says: It's like you c o u l d look at the picture a n d d o it with the picture instead of d o i n g it in your brain a n d mak ing it e x p l o d e k inda. Pricilla says: It h e l p e d instead of in your h e a d a n d get mixed up. What did the drawing do to make it easier? By looking at it a n d thinking, how m u c h I c a n cross off? Lisa says: So w h e n w e d o n ' t h a v e to d o it in our heads a n d d o i n g it in our heads is harder. Pictures as icon ic representations Lome says: I c o u l d like d r a w math a n d stuff, like circles, e x c e p t . . . t h e y were cook ies . Like if there were 15 houses..., I c o u l d p robab ly d o that, only like I wou ldn ' t d r a w full houses with so m u c h detai l , I'd use up all my time just using o n e house. . . I actual ly just d r a w circles for like everything. Interview data: Summary of the findings Chi ldren were incl ined to d r a w in responding to e a c h of these problems. This was a n indicat ion that students c o u l d see that using pictures w o u l d b e a purposeful activity that wou ld support them in m e a n i n g - m a k i n g . Their c a p a c i t y to descr ibe the usefulness of the pictures or the a c t of d rawing itself var ied with the chi ld's ability to a c c e s s l a n g u a g e to descr ibe these relationships. Several were a b l e to explain not only h o w their pictures h e l p e d (in al lowing them something to a c t on or a n ob ject to distribute, as a memory support) but also h o w drawing itself m a d e problem-solv ing more m a n a g e a b l e (using a system of el imination). Chapter 5 - Discussion Overview This c h a p t e r provides the forum to examine the findings a n d to discuss the main research questions, namely : " H o w d o young chi ldren a c c e s s visualization a n d imagery in their resolution of m a t h e m a t i c a l tasks? How does drawing as problem-solving support students in mak ing sense of a n d reasoning through a m a t h e m a t i c a l p r o b l e m ? " Results from the study reported in the previous c h a p t e r , as well as the chi ldren's interview responses a n d the app l icat ion of the f ramework for assessing a n d descr ib ing drawing as prob lem solving will b e discussed here. A n examinat ion of the condit ions contr ibut ing to successful d rawing as problem-solv ing will also b e cons ide red, a n d impl ications for c lass room p r a c t i c e will fol low. Reflecting on the problems as illustrative of drawing as problem-solving: A comparison of the Cookie Problem and the Wheels Problem O n e quest ion I sought to answer involved the d e g r e e of complex i ty of the problems presented for solution through drawing. Specif ical ly, h o w c o m p l e x d id a prob lem h a v e to b e in order for a chi ld to pick up a penc i l a n d d r a w to reason towards a solution? Questions re lated to the complex i ty of the p rob lem a n d the prob lem structure (What word(s), situations or act ions in a m a t h e m a t i c a l situation l e a d a chi ld to pick up a penc i l a n d sketch that act ion?) are addres sed be low. In a n at tempt to clarify these issues, the fol lowing presents a n examinat ion of the C o o k i e a n d Wheels Problems, both in terms of their structure a n d difficulty level, a n d students' responses to those problems. The C o o k i e Problem, by virtue of the structure of the p rob lem a n d the a c t i o n of division as sharing a l l o w e d for a great d e a l of observab le d a t a in terms of d rawing as problem-solv ing. Chi ldren h a d to share out the cook ies , a n d in so d o i n g h a d to mirror a phys ical process of distribution through d rawing . S ince 6 of the 18 cook ies h a d to b e split in half, the a c t i o n of cutt ing was rep l icated in the chi ldren's process ing of the p rob lem through drawing. Overal l , the act ions obse rved in the students' p e r f o r m a n c e i n c l u d e d : count ing a n d partit ioning (12 cook ies a n d 6 cookies) , splitting remaining cook ies a n d distributing who le a n d part ial cook ies to chi ldren. These act ions, speci f ic to division or sharing problems, p rov ided rich observab le d a t a from a research perspect ive. As wel l , the complex i ty of the p rob lem (18*12) was such that chi ldren c o u l d not easily solve the p rob lem without some sort of scaffo ld in the form of virtual manipu lat ive or r e c o r d e d system. Drawing p rov ided chi ldren with this support for thinking, al lowing them to k e e p track of the elements of the p rob lem by operat ing on smaller sets of objects or by using a system like the process of el imination. In contrast, the Wheels Problem d id not prov ide as rich a n array of d rawing as problem-solv ing strategies to observe. The nature of the wheels p rob lem in terms of c o n c e p t u a l understanding a n d complex i ty (finding multiple a d d e n d s to m a k e 18) l e a d these chi ldren to use drawing as a memory support a n d a w a y to record their solution, as o p p o s e d to using drawing as a w a y to think a b o u t the p rob lem. In the Wheels Problem, students were more often drawing their solutions rather than employ ing drawing to process information as they h a d in the C o o k i e Problem. Consequent ly , I d id not observe as m u c h of the chi ldren's thinking - or as divergent a range of thinking - as I h a d d o n e earlier in the study. Al though the p rob lem was o p e n - e n d e d a n d featu red m a n y possible answers, the number of d rawing as problem-solv ing processes for f inding a n answer were l imited. M a n y chi ldren used pictorial representations -d e t a i l e d , realistic pictures for their solution - that in some cases d i s t racted chi ldren from the m a t h e m a t i c a l thinking. Trent a n d others (Orson, Lome) were c o n c e r n e d a b o u t the quality of their pictures, c o m m e n t i n g , "I d o n ' t know h o w to d r a w a ". These c o m m e n t s suggest that physically d rawing the veh ic le was c reat ing a more difficult p rob lem for these students, compromis ing their thinking a n d gett ing in the w a y of the m a t h . A l though no chi ld was a b l e to solve this p rob lem without d rawing I w o u l d suggest this was not b e c a u s e of the complex i ty of the m a t h e m a t i c a l c o n c e p t invo lved, but rather the size of the number (18) be ing broken up into p ieces a n d reassembled. What distinguished the Wheels Problem from the C o o k i e Problem in terms of drawing as problem-solv ing was the lack of a c t i o n required to solve it. Chi ldren d id not n e e d to partit ion, split or distribute wheels as they h a d with the cook ies . Instead, they h a d to select a n d assign wheels to vehicles in groups, effect ively count ing up to 18 in chunks a n d record ing their solution as a cumulat ive total . This w o u l d imply that certain types of problems are more suited to drawing as problem-solv ing, providing for more observat ional d a t a with w h i c h to assess thinking. Problems that feature a c t i o n , are c o m p l e x in terms of number of steps, difficulty of the m a t h e m a t i c a l c o n c e p t a n d / o r the quantity be ing m a n i p u l a t e d lend themselves more readily to drawing as problem-solv ing. The m a t h e m a t i c a l reasoning involved in this kind of p rob lem requires a tool for thinking through it, a n d the c o m b i n a t i o n of virtual manipulat ive use, imagery a n d systems is well suited to solution f inding. In des igning opportunit ies for chi ldren to demonst rate a n d e x p e r i e n c e this type of m a t h e m a t i c a l process, the d a t a suggest o n e w o u l d n e e d to ensure that the problems: a) inc lude a n a c t i o n in the c o n c e p t u a l structure (joining, separat ing, sharing, distributing, c o m p a r i n g ) , b) are sufficiently c o m p l e x (multi-step, large numbers, multiple a d d e n d s or factors) to require the app l icat ion of a system. A task that is too simple for the solver c a n b e solved by similar menta l processes (a more d e v e l o p e d evolut ion of the drawing as p r o b l e m -solving schema) a n d does not require the student to man ipu late the elements of the p rob lem through representat ion. While the internalizing of these drawing as problem-solv ing processes is not a n undesi red o u t c o m e , in order to prov ide opportunit ies for chi ldren to p r a c t i c e , talk a b o u t a n d reflect on h o w drawing c a n support their m a t h e m a t i c a l thinking, c a r e must b e taken with the des ign of the problems p o s e d to inc lude a c t i o n a n d complex i ty . Act ion in the prob lem structure requires a c t i o n in the drawing; complex i ty in the prob lem requires a system for keep ing track of the elements of the prob lem during problem-solv ing. In this w a y , a student's d rawing as problem-solv ing has a purpose - the e laborat ion a n d determinat ion of a solution. What is drawing as problem-solving? Creating a definition In d e v e l o p i n g my research quest ion a n d pursuing clar i f icat ion through research, I h a d to c r e a t e some l a n g u a g e to descr ibe what it was I h o p e d to observe. With a focus on h o w students solve m a t h e m a t i c a l problems through the a c t of d rawing, I n e e d e d to distinguish b e t w e e n during problem-solv ing a n d after problem-solv ing d a t a ; focus ing on the a c t of d rawing, not the drawing itself as a measure of m a t h e m a t i c a l reasoning. I d id not w a n t to assess student drawings after they h a d so lved a p rob lem a n d m a k e assumptions a b o u t their thinking during the task. This type of assessment wou ld not prov ide m e with observat ional or p e r f o r m a n c e d a t a , only artifacts. Instead, I w a n t e d to b e a b l e to observe a n d descr ibe student thinking whi le it h a p p e n e d , in part icular the a c t of d rawing , a n d c o n n e c t that observab le a c t i o n to a m a t h e m a t i c a l thinking process or line of reasoning. As suggested by Tall (1994) I w a n t e d to observe chi ldren doing a n d thinking a b o u t the math - I w a n t e d to observe students in the a c t of conver t ing their visual images to representations a n d then reasoning through them. The term I c h o s e to use - d rawing as problem-solv ing - g rew from my research study a n d carefu l process ing of the study's d a t a . The definit ion itself e v o l v e d over t ime, a n d p r o v e d helpful in refocusing my attent ion on process as o p p o s e d to p roduct in terms of student thinking. Assessing drawing as problem-solv ing requires v ideo f o o t a g e or first h a n d observat ion; assessing drawing of a solution requires only student artifacts. The term drawing as problem-solv ing served as a reminder that it was the p rocess -based observat ional d a t a that I sought to explore. no The operat iona l definit ion that informed a n d s h a p e d my observations, then, is as follows. Drawing as problem-solv ing descr ibes the process of thinking a b o u t a p rob lem, its context a n d elements a n d process ing that information through the a c t of d rawing. Drawing as problem-solv ing c a n result in a c o m p l e t e solution, with all parts of the p rob lem d e s c r i b e d a n d e l a b o r a t e d - a finished product that a c c o u n t s for both the process a n d product of the thinking. Alternately, the drawing itself c a n b e only partially f leshed out; in this situation reasoning is suppor ted through the a c t of putt ing penci l to p a p e r a n d a response dete rm ined without a c o m p l e t e l y d rawn solution. As a process, however , the reasoning through pictures is c o m p l e t e , e v e n if the solution itself is not fully d r a w n . For e x a m p l e , e a c h of the fol lowing students used drawing as a means to solve the C o o k i e Problem, but the d e g r e e to wh ich the students m a n i p u l a t e d the i m a g e on the p a g e var ied. In d rawing to solve the p rob lem, chi ldren c r e a t e d a part ial or c o m p l e t e picture as p roduct . In Anthony's work (Figure 32 - Anthony's drawing), the process of solving the p rob lem was ev ident in the lines on the p a g e . He used drawing as a means of c o n n e c t i n g cook ies to chi ldren, a n d his work illustrates a c o m p l e t e solution - both in process a n d product . Anthony ' s use of d rawing as problem-solv ing cont inued until he h a d distr ibuted all the cook ies a n d part ial cookies to the 12 chi ldren. Anthony so lved the p rob lem by d rawing. S h a u n a , however , so lved the prob lem while d rawing (Figure 3 3 - S h a u n a ' s drawing) . Within the process of d rawing sets of cook ies a n d chi ldren a n d cons ider ing the elements of the c o o k i e p rob lem, Shauna so lved the p rob lem. The d rawing was a condui t for her thinking, a m e a n s to a n e n d ; she p rocessed the information in the prob lem a n d r e c o r d e d her answer in numbers a n d words. Figure 32 - Anthony ' s d rawing Figure 33 - Shauna' s d rawing Cookies and Kids Ptytf Thete are 18 cookies on a ptate and 12 children who want to share them. How can the children share the cookies? How much will everyone get? i n f f l Cookies and Kids There ore 18 cookies on a plate and 12 children who