cognitive tech maths edu

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4 Cognitive Technologies forMathematics EducationRoy D. PeaEducational Communication and TechnologyNew York University

This chapter begins with a sociohistorical perspective on the roles playedby cognitive technologies as reorganizers rather than amplifiers of mind.Informed by patterns of the past, perhaps we can better understand thetransformational roles of advanced technologies in mathematical thinkingand education. Computers are doing far more than making i t easier orfaster to do what we are already doing. The sociohistorical context mayalso illuminate promising directions for research and practice on compu-ters in mathematics education and make sense of the drastic reformula-tions in the aims and methods of mathematics education wrought bycomputers.The chapter then proposes an heuristic taxonomy of seven functionswhose incorporation into educational technologies may promote mathe-matical thinking. It distinguishes two types of functions: purpose func-tions, which may affect whether students choose to think mathematically.and process functions. which may support the component mental activi-ties of mathematical thinking. My hope is that the functions falling intothese two categories will apply to all cognitive technologies, that they willhelp students to think mathematically. and that they can be used bothretroactively to assess existing software and proactively to guide softwaredevelopment efforts. Definitions and examples of software are providedthroughout the chapter to illustrate the functions.

The central role that mathematical thinking should play in mathematicseducation is now receiving more attention, both among educators and inthe research community (e.g. Schoenfeld, 1985a: Silver, 198.5). As


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problems in mathematics" (1985a, p. 2). The growing alignment ofmathematics learning with mathematical thinking is a significant shift ineduca t ion .

THEMES OF CHANGE IN THE AIMS AND METHODS OFMATH EDUCATIONThere is no question but that information technologies, in particular thecomputer , have radical implications for our methods and are alreadychanging them . B ut , perhaps more impor tant ly , we are a lso coming to seetha t they are changing our a i m s and th ereby what we consider the goals ofma them atical understanding an d thinking to which our educational proc-e s s e s a r e d i r e c te d .

Mathem at ics educators , represented by such organizations a s NCT M ,a re fundam entally rethinking their aim s and m eans. In part icular, mathe-matics activities are becoming significant in a much wider variety ofcontexts than ever before . The reason for th is expans ion is the wide-sp re ad availability of powerful ma them atical tools that simplify numericalan d symbolic calculations, graphing and modeling, and many of theme ntal operations involved in mathem atical thinking. For exam ple, manyclass ro om s now have available programmable calcula tors, compu ter lan-gu ag es, s imulation and modeling languages, sprea dshee ts , algebraic equa-t ion so lvers such as TK!SOLVER ,ymbolic manipulation packages andsoftware for data analysis and graphing. The drudgery of rememberingan d practicing cumbersome algorithms is now often supplanted by activi-t ies quite different in nature: selecting appropriate computer programsa n d d a t a e n tr y .Why have these revolu t ionary changes occurred? How can we useth em as a guide in the design, tes t ing, and use of the new technologies , s othat we c an enhance both the processes of mathemat ics educat ion and ou runde rstandin g of how i t occu rs? In oth er word s, what are the beacons thatwill help light the way as we co nsid er th e role of cognitive technologies inmathem at ics education?

AN HISTORICAL APPROACH TO MENTAL ROLES FORCOGNITIVE TECHN OLOGIESA n his torical approach will help us consider how the powers of informa-tion technologies can best serve mathematics education and research. I twill he lp us look beyond the inform ation age to understand the transfo r-mational roles of cognitive technologies and to illuminate their potential


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4. COGNITIVE TECHN OLOG IES FOR M ATHE MATICS EDU CATION 9

as tools of mentation. L ong before comp uters appeared, technical instrume nts such a s written language expand ed hum an intelligence to a re m ar kable extent . I take as axiomatic that intelligence is not a quality of thmind alone, but a product of the relation between m ental st ru ct ur es andthe tools of the intellect provided by the culture (Bruner, 1966; Cole &Griffin, 1980; Lu ria , 1976, 1979; O lso n, 1976, 1985; Olson & Bruner, 1974Pe a, 1985b; Vyg otsky, 1962, 1978). L et us call these tools c ogn itivetechnologies.A cognitive technology is any medium that helps transcen d th e limitations of the mind (e .g. , attention to goals, short- term mem ory sp an ) ithinking, learning, and problem-solving activities. Cognitive tech nolog iehave had remarkable consequences on the varieties of intelligence, th

functions of human thinking, and past intellectual achievements (e.gCa ssire r, 1944; Goo dm an, 1976). Th ey include all symbol sy stem s, including writing systems, logics, mathematical notation systems, modelstheories, film and other pictorial media, and now symbolic computelanguages. Th e technologies that ha ve received perhaps the m ost att ention as cognitive tools are written language (Goody, 1977; Greenfield1972; Olson, 1977; Ong, 1982; Scribner & Cole, 1981), and s y s t em s omathematical notation, such a s algebra or calculus (C assirer, 1910, 1957K ap ut , 1985, in press; Kline, 1972) and num ber symbols (Me nni ng er1969).Contrast for a moment what i t meant to learn math with a chalk andboard, where one erased after each problem, with what i t meant to usepaper and pencil , where one could save and inspect one's work. Thiexam ple reminds us that under the broad rubric of the "cognitive tec hn ologies" for mathematics, we m ust include entities as diverse as th e cha lkand bo ard, the pencil and pap er, the computer and screen, and the sym bosystems within which mathematical discoveries have been made and thahave led to the creation of new symbol systems. Each has transformedhow mathematics can be done and how mathematics educat ion can beaccomplished . It would be interesting to explore, if space allowed, theparticular ways in which mathematics and mathematics educationchanged with the introduction of each medium.A com mo n feature of all the se cognitive technologies is that the y m akeexternal the intermediate p rod uc ts of thinking (e.g., outpu ts of com ponent s tep s in solving a comp lex algebraic equation), which can th e n beanalyzed, ref lected upon, and discussed. Transient and private thoughprocesses subject to the distortions and limitations of attention and

mem ory are "captured" and embodied in a communicable m edium thapersists, providing material re co rds that can become objects of an aly sis inthe ir ow n right-conceptual building blocks rather than shifting sa nd s


