‘~ :1 in .. 3.... 5 3.. i .. EU: -‘ . . . A a: a: . c , . ~34 .‘ h. «E a“ o)‘. x3 . 1.3.: ’1 .91 5.9 New? .. s .I p (t... a: w a... .v . c. .s. a... _ I I. 2:. . , - ‘Jufiku €51.91 .m v i 1.11“": 1.» 2 t? in J . , HF: a”; d .713... .. .xs..1u.xHM..«1 . “3 v. I :2. pa | . . .93.”, VB f , r1 gnu? :9.\ I r . $1323.... .... . r . )3). L.. 94.23.... $5.2. 9|... :- y. 7; . s 334m“ .. :flm‘rn. ,1 . P: .131... 1!.— Q: . .D‘C‘Ion.‘- .vrkx ,:.!‘u.L:. .gwwmmfifigfia 1w $43.2. I THESE 2.00\ This is to certify that the dissertation entitled TODDLERS' SYMBOLIZING AND ITS MATHEMATICAL POTENTIAL presented by Helene Alpert Furani has been accepted towards fulfillment of the requirements for ‘ Ph . D . degree in Meat ion Dos 3 [9‘wa Major professor Date /"--/l [/00 MS U is an Affirmative Action/Equal Opportunity Institution 0- 12771 LIBRARY Mlchigan State Unlverslty PLACE IN RETURN BOX to remov To AVOID FINES return MAY BE RECALLED with ear e this checkout from your record. on or before date due. lier due date if requested. DATE DUE DATE DUE DATE DUE 11m Wasp.“ TODDLERS’ SYMBOLIZING AND ITS MATHEMATICAL POTENTIAL By Helene Alpert Furani A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Teacher Education 2000 ABSTRACT TODDLERS’ SYMBOLIZING AND ITS MATHEMATICAL POTENTIAL By Helene Alpert Furani This dissertation provides a detailed, ethnographic account of three toddlers’ symbolizing. Gathered through participant-observation in home settings, data include the words, gestures and play of three boys between 16 and 22 months of age. The toddlers’ activity around naming experience, symbolizing imagined situations and playing with systems of symbols is described and analyzed through a mathematical lens. The toddlers’ symbolizing then serves as a springboard for exploring mathematical symbolizing and oommonalties between the two. Implications are made with respect to such issues as the action based origins of symbolizing, the ‘symbolic continuum,’ varieties of symbolic attention and expression, the role of abstraction, the role of playfulness and the impact of individual differences. Copyright by HELENE ALPERT FURANI 2000 .lclnorrlfdg‘" l rush to lrgm l‘j tic). are. for mi; and mtlrnut xx ha ‘1 their parents. lim Miles and Jun 1 lanes and sch es Natl rush to II ranch this p Will lll all cnmumgu mentally Jame pm my mmmit Sltlpe the work ' FtaLllcmnnc an. prm ided ram role of chart on. 0i Alberta 165mm). raj when l land ah the Cr 1 $qu?“ r. #411713 hat. Acknowledgements I wish to begin by thanking the toddlers of this study for being the joyful and brilliant souls they are, for making me their playmate, for giving me the opportunity to learn so much, and without whom none of this would have been possible. I am also deeply greatful to their parents, Jim Dearing, Joe Eisenmann, Beth Herbel-Eisenmann, Sam Larson, Mary McVee and Jian Zhang for their friendship and for allowing me access to their children, homes and selves and assisting me as co-researchers of their children’s minds.’ Next, I wish to thank the faculty and students who have helped me in so many ways to reach this point. In their courses, Bill Rosenthal, Glenda Lappan and John P. (Jack) Smith 111 all encouraged me to pursue the interests that led to my dissertation research and eventually joined my dissertation committee. I was so very fortunate to have David Pimm join my committee as my dissertation director after the study was underway. He helped to shape the work in key ways that I believe never would have happened without him. Helen Featherstone and Sandra Creepo graciously joined my committee at key junctures and provided valuable input both before and after their formal participation. Helen took on the role of chair once David could no longer of frcially serve due to his move to the University of Alberta. I especially value David and Bill’s commitment in sticking by me across great distances when I (and also they) left campus. They maintained close contact and helped me manage the colleague-less isolation as I plugged away far away. Deborah Ball was my first dissertation chair, but left my committee following her move to the University of Michigan. She stayed on as long as possible and helped me through my dissertation proposal and at other crucial stages in my graduate career. I value her continued friendship. I The parents have given me permission to thank them using their real names. irpprccult- me .; arlc m offering '. pieces. I hope It. arlrrou ledgmg '. arlidcncc the} “arm and am: slimmih xx 1.: team m3 mcr elubllllg me [0 \ ”15hr: Khalil; : 35386l’lrrlar_ I Ci Research on my 0“ mlnf’: Gdrl'k: Elsenmlnn, Cm also mm; m Ci mam supp?” pm'lded Plum: I appreciate the assistance of all my committee members. They frequently went the extra mile in offering me ideas, suggestions for reading and even mailing me copies of helpful pieces. I hope they will forgive me for drawing liberally on their ideas without always acknowledging their contribution. Most of all, I appreciate their moral support and the confidence they showed in me. They were true teachers in the best sense of the word, warm and caring, friends and guides, supporting me as a person as well as a student. I additionally wish to thank Prentice Starkey at the University of California at Berkeley, who became my mentor for the Spencer Fellows program and whose input was crucial in enabling me to continue traveling down the research path I had begun. I wish to thank my fellow students, who formed a supportive community that nurtured me as a scholar. I especially wish to thank those participants of the Mathematics