want to share them. How can the children share the cookies? How much wHI everyone get? /: yog While Anthony's thinking is explicit in his d rawing {connect cookies to children with a line, distribute all the whole cookies then cut up all the leftovers and distribute them), Shauna's thinking is implicit in terms of w h a t she r e c o r d e d on the p a g e (Figure 33). Shauna drew, n u m b e r e d a n d then c o m p a r e d sets, found 6 cookies remaining a n d split those 6 cook ies in half. She mental ly c a l c u l a t e d o n e a n d a half cook ies per ch i ld , taking into a c c o u n t the w h o l e a n d partial cookies . She e x p l a i n e d : "I tr ied o n e c o o k i e e a c h but there was 6 left, a n d then I tried 2, so I just tried o n e a n d a half a n d then it w o r k e d . " Shauna is less reliant on d rawing as p r o b l e m -solving than Anthony; she was a b l e to use the a c t of d raw ing to scaf fo ld her thinking a n d mental ly distribute the cook ies , while Anthony worked through all of the elements of the C o o k i e Problem by physical ly mov ing them. Drawing as problem-solv ing: What it is, what it isn't Drawing as problem-solv ing is not i n tended to focus on the analysis of a chi ld's representat ion of the answer to a p rob lem after a chi ld has c o m p l e t e d the thinking process. It is not i n tended to descr ibe the examinat ion of the details i n c l u d e d in the d rawn solution or the a c c u r a c y of that solution. Drawing as p rob lem solving is the process of thinking through representat ion that leads to a solution - a process w h i c h might only b e partially r e c o r d e d on p a p e r . Drawing as problem-solv ing strategies also inc lude a n d are supported by visualization a n d menta l i m a g e - m a k i n g ; for some young chi ldren in this study these latter strategies were the only ones used in the solution of the problems presented. New understandings given video observations: From "drawing the solution" to "drawing thinking" Before beginn ing this study, I h a d e x a m i n e d student work a n d interpreted this p roduct for a range of tasks - whether that finished p roduct was in the form of a d rawing, writing, numbers or a c o m b i n a t i o n of all three. Like D iezmann (2000), Smith (2003) a n d Wo leck (2001), I unders tood that d rawing suppor ted chi ldren in descr ib ing their process for solving a p rob lem, but i m a g i n e d that a student's d rawing was a w a y to represent that thinking after the fact . I d i scovered , through the analysis of the v ideo d a t a a n d through interviews with the students, that in fact a great d e a l of process ing of information h a p p e n s while chi ldren put penc i l to p a p e r - so m u c h so that the complex i ty of the process itself c a n n o t truly b e seen in the student's marks on the p a g e . Like the notion of a p r o c e p t c o i n e d by Tall (1994), d rawing as problem-solv ing provides chi ldren with both a w a y to d o the math a n d a w a y to think a b o u t the m a t h . As d e s c r i b e d in the drawing as problem-solv ing f ramework, student thinking a n d process ing of the p rob lem - a c t i n g on it - c a n b e obse rved through a variety of behaviours (creat ing sets, c h e c k a n d r e c h e c k , distribution, visualization) whi le chi ldren are act ive ly e n g a g e d in thinking through the p rob lem. Some of these behaviours l e a d to pictor ial d rawing (e laborate a n d d e t a i l e d pictures); some lend themselves to icon ic or abst ract representations (tallies, lines, dots); a n d still other behaviours d o not p r o d u c e a n y phys ical representations at all, but rather menta l images . Regardless of the p roduct (pictorial, icon ic or menta l images) , d rawing as problem-solv ing, (or reasoning through representation), is a cont inuous process as a chi ld works through a m a t h e m a t i c a l task. At its most eff icient, this is a flexible process involving m a n y over lapp ing processes or strategies. A n d , at its most e v o l v e d level, these processes take p l a c e mental ly. After problem-solv ing to during problem-solv ing Drawing as problem-solv ing h a p p e n s while a chi ld is e n g a g e d in the a c t of reasoning through the p rob lem - working with the elements of the p rob lem to m a k e sense of it. It is a "during problem-so lv ing" strategy, a n d must b e obse rved as a process or p e r f o r m a n c e . The f ramework is i n t e n d e d to prov ide a series of observable behaviours to support teachers a n d researchers in seeing these strategies as they h a p p e n , to assess student thinking during problem-solv ing rather than assessing student solutions after problem-solv ing. Assessing during problem-solv ing gives us formative p e r f o r m a n c e d a t a b a s e d on student understanding a n d strategic thinking; it is d y n a m i c a n d a c t i v e by definit ion. Affer p r o b l e m -solving assessment provides a static representat ion of the solution; it allows for eva luat ive d a t a gather ing afterwards. The Drawing as Problem-solving Framework Some questions and reflections The drawing as problem-solv ing f ramework as out l ined in C h a p t e r 4 highlights the strategies students used a n d character ist ics of their drawings as chi ldren worked through both the division as sharing p rob lem (the C o o k i e Problem) a n d the Wheels Problem. The intention of the f ramework is to identify a n d descr ibe a genera l list of drawing as problem-solv ing behaviours a n d character ist ics. In d e v e l o p i n g the framework, several questions were raised for m e , especial ly a round the b r o a d e r appl icabi l i ty of the descriptors b e y o n d these two problems a n d the range of students' d e v e l o p m e n t a l a p p r o a c h e s to drawing as problem-solv ing. These questions, regard ing the use of virtual manipulat ives, the notion of a d e v e l o p m e n t a l cont inuum to descr ibe drawing as problem-solv ing, a n d the relative sophist ication of students' representations will b e raised in e a c h subsequent sect ion. Virtual Manipulatives Through observat ion during p rob lem solving I establ ished that chi ldren were using their drawings like phys ical objects . The notion of a c t i o n was key in the descr ipt ion of virtual manipulat ives as a set of behaviours in the framework; as d o c u m e n t e d in the work of C a r p e n t e r a n d Fennema' s Cognit ive ly G u i d e d Instruction (Carpenter , F e n n e m a & Franke, 1996), chi ldren's solution finding was supported by mode l ing the a c t i o n in the p r o b l e m . Like manipulat ives, drawings of the elements of the p rob lem served a purpose in scaffo lding m a t h e m a t i c a l thinking. Chi ldren were a b l e to o p e r a t e on their drawings in m u c h the s a m e w a y they w o u l d with a b lock or counter , mov ing, count ing a n d distributing t h e m . This is consistent with the work of Wo leck (2001) a n d Smith (2003) w h o d e s c r i b e d their students' use of pictures as manipulat ives. With the support of their pictures a n d a strategy for organiz ing their d rawn counters into sets, students were a b l e to mirror the act ion in the p rob lem to solve it. Even the most sophis t icated thinkers (those w h o a c c e s s e d menta l imagery or visualization) used virtual manipulat ives to support their solution finding -only the a c t i o n per fo rmed on the manipulat ives was menta l rather than phys ical . (Iris, Shauna) From physical models to mental images I be l ieve that this a c t i v e use of drawings as a virtual manipulat ive m a y b e a precursor to d y n a m i c imagery. Dynamic imagery (Gray, Pitta, & Tall, 2000; Owens & C l e m e n t , 1998; Owens , Mi tche lmore, Outh red, & P e g g , 1996; Wheat ley & C o b b , 1990) is a c o m p l e x thinking process that involves the c reat ion of a menta l i m a g e a n d the manipulat ion of that menta l i m a g e . In m a t h e m a t i c a l research, the c a p a c i t y to c r e a t e a n d o p e r a t e on a menta l i m a g e is touted as a powerful m a t h e m a t i c a l problem-solv ing tool (Thomas, Mul l igan, a n d Go ld in , 2002). Deve lop ing d y n a m i c imagery requires that the user has h a d meaningfu l exper iences with a variety of phys ical models , a n d that these models b e c o m e internalized as visual models or images . In the progression from c o n c r e t e to abst ract - from phys ical manipulat ives to menta l images for them - virtual manipulat ives m a y prov ide a n intermediary step towards the construct ion of a menta l representat ion of a c o n c e p t or p r o c e d u r e . Chi ldren's drawings of cook ies or wheels a n d the manipulat ion of those drawings through c o n n e c t i n g lines, sl ice marks, circles a n d boxes s e e m to support chi ldren in transitioning from phys ical models to a more abst ract form of i m a g e r y - b a s e d thinking. Talking a b o u t these images , their m o v e m e n t on the p a g e a n d h o w they m a t c h a n d c o n n e c t to the images chi ldren form in their heads c a n prov ide a n important next step instructional^, s ince chi ldren's l a n g u a g e for the images they c r e a t e falls short of descr ib ing the visualization that takes p l a c e . Interview results support this; a l though chi ldren a c k n o w l e d g e d that drawing h e l p e d them to solve these problems, few were a b l e to art iculate how, a n d m a n y l a c k e d the l a n g u a g e necessary to descr ibe the process at all. Anthony was a n e x c e p t i o n ; he said that drawing was "l ike a c t i n g what ' s h a p p e n i n g . " More recogni t ion is be ing p a i d to the power of imagery in m a t h e m a t i c a l problem-solv ing. By support ing chi ldren in c o n n e c t i n g metacogni t ive ly to their menta l images , they will b e better a b l e to understand both the structures a n d processes necessary for more abst ract - a n d therefore more genera l i zable - m a t h e m a t i c a l thought. Sophistication of the drawing The fol lowing is a compar i son of the relative sophist ication of the representations m a d e by the students as they so lved the C o o k i e a n d Wheels problems. A rat ionale for support ing chi ldren in d e v e l o p i n g structures a n d systems for their images follows. Drawing in math class poses issues for c lassroom teachers , w h o are c o n c e r n e d that too m u c h student emphasis on highly deta i led drawings a n d artistic representations will det ract from the mathemat ics . Conversely, a chi ld's icon ic representations are cons ide red more ind icat ive of higher- level thinking; s ince the level of abst ract ion is greater , i con ic representations ind icate more d e v e l o p e d m a t h e m a t i c a l notions. How then to support chi ldren in mak ing the most of d rawing as p r o b l e m -solving? How c a n student representations support reasoning most effect ively? In this study, chi ldren's drawings demons t ra ted a range of sophist ication, from highly deta i led pictor ial representations to abst ract icon ic sketches. Chi ldren w h o d rew c o m p l e x a n d deta i led drawings - not surprisingly -spent more t ime on the pictures than those w h o m a d e tallies or circles. As a result, chi ldren w h o labou red over their drawings h a d less t ime to man ipu late those drawings as virtual counters, a n d consequent ly spent less t ime e n g a g e d in the math of the p rob lem. If a m a t h e m a t i c a l c o n c e p t is exp lored a n d the p rob lem resolved through the manipulat ion of the images on the p a g e , then it follows that chi ldren's t ime on task should b e spent in the manipulat ion of their images rather than on the construct ion of them. Chi ldren w h o c o n c e n t r a t e d on the d rawing of their images (Livena, etc) d id not h a v e the s a m e d e g r e e of success as those chi ldren w h o were more abst ract in their representat ion. My quest ion is whether chi ldren resort to pictorial drawings w h e n they are unc lear on the mathemat ics n e e d e d (Avril's solution) or whether chi ldren get lost in the a c t of d rawing a n d are dis t racted from the mathemat ics . Which of these (or both) is responsible for the g a p in a c h i e v e m e n t of those chi ldren w h o d r e w pictorial ly? D iezmann (2000) suggests that students w h o e x p e r i e n c e difficulty in representing their thinking d iagrammat ica l l y m a y also struggle with spatial a n d / o r number sense. In terms of this study a n d drawing as problem-solv ing, d id either of these h a v e a role to p lay? Is the drawing of e l a b o r a t e pictures a response to a p rob lem that is simply too hard, or a means of d o o d l i n g towards a n answer? I a m curious a b o u t the val id reasoning a n d sense-making that occurs while chi ldren d r a w - e v e n pictorially. Woleck (2003) a n d Smith (2001), in their c las s room-based research, speak to the d e v e l o p m e n t a l a n d individual ized use of representat ion in their studies of students' pictures as solutions, stating that the i m a g e on the p a g e was a per fect m a t c h for the chi ld in terms of meaningfu l representat ion. They d rew what m a d e sense for them, at a level of