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92 PEAthought," because they help organize thinking outside the physical con-fines of the brain.We are now seeing, in ways described throughout the chapter, howcomputers are an especially potent type of cognitive technology forlearning to think mathematically: They can operate not only with num-bers , but also with symbols-the fundamental currency of humanthought. Computers are universal machines for storing and dynamicallymanipulating symbols. Capable of real-time programmable interactionswith human users, computers may provide the most extraordinary cogni-tive technologies thus devised. But what can we learn from the history ofnoncomputer-based cognitive technologies that will inform our currentinquiries?Cognitive technologies, such as written languages, are commonlythought of as cultural amplif iers of the intellect, to use Jerome Bruner's(1966, p. xii) phrase. They are viewed as cultural means for empoweringhuman cognitive capacities. Greenfield and Bruner (1969) observed thatcultures with technologies such as written language and mathematicalformalisms will "push cognitive growth better, earlier, and longer thanothers" (p. 654). We find similarly upbeat predictions embodied in awidespread belief that computer technologies will inevitably and pro-foundly amplify human mental powers (Pea & Kurland, 1984).This amplifier metaphor for cognitive technologies has led to manyresearch programs, particularly on the cognitive consequences of literacyand schooling (e.g. on formal logical reasoning) in the several decadessince Bruner and his colleagues published Studies in Cogni t ive Growth(e.g., Greenfield, 1972; Olson, 1976; Scribner & Cole, 1981). The meta-phor persists in the contemporary work on electronic technologies byJo hn Seeley Brown of Xerox PARC, who, in a recent paper, described hisprototype software systems for writing and doing mathematics as "ideaamplifiers" (Brown, 1984a). For example, AlgebraLand, created byBrown and his colleagues (Brown, 1984b), is a software program in whichstudents are freed from hand calculations associated with executingdifferent algebraic operations and allowed to focus on high level problem-solving strategies they select for the computer to perform. AlgebraLand issaid t o enable students "to explore the problem space faster," as theylearn equation solving skills. Although qiraiztitative metrics, such as theefficiency and speed of learning, may truly describe changes that occur inproblem solving with electronic tools, more profound changes-as 1 willla ter describe for the AlgebraLand example-may be missed if we confineourselves to the amplification perspective.There is a different tradition that may be characterized as the cultrirul-h i s tor ica l study of cognitive technologies. This perspective is most famil-iar to psychologists and educators today in the influential work of Vy-gotsky (1978). Vygotsky offered an account of the development of higher


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4. COGNITIVE TECHNOLOGIES FOR MATHEMATICS EDUCATION 93

mental functions, such as planning and numerical reasoning, as beingbae d on the "internalization" of self-regulatory activities tha t first ta keplace in the social interaction b etwee n children and adul ts. The historicalroo ts of Vygotsk y's orientation provide an illuminating framew ork for th eroles of computer technologies in mathematical thinking and learning.Influenced by the writings of Vico, Spinoza, and Hegel, Marx an d E ng elsdeveloped a novel and powerful theory of society now described ashistorical , or dialectical, materialism. According to this theory, humannat ure is not a produc t of environm ental forces, but is of o ur ow n mak inga s a society and is continually in the process of "becoming." Hu m ank indis reshaped through a dialectic, or "conversation," of reciprocal influ-ences : Our productive act ivi ties chan ge th e world, thereby changing th eway s in which the world c an change us . By shaping nature and how ou rinteractions with it are mediated, we change ourselves. As the biologistSte ph en Jay Gould observes (l980 ), such "cultural evolution," in co ntra stto Darw inian biological evo lutio n, is defined by the transmission of skills,knowledge, and learned behavior acro ss generations. I t is one of the wa ysthat we a s a species have transcended natu re.See n from this cultural-historical perspective, labor is the factor m edi-ating the relationship of human beings to nature. By creating and usingphysical instruments (such as machinery) that make our interaction withnature less and less direct , we reshape our own, human nature. Thechange is fundamental: Using different instruments of work (e.g. a plowrather than the hand) changes the functional organization, or systemcharacteristics, of the human relationship to work. Not only is the workfinished more quickly, but the actions necessary to accomplish therequired task hav e changed.In an attem pt to integrate accou nts of individual and cu ltural chan ges ,the Soviet theorists L . S. Vygotsky (e.g., 1962, 1978) and A. R . L ur i a(1976, 1979) generalized the historical m aterialism that Ma rx and En ge lsdeveloped for physical instruments. They applied it to an historicalanalysis of symbolic tools, such as written language, that serve asinstruments for redefining culture and human nature. What Vygotskyrecognized was that "mental proc esses always involve signs, just a saction on the environment alway s involves physical instrum ents (if only ahuman hand)" (Scribner & Co le, 1981, p . 8). A similar instrum ental an ddialectical perspective is reflected in recent studies of the "child as acultural invention" (Kessel & Siegel, 1983; Ke ssen , 1979; Wh ite, 1983).Tak e, for instance, Wartofsky's (1983) description of the shift in pe rspe c-tive:

Children are , or become. w hat they a re taken to be by othe rs , and what theycom e to take themselves to be. in the course of their social communicat ion


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historical kind, rather than a natural kind, and therefore also a constructedkind rather than one given, so to speak , b y nature in some fixed or essentialform. (p . 190)Applied to mathematics education, this sociohistorical perspectivehighlights not the constancy of the mathem atical understandings of whichchi ldren are capable a t par t icular a ges , but how what w e take for granteda s l imits are redefined by the child 's use of new cognitive technologies f orlearning and doing mathematics. Similarly, Cole and Griffin (1980) notedhow symbolic technologies qualitatively change the s tructure of thefunct ional sys tem for such mental activit ies as problem solving or mem-o r y .T he term "amplify" has other implications. It m ean s t o m ak e m o r epow erful, and t o amplify in the scientif ic sense "refers specifically to th e

intensification of a signal (ac ou stic , electro nic), which does no t undergoc h a n g e in its basic structure" (Cole & Griffin, 1980, p. 349). Thus, theamplifier metaphor for the roles of technologies in mathematical thinkingleads one to unidimensional, quantitative theorizing about the effects ofcognitive technologies. A pencil seems to amplify the power of a s ixthgra de r 's mem ory for a long list of wo rds when only the o u t c o m e of th e l istlength is considered. But i t would be dis tortive to go on to say that them en ta l process of remem bering that leads to the o utcome is amplif ied bythe pencil . The pencil does not amplify a f ixed mental capacity calledmemory; i t res t ructures the funct ional sys tem of remember ing andthere by leads to a more powerful outcom e (at leas t in terms of the n um berof i tems memorized). Similar preoccupations with amplif ication led re-searchers to make quant i ta t ive compar isons of enhancements in thelearn ing of basic math facts that a re brought abou t by softw are and printme dia , ra ther than to cons ider the fundamental changes in ar i thmet icalthinking that accompany the usage of programmable calculator functions(Confe rence Board of the Mathematical S ciences , 1983; Fey , 1984; Na-tional Sc ienc e Bo ard, 1983).