Learning Research Group (MLRG) who taught me by letting me see their work and giving feedback on mine: Garnet Hanger, Melissa Dennis, Angela Krebs, Tat-Ming Sze, Beth Herbel- Eisenmann, Candy Baguilat, Dara Sandow, Kyle Ward, Faaiz Gierden and Jan Gormas. I also thank my dear friend and colleague, Fernando Cajas. I wish to thank my parents, Joan and Hugh Alpert, who provided ever needed moral and material support and encouragement that only parents can give. My mother additionally provided painstaking and expert copy-editing on final drafts. My husband, Khaled Fulani, was always by my side and gave assistance in every way possible, including frequently needed boosts to my confidence and even valuable references from time to time. This document is nearly as much his accomplishment as it is mine. My daughter, Mysoon was born in the middle of it all and might be seen by some as having delayed or hindered my progress. To the contrary, her embodiment of love and joy (and not to mention, cuteness) enabled me to persist in lonely New York, so far from my scholarly community. Her smile rs alien cnou hsfptul pcrxpn next subject a [rush to tltml :rsprauon an. was often enough to get me started each day. Watching her grow since birth has given me helpful perspectives on the three boys I began studying as toddlers. She may become my next subject as all have anticipated. I wish to thank the Eternal Spirit that can have no single name, for health, strength and inspiration and allowing me to see this day. vi Table of Con! Gupta 1: lntrtt: Charter ll: 0n 1. Ctrptcrlll: .\lc: (flatter ll': Xi" 0.21pm V: Syn Chapter Vl: Sr . Claptcr \'ll: C -. References Table of Contents Chapter I: Introduction p. 1 Chapter II: On Learning, Language and Mathematical Knowledge p. 12 Chapter III: Methods p. 46 Chapter IV: Naming p. 72 Chapter V: Symbolizing the Imagined p. 120 Chapter VI: Systematizing and Playing with Symbols p. 157 Chapter VII: Conclusions p. 203 References p. 215 vii Chapter I: ln As he inlet .1; young tttldlcrs young chldrcn and he clopnn research that p mathematml ' and analyzes 1 “11h an e} c in talents for 1 In this llllttklt minted an, them“ and Ur remaining ch; All” glans Mahdi“ jur ‘llindlul‘ of; l). 34), NO[ 01 realm or [he I. (Impumrs an W M W rm Chapter I: Introduction As the title implies, this study is essentially a mathematically informed interpretation of young toddlers’ symbolizing. As such, it is unique. Although aspects of symbolizing in young children have been studied by linguists (e.g., various aspects of language learning) and developmental psychologists (e.g., symbolic or pretend play), I have not come across research that places toddlers’ symbolic continuum within a single analysis, let alone from a mathematical viewpoint. This is what this study attempts. It carefully examines, explicates and analyzes the symbolizing activity of three young toddlers in its various expressions, with an eye to understanding the underlying cognitive processes and their potential relevance for doing mathematics. In this introductory chapter, I offer personal assumptions and experiences that have motivated and shaped this study, address preliminary methodOIOgical concerns and indicate themes and undercurrents that run throughout. I follow this with a brief overview of the remaining chapters. At first glance, and even second and third, toddlers and mathematics appear to be an outlandish juxtaposition. Mathematics is a cool, calculating, abstract activity practiced by a ‘handful’ of adults, while in young children “the blood still runs warm” (Donaldson, 1978, p. 24). Not only is mathematics not for children, but to many adults, it approaches the realm of the inhuman It belongs to computers and robots and people who think like computers and robots. No wonder countless recoiled in near horror when I answered their cocktail party question with, “I’m a math major.” Although I majored in math as an undergraduate, I came about it rather circuitously, and for a long time, I too shared the common view of mathematics as being somehow inhuman til the it orld . mat-om} b} tut those tt ho ml? 1 rent through school I beg major. but I St success. 1 ml; ‘nath men.‘ it On the other l of Chuck‘s tn symbol and p mama: the Sdtcnl, he be, Hf Strand ti Gardiner b“. lulllt)! hl‘llt St Comes. He \ compared to mathematics glide lfi’it‘hc; 9113th me 21 H0“ Clef. (ill ’1'? s r. Ll‘itk dlll CW" ‘0 and the world divided into ‘math people’ and ‘non-math people,’ the latter forming the majority by far. ‘Math people’ were even further categorized as ‘truly math people’ and those who managed to fake it. I was someone who could fake it. I went through school mathematics without truly understanding it, particularly in high school. I began to understand and even enjoy mathematics in college, hence my choice of major, but I suspected that my attending a woman’s college had something to do with my success. I only began to believe I could hold my own among the genuine virtuosos, a.k.a. ‘math men,’ when I did well in a math class at a co-ed university. On the other hand, my younger brother Chuck was born a ‘math person.’ ‘Two’ was one of Chuck’s first words. He would toddle around with a plastic, magnetic version of the symbol and proudly name it for anyone in sight. Chuck could count to one hundred and recognize the corresponding symbols before the age of two and a half. In elementary school, he began to calculate batting averages in his head while watching baseball games. He squared the numbers of license plates as we rode around town and read Martin Gardiner books (of mathematical puzzles) for fun. In sixth grade, Chuck was bussed to the junior high school to take algebra and in eleventh grade, to the university for post-calculus courses. He was a star on the county math team. Compared to Chuck, I was a mere impostor. Not that I minded being ‘lesser’ in mathematics. I had other interests and talents and never really ‘got into’ math. My sixth grade teacher ‘recognized my abilities’ and based on his recommendation, my mother pushed me ahead of my grade. As I said before I got by, but without really understanding. However, once in a while I would really try to puzzle out a problem and eventually manage to understand it. When that happened, I recognized that l was thinking about the problem quite differently from how the teacher or textbook presented it. mite cxpt‘flC-‘V clatmm '0“ " tomtictuttn 0" neat-IOU“) \\ I. these t\\ 0 pt ‘li ‘ trident MW t llell into tear? it was b} accrd populated by h ‘bttsr'cs.’ and l mandates. star one questionc \l\ mixing] 1 Stttdcnts‘ idc: SEED a pro by helping st Those experiences planted the kernel of an idea that remained dormant until I entered the classroom ‘on the other side of the desk,’ but it was there nonetheless and received gradual fortification over the years. Rather than the world being split dichotomously into those near-robots who ‘got math’ and those who did not, or even along a continuum between these two poles, perhaps people thought about math differently; but different ways of thinking were not fostered or even accepted in school. I fell into teaching by accident just as I had become a mathematics major. Perhaps because it was by accident, I had a terrific first experience. I taught mathematics in a private school populated by home-schooled children. This meant that parents were responsible for all the ‘basics,’ and I was responsible for anything extra that I chose. I was free from curriculum mandates, standardized tests, and departmental oversight. I was considered the expert; no one questioned what I did. My personal experiences with math perhaps pre-di sposed me to be open—minded about students’ ideas, but I also had the fortunate opportunity to receive training from Project SEED, a program that seeks to build student self -esteem in disadvantaged school districts by helping students experience success with challenging mathematics. SEED promotes student success through such means as classroom discussion, active engagement by all students, bodily participation (through use of gestures for ‘agree,’ ‘disagree,’ showing answers with fingers and the like), unconventional symbolism that helps to connect new ideas to ones students already know, step by step development of ideas, fostering of disagreement that must be worked out by argument and proof, deliberate errors by teachers that students must watch for and correct, and the posing of challenging and exciting problems with an attitude of great excitement as well. However, what most influenced me at the time was SEED ’3 teaching that students thought logically. Whatever ‘errors’ studfnb {Twink up}n In \IIJCI pitted :1 cruel mrthcmattcs. ltork lhlS '90 thought :51 I also I ound st" that theirs M attested trot students had link‘ll’li‘dgt‘ (1 times the ma PhllOSOphcr' ’Smable git llIl (firm; Ol- allSlVerS On ‘ about ”lathe students made had a logical foundation that was worth uncovering, honoring and building upon in order to reach ‘correct’ understanding. Honoring children’s ways of thinking played a crucial role in enhancing their self-esteem and helping them achieve success in mathematics. I took this perspective into my first classroom. I looked for the logical ways students thought as I introduced them to interesting mathematics. And logical thinking I found, but I also found something else. In my attempts to uncover students’ understandings, I learned that theirs were not only frequently different from my own, experienced view, as might be expected from young learners, but their understanding differed from one another’s. The students had different entry points, different ways of making sense, different prior knowledge and different dispositions. And they challenged my preconceptions. There were times the math-enamored ‘calculator’ missed the mark, while the ‘head in the clouds philosopher’ suddenly came down to earth and broke the problem wide open. The ‘sociable girl’ gave the clearest, most thorough explanations, and the ‘quiet, least-prepared’ (in terms of prior education) boy showed he was fully keeping up by giving correct answers on his weekly ‘fun sheets.’ Different students took different paths and thought about mathematics differently. I began to formulate the opinion that far from being inhuman, math was decidedly human and being human had multiple forms of expression. Math connected organically to all my students’ minds but did so in various ways. Perhaps society’s problem with mathematics lay in an intolerance of this multiplicity, in the uniformity in which mathematics is presented in school and used in the wider society. Mathematics is treated as a monolith, a fait accompli, not as something living, changeable and variegated. Children write stories and poems. They make their own drawings, but not their own mathematics. hl} tlasm rm ' ldid not es en t of send}. The s truly hear therr predrsmsed m. about \t hat m} This stud} has Students. 