complex i ty a n d sophist ication that m a t c h e d (Woleck, 2001, Smith, 2003). In this study, students m a d e sense through their d rawing of pictures: Avril's storytelling with the images on the p a g e , Jason's stroke marks, Anthony's st ickmen all m a d e sense for that individual ch i ld - a l though not all of these l e a d to a response of 1 a n d a half cook ies . Chi ldren in this study r e c o g n i z e d that it was not necessary to d r a w d e t a i l e d pictures in order to b e understood or to solve the p rob lem. A l though I g a v e no di rect ion a round the d rawing, chi ldren h a d a sense that there were short cuts that c o u l d b e taken in the representat ion. Some h a d b e e n taught strategies in their classrooms to support them (see Figure 35 - Pricilla's lines) a n d substituted this strategy. Other chi ldren suppor ted o n e another to simplify the task: Lome suggested to C a t h y that she c o u l d "just d r a w f a c e s " a n d Norman of fered his a d v i c e to a peer : "Try to d o it with sticks". Jason monitored this process for himself. He addres sed the c o o k i e p rob lem by writing his ideas a b o u t the distribution of the first who le c o o k i e . Then he b e g a n to d r a w 12 chi ldren - sketching their faces only -to o p e r a t e on them. After 5 faces , he g r o a n e d , crossed them out a n d sw i tched to stroke marks to represent chi ldren. (Figure 34 - Jason switches) Jason's g r o a n i n d i c a t e d his frustration with drawing all the elements, a n d the switch to icon ic representations m i d w a y through his solution-finding inc reased his ef f ic iency in solving the p rob lem. Figure 34 - Jason switches * ' 1 1 1 H j I m Two things are of note here. First, a l though chi ldren knew they d id not h a v e to d raw d e t a i l e d pictures to represent their thinking a n d g a v e others a d v i c e to this ef fect , they d rew nonetheless (eg. Lome, Norman) . S e c o n d , a l though some chi ldren h a d b e e n taught a l ternate strategies for representing m a t h e m a t i c a l thinking, they c h o s e not to a p p l y that strategy, or, in the c a s e of Pricilla, the app l icat ion of the strategy d id not help. D iezmann a n d English (2001) a n d Pantziara et al (2004) r e c o m m e n d that students b e explicitly taught a n d d e v e l o p a n icon ic system of representat ion over t ime a n d with p r a c t i c e ; that in so d o i n g chi ldren will c o m e to a more sophis t icated sense of understanding - both of the p rob lem itself a n d the m a t h e m a t i c a l c o n c e p t s beh ind it. I be l ieve that chi ldren m a y get dist racted from the m a t h e m a t i c s by over-attent ion to detai l in their representations, but I w o u l d stress that chi ldren's d e v e l o p m e n t b e careful ly cons ide red with regards to the introduct ion of more icon ic images . Like the introduction of a n algor i thm, the premature t e a c h i n g of icon ic images for m a t h e m a t i c a l situations c a n l e a d to misconcept ions on the part of students - a n d false assumptions of understanding on the part of teachers observing t h e m . Pricilla provides a n interesting e x a m p l e (see Figure 35 - Pricilla's lines). While solving both the C o o k i e Problem a n d the Wheels Problem, she a t t e m p t e d to use the strategy of " m a k e tallies a n d cross them off" as she h a d b e e n taught by her c lass room t e a c h e r . Pricilla's response was to over-genera l i ze a strategy better a p p l i e d to a situation involving subtract ion. After several attempts , she a b a n d o n e d the strategy a n d returned to writing as her preferred m o d e of information process ing. When I asked Pricilla if d rawing h a d h e l p e d her, she r e s p o n d e d in a w a y that conf i rmed her re l iance o n this part icular strategy a n d her understanding of its use: It h e l p e d instead of in your h e a d a n d get mixed u p . What did the drawing do to make it easier? By looking at it a n d thinking h o w m u c h I c a n cross off. Figure 35 - Pricilla's lines ... , "I-. C o o k i e s a n d K i d s Perhaps rather than valuing certa in types of drawings over others or int roducing chi ldren to icon ic representations too early, (as sugges ted by D iezmann a n d English, 2000; Novik, Hurley a n d Francis, 1999) it w o u l d b e worthwhi le to support chi ldren in pair ing up their drawings with a system for operat ing on t h e m . Lome suggests this very i d e a w h e n he ta lked a b o u t h o w drawing h e l p e d him: What's easier? Pictures or no pictures? They're both the s a m e . I thought not drawing was like 100 - like 98% hard , but actua l l y it's fairly easy. Like I c a n ' t tell - m a y b e this is slightly harder, that is slightly harder. It sure is shorter d o i n g this ( indicates number sentence) . But I n e e d pictures to g u i d e m e . So it's best if I t e a m up. ( indicates pictures a n d number sentence) . In the solution of the p rob lem, Lome speaks to his use of multiple strategies to support his solution-finding. A meta-cogn i t i ve a n d verba l ch i ld , Lome is a w a r e of a n d c a n art iculate his thinking a b o u t pictures, systems, numbers a n d the interactivity b e t w e e n them. Asking chi ldren to stop a n d talk a b o u t w h a t they ' re d o i n g , why they are d o i n g it a n d h o w that a c t i o n is help ing them are powerful prompts for teachers to use during the a c t of problem-solv ing - particularly if w e want to e n c o u r a g e c o n n e c t i o n -mak ing b e t w e e n the picture on the p a g e a n d the d y n a m i c i m a g e in the brain for use in m a t h e m a t i c a l thinking. Gifford (2005) suggests that rich networks of visual images , a n d a mult i -modal a p p r o a c h support chi ldren in d e v e l o p i n g menta l structures for problems-solving. " Teaming u p " (to q u o t e Lome) is i n d e e d a powerful strategy. What makes for successful drawing as problem-solving? In address ing the research quest ion namely , " H o w does drawing as problem-solving support students in making sense of a n d reasoning through a m a t h e m a t i c a l p r o b l e m ? " it was important to identify the behaviours d i sp layed by chi ldren while they reasoned through the problems. The drawing as problem-solv ing f ramework organizes a n d provides examples for these strategies. Determining the character ist ics of ef fect ive use of drawing as problem-solv ing helps to descr ibe how chi ldren use these strategies to a d v a n t a g e . I used the f ramework a n d speci f ic students as exemplars to descr ibe success in drawing as p r o b l e m -solving. Knowing what it looks like w h e n chi ldren are exper ienc ing success in this a r e a m a y support teachers in assessment. The active use of virtual manipulatives Students w h o used their pictures as virtual manipulat ives were a b l e to solve the problems, p rov ided there was a c t i o n a p p l i e d to those manipulat ives. The a c t of moving the images or operat ing on these pictures in some w a y was crit ical to success with the problems p o s e d . While the drawings c o u l d b e either icon ic or pictor ial , the important contr ibutor to success with the problems was the a c t of m o v e m e n t of the images c r e a t e d . Act ions like distributing, using the pictures as counters, a n d creat ing sets are examples . A l though students w h o d r e w more icon ic , less d e t a i l e d representations were more eff icient with their solutions, those w h o d rew pictorially still e x p e r i e n c e d success with the problems, g iven t ime. Figure 36 - Virtual manipulat ives - student samples Figure 36a John distributes half cook ies to e a c h person (represented by a stick). He sees there are 36 halves to h a n d out. Figure 36b Jason distributes wheels to vehicles using a n el imination system, crossing off wheels used. Figure 36c J o a n n a connects p e o p l e to cook ies using lines to m a t c h chi ld to who le a n d part cook ies . Figure 36f J o a n n a uses her eraser to point a n d count cook ies to ensure they h a v e b e e n m a t c h e d . Martin circles to create sets of wheels into virtual vehicles, then draws his ideas. Ruby numbers cook ies to 12 then isolates a set of 6 remain ing cook ies with her fingers.  In m u c h the s a m e w a y as a manipulat ive benefits chi ldren in working through a p rob lem, virtual manipulat ives support chi ldren in emulat ing the structures of a p rob lem (creating sets, joining, separat ing) a n d in process ing its e lements (splitting, distributing, count ing , est imating). Like a manipulat ive, though, these images must b e used with m e a n i n g a t t a c h e d ; that is, they must b e c o n n e c t e d d e v e l o p m e n t a l l y to the students' o w n menta l i m a g e of the prob lem. This includes the chi ld's construct ion of the problem's elements, shown in the i c o n i c or pictor ial form in wh ich the chi ld elects to d r a w them. The representat ion of the elements must m a t c h the chi ld's level of abst ract thinking to m a k e the use of virtual manipulat ives a meaningfu l a n d useful strategy. Unlike Pricilla's a t tempt to e c h o her teacher ' s more abst ract w a y of representing cook ies a n d wheels with tallies, w h i c h ultimately confused her, the symbol ic representat ion with w h i c h students d o the math a n d through w h i c h they think about the math (Tall, 1994) must then b e cognit ively a n d deve lopmenta l l y a l igned to the student's unders tanding. So too must the chi ldren man ipu late their pictures to m a k e the effort of d rawing them worthwhile a n d meaningfu l . It is cr it ical here that the manipu lat ion of the i m a g e b e c o n n e c t e d to the chi ld's expe r ience . In this study, 3 chi ldren d r e w the cookies a n d the chi ldren, but h a d not m a d e sense of the p rob lem as sharing e q u a l parts. As such, they d id not man ipu late the images (split a n d then distribute them) a n d therefore d id not solve the p rob lem. Virtual manipulat ives, then, like other manipu lat ive representations require that a chi ld has exper iences model ing p rob lem situations, a n d that these situations are c o n n e c t e d to both a menta l i m a g e of the objects (ranging from d e t a i l e d , real objects to abst ract representations) a n d a n a c t i o n to b e per formed on t h e m . M e a n i n g -mak ing is key. Smith's research r e c o m m e n d a t i o n s (2003, p.273) are consistent with this. He indicates that in order to br idge from idiosyncrat ic (pictorial or case- spec i f ic representations) to m a t h e m a t i c a l (more general izable) representations in a deve lopmenta l l y appropr ia te w a y , students should a c c e s s l a n g u a g e , a n d h a v e the opportunity to c o m m u n i c a t e their m a t h e m a t i c a l ideas by sharing, discussing a n d analyz ing peers ' solutions. The flexible use of strategies Having a set of counters d rawn on the p a g e a f fo rded students the s a m e scaf fo ld as a phys ical manipulat ive w o u l d ; that is, students were a b l e to count , c h e c k a n d r e c h e c k their ideas, try out solutions a n d then self cor rect errors if they o c c u r r e d . It also a l l o w e d students the opportunity to m o v e b e y o n d the virtual manipulat ive a n d use a visualization strategy. Some, like Iris a n d Dav id , d rew only o n e e lement of the p rob lem (only the cookies) a n d visualized the chi ldren w h o wou ld rece ive t h e m . This scaf fo ld of hav ing o n e e lement of the prob lem represented a n d visible suppor ted chi ldren in solution-finding. While c reat ing virtual manipulat ives to solve the p rob lem was helpful, be ing a b l e to use these manipulat ives in flexible ways was more so. Sara's work shows this flexibility clear ly. (Figure 3 7 - S a r a ' s flexibility) Sara b e g a n by c reat ing a set of chi ldren (drawn pictorially in Figure 37a) on the left h a n d side of her p a g e , then d rew a line to div ide it from her 19 cook ies . She re-drew a c o o k i e bes ide 6 of the chi ldren a n d then lost t rack of h o w m a n y she h a d distr ibuted, so she b e g a n to " X " out cook ies , el iminating the ones she h a d g iven a w a y so far (Figure 37b). Next, she used her virtual manipulat ives as counters to c h e c k , establishing that she h a d g iven out six a l ready (Figure 37c). When e a c h chi ld h a d a c o o k i e next to them, (Figure 37d) she c o u n t e d a g a i n to b e sure her system h a d a c c o u n t e d for them all; at this point she d i scovered a n error a n d erased o n e extra c o o k i e (Figure 37e a n d 36f). Figure 37 - Sara's flexibility Mak ing a set. Note the line d rawn in the middle of the p a g e as a separator . Sara has b e g u n to assign cookies to chi ldren. The first 6 kids h a v e a ci rc le d rawn next to them. N o w Sara a d o p t s a system for keep ing track, crossing off 1 cook ie . . . . . .and then m a t c h i n g it to a ch i ld . This is o n e p h a s e of c h e c k a n d r e - c h e c k -note that the c i rc le representing the c o o k i e is a l ready present.  