Olson (1976) makes s imilar arguments about the capacity of writ tenlanguage to res t ructure th inking processes . For example, wr i t ten lan-gua ge facil itates the logical analysis of a rgumen ts for consistency-contra-diction because print provides a means of s toring and communicatingcultural knowledge. I t transcends the memory limitations of oral lan-gua ge. What this means is that tech nologies do not s imply either a m p l i ' ,l ike a radio amplif ier , the mental powers of the learner or speed up andmake the process of reaching previously chosen educational goals moreefficient. Th e standard image of the cognitive effects of comp uter use isone-directional: that of the child seated at a computer terminal andundergoing certain changes of mind as a direct function of interactionwith the machine. The relatively small number of variables to measure


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4. COGNITIVE TECHNOLOGIES FOR M ATHEM ATICS EDUCATION 95mak es this image seductive for the researcher. But since the technologieschange the system of thinking activities in which the technologies play arole, their effects are much more complex and often indirect. Like print,they transcend the memory limitations of oral language. Complicatingmatters e ven m ore is that the specific restructurings of cognitive tech nol o-gies are seldom predictable; they have emergent properties that arediscovered only through experimentation.I es po us e a qu ite different theory abou t th e cognitive effects of comp u-ters than tha t just describe d. My the ory is consisten t with questions basedin a two -direction al image that other m athem atics educators and research-ers ( e.g . , Ka pu t, 1985, in press) are posing, such a s: What are the newthings you can d o with technologies that you could not do before o r thatweren' t practical to do? Once you begin to use the technology, whattotally new things do you realize might be possible to do? By "two-direction al image." I mean that not only d o comp uters affect people , butpeople affect computers. This is true in two senses. In one sense, we allaffect computers and the learning opportunities they afford students ineducation by how we interpret them and by what w e define as appropr iatepractices with t he m ; as these interpretat ions change over t ime, we chang ethe effects the computers can have by changing what we do with them.(Co nsider how we began in sch ools, with drill and practice and co m pu terliteracy activities, and now emphasize the uses of computers as tools,such as word processors , spreadshe ets , da tabase management sys tems. )In ano the r se ns e, we affect com puters w hen we study their use, reflect onwh at we see happ ening , and then act to chan ge it in ways we prefer or se eas necessary to get the effects we want. Such software engineering isfundamentally a dialectical process between humans and machines. Wedefine th e e duca tional goals (either tacitly o r explicitly) and then cre atethe le arning activities that work toward the se goals. We then try to createthe appro priate softw are. We experiment and test , experiment and test ,until we ar e satisfied . . . which we tend ne ver to be. Experimen tation is aspiral proc ess tow ard the unknow n. Through experim entation, new goalsand new ideas for learning activities emerge. And so on it goes-wecreate our own history by remaking the tools with which we learn andthink, an d w e simultaneously chang e ou r goals for their use.

COGNITIVE TECHNOLOGIES IN MATHEM ATICSEDUCATION

How does the idea of cognitive technologies relate to mathematicseducat ion? A few historical notes pre pare the stage. We may recall Er ns tM ac h's (189311960) state m ent, in his seminal work on the science of


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96 PEAto save mental effort. Thus arithmetic procedures allow one to bypasscounting procedures. and algebra substitutes "relations for values, sym-bolizes and definitively fixes all numerical operations that follow the samerule" (p . 583). When numerical operations are symbolized by mechanicaloperations with symbols, he notes, "our brain energy is spared for moreimportant tasks" (p . 584), such as discovery or planning. Although overlyneural in his explanation, his point about freeing up mental capacity bymaking some of the functions of problem solving automatic is a centraltheme in cognitive science today.Whitehead (1948) made a similar point: "By relieving the brain of allunnecessary work, a good notation sets i t free to concentrate on moreadvanced problems, and in effect increases the mental power" (p. 39). Henoted that a Greek mathematician would be astonished to learn that todaya large proportion of the population can perform the division operation oneven extremely large numbers (Menninger, 1969). He would be moreastonished still to learn that with calculators, knowledge of long divisionalgorithms is now altogether unnecessary. Further arguments about thetransformational roles of symbolic notational systems in mathematicalthinking are offered by Cajori (1929a, 1929b), Grabiner (1974), and,particularly, Kaput (in press).Although long on insight, Mach and Whitehead lacked a cognitivepsychology that explicated the processes through which new technologiescould facilitate and reorganize mathematical thinking. What aspects ofmathematical thinking can new cognitive technologies free up, catalyze,or uncover? The remainder of this chapter is devoted to exploring thiscentral question.A historical approach is critical because it enables us to see howlooking only at the contemporary situation limits our thinking about whatit mean s to think mathematically and to be mathematically educated (cf.Resnick & Resnick, 1977, on comparable historical redefinitions of "liter-acy" in American education). These questions become all the moresignificant when we realize that our cognitive and educational researchconclusions to date on what student of a particular age or Piagetiandevelopmental level can do in mathematics are restricted to the stat icmedium of mathematical thinking with paper and pencil.' The dynamicand interactive media provided by computer software make gaining anintuitive understanding (traditionally the province of the professionalmathematician) of the interrelationships among graphic, equational, andpictorial representations more accessible to the software user. Doors tomathematical thinking are opened, and more people may wander in.

'This argument is developed mo re fully wi th respec t to cogn ~t iv e eve lopm ent in genera lwith new technologies In Pea (1985a).


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4. COGNITIVE TECHNOLOGIES FOR MATHEMATICS EDUCATION 97Thus, the basic findings of mathematical education will need to berewritten, so that they do not contain our imagination of what studentsmight do, thereby hindering the development of new cognitive technolo-gies for m athematics education.

TRANSCENDEN T FUNC TIONS FOR COGNITIVETECHNOLOGIES IN MATHEMATICS EDUCATION

Rat ionaleWhat strategy shall we choose for thinking about and selecting amongcognitive technologies in ma them atics edu cation? I argue for the need t omove beyond the familiar cookbooks of 1,001 things, in near randomorder, that on e ca n d o with a comp uter . S uch lists are usually so vast a s tobe u nusab le in guiding the current c hoice and th e future develop me nts ofmathematics educational technologies. Instead, we should seek out highleverage aspects of information technologies that promote the develop-ment of mathematical thinking skills. I thus propose a list of "transcen-de nt functions" for cognitive technologies in m athematics education.W hat is the status of such a list of functions? Incorporating them into apiece of so ftware would certa inly not be sufficient to promo te mathemati-cal thinking. T he strategy is m or e probabilistic-other things being equ al,more students are likely to think mathematically more frequently whentechnologies incorporate these functions. Som e few students will becom eprodigious m athematical thinkers, whatev er obstacles must be overco mein the ma them atics ed ucation they face.2 Ot he rs will not thrive without aricher environ me nt for fostering mathem atical thinking. This taxonom y isdesigned t o serve as a heuristic, or guide. Assessm ents of w hether i t isuseful will emerge from empirical research programs, not from intuitiveconjecture. Indeed, until t ighter connections can be drawn betweentheory and practice,3 the list can only build on what we know fromresearch in the cognitive scie nce s; it should not be limited by thatresearch . -

2It is more com monly t rue that prodigious mathem atical thinkers have had a rema rkab lecoalescence of suppor t ive envi ronmenta l condi t ions for the i r l earning ac t iv i t i es , e .g . ,sui table models , r ich resource environment of learning materials , community of peers , andpr iva te tu toring (e .g . , as descr ibed by Feldm an, 1980).