100'; alternatrxes. lt mathematrts r as a teacher ar gtt'e mice to ‘ differences 3: ’Oting [mulle- My classroom was sadly no exception. Although I was flexible, I was not flexible enough. I did not even imagine students creating their own mathematical terms, symbols and objects of study. The students did a lot of talking while I stood by with a poker face, but I did not truly hear them. I could not hear much beyond what my mathematics training already predisposed me to hear. I knew too little outside the bounds of my education, too little about what my students might actually have been thinking. This study has been an attempt to overcome this deficiency -- to enable me to actually hear students, to open my mind to multiple ways of drinking mathematically by uncovering alternatives. It has been a step towards understanding cognitive aspects of doing mathematics in their purest, most human, least adulterated form. Naturally, my experiences as a teacher and a learner of mathematics have guided my inquiry throughout. My desire to give voice to varied student thinking in the mathematics classroom has led me to search for differences among the toddlers in this study and begin considering how those differences might inform cuniculum and pedagogy. Young toddlers were the ideal subjects for many reasons. Since they had not yet begun formal schooling of any kind, they were free from its ‘indoctrinating’ force. While they were beginning to express themselves verbally and physically, they were still at the ‘cusp’ of language, choosing words and meanings that meshed with their thoughts, rather than having language take a dominating role.l Being young, inexperienced, playful and wildly curious about all sorts of things, their activity was still relatively simple, limited in scope, 1 This view of language as taking a dominant position as a mediator of thought has gained increasing popularity in recent years, known as the ‘Sapir-Whorf hypothesis.’ Gee (1999) describes language as “simultaneously reflectfing] reality (‘the way things are’) and constructfing] (construflngl) it to be a certain way” (p. 82). He goes on to say, “Different sign systems and different ways of knowing have, in turn, different implications for what is taken as the ‘real’ world, and what is taken as probable and possible here and now, since it is only through sign systems that we have access to ‘reality.”’ (p. 83). and amiabl- ‘ l mixers he lbegan IlllS that my esp} domrrunt at" or ‘symh tltt‘ such a one; nitration m. This studx rs mmCUllI Cr“ HPcttences a mamfimauts This Study Ca “filler. “ln u maihemarm through \t hm. expllCll). The h" mph” car am“ that and available to deep analysis of a sort impossible with their school age counterparts. The toddlers had much to teach me. I began this surdy wanting to learn about ‘mathematical thinking.’ I was not at all aware that my explorations would lead to a focus on symbolizing. But symbolizing emerged as a dominant activity in the toddlers’ lives, from their encounters with language, to pretending or ‘symbolic play,’ to playing with symbols of a more abstract sort. And symbolizing is such a crucial aspect of doing mathematics, one still largely ignored in mathematics education research and practice.2 This study is both empirical and theoretical. I analyze data gathered on three toddlers with particular concerns in mind. I also draw connections to mathematics based on my experiences as a doer of mathematics, a teacher of mathematics and a participant in the mathematics education research community. This study can be seen as an extensive application of metaphor. I continually ask and try to answer, “In wlmt ways does this activity (of one of the toddlers) resemble a particular mathematical activity?” from an underlying, cognitive perspective. Mathematics is a lens through which I view the toddlers’ actions (although I do not necessarily make this explicit). The resulting analysis in turn offers fresh views into doing mathematics. Metaphor can be understood as crossing domains (Donnelly, 2000), and indeed I crossed domains that appear f ar-reaching if not outright bizarre. Toddlers and mathematics? Yet, in 2 This situation is beginning to change. More researchers have taken up questions of language and mathematics in recent years, e.g., English (1997), but the area is still very sparse. In NCT M’s (2000) recent standards document, a new process strand has been added called ‘representation.’ While NCI'M touches eloquently on some of the issues explored here, cognitive aspects and the role of symbols in mathematics itself (i.e., that there is no mathematics -- that is, the cultural artifact taught in schools -- without symbols) are largely missing. addition to m} | uords of Pot nt‘ [llt is l‘ pro-g tlm a l’\ p. 3'78, TC“ One may \\ on. these toddlers (ICOUTSC l d). 0i the term h hlEIhL‘maUCS ‘ m0“? objects “5- H0“ eye. prOCeSses ssh pit—fillets tau; WWSSQS‘ 21F the “odd Ill mmd; I am rl thinking abnl addition to my own powerful learning from this undertaking, I am encouraged by the words of Poincare, who said of science: [I]t is by unexpected union between its diverse parts that it progresses....Among the chosen combinations the most fertile will often be those formed of elements drawn from domains which are far apart (1982, p. 378, 386). One may wonder how I could answer the metaphor question as delineated above. Did I see these toddlers write arithmetic problems? Did they solve equations for x and y? Well, no. Of course I did not actually see the toddlers doing mathematics by any current definition of the term, but I did see them involved in symbolizing, in some of the central processes required to engage in mathematics. Mathematics is often viewed as a collection of objects and tools, and topics of study around those objects and tools: numbers, shapes, formulae, algorithms, measurement, geometry, etc. However, these are cultural products of certain processes of the human mind, processes which are also considered mathematical (see e. g., NCT M, 2000). The cultural products taught in school are not the inevitable and only outcomes of mathematical processes, and while I believe the processes are essentially human ways of interacting with the world, the products now subsumed under the title ‘mathematics’ may not ‘fit’ all minds. I am here referring to experiences of ‘mis-match’ between individual ways of thinking about mathematics and those embodied in canonized concepts, symbols and procedures, which I have observed in myself and in students, as well as the knowledge that alternatives can exist Researchers in ethnomathematics, the study of mathematics practices in other, frequently non-literate cultures, have catalogued numerous mathematical products that differ from those canonized in the Western mathematics tradition (which also has decidedly non- ‘ Western {001 Bid-1"? l l9.“ t .. l . ptiyng tint. t SOfUIlg. 1116;}: uncox enng 3 another. Rest demonstrate lI learning m the products. mathematics Comm mOR Western roots). Some have even identified cultural processes that give rise to mathematics. Bishop (1988) includes among these processes: counting, locating, measuring, designing, playing and explaining. D’Ambrosio (1994) includes observing, counting, ordering, sorting, measuring and weighing. Ethnomathematics is one area of research that leads to uncovering alternative mathematical processes and products. History of mathematics is another. Research on young children is yet a third fruitful avenue as this study demonstrates. If learning mathematics were more about developing the processes and less about adopting the products, there could possibly be more widespread success. There might truly arise a ‘mathematics for all,’ and an empowering mathematics at that, mathematics that would connect more organically to children’s minds, enabling them to think in powerful ways.3 Regarding the question of toddlers and mathematics, no, I did not see the toddlers make use of mathematical products much at all, but they did make heavy use of certain mathematical processes, including many that I had never before considered. Of course, the problem remains as to how I could identify mathematical processes without the products attached. This challenge did indeed pose some difficulty, but the toddlers offered me a fruitful solution. As they cleariy engaged in symbolizing -- using and creating meaningful words and gestures, playing make-believe, playing games with patterns and symbols as place-holders -- I was able to conduct a deep analysis of their activity and speculate as to its relevance to mathematical symbolizing and to learning mathematics in the classroom. 3 By ‘mathematics for all’ I am referring to what NCTM is currently stating as its ‘equity principle’: “Equity requires high expectations and worthwhile opportunities for all” (2000, p. 12). While meeting this principle and other stated goals, NCTM talks about processes of doing mathematics (1989) and teaching mathematics (1991). However, NCTM has yet to emphasize processes over products or to consider that through developing the mathematical processes of all children, the content of curriculum could itself change. Men I s. ”a thought uh researchers] child and ser only reeours education re inten'teu'cd beginning It" d0ll1g some; OUI‘wafd be} Tfildlers an “ordst thet D’thelh t1 hm Pimm SOns‘ an“ “Wmmu v“Odds. When I speak of mathematical processes, I am in many ways referring to ‘processes of thought,’ which appears to be an ephemeral object of study. Every teacher, every researcher of children would probably love to just (fr guratively) open the brain of every child and see what they are thinking and know their thought Since this is impossible, the only recourse is to infer thought from what children say, do or write. In mathematics education research, children are generally posed tasks, problems to solve, or are interviewed. With toddlers, however, these methods are impossible. Even though they are beginning to use conventional language, they could not possibly explain why they are doing something or what they are thinking. Hence, the demands for inference from outward behavior are even stronger. However, this situation need not pose a problem. Toddlers are incredibly physically active, and through their actions (including actions with words) they express their thoughts. Donnelly (1998) observed the actions of her sons from their earliest movements and noted how particular ones held their interest to the point of compulsion. As they grew older, her sons’ actions manifested themselves in certain patterns of thought, ways of interacting with the world that came out in how they made sense of events, told stories and created fantasy worlds. In childhood, the capacity to create metaphors is manifested in play and it is the gesture or action that forms the link between one object and another object or idea. Gradually these metaphor-actions become internalized and become visual or verbal metaphors and form the basis for our conceptual understanding of the world (Donnelly, 2000). Toddlers’ actions can thus be ‘read’ for meaning as with any text. They offer a relatively ‘transparent’ view of thought that at later ages becomes internal, requiring a degree of excavation in order to be seen. Before I lIIlIl ti assumptions .1. sensem '1 .. character from large!) rigid a. ngadrry ms 0h mdmdual drf: Before I introduce