Figure 37d Figure 37e Figure 37f Crossing off the next c o o k i e in the set to el iminate it from the set. O n c e d o n e , Sara re-counts her set of cookies to ensure she has e l iminated 12... . . .and finds a n error, w h i c h she erases. When she saw all 12 h a d b e e n distr ibuted, she c o u n t e d leftover cook ies a n d said, " N o w I c a n split some. . . . That's go ing in a half c o o k i e . " She split up the remaining 6 cook ies by drawing a line through e a c h o n e , (Figure 37g) then re-drew a half c o o k i e bes ide e a c h ch i ld , showing clear ly that e a c h chi ld wou ld get o n e who le cook ie a n d a half c o o k i e (Figure 37i). While she d id not systematical ly el iminate these half cook ies , she d id distribute them in p ieces to e a c h ch i ld . (Figure 37h) Dividing the 6 Sara's cookies - all Showing pictorial ly the remaining cook ies in used up a n d a m o u n t e a c h chi ld will half with a line a c c o u n t e d for; 12 get . (One a n d a half g iven a w a y , 6 cut in cookie.) hajf. | In Sara's a p p r o a c h w e see several things: first, the flexible use of multiple drawing as problem-solv ing strategies, a n d the repetit ion of these strategies over the course of resolving the p rob lem; Sara's incl ination to c h e c k a n d r e - c h e c k her work for a c c u r a c y , w h i c h a l l o w e d her to c a t c h a n error in her d rawing ; a n d the system she a d o p t e d in order to k e e p track. As Sara a p p l i e d these strategies a n d m o v e d flexibly b e t w e e n them, she both d r e w to solve the prob lem a n d d r e w her solution concurrent ly - showing e a c h chi ld's portion of the shared cook ies . To v iew Sara's flexible d rawing as problem-solving work, p lease see V i d e o Cl ip 5, "Sara's Flexibility". Strategic c o m p e t e n c e a n d flexibility in m a t h e m a t i c a l problem-solv ing are powerfu l thinking tools. The s a m e is true in terms of d rawing as p r o b l e m -solving strategies. In this study, f luency was demons t ra ted in knowing w h e n to a p p l y a part icular strategy (distribution, c reat ing sets), using self-monitor ing scripts ( c h e c k a n d re -check , keep ing track) a n d recogniz ing the n e e d to shift to a n e w strategy (elimination, icon ic representat ion). Chi ldren in this study w h o e m p l o y e d multiple d rawing as problem-solv ing strategies a n d w h o were flexible in their app l icat ion were a b l e to solve the p rob lem more efficiently. The incl ination to m o v e b e t w e e n strategies a n d to shift from o n e to another p revented chi ldren from gett ing stuck, over-emphas iz ing or over-relying on a part icular m e t h o d . The power of a system These problems, desp i te the d e g r e e of student success, w e r e not easy to solve. Chi ldren struggled with the complex i ty a n d the large numbers in both of the problems. What was observed over t ime was that students w h o used a system to support their drawing as p rob lem solving were more likely to solve the p rob lem. Norman's work is a g o o d e x a m p l e of this (Figure 38 - Norman' s system). Norman persisted with the c o o k i e prob lem for over 11 minutes. He b e g a n by construct ing sets of virtual manipulat ives on his p a g e - a set of cook ies a n d a set of chi ldren (Figure 38a, 38b). Then he c o u n t e d a n d a t t e m p t e d to distribute the pictures of cook ies to the chi ldren, touch ing a single c o o k i e a n d then 2 chi ldren (Figure 38c). He s e e m e d to b e div iding the cook ies up into 2 parts in order to share them out, but was d o i n g so without physically splitting them - rather he s e e m e d to split t h e m mental ly, using d y n a m i c imagery . Figure 38 - Norman's system Set of cook ies a d d e d . Using penc i l to m a t c h c o o k i e to 2 f a c e s . Next, he b e g a n to t o u c h , a n d then d raw faint lines to a t t a c h who le cook ies to chi ldren (Figure 38d, 38e, 38f), then p a u s e d a n d a s k e d for help. He told m e what he was trying to d o : "I was trying to d o a half a n d then trying to m a t c h w h i c h p ieces g o to w h i c h " , so I re i terated the steps he h a d fo l lowed to that point, a n d he asked for some more t ime to think: " C o m e b a c k for 5 minutes..." (Figure 38g) Figure 38d Touching o n e c o o k i e . Figure 38e .to o n e f a c e . Drawing faint lines to c o n n e c t a single c o o k i e to a single ch i ld . Lines d r a w n for 12 cook ies . " I 'm stuck. I n e e d some he lp . " I g a v e him 3 minutes of think t ime... ...then a n e w m e t h o d -number ing was a p p l i e d . Numbers 1-18 in the cookies a n d numbers 1 -12 over the kids. Note that Ch i ld 1 a n d Chi ld 2 h a v e 1/2 written over them. After approx imate ly 2 minutes, Norman d e v e l o p e d a more de l iberate m e t h o d - number ing his cook ies from 1-18 a n d the cook ies from 1-12. (Figure 38h) He r e c o g n i z e d that there were go ing to b e s o m e cook ies left over. In his h e a d , he visualized c o o k i e 13 broken into two p i e c e s a n d wrote 112 over Ch i ld 1 a n d Chi ld 2 (Figure 38i) - but after trying to d o the s a m e with c o o k i e 14 (Figure 38], 38k) a n d 15, Norman b e c a m e c o n f u s e d , not recal l ing w h i c h chi ld h a d b e e n al lotted a half c o o k i e . (Figure 38m, 38n) Hal fway through his menta l distribution, he e x c l a i m e d , "I lost c o u n t ! " Norman's flexible use of strategies - a n d his positive sense-making disposition - resur faced here as I w a l k e d a w a y . Figure 38j He explains why the 1/2 over Chi ld 1 a n d Chi ld 2. "This o n e , this o n e , those 2 c r a c k e d in half." (pointing to #13) When asked for clar i f icat ion, Norman says, "It means this o n e a n d this o n e ( indicating only #14) goes . . . "  " . . .here . " No rman splits the cook ies a n d distributes them mental ly. I returned several minutes later w h e n Norman c r ied out "I d id it! I so lved the p r o b l e m ! " He h a d finally a p p l i e d a system of el imination to the task by colour ing, a n d s u c c e e d e d in finding a n answer. Norman h a d c o l o u r e d over cook ies 13-18 as he distributed 112 of e a c h c o o k i e to the next 2 chi ldren in his n u m b e r e d set. He exp la ined his i d e a : "These are just colour ing 'cuz if these are all white I k e e p losing tracks a n d so I m a k e a g o o d decis ion to co lour it so that's why I d o n ' t get lost t rack. " (Figure 38q) "Then this o n e (#15) goes . . . " Hesitates... " O h ! I lost c o u n t ! " No rman d e v e l o p s a system of el iminat ion, colour ing in the " u s e d " cook ies . Norman demons t ra ted at least 8 different d rawing as problem-solv ing strategies in the solution of this p rob lem. A l though he was flexible in app ly ing different strategies w h e n he e n c o u n t e r e d a r o a d b l o c k (and also asking for think t ime to b e a b l e to d o that important m e t a - c o g n i t i v e work), Norman's drawings d id not support him in f inding a solution until he d e v e l o p e d a system - namely el imination - for negot iat ing of the last e lements of the p r o b l e m . To v iew Norman's d rawing as problem-solv ing work involving a system, p lease see V i d e o Cl ip 6, " N o r m a n ' s system". Figure 39 - Norman's final work Norman explains: Cookies and Kids There are 18 cookies on a plate and 12 children who want to share them. How can the children share the cookies? How much will everyone get? \ f t * ! , ; , g % AY, ,;,, "1 d r a w 18 cook ies a n d 1 a c c i d e n t a l l y d r a w 22 so 1 crossed it out, a n d 1 d r a w 12 kids a n d 1 put every number in the cook ies a n d then 1 put the s a m e a m o u n t of cook ies a n d w h e n 1 got to 12, . . . a n d then 1 c r a c k two a n d 1 got o n e half... a n d 1 s h a d e it in like this." How did the shading in help you? "It doesn ' t m a k e m e lost c o u n t . . . " Systems such as number ing, keep ing track, a n d el imination were very support ive of chi ldren in successfully solving the problems. Like N o r m a n , several other chi ldren a c c e s s e d systems (Cathy, Sara, Karl) a n d were a b l e to arrive at a n answer b e c a u s e of the structure these strategies a l l o w e d t h e m . Processing the elements of the prob lem in a systematic w a y (eliminating used wheels , crossing off who le cook ies that h a d b e e n distr ibuted, colour ing in cook ies broken up a n d h a n d e d out) f o r c e d chi ldren to a t t e n d to e a c h small p i e c e of the p rob lem a n d record their act ions. A l though many h a d a system in mind, those w h o drew, m a d e marks or phys ical representations on the p a g e were more successful in c o m p l e t i n g the p rob lem. Their pr inted representations a l l o w e d them to r e d u c e the l o a d on their working memory a n d p rov ided them with something to a c t upon in terms of a manipulat ive. What about children who don't draw? Successful problem-solv ing d id not necessari ly require that students d raw. Within this study, there were several chi ldren w h o d id not d r a w at all to solve the problems, but m a n a g e d to solve them successfully. These chi ldren, w h o m I ca l l "visualizers", are spatial thinkers, w h o s e e m to b e a b l e to h a n d l e a large a m o u n t of information a n d man ipu late that information mental ly. Interestingly, it was n o t e d that some visualizers felt the n e e d to d r a w something after the fact ; that is, a l though they so lved the p rob lem mental ly they felt c o m p e l l e d to d r a w a picture to show their solution. This w o u l d b e classif ied as drawing of problem-solv ing, rather than drawing as p rob lem solving, s ince the a c t u a l process took p l a c e mental ly, rather than on the p a g e . This group of students in part icular h e l p e d m e to address my s e c o n d research quest ion, to unders tand h o w young children access visualization and imagery in their resolution of mathematical tasks. The only w a y to a c c e s s this kind of menta l i m a g e mak ing was to observe careful ly a n d to ask questions of the chi ldren. For those a b l e to explain what they were seeing in their images , it was ev ident that multiple processes were h a p p e n i n g simultaneously. The complex i ty of the thinking required not only visual spatial c a p a c i t y but also strong working memory . Sets were c r e a t e d , cook ies a n d chi ldren c o u n t e d a n d c o m p a r e d , cook ies broken up a n d distr ibuted while students kept both the number in e a c h set, then number distr ibuted a n d the half p ieces a l lot ted to chi ldren separate in their heads . Clear ly these chi ldren h a d a p lan , a process a n d a system for solving the prob lem - but b e c a u s e nothing was written on the p a p e r , these processes were invisible. It is easy to misinterpret this lack of print on the p a g e . While filming the very first g roup of chi ldren I m a d e this error myself, with Mark. Mark a n d the others at his tab le were presented with the C o o k i e Problem. While all the others b e g a n to d r a w a n d talk a b o u t their ideas, Mark d id not. He p i c k e d up his penc i l several times but s e e m e d , at first g l a n c e , to b e c o n f u s e d a b o u t where to start (Figure 40a). I f i lmed him for a while, w a t c h i n g him mak ing false starts, but then m o v e d on to other chi ldren. A b o u t 3 minutes into the p rob lem I asked if he unders tood w h a t he was supposed to d o , w h i c h he assured m e he d id - e v e n telling m e he was "...trying to count by like stuff - 3's a n d 2's a n d ones " so I left him a g a i n . When another 5 minutes h a d passed a n d Mark still h a d nothing on his p a g e , I of fered him a simpler p rob lem to solve, thinking he was not a b l e to process the quest ion. Fortunately he ignored m e a n d persisted with the original task. I was startled, at the e n d of 12 minutes of work t ime, w h e n Mark orally presented a n e l a b o r a t e , c lea r a n d cor rect solution for the p rob lem. He explains: If you cut six cookies in half everyone gets half a c o o k i e , but if you cut six cook ies three times (cuz 6 plus 6 plus 6 is 18), so you cut six cook ies in half a n d you cut 6 cookies in half a n d you cut 6 cook ies in half so e v e r y b o d y gets 3 halts of a c o o k i e . Where did you get that idea from? It just c a m e in my h e a d . Intrigued, I w a t c h e d the v ideo a g a i n a n d observed the fol lowing postures a n d behaviours whi le Mark r e a s o n e d through the p rob lem. These images i n d i c a t e d to m e that Mark was not only thinking, but thinking d e e p l y a b o u t the p rob lem - his pers istence a n d multiple attempts to process the information are c lea r in the m o v e m e n t of his eyes, fingers a n d mouth . Figure 40 - Mark uses imagery Everyone writing, Mark without his penc i l in h a n d . Looking up, finger count ing, w a t c h i n g cookies a n d chi ldren. Leaning b a c k from the tab le , g roup ing of 2's a n d 3's with his fingers (note they are held together in sets).  