3This s i tuat ion is the rule in theory-pract ice relat ions in educat ion (Champagne &Chaikl in, 1985; Su pp es, 1978). Fo r this reason , I hav e recent ly proposed (Pea. 1985b) thenee d for an act ivist research paradigm in educat ional technolog y, with the goal of s imultane-ous ly crea ting and s tudying changes in the processe s and outcom es of human learning wi th


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Final ly , why should we focus on t ranscendent funct ions? There aretwo major reasons. We would l ike to know what functions can becom mon to all mathematical cognitive technologies, so that each technol-ogy need not be created from the ground up, mathematical domain bymathematical domain. We would like the functions to be transcendent inthe s ense that they apply not on ly to arithmetic, or algebra, o r calculus,but potentially across a wide array, if not all, of the disciplines ofmathematical education, past , present, and future. The transcendentfunctions of mathematical cognitive technologies should thus survivechang es in the K-12 math curricula, s in ce they exploit gen eral featu res ofwhat i t me ans to think mathematically-features that are at the core ofthe psychology of mathematics cognition and learning. These functionsshould be central regardless of the career emphasis of the s tudents andregardless of their academic future. Lessons learned about these func-tions from research and practice should allow productive generalizations.The transcendent functions to be highlighted are those presumed tohav e great impact on mathematical thinking. They neither begin nor en dwith the computer but arise in the course of teaching, as part of humaninteraction. Educational technologies thus only have a role within thecontexts of human action and purpose. Nonetheless , interactive mediamay offer extensions of these crit ical functions. Let us consider whatthese extensions are and how they make the nature or variety of mathe-matical experience qualitatively different and more likely to precipitatemathematics learning and dev elopm ent .

These functions are by no means independent, nor is i t possible toma ke them s o. They define central tendenc ies with fuzzy bounda ries , l ikecon cep ts in general (Rosch & Mervis , 1975).They are also not presentedin order of relative importance. I will illustrate by examples how manyoutstanding, recently developed mathematical educational technologiesincorporate m any of the fun ction s. But very few of these programs reflectall of the functions. And only rare examples in classical computer-assisted instruction, where electronic versions of drill and practice activi-t ies have predominated, inco rporate any of the functions.One could approach the question of technologies for math e ducation inquite different ways than the one proposed. One might imagine ap-proache s that assume the do minan t role for technology to be amplif ier: togive stud ents more practice , mo re quickly, in applying algorithms that canbe carr ied out fas ter by com puters than otherwise . One could discuss thebest ways of using computers for teacher record-keeping, preparingproblems for tests , or grading tests . In none of these approaches, how-ev er, can com puters be considered cognitive technologies.A different perspective on the roles of computer technologies inmathematics education is taken by Kelman et al . (1981) in their book,Computers in Teaching Mathematics . They describe various ways soft-


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ware can help create an effective environment for student problemsolving in m athem atics. Their comprehen sive book is organized acc ord-ing to traditional software categories and curriculum objectives: compu-ter-assisted instruction, problem solving, computer graphics, appliedmathematics, computer science. programming and programming lan-guages. T h e spirit of their recomm endations is in harmony with the ske tchI propose in this chapter, although their orientation is predominantlycurr icular rather than cognitive. My stre ss on transcendent functions isthus a complementary approach, taking as a s tar t ing point the root orfoundational psychological processes embodied in software that engagesmathematical thinking.In my choice of software illustrations I have leaned heavily towardcases that manifest most clearly the specific loci supporting the sevenPurpose or Process functions. Although programming languages, spread-sheets, simulation modeling languages such as MicroDynamo (Addison-Wesley) , and symbolic calculators such a s muM ath (Microsoft) andTK !So lve r (Software Arts) can be central to thinking mathematically inan information age (e.g . , Elgarten , Posam entier , and Mo resh, 1983), Ihave seldom chosen them as examples. Although I take for granted theutility and power of these types of tools in the hands of a personcommitted to problem solving, their usefulness stems in part from theextent to which they incorporate the purpose and process functions. Fo rexample, Logo graphics programming provides the different mathemati-cal repre sen tatio ns of procedural text instructions and the graphics draw-ing i t creates (Process Function 3); and simulation modeling languagesand spreadsheets are excellent environments for mathematical explora-tion (Process Function 2), since hypothesis-testing and model develop-ment and refinement are central uses of these interactive software tools.But othe r env ironm ents in which thes e tools are used-for exa mp le, drilland practice on programming language syntax or abstract exercises towrite programs to create fibonnaci number series need not offer muchencouragement for mathematical thinking. In other words, the intrinsicvalue of su ch tools in helping stud ents th ink m athematically is not a given.Th e s t ress on Funct ions remains centra l .

A GUIDING DICHOTOMY: PURPOSE AND PROCESSFUNCTIONS FOR COGNITIVE TECHNOLOGIESHow can technology support and promote thinking mathematically? Inbroad str ok es , what a ppe ar to be the richest loci of potential cognitive andmotivational support of technologies for math education?


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side-will students choose to commit themselves to learning to thinkmathematically? Mathematics educators have to some extent neglectedthe concepts of motivation and purpose (e .g ., McLeod, 1985): that neglectmay help explain girls' and minorities' documented lack of interest inmathematics. What students learn also depends on the co g n i t i ve s u p p o r tgiven them as they learn the many problem-solving skills involved inthinking mathematically.My perspective on the functions necessary for cognitive technologiesthus has two vantage points. First, students are purposive, goal-directedlearners, who have the will (on any given occasion or over time) to learnto think mathematically or not. Then once they have embarked onmathematical thinking, they may be aided by technologies in mathemati-cal thinking. For simplicity of exposition. we thus divide function typesbetween: (a) those which promote PURPOSE-engaging students tothink mathematically; and (b) those which promote PROCESS-aidingthem once they do so.Purp ose Functions in Cognitive TechnologiesWhat lies at the heart of cognitive technologies that help make mathemati-cal thinking purposeful and help commit the learner to the pursuit ofunderstanding? Cognitive technologies that accomplish these goals arebased on a participatory link between self and knowledge rather than anarbitrary one. This organic relationship was central to John Dewey'spedagogical writings and integral to Piaget's constructivism: We mustbuild on the child's interests, desires and concerns, and more generally,on the child's world view. But what exactly does this mean?The key idea behind purpose functions is that they promote theformation of promathematics belief systems in students and thus ensurethat students become mathematical thinkers who participate in and ownwhat is learned. Students benefiting from purpose functions are no longermere storage bins for or executors of "someone else's math." Theimplication is that technologies for mathematics education should be toolsfor promoting the student's self-perception as mathematical "agent," assubject or creator of mathematics (Papert, 1972, 1980). For example,Schoenfeld (1985a, 1985b) argues that the belief systems an individualholds can dramatically influence the very possibilit ies of mathematicaleducation:

Students abs t rac t a "m athema t ica l wor ld view" both f rom thei r exper ience swith mathematical objects in the real world and from their classroomexper iences wi th mathem at ics . . . . These perspect ives af fec t the wa ys thats tudents behave when co nfronted wi th a mathemat ica l problem . both


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4 . COG NITIVE TECHNOLOGIES FOR MATHEM ATICS EDUCATION 101influencing what they perceive to be important in the problem and what setsof ideas , or cognitive resources, they use (Schoenfeld, 1985, p. 157).Although Schoenfeld's focus is broader than the point here, the stu-den t's m athematical world view includes the self: W hat am I in relation tothis mathematical behavior I am producing? If s tudents do not viewthem selves a s mathematical thinkers, but only a s recipients of the "ine rt"mathem atical knowledge that others po ssess (Whitehead, 1929), thenmath education for thinking is going to be problematical-because theagent is missing.In the prototypical educational setting, we often erroneously presup-pose that we have engaged the student 's learning commitment. But the

stu den t rarely see s significance in the learning; som eone else has m ade allthe decisions about scope and sequence, ab out the lesson for the day. Th elearning is m eant to deal not with the stu de nt 's problem or a problematicsituation the teacher has helped highlight, but with someone else's. Andthe knowledge used to solve the problem is someone else 's as well ,something that someone else might ha ve found useful at som e other t ime.E ve n that past utility is seldom conv eyed : s tudents a re almost never toldhow me asurem ent activities were es sential to building projects o r makingclothe s, or how numeration system s were n ecessary for trade (McLellan& Dewey, 1895).Acc ording to D ewey's (1933, 1938) sch em e for the logic of inquiry, theprototyp ical sy stem of delivering mathe ma tical facts leaves out the nece s-sary 3 r s t step in problem solving: the identification of the problem, thetension that ar ises between what the stud ent already knows and what heo r she nee ds to know that drives subse quen t problem-solving processes.It is interesting that Polya (1957) also omits this first step; in otherrespects his phases of problem solving correspond to Dewey's seminaltrea tm ent : problem definition, plan cre atio n, plan exe cution, plan evalu a-tion, and reflection for generalization of what can be learned from thisepisod e for the future (c f. , Noddings, 1985). Perha ps the ex pert ma them a-tician tak es this first step for granted : Fo r wh o could not notice mathem at-ical problems? The world is full of them! But for the child meeting theformal systems that mathematics offers and the historically accruedproblem-solving contexts for which mathematics has been found useful,the first st ep is a giant one , requiring su pp or t.Purp ose functions that help the stude nt become a thinking subject ca nbe inco rporated into mathematically oriented educational technologies inmany ways. Here, we go beyond Dewey to suggest o ther componentfeature s of mathematical agency:1 . Ownership. Agency is more likely when the student has primary


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102 PEAar y ow nership, i .e . , identif ication with the acto r in the problem sett ing, ina n "as if it were m e" simu lation). A central pedagogical con ce rn is to findwa ys to help people "own" their own thoughts and the problems throughwhich they will learn. Kaput (1985) and Papert (1980) have providedsuggestive examp les from softw are mathem atics discovery env ironm entswh ere the "epistemological con tex t" is redefined: Auth ority for wha t isknown must res t on proof by either the s tudent or the teache r; it must notrest exclusively with the teacher and the text . Students can offer newproblems to be solved, and the y can also create new kno wledge.2. Se l f - w or th . It is hard for stu den ts to be mathem atical agents if theyview opportunit ies for thinking as occasions for fai lure and diminishedself-worth. Student performance depe nds part ly on self-concept and self-evaluation (Harter , 1985). Research on the motivation to achieve byDweck and col leagues (e .g . , Dw eck & Elliot , 1983) indicates that s tude ntstend to hold one of two dom inant views of intel l igence, and that t he o neheld by each part icular s tudent he lps determine his or her goals . On o nehand, if the child views intelligence as an entity, a given quantity ofsomething that one either has or has not, then the learning eventsar ranged a t school become oppor tuni t ies for success or occas ions forfailure; if the child looks bad. his or her self-concept is negativelyaffected . On the oth er han d, if the child views intelligence a s "inc rem en-tal ," then these same learning events are viewed as opportunit ies foracquiring new understanding. Although l it tle is known a bout t he ontogen -esis of the detrim ental entity view , it is appa rent that this belief can hin derthe possibility of mathematical agency and that software or thinkingpractices that foster an incremental world view should be soug ht.3 . K now le dge for ac t ion . A third condition for promoting mathemati-cal agency is either that the mathematical knowledge and skills to beacquired have an impact on s tuden ts ' o wn l ives or future careers or thatknowledge actually facilitates their solution of real-world p roblem s. N ewknowledg e, whether problem-solving skil ls or new m athematical ideas ,should EMP OW ER children to u nderstand o r do something better thanthey co uld prior to its acqu isition. Th at this condition is imp ortant is cle arfrom research on the transferability of instructed thinking skills such asmem ory s t ra tegies (e .g . , Cam pione & Brow n, 1978). Th is resea rch indi-cates easier transfer of the new skills to other problem settings if onesimply explains the benefits of the skill to be learned, that is. that morematerial will be remembered if one learns this strategy.Technologies for mathematical thinking that incorporate these Purposefunctions should make clear the impact of the new knowledge on thestudents ' l ives .TO summ arize: In characterizing the general catego ry of P urpos efunctions for cognit ive technologies . I have focused on the im portan ce of


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4 . COGNITIVE TECHNOLOGIES FOR MATHEMATICS EDUCATION 103

linking th e child-as-agent with the knowledg e t o be acquired instead of onth e alleged motivational value ( e. g. . Le ppe r. 1985; Ma lone, 1981) ofmathe matical educational technolog ies. 1 have done so because it isinap prop riate to think about technologies as artifacts that mechanisticallyinduce motivation. That perspective has led to the extrinsic motivationcharacterizing most current learning-game software: bells and whistlesar e added that serve no function in the stud ent 's mathematical thinking.Fu rth er m or e, these extr insic motivational features are not proagentive inthe sense described earlier. Incorporating the purpose functions I havedescribed into educational technologies could help strengthen intrinsicmotivation. Thi s can be don e by building ed ucational technologies basedin specific types of functional and social environments.