the rest of the chapters in this text, I will summarize some of the primary assumptions and motives that have guided this study. Mathematics is essentially a human sense-making activity, based in human mind and action, which derives its content and character from these foundations. However, as taught in schools today, mathematics is largely rigid and constraining of thought and creativity. A possible means out of that rigidity involves attending to the processes that give rise to mathematics, including to individual differences in these processes, both in research and classroom practice. Rather than primarily enculturating children into the mathematical canon, perhaps children’s ways of thinking should additionally guide what happens in mathematics classrooms. Children might then create mathematics that ‘fit’ their minds, thus leading to enhanced self -esteem and true ‘mathematical power.’ I see all of these views as resting on a fallibilist mathematics epistemology, one that regards mathematics as any other form of human knowledge, a social historical product that is highly valuable, but does not represent ‘truths’ about the world outside of human attempts to understand it. By stepping outside of the ‘cult of objectivity,’ which holds that objective knowledge is possible, multiple and untold forms of thought and expression may be permitted and encouraged.4 In Chapter II, I review scholarship from a wide range of fields and carve out my own position on what is involved in learning mathematics, the relationship between thought and language (which naturally includes mathematical language), and the nature of mathematical knowledge. I conclude the chapter with a section that clarifies terms central to the data analysis. Here I define ‘symbolizing’ and begin to sketch out both the role of symbolizing 4 Gill (1993) explains how the ‘cult of objectivity’ rests on the fallacy that knowledge can be separated from human knowing that must take place within human bodies and actions. “The contention is that only that which is purged of all ‘subjective’ elements can qualify as genuine knowledge” (p. 50). 10 in doing ma naturallx Lil In Chapter I there. it h} and discuss Chapters l\' the tojdlers‘ examples gr. mldmlhg ll Chapter \‘tr direcuom f0 in doing mathematics and particular features of mathematical symbolizing. These issues are naturally taken up in greater depth in the data chapters. In Chapter III, I describe the study’s Methods. I include the usual details on what, when, where, why and how of gathering data for the study. I also explicate the study’s questions and discuss why and how these questions, primary analytical tools and methods evolved. Chapters IV, V, and VI are the ‘data chapters.’ Each is f ramed around particular aspects of the toddlers’ symbolizing activity, which are described, analyzed and illustrated with examples great and small. Each chapter also contains a section on mathematical connections that draws links between the data analysis and aspects of symbolizing in mathematics. Chapter VII revisits the primary concepts developed in the data chapters and offers further directions for research. 11 Chapterll One mghtt open on cl" trth mather t'er) stmtlar here “1th Pr throughout 1 reletance an knot led 2e, Rfisearch cor 01‘ Piaget's n PUtPUrted- in 5“? Donald» hlS theonce Piaget‘s pUts lnl'estj Either regard )‘OUn the qUite art “Elle“ tint asd7y, ‘Clm‘scnalp language Is 1'1973) Chapter II: On Learning, Language and Mathematical Knowledge One might expect me to begin with Piaget, widely regarded as the “father” and foremost expert on child development. Not only am I dealing with cognition in young children and with mathematical processes, particular foci of Piaget’s, but I also studied toddlers in a very similar fashion to how Piaget famously studied his own children. Indeed, I do begin here with Piaget, or more accurately with challenges to his work, and I return to him throughout the chapters in dialogue. However, key aspects of Piaget’s work are losing relevance and scholars from alternative perspectives have much insight to offer on human knowledge, learning and development I discuss their views as well. Research conducted from the 1970’s onward has challenged the construct validity of some of Piaget’s most famous experimental tasks (i.e., whether the tasks measure what they are purported to measure, e.g., ‘object concept,’ ‘egocentrism,’ ‘number conservation,’ etc.; see Donaldson, 1978; Hughes, 1986; Starkey, 1992) as well as foundational premises to his theories. These studies and a subsequent widening of the field endorses deviation from Piaget’s pursuit of universal, invariant characteristics among children, and the converse investigation of varying manifestations of fundamental cognitive processes. Rather than regard young children as ‘egocentric’ and ‘illogical’ (Hughes, 1986), I work to uncover the quite awesome cognitive powers of children. I review one famous ‘finding’ of Piaget’s by way of example, the ‘finding’ that before the age of 7 young children cannot ‘conserve number.’ The Piagetian task used to determine ‘conservation of number’ involves a researcher asking a child questions. This use of language is one aspect that has called for reexamination. In the words of Donaldson (1978), 12 [Piaget lllt‘ Clu‘. ll hen for sin of this The standard linear texts or lhe e\tenme' the Child ogre. apart and rent to“ no“ h,» Elle)” are appn They ar e belt rather than or Treaty, tr. “’Ctultl be drf “Memes ludsmem St the e-‘Penmt 50inch” Str the (roam—m [Piaget] is sensitive to differences between what language has become for the adult and what language is for the child in the early stages. However, when he himself as an experimenter, uses language, as part of his method for studying children’s thinking, he appears to lose sight of the significance of this issue (p. 61). The standard number conservation task involves an experimenter displaying for a child two linear rows of objects, each of the same number, placed in a one-to—one correspondence. The experimenter then asks the child if there are the same number of objects in each row. If the child agrees, the experimenter lengthens one of the rows by spacing the objects further apart and repeats the question. Children 6 years and under typically respond that the longer row now has more. Children are thereby deemed unable to ‘conserve’ number because they are apparently unaware that moving objects in space does not change their number. They are believed to rely on visual cues that might indicate ‘more’ (e. g., a longer row) rather than on logical knowledge about the stability of number through spatial displacement. Typically, the rows in number conservation tasks contain from 6 to 22 objects each. It would be difficult even for adults to determine the absolute number or compare the numerosities of sets of this size without counting or relying on some other means for judgment, such as one-to-one correspondence, length or density.l How do the children in the experiments know that the researchers only moved the objects around and did not somehow surreptitiously add to them? If nothing signifieant has changed, why then repeat the question? l Dehaene (1997) reports on studies in which adults were able to quickly and correctly recognize the numerosities of sets of dots up to three without counting. However, both response time and inaccuracies increased linearly with numerosities beyond three. The process by which people (and animals) recognize small quantities immediately, ‘apprehend’ them without counting, is called ‘subitizing.’ 13 tl'r’lerdtne t questions as ‘ tip. 48L The t sense to chili possrhtltr} fr Winch: the I in Lh Hen. temp 1‘ ”‘33 be it. mtfnttom a the reSearch longer rots “lien mim mum to t: ample, } CODSEncd CliScuSSCS c rather Uta: “Ere {Olin Walkerdine (1988) reports how, “[C]hildren when asked such apparently nonsensical questions as ‘Is yellow bigger than green?’ would search for objects to justif y an answer” (p. 48). The questions experimenters ask in conservation tasks might similarly not make sense to children and yet children provide the answers they think are desired. This possibility fits with findings on number conservation tests discussed by Dehaene (1997) in which: the youngest children, who were about two years old, succeeded perfectly in the test....Only the older children failed to conserve [in one of the tasks]. Hence, performance on number conservation tests appears to drop temporarily between two and three years of age (p. 45). It may be that after age two, children make greater attempts to understand the researchers’ intentions and, as a result, mis-interpret what is said. In other words, children may believe the researchers want a different answer than the one first given and that they want the longer row to be identified. When number conservation tasks are administered differently from the standard, in ways meant to take into consideration potential difficulties, children succeed at greater rates. For example, Hughes (1986) reports on findings in which children as young as three conserved when the quantities involved were small (under four objects). Donaldson (1978) discusses experiments in which a ‘naughty teddy’ accidentally messed up the situation, rather than a researcher performing deliberate manipulations, and again young children were found to conserve. In experiments conducted by Starkey (1992), rather than compare two rows of objects using sight only and in response to questions, children displaced singular sets of balls themselves into an opaque ‘searchbox’ and then retrieved them. Balls were secretly removed from the box in the interim. Children from 18 to 48 months of age demonstrated surprise when they could not find the identical number of balls they had placed inside the box and were, therefore, found to ‘conserve number.’ 14 Drscreprncre importmt iss atsolutely \\ matters. and matters. Put it is a fun ‘g light of a st: “abstracted D‘ll‘iiltlxt‘in QWSUOnS, f “0’35 are u l‘l'flpts‘tatir the SIWitter m“smiths. ‘ (161111“ “Dr, rematch hah'e led Cl“ Atop-mgh t «IC'llabIc‘s ‘ Discrepancies among success rates with various number conservation tasks reveal several important issues: absolute number matters (i.e., a number small enough to perceive absolutely without counting), children’s interpretation of the researcher’s words and intent matters, and in the words of Donaldson (1978) that the situation make “human sense” matters. Putting balls in a box and then looking for them makes ‘human sense’ to children; it is a fun ‘game,’ similar to hide and seek. Comparing long rows of vases and flowers in light of a strange adult’s curious movements and questions does not . Such a task is “abstracted from all basic human purposes and feelings and endeavors” (p. 17). Donaldson (1978) sheds further light on how a child could misinterpret a researcher’s questions. She discusses how children first try to ‘understand the situation’ in which words are uttered, then the people and their intentions and lastly the words. When a child’s interpretation of the meaning of a situation and of the words uttered in the situation conflict, the situation might take precedence. Children initially learn word meanings by interpreting situations. “It is possible to figure out what words mean because they occur together with certain non-linguistic events” (p. 37). Therefore, although in conservation tasks researchers may have asked children to compare the number of objects, the situations may have led children to believe they were being asked to compare length. Although the traditional Piagetian number conservation tasks might remain highly ‘reliable,’ children’s success with variations on the tasks suggest that the tasks fail to measure what they purport to measure. In other words, even though young children fail Piaget’s tasks, they can ‘eonserve number.’ This fact naturally calls into question Piagetian stage theory and the accompanying view that correct knowledge of number is built up over the early years through ‘logico-mathematical experience,’ although not completely. The studies cited above show that seven years is not required to construct number knowledge, 15 but the} stilt trorr mg er rdt Another strrtiJ 8 months of ; sounds. Dd): distinguish q notabl} rn lrr structures rh, m3“ led 90. iii}; bet; l’C'pr Pitt rm g Q\ Q pro Regard," Al (70 km)“ ch‘lt‘ \ “3m, 19 ”005386 hwledg TthE ma m” Imph but they still allow for the possibility that 18 months is necessary. However, there is growing evidence that this is not the case; the ‘number concept’ may be innate. Another study by Starkey (Starkey, Spelke & Gelman, 1990) found that infants from 6 to 8 months of age could distinguish between the numerosities of 2 and 3 in images and sounds. Dehaene (1997) reports on another study in which 4 day old infants could distinguish quantitatively between 2 and 3 syllable words. These and other studies, most notably in linguistics and animal cognition, point to the possibility of innate cognitive structures that guide attention to certain aspects of experience. People are born with a basic knowledge. Learning begins from there: [H]uman cognition...is a collection of specific capabilities which evolved because they contributed in specific ways to survival and reproduction... [C]ognitive development is not a completely general process that equips the child to learn anything at all to which he or she might be exposed; rather, it is a collection of adaptive systems that have evolved to enable children to acquire specific kinds of knowledge that have proved valuable evolutionarily (Sophian, 1995, p. 18). Regarding mathematics, studies indicate that a number concept and perhaps basic knowledge of addition and subtraction and certain spatial abilities may be innate (see e. g. Wynn, 1992; Hermer & Spelke, 1996). It is upon the human species’ innate constructs and processes that mathematics has been constructed. While this nativist view of species-wide knowledge may seem to conflict with a perspective on individual differences, it need not. There may be some common cognitive inheritance to all people, but this hypothesis does not imply that all people must think alike. For example, all humans may possess the same innate cognitive structures that guide the learning of language, but these structures do not prevent children from constructing the various different grammars that govern different languages (see e. g., Brown, 1973; Hirschfeld & Gelman, 1994). Regarding mathematics, 16 some basic C01 ran 2 great tit tusterrt arm in; schml mdrhcr Tht‘U‘t' n‘mih before rntuztr‘ archaic dritrcu rnturtr' Tris coma] rr 313. Some ch11 sersrcal. Sam that “hates m these muons whites. empiv Versus hr)\\‘ m Yemamg)_ Thr some basic concepts may be present in all children, but where children go from there could vary a great deal. This possibility is evident in the wide range of mathematical practices existent among various cultures as well as the difficulty most children face with much of school mathematics. As Dehaene (1997) puts it, Though a few years of education now suffice for a child to learn digital notation, we should not forget that it took centuries to perfect this system before it became child’s play. Some mathematical objects now seem very intuitive only because their structure is well adapted to our brain architecture. On the other hand, a great many children find fractions very difficult to learn because their cortical machinery resists such a counter- intuitive concept (p. 7). The cortical machinery of many children may resist certain concepts, but not necessarily all. Some children might find fractions completely obvious, whereas others find them non- sensical. Some children may have difficulty with the idea that all parts must be equal, or that wholes must be equal when comparing fractions. Others may have no problem with these notions and yet find it more sensible to compare parts to parts rather than parts to wholes, employing a ratio rather than a fraction understanding (e. g., how much pizza left versus how much pizza already eaten, rather than what portion of the original pizza remains). Thus, ‘cortical machinery’ can differ from child to child. Thus far, this discussion on nativism has focused on mathematical concepts and objects, but not on processes or ways of thinking. These too might be innate and may also differ in character from person to person. For example, processes of organizing perception, generalizing and symbolizing may be part of the human inheritance that permits the construction of mathematics from basic knowledge. However, the ways in which people engage in these processes can vary, as the three toddlers in this study demonstrate. 17 Kant rsm m3 :1 hart-ratl r