Figure 40d I I •I II I! ^^^^^^^ Figure 40f M a k i n g a set of three. Then dismissing that i d e a . Still, persisting to solve the prob lem. Count ing o n e group of 6 cook ies , then count ing to 12, then 18 in sets of 6's. Figure 40g Figure 40h Figure 40i S j Looking up a n d finger count ing Scann ing his menta l i m a g e while finger count ing, h e a d b o b b i n g  Explaining his solution a l o u d , by looking up to descr ibe the 3 sets of 6 cook ies cut in half. To v iew Mark's visual working through this p rob lem, p lease see V i d e o Cl ip 7, "Mark ' s use of i m a g e r y " . There were m a n y behaviours here that I h a d not a t t e n d e d to during the process itself. Mark was using his visual i m a g e of 18 a n d 12, a n d trying to determine h o w many p ieces he n e e d e d to break e a c h c o o k i e into in order to g ive everyone a n e q u a l share. Unlike the other chi ldren in this study (with the e x c e p t i o n of John), Mark split all of the cook ies in half a n d distr ibuted those halves fairly. Imagine the complex i ty of m a n a g i n g 36 half cook ies mental ly! Mark's finger count ing p rov ided him with a n important scaf fo ld ; but surprisingly, he d id not a lways look at his fingers whi le touch ing them (Figures 40b, 40c, 40g). Instead, he looked upwards , toward his menta l picture a n d t o u c h e d his fingers simultaneously (Figure 40h). The visual i m a g e was the key p i e c e here, a n d his fingers o p e r a t e d like Sara's X's - providing him with a system for keep ing track of the cook ies he h a d split. This is a n important behav iour to note. In order to visualize effect ively, students must also e m p l o y a system. In this study, however , that visual system was not a c k n o w l e d g e d as it was not " d r a w i n g " per se. There was no written tally or el imination of items on the p a g e that g a v e a phys ical indicator of his process ing. Mark's p e r f o r m a n c e a n d solution g a v e m e a great d e a l to w o n d e r a b o u t , a n d to quest ion a b o u t my o w n p r a c t i c e . What supports a ch i ld in c reat ing visual images strong e n o u g h to solve a c o m p l e x p rob lem without p a p e r a n d penci l? What systems are the most ef fect ive for students w h o are visualizers? A n d what types of problems are too difficult to b e unders tood a n d processed in this w a y ? Most importantly, this p i e c e of v i d e o a n d my o w n react ion to it raised the quest ion of c lassroom p r a c t i c e . If I was intently f o c u s e d on the chi ldren a n d d i d not not ice Mark's thinking, h o w is a classroom t e a c h e r to b e c o m e a w a r e of this kind of menta l activity? What about children for whom drawing is not helpful? Not all chi ldren were successful in solving the problems. Below are reflections after observing v ideo f o o t a g e of a group of 3 chi ldren w h o were not a b l e to solve the C o o k i e Problem. My c o n c e r n with this g roup of students was that perhaps drawing was too abst ract for them, that the lack of a menta l i m a g e for the process of dividing a n d sharing m a d e drawing inaccess ib le - or unhelpful - to them as a problem-solv ing strategy. M o n a , for ins tance, d rew two sets of pictures on her sheet of p a p e r - 18 cook ies a n d 12 chi ldren - a n d then a d d e d the two sets together to arrive at 30 as a response. When asked , "What does 30 m e a n ? " M o n a r e s p o n d e d , "It's what the answer is for the who le th ing. " Avril d rew two tables - o n e with cook ies on it a n d o n e with glasses of milk, d iv ided 9 of the 18 cook ies in half a n d distributed only those halves to the chi ldren. "Everyone has half of a c o o k i e " , she wrote on her p a g e - a l though she c r e a t e d 18 halves a n d there were 12 chi ldren. R e b e c c a struggled with multiple attempts , erasing her pictures a n d starting over e a c h t ime. She b e g a n with: " I 'm giving everyone 18 cook ies . " A n d in the e n d , she d rew 18 cook ies , cut e a c h o n e in half a n d wrote: "They are e a c h go ing to get a half a c o o k i e " . When asked if d rawing h e l p e d her, R e b e c c a r e s p o n d e d : ...not really, b e c a u s e w h e n I tried to d raw it really d idn ' t m a k e sense, but w h e n I wrote it, it really m a d e sense. I only wrote pictures, not all those chi ldren.. . A l though e a c h of these chi ldren d rew while process ing the p rob lem, their drawings d id not help them. This raised questions for m e regard ing both their b a c k g r o u n d k n o w l e d g e a n d the appropr iateness of d rawing as a problem-solv ing strategy for these students. Was it just too early to in t roduce them to a p rob lem like this without manipulat ives? A n d w o u l d they h a v e b e e n a b l e to solve it e v e n with phys ical models? Students' m a t h e m a t i c a l understanding is supported by the opportunity to m o d e l problems a n d talk a b o u t their reasoning. I suspect that a lack of c o n n e c t e d m a t h e m a t i c a l exper ience might h a v e i m p a c t e d both their ability to reason through this p rob lem a n d their c a p a c i t y to use drawing as a problem-solv ing strategy to a d v a n t a g e . D iezmann (2000) states that the inability to visualize a m a t h e m a t i c a l situation affects a student's ability to represent that situation in a d i a g r a m . In address ing this a r e a of n e e d , Gifford (2005) r e c o m m e n d s instructional support, particularly for students with dysca lcu l ia , to c r e a t e networks of images a n d vivid associations to support memory a n d m e a n i n g - m a k i n g . While I a m not suggest ing these three students suffer from dysca lcu l ia , I w o u l d e c h o that for all students, l a n g u a g e , visualizing a n d the d e v e l o p m e n t of number sense are inter-relatedness processes. Drawing as problem-solv ing (eg. using virtual manipulatives) c a n only support m a t h e m a t i c a l p rob lem solving w h e n c o n n e c t e d to meaningfu l prior exper iences , a menta l i m a g e for the c o n c e p t involved a n d a n a c t i o n to b e per fo rmed on the representat ional ob ject . Gifford suggests providing visual reasoning support to students w h o struggle to ca l l up a menta l i m a g e of the process; a l lowing them to d o the math a n d think a b o u t the math through their representations. Dispositions and mathematical sense making: How they influenced drawing as problem-solving behaviours Asking permission to draw During the study, e a c h of the problems was p o s e d of the chi ldren orally, a n d a blank p a p e r with only the p rob lem written on it was presented to the students. There were no manipulat ives on the tab le , a n d no directives g iven by m e for solving the p rob lem. What was surprising was that, within a very few seconds of be ing presented the p rob lem sheet, chi ldren asked permission to use pictures as a problem-solv ing strategy. " C a n w e d r a w ? " , "Is it ok if w e use p ictures?" or other similar phrases were asked in 6 of the 9 problem-solv ing groups. It was interesting to cons ider the chi ldren's queries a b o u t drawing as a n a c c e p t a b l e form of p r o b l e m -solving. I h a d thought that a blank p a g e (as o p p o s e d to o n e with lines on it) w o u l d suggest f r e e d o m to d r a w or sketch a response to the p rob lem, but chi ldren's prior e x p e r i e n c e or some notion of the inappropr iateness of pictures in math p revented them from employ ing the strategy without a p p r o v a l . Rather than be ing lead ing , my response was either, " W o u l d drawing help y o u ? " or " D o what wou ld help y o u . " O n c e drawing was d e e m e d a c c e p t a b l e , almost all of the chi ldren c h o s e drawing as a strategy for solving the problems. Those chi ldren w h o d id not d r a w to solve the p rob lem (the visualizers) found themselves in a n a w k w a r d position. S ince they d id not d r a w to process the information but used menta l imagery instead, they d rew afterwards, record ing their response in numbers, pictures or words. Mark a n d Trent wrote out their ideas in long form, descr ib ing their p roduct (3 halves e a c h ) a n d their process respectively (I c o u n t e d by ones to 12...). Char l ie a n d Stephanie c r e a t e d pictures to m a t c h the solution they h a d g e n e r a t e d in their heads - d rawing their solution rather than drawing to solve. For the first g roup of chi ldren, there was a sense that d rawing might not b e a n appropr ia te means of solving the p rob lem. For the visualizers, d rawing was not n e e d e d , but in w a t c h i n g the other chi ldren it was establ i shed as the cor rect or desi red w a y to record a response. Class room e x p e c t a t i o n a n d norms - e v e n a chi ld's construct ion of " w h a t is mathemat ica l l y a p p r o p r i a t e " - clear ly p l a y e d a part in students' response to the p rob lem. "But what's division answer going to be?" Classroom culture and student expectations Lome a n d Norman h a d a fascinat ing discussion a b o u t " w h a t is mathemat ica l l y a p p r o p r i a t e " after solving the C o o k i e Problem. Both boys w o r k e d hard to solve the prob lem. Lome a p p l i e d a range of strategies (set c reat ion , count ing , distribution, ci rcl ing, drawing lines) a n d Norman persisted through several attempts, finally s u c c e e d i n g u p o n a p p l i c a t i o n of a system ("I got it I got it! I so lved the prob lem! " ) . Both boys found 1 a n d a half cook ies as a response, a n d c o m p a r e d their ideas afterwards. Lome saw that they h a d the s a m e answer, yet asked Norman, "But what ' s the division answer go ing to b e ? " He po in ted to his p a g e , where he h a d earlier r e c o r d e d 18-5-12. Norman looked over at Lome's work a n d repl ied "12 of 18. So hard. Try to d o it with sticks." A l though the boys h a d successfully - painstakingly - so lved this p rob lem, they d id not see the c o n n e c t i o n b e t w e e n the solution they g e n e r a t e d (1 a n d a half cook ies e a c h ) a n d the algor ithm they were so keen to write a n d respond to. In fact , w h e n I asked Lome to tell m e a b o u t his equat ion on the p a g e , he said, "That's just for extra work. Like if I figure that out first (pointing to his d rawing of 18 cookies a n d kids) I c a n just write the answer. " Even though he g e n e r a t e d the equat ion to solve for "ext ra work" a n d saw that by drawing to solve the prob lem he c o u l d "just write the answer " , Lome c o u l d not c o n n e c t b a c k to that equat ion a n d saw it as a n e w prob lem to solve rather than a representat ion of the s a m e p rob lem. Norman's response to his cogni t ive d i s sonance was to request a b a t h r o o m break a n d to ask the custod ian for help.. . ! "What ' s 18 * 12?" Norman asked . The cus tod ian s c r a t c h e d his h e a d . "18 -s- 12?! What g r a d e are you in?" Lome a n d Norman c o u l d work through the p rob lem with drawings, but still sought a " rea l m a t h " w a y to find a n d illustrate the solution. Even in the s e c o n d g r a d e , norms for what makes " rea l m a t h " h a v e b e e n cons t ructed a n d unders tood by students in our classrooms. How chi ldren interpret a n d a p p l y this understanding c a n b e e x a m i n e d in part in the analysis of the interview d a t a that follows. Understanding how drawing helped: Insights from the interview results In order to address the research questions: " H o w d o young chi ldren a c c e s s visualization a n d imagery in their resolution of m a t h e m a t i c a l tasks?", a n d " H o w does drawing as problem-solving support students in mak ing sense of a n d reasoning through a m a t h e m a t i c a l p r o b l e m ? " , I h a d to actua l ly ask the chi ldren what they thought. Interview d a t a p r o v e d extremely i l luminating a n d suppor ted insights g a i n e d from examinat ion of the f ramework. Chi ldren d e s c r i b e d a variety of ways in wh ich drawing h a d h e l p e d them to solve the p rob lem. Several of their reasons were well thought out a n d wel l a r t icu lated. Themes in their responses highl ighted the notion of the drawing as a virtual manipulat ive to b e o p e r a t e d on ("It helps b e c a u s e I c a n try different things"; "I saw the 6 leftover a n d I knew that I h a d to cut them into p i e c e s " ; " Y o u c a n d r a w faces a n d d r a w cook ies to the f a c e " ) , or as a memory a i d ("It doesn ' t m a k e m e lost c o u n t " ; " . . . in s tead of d o i n g it in your brain a n d mak ing it e x p l o d e k inda. " ; " D o i n g it in our heads is harder " ) . These chi ldren h a v e m a d e the c o n n e c t i o n b e t w e e n their drawings a n d the m a t h e m a t i c a l purpose for them. Like a phys ical manipu lat ive or other tool , these students unders tand that their drawings assist them with problem-solv ing. John spoke a b o u t the parts of his drawing a n d exp la ined their purpose. "The lines...tell you how m a n y you n e e d a n d the lines that c o n n e c t tell you wh ich cook ies you g e t . " His response to the p rob lem i n c l u d e d cutt ing all the cook ies in half a n d distributing halves to e a c h ch i ld : "[Y]ou break all the cook ies in half in the middle a n d it gives you 36 halts. Then you give the 36 halts to the chi ldren until there's no more . " After d rawing lines to