Func t i ona l Env i ronments Tha t P rom ote Mathemat i ca lT h i nk ingThese are environments that help motivate students to think mathe-matically by providing ma thema tics activities whose purposes go beyond"learning math." Whole problems, in which the mathematics to belearned is essential for dealing with the problems, are the focus. Themathematics becomes f i inctional, since the technologies prompt the

development of mathematical thinking as a means of solving a problemrathe r tha n a s an end in itself. Syste m s that provide a functional en viron-ment help students interpret the world mathematically in a problem-solving con tex t. Just a s in real-life p roblem solving, associated curriculaar e not disembodied from purpose (L es h, 1981). In other words. s tudentsse e that the mathem atics used has a point and can join in the lea rningactivities that pursue the point.An exa m ple is provided in a three-st age approach t o algebra edu catio nusing new technologies outlined in the recent Computing and EducationR ep o r t ( F e y , 1984, p. 24). In Stage 1 , students begin with "problemsituations for which algebra is useful." These types of problem situa-tions-such as scien ce problems of projectile motion and nonlinear profito r co st functions--offer "the best possible motivations." In Stage 2, theylear n how t o solve such problems using guess-and-test successive app rox-imations-by ha nd , by graph , and by computer-as well as by m ea ns offorma l compu ter tools such as TK !Solver and muM ath. In Stage 3 . theylearn more formal techniques for solving quadratics, such as factoringformulas and the number and types of possible roots. Through such ase qu en ce , studen ts begin by seeing several applications immediately, notby learning techniques whose application s they will see only later. Similarsequences developed from mathematically complex musical or ar t ist ic


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Although such functional environments for learning mathematics canbe created without computers. computers widen our options. Softwaremay provide innovative, adventurelike problem-solving programs forwhich mathematical thinking is required of the players if they are tosucceed. The five programs in Tom Snyder's (1982) Search Ser ies (Mc-Graw-Hill) , for example, encourage group problem solving. In EnergySearch, students manage an energy factory. collaboratively making inter-dependent decisions to seek out new energy sources. Geography Searchsets students off on a New World search for the Lost City of Gold:climate, stars, suns, water depth, and wind direction, availability ofprovisions, location of pirates, and other considerations must figure intheir navigation plans and progress.In Bank Street's multimedia "Voyage of the Mimi" Project in Scienceand Math Education (Char. Hawkins. Wootten. Sheingold. & Roberts.1983) video, software, and print media weave a narrative tale of youngscientists and their student assistants engaged in whale research. Scienceproblems and uses for mathematics and computers emerge and aretackled cooperatively during the group's adventures. One of the softwareprograms, Rescue Mission (also created by Tom Snyder). simulatesnavigational instruments-such as radar and a direction finder-used onthe Mimi vessel and the realistic problem of how to use navigation to savea whale trapped in a fishing net. To work together effectively during thissoftware game, students need to learn how to plan and keep records ofemerging data, work on speed-time-distance problems, reason geometri-cally, and estimate distances. It is in the context of needing to do thesethings that mathematics comes to serve a functional role.Sunburst Corporation has also published numerous programs thathighlight simulations of real-life events in which students use mathematicsskills as aids to planning and problem solving in real-world situations. Forexample, Survival Math requires mathematical reasoning to solve real-world problems such as shopping for best buys, trip planning, andbuilding construction, and The Whatsit Corporation requires students torun a business producing a product. While problems such as these can besolved on paper, the interact ive, model-building features of the computerprograms can motivate mathematical thinking much more effectively.

Social Environments for Mathematical ThinkingSocial environments that establish an interactive social conte xt fordiscussing, reflecting upon, and collaborating in the mathematical think-ing necessary to solve a problem also motivate mathematical thinking.Studies of mathematical problem solving, for example, by Noddings(1985), Pettito (1984a, 1984b), and Schoenfeld (1985b) indicate how useful


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4. COGNITIVE TECHNOLOG IES FOR MAT HEM ATICS EDUCATION 105

dialogues am ong mathem atics problem solvers can be in learning to thinkmathematically. Small group dialogues prompt disbelief, challenge, andthe need for explicit mathematical argumentation; the group can bringmo re previous experience to bear on the problem than can any individual;and t he need f or an o rderly problem-solving proc ess is highlighted (N od-dings, 1985). Coo pera tive learning re sea rch in ot her disciplines of sch ool-ing ( e. g. , Slav in, 1983; Slavin et a l. , 1985; Stodolsky, 1984) and the n ewfocus in writing composition instruction that emphasizes thinking-aloudactivit ies (Bereiter & Scardamalia, 1986) also focus on social environ-ments . The computer can serve as a fundamental rnediational tool forpromoting dialogue and collaboration on mathematical problem solving.In mathem atical lear ning , as in writing p roce ss activities (Grav e & Stuar t ,1985; M eha n, Moll, & Riel, 1985), social contex ts can ope n up opportuni-ties for the child to develop a distinctive "voice" and to internalize thecritical thinking processes that get played out socially in dialogue.To date, computers have rarely been used to facilitate this functionexplicitly. The record-keeping and tool functions of software could,how eve r, effectively suppo rt collaborative processes in mathe matics, justas they have in multiple text authoring environments (Brown, 1984b).This function is usually exploited only implicitly, as in Logo program-ming, where students often work together to create a graphics program.In doing s o, they argue the com parative merits of strategies for solving themathematics problems that are involved in the programming (Hawkins,Hamolsky , & H eid e, 1983; W ebb, 1984). T h e public nature of the c om pu-ter sc reen a nd the eas e of revision fu rther encou rage collaboration am ongstudents (Hawkins, Sheingold, Gearhart , & Berger, 1982). Self-esteemcan also grow in a collaborative context w hen students view one of theirpeers as expert . There have been some instances in Logo programmingresearch where students with little previous peer support and low self-es teem h ave emerged as "exper ts" (Sheingold , Hawkins , & Char, 1984;Pape r t , Watt , d iSessa, & Weir, 1979).M ath em ati cs is often a social activity in the world . Explicitly recogniz-ing and encouraging this in mathematics education would not only beeducationally beneficial and mo re realistic, but would also make m athe-matics m ore enjoyable-sharable rath er than sufferable. Ma thema ticseducators should provide better tools for collaborating in mathematicsproblem solving and work towards promoting more instructionally rele-vant p ee r dialogue around mathe matical thinking activities.An e xa m ple of a program that d oe s jus t that is part of the Voyage of theMimi Project in Science and Math Education at Bank Street (Char,Haw kins , W ootten , Sheingold , & R ob er ts , 1983). a line of softwa re calledthe Ban k S treet L ab , developed in conjunction with TER C. I t is com-


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106 PEAinvolving Probe, a hardware device that plugs into the microcomputer andcan measure and graph changes in light, heat, temperature, and soundover time. Students work together taking measurements and designingand carrying out experiments. Supplementary teacher materials suggestactivities where students work in teams to apply mathematical thinking inmaking scientific discoveries.Tom Snyder's programs (1982) also allow for small groups of studentsworking cooperatively o r competing against other groups.

Process Functions in Cognitive TechnologiesA second set of categories of functions are those which help studentsunderstand and use the different mental activities involved in mathemati-cal thinking. Although our understanding of the psychology of mathemat-ics problem solving and learning is continually evolving, there are fivedifferent general categories of Process Functions that can be clearlyidentified for cognitive technologies in math education. Each providesimportant cognitive support:

tools for developing conceptual fluencytools for mathematical explorationtools for integrating different mathematical representationstools for learning how to learntools for learning problem-solving methods.

I will briefly define and illustrate each of these categories of functionsas they may appear in mathematics software.1 . Conceptual Fluency Tools . Fluency tools are programs that free upthe component problem-solving processes by helping students becomemore fluent in performing routine mathematical tasks that could belaborious and counterproductive to mathematical thinking. Computertechnologies can promote fluency by allowing individually controlledpractice on routine tasks and thus freeing up students' mental resourcesfor problem-solving efforts.There is ample room for debating what these component skills are insecondary school mathematics (e.g. Fey, 1984; Pollak. 1983) and in highschools (Maurer, 1984a. 1984b; Usiskin. 1980). Many software programsroutinize irrelevant skills such as practice on long division algorithms.And the issue is made more complex by the fact that there are manymental functions, such as the component operations of numerical andsymbolic calculation, that can now be entirely carried out by mathematicssoftware (e.g. , Kunkle & Burch, 1984; Pavelle, Rothstein, & Fitch. 1981;


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4. COGNITIVE TECHNOLOGIES FOR MATHEMATICS EDUCATION 107Wilf, 1982; William s, 1982). We need to fo cu s on determining the skillsand knowledge required to design the inputs and understand the outputsof these m athematical tools.

There are routine mathematical tasks that one should be able to doeasily to m ake progress in mathematical thinking. Fo r example, informa-tion technologies could improve the fluency of the estimation skills thatare at the core of a revised early mathematics curriculum. There aregames involving arithmetic estimation activities at which students be-co m e quickly proficient (e .g . Pet tito, 1984a). T he routinization of certainmathematical skills is equally appropriate at the highest level of mathe-matical achievement. What is at the fringe of mathematical thinking andcreativity toda y is the slog work of tom orrow (Wilder, 1981). Wh at is acreative invention at one point, such as Leibniz 's calculus or Gauss 'sdev elop me nt of complex num bers , is likely to becom e s o routine later thateffective instruction makes it widely accessible.The appropriate roles for such fluency tools is the subject of muchcurrent deb ate (Cole, 1985; Mehan et a l . , 1985; Patterson & Smith, 1985;Resnick, 1985). Schools frequently establish a two-tiered curriculum, inwhich basic com putational skills are presumed to be necessary prerequi-sites for engaging in more complex, higher order thinking and problemsolving. Limited to activities with little motivational significance in thefirst tier of this cu rricul um , many stud ents n eve r engage in the mathemati-cal thinking chara cteris tic of the seco nd tier (Co le, 1985). But recent workin writing (in which s uch a two-tiered app roac h is commo n: Mehan et al. ,1985; Simmons, in press) implies that so-called basic skills can be ac-quired in the context of more complex mathematical thinking activities.When a child's conceptual fluency hampers complex thinking. a func-tional contex t is established and the child realizes the need for practice.Thus, practice is self-motivated. This contrasts with the two-tiered ap-proach, in which the child is trained to some threshold skill level beforebeing let loose t o solve problem s. Drill an d practice software for fluencyin "basic mathe matics" seem s to work better as a fallback rather than as astar tup activity.2. Mathematical Exploration Tools. Ma them atics education has longemphasized discovery learning, particularly in the primary school withthe use of manipulatives such as Dienes blocks, Cuisenaire rods, andpattern blocks. M uch m ore complex concep tions and mathematical rela-tionships, such a s recursion and variables, ca n also be approached at amore intuitive level of understanding without abstract symbolic equa-tions.The computational discovery learning environment provides a richcon text th at helps stu den ts broaden their intuition. Logo programming is


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108 PEA

assumption that one can recognize patterns and make novel discoveriesabout properties of mathematical systems through self-initiated search ina well-implemented domain of mathematical primitives (Abelson & Di-Sessa, 1981; Papert, 1980). Nonetheless. recent findings indicate thatstudents encounter conceptual difficulties-for example. with recursion.procedures, and variables-in Logo and find it hard to ~mders tandhowLogo dictates flow of control for command execution (Hillel& Samurcay.1985; Kurland & Pea, 1985; Kurland, Clement, Mawby, & Pea, in press;Kurland. Pea. Clement. & Mawby. in press: Kuspa & Sleeman. 1985:Mawby, in press; Pea, Soloway, & Spohrer, in press: Perkins. Hancock,Hobbs, Martin, & Simmons. in press; Perkins & Martin, 1986). There isconsequently much debate about the extent and kinds of structure neces-sary for successful discovery envi ronments. The question of how well litthe paths of discovery need to be remains open.

Many software programs now offer structured exploration environ-ments to help beginning students over some of these difficulties. Programssuch as Delta Draw (Spinnaker) and Turtle Steps (Holt. Rinehart &Winston) are recommended as preliminary activities to off-the-shelfLogo. There are in fact several dozen programs that allow students to usea command language to create designs and explore concepts in planegeometry, such as angle and variable, in systematic ways.The Geometric Supposers (Sunburst Corporation: for Triangles. Quad-rilaterals, and Lines; see Kaput, 1985; Schwartz & Yerushalmy, in press)are striking examples of a new kind of discovery environment. Usingthese programs, students make conjectures about different mathematicalobjects in plane geometrical constructions-medians. angles, bisectors.Intended for users from grade 6 and up. i t is designed so that students canexplore the characteristics of triangles and such concepts as bisector andangle. In this way, students can discover theorems on their own. Theprogram is an electronic straight-edge and compass. I t comes with "build-ing" tools for defining and labeling construction parts (like the side of atriangle or an angle) and measurement tools for assessing length of lines.degrees of angles, and areas. Most significant. it will remember a geomet-ric construction the student makes on a specific object (such as an obtusetriangle) as a procedure (as in Logo) and allow the student to "replay" iton new, differently shaped objects (e .g . ,equilateral triangles). The excit-ing feature of the environment is that interesting properties that emerge inthe course of a construction cry out for testing on other kinds of triangles.and students can follow up. Their task is simplified by the labeling andmeasurement tools provided for mathematical objects in the construct ion.and experimentation is encouraged. In fact, several students have discov-ered previously unknown theorems with the discovery tool. Students findthis program an exciting entry into empirical geometry (induction). and i t


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4. COGN ITIVE TECHNOLOGIES FOR MATHEMA TICS EDUCATION 109can be used to complement c lassroom work on proofs (deduction). Kapu t(1985) has highlighted the major representational breakthrough in theSupposers: they allow a particular construction to represent a generaltype of construction rather than just i tself.3 . Representat ional Tools. These tools help students develop thelanguages of mathematical thought by linking different representations ofmathem atical co nce pts , relationships, and pro cesses . Th eir goal is to helpstudents understand the precise relationships between different ways ofrepres enting m athematical problems and the w ay in which changes in on erepresentation entail changes in others. The languages of mathematicalthought, which become apparent in these different representations, in-clude:

Natu ral language description of mathematical relations (e .g ., l inearequa t ions) .Eq uation s com posed of mathematical symbols (e .g . , l inear equa-t ions) .Visual Cartesian coordinate graphs of functions in two and threedimensions .Graphic representat ions of objects (e .g . , in place-value sub tract ion ,the use of "bins" of obj ects repres enti ng different typ es of placeunits) .