c o n n e c t e a c h half c o o k i e to a line representing a ch i ld , (Figure 41 -John distributes) John c o n c l u d e d that e a c h person w o u l d get " o n e who le a n d o n e half" c o o k i e , (circling with his penci l tip, in Figure 42 - John explains) For John , the a c t of d rawing a l lowed him to c o n n e c t his i d e a of break ing up the cook ies to a final solution. He descr ibes not only his drawings (the cook ies cut in half) but also h o w the act of drawing suppor ted him in f inding a n answer. Another set of chi ldren spoke explicitly a b o u t the a c t of d raw ing , a n d exp la ined how the a c t itself assisted them. Anthony said that d rawing was "like a c t i n g what ' s h a p p e n i n g . " A n d Orson a g r e e d : "It's like a c t i n g b e c a u s e it's like w h a t p e o p l e d o . " In their reasoning through the p rob lem, these boys saw that the a c t of d rawing was mirroring the a c t of dividing a n d giving out cookies - sharing them equal ly a m o n g chi ldren. They were a b l e to c o n n e c t a menta l act ion to their representat ional a c t i o n , reasoning through their drawings in a d y n a m i c w a y . They referred to a c t i o n in telling a b o u t drawing to solve problems (what's h a p p e n i n g a n d w h a t p e o p l e do) a n d their solutions themselves c o m p l e m e n t e d this thinking. Anthony ' s solution to the c o o k i e p rob lem i n c l u d e d lines to distribute the cook ies (act ing what ' s h a p p e n i n g ) ; Orson's solution i n c l u d e d groupings of cook ies , c i rc led together with the chi ldren w h o w o u l d eat them (what p e o p l e do) . In their responses, these boys e c h o e d Tail's notion of a p r o c e p t (1994) - for them, their drawings were a w a y of doing a n d thinking a b o u t the math . Figure 43 - Jessie works it out Cookies and Kids Th&earc 18 cookies pr< opiate and 12 children who want to shore them. Hew can ifte cbiidten show the cookies? How much witf everyone get? Students spoke a b o u t math mov ing a n d hav ing a c t i o n as expressed through their drawings. Act ions like " tak ing a w a y " , "g iv ing to p e o p l e " , "cross ing off" a n d " d r a w i n g cook ies to the f a c e s " were all interpreted by the chi ldren as m a t h e m a t i c a l events. Jessie said that " d r a w i n g is sort of like d o i n g the math. . . like drawing the m a t h " a n d c i t e d group ing a n d t a k e - a w a y s as examples of the math he was d rawing . In solving the prob lem (see Figure 42 - Jessie works it out), you c a n see partially e rased attempts at the p rob lem before Jessie settled on half cook ies for t o d a y a n d who le cook ies for tomorrow. In working through the C o o k i e Problem, Jessie c o u n t e d , c r e a t e d sets, d iv ided, a n d c h e c k e d a n d r e c h e c k e d his thinking. The a c t of drawing for him was synonymous with solving the p rob lem - he was drawing the math . This notion is reminiscent of Tail's p r o c e p t , in w h i c h students' symbol ic representat ion encapsu la tes both the c o n c e p t a n d the process - al lowing the interpreter of that symbol to d o the math a n d think about it concurrent ly. For Jessie, d o i n g the math a n d thinking a b o u t it h a p p e n e d through the a c t of d rawing. Not all chi ldren i n d i c a t e d that drawing h a d h e l p e d them to solve the problems, but rather there were some chi ldren w h o r e s p o n d e d that d rawing h a d not h e l p e d them. Interview statements for this g roup of chi ldren r e v e a l e d either confus ion a round the prob lem or the purpose of the pictures as a problem-solv ing tool . Acces s ing the potent ia l for reasoning through the drawings s e e m e d difficult. M o n a c o m m e n t e d , "I knew it d idn ' t work a n d I started starting to the pictures a g a i n a n d a g a i n " Sammy said, " W h e n e v e r I tried to m a k e 18 it just went to 32." Unlike the other chi ldren w h o exp la ined that drawing h a d h e l p e d , these chi ldren were not a b l e to a c t on their representations in a w a y that a l l o w e d them to reason through the pictures they c r e a t e d . What the visualizers reported There were 4 chi ldren (Trent, Mark, Char l ie a n d Stephanie) w h o d id not d r a w to solve the p rob lem, but rather d rew their solutions or wrote a b o u t their thinking. As such, Trent a n d Mark d id not answer the interview quest ion at all. They a c k n o w l e d g e d that in fact they h a d not d r a w n to solve the prob lem a n d so c o u l d not say that drawing h a d h e l p e d . Char l ie tried though, a n d g a v e a n interesting response. He said that d rawing h e l p e d "...so I c o u l d r e m e m b e r the answer. " For Char l ie, solving the p rob lem was a menta l process involving imagery. In fact , he r e c o r d e d the answer "1 a n d a half" before ever d rawing anyth ing. Drawing the cook ies wasn ' t helpful in solving the p rob lem - only in representing the solution he h a d arr ived at in his h e a d . This was the point at w h i c h I b e g a n to distinguish b e t w e e n drawing to solve a n d drawing a solution, recogniz ing that visualizers d o not record their images in order to o p e r a t e on them. Instead, visualizers o p e r a t e on the images in their heads a n d then translate that menta l a c t i o n to pictures or words after the fact . I w a t c h e d Stephanie do ing this very thing, so I asked her " D o you ever see things in your h e a d ? " S tephanie n o d d e d . "What d o you see in your h e a d w h e n you d o this p r o b l e m ? " I a sked . "I just see like cook ies a n d kids... a n d see like h o w c a n you share them in your h e a d . " S tephanie a n d I h a d a separate conversat ion a b o u t problems in math a n d her representations of t h e m . She a d m i t t e d that she knew the answers long before she r e c o r d e d anyth ing on the p a g e , a n d that in fact she only r e c o r d e d to satisfy the t e a c h e r . The responses from the visualizers in the group distressed m e . A g roup of c a p a b l e mathemat ic ians a n d flexible thinkers, these chi ldren s e e m e d reluctant to share their k n o w l e d g e a n d thinking - e v e n purposefully hiding their processes. When Trent pul led m e a w a y from the c a m e r a to ask if his i d e a of count ing was ok I was a m u s e d at first - then was c o n c e r n e d to think that he should quest ion his d e v e l o p e d a n d sensible m e t h o d . I unders tood better why chi ldren might disguise their thinking w h e n I misinterpreted Mark's attempts at p rob lem solving a n d of fered him a simpler p rob lem. My o w n assumptions a b o u t the w a y to solve the p rob lem through pictures l e a d m e to be l ieve that Mark was struggling w h e n in fact he was act ive ly problem-solv ing. S tephanie a n d Char l ie 's incl ination to d r a w the solution rather than to d r a w to solve speaks to a n implicit p r o d u c t - b a s e d e x p e c t a t i o n as wel l . A n d yet if I c o u l d n ' t see them thinking, h o w c o u l d I recognize their cogni t ive work? I return then to the quest ion of h o w young chi ldren a c c e s s visualization a n d imagery in their resolution of m a t h e m a t i c a l tasks. D a t a from this study suggests that students w h o a c c e s s imagery in their reasoning through the problems (and then d r e w or d id not d r a w their solution) d id so in a visually a c t i v e w a y , mov ing images a round in their heads , manipu lat ing those images in d y n a m i c ways a n d showing outward signs of this manipulat ion through gestures a n d upraised eyes. Students w h o were a b l e to a c c e s s l a n g u a g e for these processes d e s c r i b e d likewise them in a c t i v e terms. Visualization might b e cons idered a n invisible form of d rawing as p r o b l e m -solving, the culminat ion of exper iences a n d m e a n i n g - m a k i n g in m a t h e m a t i c a l terms. As a val id a n d c o m p l e x strategy, it needs recogni t ion on the part of teachers a n d students in order to b e used conf ident ly a n d efficiently by students. Research highlighting the eff icient use of imagery a m o n g high ach iev ing students - imagery that is d y n a m i c , flexible a n d fluid in its a p p l i c a t i o n - ce lebrates this kind of menta l activity. (Gray, Pitta & Tall, 2000; Reynolds & Wheat ley , 1997; Thomas, Mul l igan & Go ld in , 2002). Class room teachers wou ld benefit from support to d o the s a m e . In observing this g roup of chi ldren, I learned to recogn i ze - a n d a c k n o w l e d g e a l o u d - the outward signs of visualization, a n d m a k e ment ion n o w of these behaviours w h e n I speak to chi ldren. ("I see you ' re looking up . You are thinking a b o u t the parts of the p rob lem. What d o you see in your h e a d ? Is anything moving? How?") I lea rned that spatial sense a n d its role in m a t h e m a t i c a l problem-solv ing should b e r e c o g n i z e d , n a m e d a n d honoured as a val id a n d powerful strategy. Further suggestions for classroom p r a c t i c e fol low. Implications for practice Educat iona l researchers h a v e a responsibility to both explore questions regard ing the e n h a n c e m e n t of learning a n d to translate research findings into access ib le instructional guidel ines a n d tasks for the classroom t e a c h e r . To this e n d , the fol lowing sect ion begins to address notions of implementat ion by cons ider ing four genera l areas: types of problems, types of learners, metacogn i t i ve a n d instructional tasks. Types of problems In order to support d rawing as problem-solv ing or reasoning through the a c t of d rawing , the tasks presented to students should meet certa in criteria a round complex i ty a n d m o v e m e n t . That is, the problems p o s e d of the chi ldren must b e c o m p l e x e n o u g h (eg. multi-step, non-routine, problemat ic) to ensure that reasoning towards a n answer requires cogni t ive effort on the part of students. In this study, o n e student (Stephanie) repor ted that often the questions she was asked to solve were too simple, a n d that she h a d arr ived at the answer long before she p i c k e d up a penc i l to represent the solution. O p e n - e n d e d tasks requiring creativity a n d generat ive thinking are examples of non-routine problems; these m a y lend themselves to d rawing as problem-solv ing a n d visualizing behaviours in students. Problems should also feature some m o v e m e n t or distribution of quantity in their structure to e n c o u r a g e drawing as problems-solving or visualizing activity. M e n t a l math tasks involving m a t h e m a t i c a l operat ions in w h i c h chunks of number are m o v e d , b a l a n c e d or set as ide a n d r e p l a c e d are g o o d examples . Types of learners Within this study there were three genera l groupings or types of learners, d e s c r i b e d by the ways in wh ich they in te racted with the problems a n d the manner in w h i c h they reasoned through their drawings. Interview d a t a a n d observations re fe renced against the f ramework i n d i c a t e d there were students w h o were not yet ready to a c c e s s d rawing as a problem-solv ing strategy. I a m unclear as to whether this was ow ing to confus ion a round the prob lem context or their difficulty is mak ing their drawings "work" for them. Nonetheless, there are students for w h o m drawing as problem-solv ing is not helpful. Their part icular needs with respect to drawing as problem-solv ing a n d the understanding of the > m a t h e m a t i c a l structures within the problems should b e r e c o g n i z e d a n d , as Gifford (2005) suggests, a mult i -modal a p p r o a c h used to build spatial thinking. Like the majority of the chi ldren involved in this study, I w o u l d suggest that chi ldren in the regular c lassroom will also respond to problems (like those d e s c r i b e d in the previous section) by drawing a n d / o r visualizing. In order to d e v e l o p a n d hone this c a p a c i t y , support chi ldren w h o reason through drawing to reflect on the a c t of drawing a n d to ask themselves - h o w does it help m e ? A n d if d rawing is not help ing, then students should b e e n c o u r a g e d to switch gears, to refine or alter their d rawing as p r o b l e m -solving strategy, or a p p l y a system to the process of reasoning. The third group of chi ldren w h o identif ied themselves through this study i n c l u d e d those chi ldren w h o d id not n e e d to d r a w to solve the p rob lem; chi ldren w h o reasoned only visually, a n d e l e c t e d if they d rew at all to record a solution to the prob lem. Students like these in the regular class room, despi te their seeming facility with problem-solv ing m a y still benefit from metacogn i t i ve explorations a n d the d e v e l o p m e n t a n d explicit naming of systems to support menta l or i m a g e - b a s e d p r o b l e m -solving (eg. finger count ing , stroke marks, scaffo ld ing tools like a number-less number line). Addressing and supporting metacognition Questions and prompts for classroom teachers Support ing students to reason mathemat ica l l y is e n c o u r a g e d by the d e v e l o p m e n t of metacogn i t ion , the ability to think a b o u t thinking a n d reflect on the menta l processes be ing used. These prompts are especia l ly ef fect ive w h e n used during the a c t of thinking itself. The fol lowing list provides some suggested questions classroom teachers might cons ider asking to both assess the menta l process taking p l a c e a n d to m a k e the a c t of reasoning through drawing explicit for the students w h o are using it. Suggested prompts a n d a rat ionale for their use inc lude: • What are you do ing? Why are you do ing it? How does it help you? (addresses metacogni t ion) • Tell m e a b o u t your p icture/menta l i m a g e . What are the parts for? (suggests the n e e d for structure in menta l imagery) • C lose your eyes. What d o you see? What 's h a p p e n i n g ? How does this help you solve the p rob lem? (focuses attent ion on the menta l i m a g e ; highlights the notion of act ion) • Stop a n d share with a partner. How are your drawings/ images al ike? How are they different? (encourages l a n g u a g e a n d expands thinking) • What does your partner's picture tell you? What c a n you see h a p p e n i n g ? (promotes read ing of pictures, representat ional act ion) • I see you looking up. What d o you see there? How are you keep ing t rack? How are all the p ieces organized? (recognizes outward signs of visualizing, e n c o u r a g e s app l icat ion a system to visual images) Classroom strategies: Clas s room-based tasks that e n c o u r a g e visual reasoning a n d capi ta l i ze on students' spat ial strengths w o u l d b e of benefit to students in their a p p l i c a t i o n of drawing as problem-solv ing strategies. Support ing students to a c c e s s mult i - layered systems (pictures, menta l models , manipulat ives, numbers, words, language) for representing a n d thinking though problems a n d e n c o u r a g i n g them to " t e a m u p " will help to strengthen networks of visual thinking a n d al low for flexibility b e t w e e n them. Teacher model ing of metacogn i t i ve talk re lated to drawing as p r o b l e m -solving pract ices through think-alouds ("I a m picturing a tree with steps on the side, a n d a squirrel c l imbing up a n d d o w n the stairs. I'll n e e d to use my fingers to k e e p track of the act ion here.. . ") is a powerful instructional t e c h n i q u e that supports the d e v e l o p m e n t of l a n g u a g e to descr ibe this form of m a t h e m a t i c a l reasoning as well as raising the awareness of its ex i s tence. After all, teachers d o n ' t "just know" the answers. C a u t i o n should b e exerc i sed so that students d o not simply e c h o the teacher s ' strategy; providing opportunit ies for chi ldren to "think a l o u d " val idates deve lopmenta l l y appropr ia te ways of reasoning through d rawing . Using dry-erase or cha lkboards during lessons a n d partner ing students over them to e n c o u r a g e sharing a n d talk a b o u t their reasoning is o n e suggested strategy. The ef fect of c lassroom culture a n d e x p e c t a t i o n c a n n o t b e underes t imated. It is important to c r e a t e a classroom culture that allows for multiple representations ( including icon ic , pictorial a n d menta l images) , flexibility of strategies (manipulat ing a n d a c t i n g u p o n drawings or menta l images) a n d the app l icat ion of systems (keeping track, el imination, c h e c k a n d d o u b l e c h e c k ) where helpful. Summary and further research directions In address ing the research quest ion, " H o w does drawing as problem-solving support students in mak ing sense of a n d reasoning through a m a t h e m a t i c a l p r o b l e m ? " , student's interview d a t a p r o v e d extremely helpful. Two purposes for drawing e m e r g e d in the students' response to the quest ion, " H o w d id drawing h e l p ? " . A c c o r d i n g to the chi ldren, d rawing h e l p e d in o n e of two ways - by providing them with something to see a n d o p e r a t e on , a n d in al lowing them to process their thinking through m o v e m e n t . For those w h o d id not d raw, I w o u l d suggest that visual images p rov ided them with exact ly the s a m e scaf fo ld - something to see a n d a w a y to m o v e the elements of the prob lem - p roduct a n d process both , a l though invisible to the observer. Observat ional d a t a g a t h e r e d a n d organ i zed in the drawing as p r o b l e m -solving f ramework suppor ted student statements during the interviews; that is, the clusters of behaviours obse rved while the students were in the a c t of problem-solv ing were consistent with what the students themselves repor ted. In cons ider ing h o w drawing as problem-solv ing supports students in reasoning through a p rob lem, the v ideo d a t a c o l l a t e d in the f ramework suggests that several behaviours e m e r g e across problems, regardless of p rob lem structure. The use of virtual manipu lat ive behaviours, including the creat ion of sets, the use of pictures as counters a n d the a c t of distribution were ways in wh ich students in this study a p p l i e d a n a c t i o n to their pictures. Through virtual manipulat ive use, students were a b l e to reason through the m a t h e m a t i c a l p rob lem by a c t i n g on their pictures - whether those pictures were icon ic or pictor ial in nature. System use e m e r g e d in student interview d a t a as wel l as v i d e o d a t a obse rved during student problem-solv ing. Systems might b e d e f i n e d as sel f -def ined strucfures through wh ich students ref ined a n d clar i f ied their thinking a b o u t the p rob lem. In the c a s e of drawing as problem-solv ing, this invo lved apply ing a system or structure to the drawings - using drawings as the ob ject of a system a n d "opera t ing on t h e m " in a systematic w a y . Elimination, c h e c k a n d r e c h e c k a n d keep ing t rack behaviours were obse rved in student work a n d were ment ioned by students dur ing the interview as a w a y in w h i c h drawing (or the drawings) h a d h e l p e d them reason through the task. The d e v e l o p m e n t of c a p a c i t y in drawing as problem-solv ing is a n issue of interest. Students' representat ional objects c a n take o n e of three forms -pictor ial , icon ic or menta l - or e v e n a c o m b i n a t i o n of the three. Support ing students in a c c e s s i n g a n d app ly ing a deve lopmenta l l y appropr ia te representat ion that provides a fit cognit ively whi le al lowing the student to a p p r o a c h drawing as problem-solv ing in a n eff icient w a y is worth pursuing. A l though not linear or h ierarchical in its des ign a n d a p p l i c a t i o n , a cont inuum of d e v e l o p m e n t of menta l imagery is sugges ted here - that the meaningfu l use of phys ical manipulat ives m a y p r e c e d e the d e v e l o p m e n t of pictor ial representations, that pictor ial , p rob lem-speci f ic representations m a y p a v e the w a y for more icon ic or genera l i zab le ones, a n d that the use of icon ic representations a n d a n explorat ion of the relationships b e t w e e n the elements of the p rob lem over t ime m a y help to c r e a t e menta l images . D a t a in this study suggest that these categor ies exist; h o w they d e v e l o p a n d are subsequently i m p l e m e n t e d by students is a n a r e a for further research. As obse rved during this study, these interrelated representat ional tools c a n b e used in problem-solv ing with success; they c a n likewise b e used in c o m b i n a t i o n to reason through problems. Lome shared his thinking a b o u t the use of drawings a n d other symbol ic representations to solve problems: "I n e e d pictures to gu ide m e . So it's best if I t e a m u p . " Lome's c o m m e n t goes a long w a y to answering o n e of the key research questions: " H o w d o y o u n g chi ldren a c c e s s visualization a n d imagery in their resolution of m a t h e m a t i c a l tasks?" These visual activities, e v i d e n c e of w h i c h was n o t e d during the v ideo analysis, supported chi ldren in reasoning through the problems; al lowing them to " t e a m u p " their d rawn representations with visual images . Further research is war ranted to fully explore this notion of interrelated a n d interact ive strategies for drawing as problem-solv ing. I p ropose a more in-d e p t h study of individual chi ldren working through c o m p l e x , multi-step problems to both va l idate the drawing as problem-solv ing f ramework as a more broadly a p p l i c a b l e tool a n d to determine the extent to w h i c h drawing as problem-solv ing strategies a p p l y to different p rob lem types a n d structures b e y o n d the 2 problems used here. I suggest that students w h o are visualizers b e identif ied a n d t a r g e t e d for inclusion in a similar study to address questions re lated to drawing as problem-solv ing - for these students w h o tend not to d r a w at all. To w h a t d e g r e e are their menta l strategies sufficient in solving c o m p l e x problems, a n d at w h a t point are supp lementa l strategies necessary? 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Appendix A - Interview data Organ i zed by d e g r e e of help p rov ided by drawing Colour c o d e d by type of help p rov ided by drawing Drawing didn't help TAPE 3B Rebecca - " . . .not really, b e c a u s e w h e n I tried to d r a w it really d idn ' t m a k e sense, but w h e n I wrote it, it really m a d e sense. I only wrote pictures, not all those chi ldren I h a d to d r a w all of those chi ldren so I thought writing is better than d r a w i n g . " ** NOT SUCCESSFUL IN SOLVING THE PROBLEM ** Mona: "It d idn ' t help for m e for m e for all the pictures b e c a u s e I c o u n t e d it all together there were 30 a n d then I f igured a n d I knew it d idn ' t work a n d I started starting to the pictures a g a i n a n d a g a i n . " ** NOT SUCCESSFUL IN SOLVING THE PROBLEM ** TAPE 4B Susie: Did the picture help you? "Not really. C u z w h e n e v e r I tried to m a k e 18 it just went to 32." What would have helped you? If you could have had anything in the room, what would have helped? "Urn... u m m m . . . " Cookies? " M a y b e . " SUMMARY TYPE OF SUPPORT NUMBER OF REFERENCES TO EACH TYPE (HOW DRAWING HELPED) OF SUPPORT Subject,of a n a c t i o n 0 Distribution 0 M e m o r y a i d 0 Elimination system 0 Pictures as icon ic 0 representations Not clear on how drawing had helped - or could not explain clearly how TAPE 1B  Susanne: "Ypy, can count like the ones that are first, then the ones that are left w e just put it d o w n there. It just helps." TAPE 2A  Larry: "He lp ing m e um make lines a n d ... a n d the kids were h a p p y a n d they h a d to get crumbs a n d c h o c o l a t e a n d 2 of them shared it." TAPE 2B  Avril: I counted h o w m a n y cook ies they should get . Drawing gives you a n i d e a . . . . A c t i n g was b reak ing. " ** NOT SUCCESSFUL IN SOLVING THE PROBLEM ** TAPE 3A Lome - " i t j h o w e d m e Jike half a n d half or who le a n d stuff. A n d i c o u l d like draw-math a n d stuff, l ike-ciVcles^eMe.pt... they-were cook ies . " Tell me more about draw the math. "Like if there were 15 houses a n d 5 probably , I c o u l d p robab ly d o that, only like I wou ldn ' t d r a w full houses with so m u c h detai l , I'd use up all my t ime just using o n e house. . . I actual ly just d r a w circles for like everything. Sometimes squares, but rarely ever . " SUMMARY TYPE OF SUPPORT (HOW DRAWING HELPED) NUMBER OF REFERENCES TO EACH TYPE OF SUPPORT Subje'QCof'ah "act ion 2 Distribution 1 M e m o r y a i d 0 Elimination system 1 Pictures as! representations 1 Drawing helped - vague responses TAPE 1B  Lisa: "I d rew f a c e s . . . . a n d cookies . I d o n ' t know why. I just w a n t e d to. So w h e n w e d o n ' t h a v e to d©ltiiSour|hedds a n d do ing it in our heads is harder . " Jason: "I just m a k e every chi ldren h a v e o n e c o o k i e a n d then there were some cook ies left a n d so I d iv ided them in 2's so everyone w o u l d h a v e o n e a n d a half." " . . . to remind y o u . " "It's easier to writent d o w n . " Do you sometimes make a picture in your head? " Y e a h . " Grant: " C u z you c a n d r a w faces a n d ^rqwl^o'k ie's l t^-the f a c e s a n d you c a n . . . instead o j jftst keep ing it in your h e a d . " TAPE 2A  John: "...it h e l p e d us not ice um h o w many . . . kids get the cook ies a n d h o w they c a n share them al l . " jHqvvdid it help you? "The lines...tell you h o w m a n y you n e e d a n d the lines tj iat c o n n e c f t e l l you w h i c h pdok ies you g e t . "  Martin: " . . . to help nrie c o u n t a n d stuff. Because like I p re tend these are like cook ies , a n d then I gf6s?o1f l s^^rcoo,k ies b e c a u s e . t h e kids eat them. Then there were 6 more left, then I crossed them out b e c a u s e 6 more chi ldren a t e t h e m . . . "  Pricilla: "It h e l p e d instead;qf : in y o u r h e a d a n d get mixecl ,up." What did the drawing do to make it easier? "By looking at it a n d thinking h o w m u c h I c a n cross off." TAPE 3A  Cathy: "it helps m e d r a w so like, uh, in c a s e I d o like four in half a n d it doesn ' t work then like sometimes it c a n help m e , so \t±so if l|hgye to cut ij in half or just l e a v e it..." Ok. Does it help in any other way? " It helps m e solve the p r o b l e m . . . "  Norman: "I d o n ' t really know. I d r a w 18 cookies a n d I a c c i d e n t a l l y d r a w 22 so I crossed it out, a n d I d r a w a 12 kids a n d I put every number in the cook ies a n d then I put the s a m e a mount of cook ies a n d w h e n I got to 1_2,... a n d then I c r a c k t w o and_[_got onejnalf . . . land I shade it in like" this..." H o w did the\ shading^in help you? "It doesn ' t m a k e m e lost c o u n t y [ f TAPE 3B  Ruby: "I d r a w e d o n e c o o k i e for e a c h chi ldren then I saw;the'6~leftover ancf I knew that I h a d to cut them into p ieces a n d I cut them into 2 p ieces . B e c a u s e then it will all g ive o n e a n d a half."  TAPE 4A  David: "So w e ; d o n ' t i d r g e t . " When did it help most? " W h e n I put the l ine." Charlie: "So I c o u l d r e m e m b e r the answer. " TAPE 4B  Hannah: "It just helpecTme count the cookies'." TAPE 5A  Shauna: What strategy did you use? "I, u m d r a w i n g . " How did it help you? " G o o d , c u z J ^ p u l d r i ' ^ d o it in my*hedd. C u z there was 18 cook ies . " "It helps b e c a u s e I 'cag^ry d i f f e r e n t . I S M ^ I l i S ^ ' ' ' . ones if that l f l f k s . " i Iris: "It h e l p e d m e h o w to get the p rob lem so lved in a n easier w a y . " What makes it easier? "Like I use the picture to help m e with the p rob lem. Like |o. |ee^ifel:.have to^fnake^some o f R e m a half."  Stephanie: D o you ever see these things in your head? " Y e a h . " What do you see in your head when you do this problem? "Just see like cook ies a n d kids a n d like a n d see like, h o w c a n you share"'them.in your - t iead. " SUMMARY TYPE OF SUPPORT (HOW DRAWING HELPED) NUMBER OF REFERENCES TO EACH TYPE OF SUPPORT Subject of a b a c t i o n 6 Distribution 3 M e m o r y a i d 6 Elimination system 3 Pictures as i c o n i c , • representations 0 Drawing Helped - Clear response TAPE 2B  Anthony: What made you want to draw? " B e c a u s e drawing helps." "It's like a c t i n g what ' s happen ing . 1 Drawing's like a c t i n g b e c a u s e it... [interrupted by Orson]" How is it like acting? VVheri you were drawing, what part were you acting? " t h e kids that take cook ies . " So it's the taking. That's what you were acting out?  Orson: "D rawing gives you more ideas to write abojut or think a b o u t . ...like h o w m u c h you g ive other p e o p l e . It's like a c t i n g b e c a u s & i f s I jk^what' p e o p l e d o . " Did you have any acting in yours, Orson? "Yes, like taking a w a y , like giving p e o p l e . " TAPE 4B  Zarah: "It h e l p e d m e so w h e n I look at it, it w o u l d b e a little bit easier. ... it's like you c o u l d look at the picture a n d d o it with the picture instead of doing' i t in your brajn a n d , m a k i n g it e x p l o d e k inda. " (smiles) So how does a picture help? "It helps you put your ideas d o w n on the p a p e r so that your brain w o n ' t feel like it's go ing to e x p l o d e . " Jessie: "It h e l p e d m e um get smarter. By gett ing harder questions." So can you work on harder questions if you can draw it? Is that what you mean? " Y e a h , by like um, the drawing is sort of like d o i n g j h e m a t h . Like d r a w i n g the m a t h . " What part of the math were you drawing, Jessie? "It's like w e ' r e d o i n g take-aways . Like I d id o n e , there's 1, 2, 3 a n d 1, 2, 3, a n d it makes six. A n d it makes 12." TAPE 3A  Cathy: What's easier? Pictures or no pictures? "Pictures are easier - cuz then you just d r a w a n d then you know h o w m u c h stuff... h o w m a n y wheels are on a toy c a r or instead of just writing writing writing a n d you take up the who le s p a c e . "  Lome: What's easier? Pictures or no pictures? "They're both the s a m e . I thought um not d rawing was like 100 - like 98% hard, but actual ly it's fairly easy. Like I c a n ' t tell - m a y b e this is slightly harder, that is slightly harder;: It sure, is shorter doing-this; ( indicates.number sentence) . ButJJ n e e d pictures to gu ide me.-So it's best if r t e a m Up.,( indicates pictures a n d n u m b e n s e n t e n c e ) . SUMMARY TYPE OF SUPPORT NUMBER OF REFERENCES TO EACH TYPE (HOW DRAWING HELPED) OF SUPPORT S u b j e c t o T d n action" 3 'Distribution 1 Memory/a id 1 (Elimination system 0 f ic lu res as icon ic 0 representations SUMMARY OF ALL RESULTS: TYPE OF SUPPORT (HOW DRAWING HELPED) NUMBER OF REFERENCES TO EACH TYPE OF SUPPORT Subject of a n a c t i o n 6 Distribution 5 M e m o r y a i d 7 Elimination system 3 Pictures as icon ic representations 1 Append ix B: Consent Forms Cynthia Nicol Department of Curriculum Studies Faculty of Education 2125 Main Mall Vancouver, B.C. C a n a d a V6T 1Z4 Telephone (604) 822-5246 FAX (604)822-4714 Email cynthia.nicol@ubc.ca M A R C H , 2 0 0 5 Dear Ms. XXXXX: This letter is to seek permission to c o n d u c t e d u c a t i o n a l research at your schoo l , XXX Elementary. As the co- invest igator in a study entit led "D rawing as problem-solv ing: Young chi ldren's m a t h e m a t i c a l reasoning through pictures" , I wou ld like to present to you the purpose a n d overv iew of p rocedures be ing p r o p o s e d . I a m currently pursuing a Masters of Arts in M a t h e m a t i c s Educat ion ; the study d e s c r i b e d b e l o w will form the basis for my g r a d u a t e research. A l o n g with my faculty advisor a n d g r a d u a t e research supervisor Dr. Cynth ia Nicol , I h o p e to invest igate h o w young chi ldren think through m a t h e m a t i c a l problem-solv ing. W e are interested in f inding out h o w chi ldren respond w h e n presented with a m a t h e m a t i c a l p rob lem to solve, w h a t kinds of pictures they d r a w spontaneously, the things they are thinking whi le they d raw, a n d h o w to support chi ldren in d e v e l o p i n g these skills in order to b e c o m e better m a t h e m a t i c a l problem-solvers. In terms of procedures , I wou ld like to work with approx imate ly twenty g r a d e 2 students from XXX, d rawn from the different g r a d e 1/2 a n d 2/3 classes. I will work with small groups of 4-5 chi ldren at a t ime, preferably in a small resource room or other quiet s p a c e that will a l low for v i d e o t a p i n g with quality a u d i o . Chi ldren will b e presented with a m a t h e m a t i c a l p rob lem to solve. I will v i d e o t a p e the small g roup whi le they solve the p rob lem a n d ask questions a b o u t students' thinking while working through the p rob lem. Afterwards, the chi ldren's work will b e c o l l e c t e d . This small group problem-solving session will last approx imate ly 45 minutes. The v ideotapes a n d student work will b e e x a m i n e d afterwards, looking for illustrative p ieces that will help to show the nature of drawing as a problem-solv ing tool in mathemat ics . addi t iona l m a t h e m a t i c a l problems. This individual problem-solv ing interview will last a b o u t 45 minutes. It will also b e v i d e o t a p e d , a n d will fol low the s a m e format as out l ined a b o v e - chi ldren will b e presented with a p rob lem, then whi le solving it, b e asked to talk a b o u t their thinking a n d d rawing . If your chi ld is se lec ted a n d you a n d your chi ld are willing for your ch i ld to par t ic ipate in this individual problem-solving interview, notif ication a n d further information will b e g iven. Throughout this project w e will b e very h a p p y to show part icipants the v ideos of themselves. The only p e o p l e w h o will see the v ideos apar t from them will b e your chi ld's c lassroom t e a c h e r a n d the investigators. No o n e else will see the tapes . Your chi ld will of course b e free to opt out of the project at any t ime a n d such wi thdrawal will not af fect his/her g rades or relationship with the school in any w a y . Students w h o d o not c h o o s e to b e part of the project will simply not b e v i d e o t a p e d . Confidentiality: Any information resulting from this project will b e kept strictly conf ident ia l . All d o c u m e n t s a n d v ideotapes will b e kept in a l o c k e d c a b i n e t . Any digital v ideo will b e kept in secure e lect ron ic files. Any report that is written c o n c e r n i n g this project will, by giving false names , preserve the c o m p l e t e anonymity of all part ic ipants a n d their schoo l . However , you a n d your chi ld are w e l c o m e to v iew at a n y t ime, u p o n request, a n y v i d e o clips that inc lude your ch i ld . O n c e the c o -investigator, C a r o l e Saundry, selects p ieces of v i d e o to c r e a t e a digital illustration of " d r a w i n g as p rob lem solv ing", w e will ask you a n d your ch i ld if you are willing for these v ideo clips to b e used in a more publ ic forum to help other teachers or researchers. But you a n d your chi ld d o not n e e d to m a k e that decis ion until after you h a v e seen the v ideo clips. Contact: If you h a v e a n y questions or desire further information a b o u t this study you m a n y c o n t a c t Dr. Cynth ia Nicol at 822-5246. If you h a v e a n y concerns a b o u t your chi ld's rights as a research part ic ipant you m a y c o n t a c t the Director of Research Services at the University of British C o l u m b i a , at 822-8598. Consent: Students' part ic ipat ion in this study is entirely voluntary a n d they m a y refuse to par t ic ipate or w i thdraw from the study at a n y t ime without j e o p a r d y to their class s tanding, grades , or relationship with the schoo l . P lease ind icate your chi ld's part ic ipat ion or non-part ic ipat ion by c o m p l e t i n g a n d d e t a c h i n g the consent slip be low. Please k e e p this descr ipt ion of the study a n d d e t a c h the consent slip b e l o w . Thank you . PLEASE KEEP FOR YOUR FILES Please check the box indicating your decision: I CONSENT to my chi ld part ic ipat ing in the v i d e o t a p i n g of smal l -group problem-solv ing sessions, a n d , if c h o s e n , I CONSENT to my chi ld part ic ipat ing in the individual problem-solv ing interview. I CONSENT to the c o p y i n g of materials p r o d u c e d during the lessons as d e s c r i b e d in the a b o v e form. I D O NOT CONSENT to my chi ld part ic ipat ing in the study as d e s c r i b e d in the form. I a c k n o w l e d g e that I h a v e r e c e i v e d a c o p y of this consent form. Name of student (please print) Date:. Signature of parent or guardian PLEASE DETACH CONSENT SLIP AND RETURN TO CLASS TEACHER Please check the box indicating your decision: I CONSENT to my chi ld part ic ipat ing in the v i d e o t a p i n g of smal l -group problem-solv ing sessions, a n d , if c h o s e n , I CONSENT to my chi ld part ic ipat ing in the individual problem-solv ing interview. I CONSENT to the c o p y i n g of materials p r o d u c e d during the lessons as d e s c r i b e d in the a b o v e form. I D O NOT CONSENT to my chi ld part ic ipat ing in the study as d e s c r i b e d in the form. I a c k n o w l e d g e that I h a v e r e c e i v e d a c o p y of this consent form. Name of student (please print) Date:. Signature of parent or guardian DETACH CONSENT SLIP AND RETURN TO CLASS TEACHER Please check the box indicating your decision: j j I CONSENT to the sharing of digital v ideo clips that inc lude my chi ld in a more publ ic forum - in t e a c h e r e d u c a t i o n coursework, g r a d u a t e student coursework, a n d at e d u c a t i o n a l a n d research or iented con fe rences . I unders tand that my chi ld m a y b e identif ied in the v ideo clips, however the school a n d classroom will not b e n a m e d . '] I D O NOT CONSENT to the sharing of digital v i d e o clips that inc lude my chi ld in a more publ ic forum. I a c k n o w l e d g e that I h a v e r e c e i v e d a c o p y of this consent form for my o w n records. N a m e of student (please print) D a t e : Schoo l : Teacher : Div: Signature of parent or guard ian Script for Introducing Study to G r a d e 2 Students Today I will be asking you to take a letter home to your parents. This letter tells about a study that a teacher is doing. She is curious about how grade 2 children think when they solve math problems. She would like to work with small groups of children in grade 2, give them some math problems to solve and ask them questions about their strategies. This teacher will videotape children while they work so that she can keep a record of their thinking. The children will work in another small room in our school where they can have some space to spread out and then talk about their ideas and strategies. She will read you the problems and give you paper to record your thinking. This letter talks about the study, and asks your parent if it is ok for you to participate. It's all right if you don't want to. This is your choice. If you don't want to work on the problems with the teacher, no one will be upset. You can stay in the classroom with the rest of the class. If you want to participate, your parent should sign the form and send it back to the school. After that, the teacher will begin to work with small groups to solve math problems. Thank you. 

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