Mathematics educators have begun to use cognitive technologies inthis way a s a result of empirical studies dem onstra ting how com petenc y inmathematical problem solving depends partly on one's ability to think interms of different representational systems during the problem-solvingproc ess. E xp er ts can exploit part icular strengths of different rep resen ta-t ions according to the demands of the problem at hand. For example,many relationsh ips that are unclear in textual d escription s, mathem aticalequations, or other tables of data values can become obvious in well-designed g rap hs (Tufte, 1983). On e can often gain insight into mathem ati-cal relations hips, l ike algebraic func tions, by seeing them d epicted graphi-cally rather than as symbolic equations ( K ap ut . 1985, in press).Interactive technologies provide a means of intertwining multiplerepresentat ions (Dick son, 1985) of mathem atical concepts and relation-ships-like gra ph s and equa tions or num bers and pictorial represe ntationsof the objects the numbers re pres ent . Th ese representat ions en hance thesymbolic tools available to the student and the flexibility of their useduring problem solving. They also have the effect of shortening the timerequired for mathematical experience by allowing, for example, manym ore gra ph s to be plotted per unit of time (D ugd ale, 1982), and they make


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possible new kinds of classroom activities involving data collection,display, and analysis (e.g.. Kelman et al., 1983: point and functionplotting; histograms).Manipulable, dynamically linked, and simultaneously displayed repre-sentations from different symbol systems (Kaput. 1985) are likely to be ofvalue for learning translation skills between different representationalsystems, although they are as yet untested in research. For example.students can change the value of a variable in an equation to a new valueand observe the consequences of this change on the shape of the graph.These experiments can be carried out for algebraic equations and graphsin the motivating context of games like Green Globs (in Graphing Equa-tions by Conduit; cf. Dugdale, 1982) and Algebra Arcade (Brooks-Cole).In these games for grades K-6, the player is given Cartesian x-y coordi-nate axes with 13 green globs randomly distributed on the graph. Theplayers have to type in equations, which the computer graphs: when agraphed equation hits a glob, it explodes and the player's score increases.Students become skilled at knowing how changes in the values of equa-tions, like adding constants or changing factors (x to 3x, for example),correspond to changes in the shape of the graph (Kaput, personal com-munication). They discover equation forms for families of graphs, such asellipses, lines, hyperbolas, and parabolas.Other examples of dynamically linked representational tools are pro-grams that give visual meaning to operations on algebraic equations (suchas adding constants to conditionally equivalent operations). Operationson equations are simultaneously presented with coordinate graphs so thatany action on an equation is immediately reflected in the graph shape(Kaput, 1985; Lesh, in press). The student can literally see that doingarithmetic, that is, acting on expressions rather than equations. does notchange graphs.Another example is provided by software programs in which differentrepresentations are exploited in relation to one another for learning place-value subtraction. We can point to Arithmekit (Xerox PARC: Sybalsky,Burton, & Brown, 1984), Summit (Bolt, Beranek & Newrnan: Feurzeig &White, 1984), and Place-Value Place (Interlearn) . In each of these cases,number symbols and pictorial representations of objects (and in the caseof Summit, synthesized voice) are used in tandem to help studentsunderstand how the symbols and the operations on them relate to thecorresponding pictorial representations. Only Place-Value Place is com-mercially available. In this program. a calculator displays the addition andsubtraction process using three different representations of number val-ues: a standard number symbol, a position on a number line., and a set ofproportionally sized objects (apples, bushels of apples. crates of bushels.truckloads of crates). As students add or subtract numbers, all three


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4 . COGNITIVE TECHNOLOGIES FOR MATHEMATICS EDUCATION 111displays change simultaneously to illustrate the operation. Kaput (1985)describes efforts underway at the Harvard Educational Technology Cen-ter to develop dynamically linked representational tools for ratio andpropor t ion reasoning act iv i ties (e .g . , m.p .g . and m.p .h . problems:Schwartz, 1984).Progra ms incorpo rating this function are excellent examples of how therapid interactivity and representational tools the computer provides cre-ate a new kind of learning experience. Stude nts can test out hypotheses,imm ediately s ee their effects, and sha pe their next hypothesis accordinglythrough m any cy cle s, perhaps through many m ore cycles than they wouldwith n onco m puter technologies.Co m put ers ar e also frequently used in displaying graphs and functionsin algebra, transformations in geometry, and descriptive statistics. Theiruse is dynamic and allows student interaction with mathematics in waysthat would not be possible in noncomputer environments. In the recentNC TM Com puter s and Mathemat ic s Repor t (F ey , 1984), the im portanc eof multiple representations in mathematics education is highlighted. Inparticula r, the a uth ors note the rich possibilities for the dynamic study ofvisual co ncep ts, such as sym metry, projection, transformation, vectors,and for deve loping an intuitive sen se of sh ape and relationship to nu mb erand more formal concepts .

4. Tools for learning how to learn. This category refers to softwareprograms th at prom ote ref lective learning by d oing. They star t with thedetails of specific problem-solving experiences and allow students toconso lidate wha t they have learned in episod es of mathematical thinking.Th ey foc us on w hat both D ewey (1933) and Polya (1957) describe as th efinal step in problem solving, reflection that evaluates the work accom-plished and assesses the potential for generalizing methods and results(Brown, 1984b). These programs also make possible, in ways to bedesc ribed, new activities for learning how to le arn .The programs leave traces of the student 's problem-solving steps.Tools based on this function provide a more powerful way of learningfrom e xperie nce , because they help studen ts relive their experience. T heproblem-solving tracks that students leave behind can serve as explicitmaterials for s tudyin g, moni tor ing, and assessing partial solutions to aproblem as they eme rge. They can help s tudents learn to control theirstrategic knowledge and activities during problem-solving episodes(Schoenfeld, 1985a, 1985b).T he crucial featu re of such systems is that stude nts have access to

trace records of their problem solving processes (e.g . , the network ofsteps in geom etry proofs: Geom etry Tutor [Boyle & A nd er so n, 19841; th eseque nce of operations in algebra equation-solving: AlgebraLand [Xero x


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112 PEA

FIG. 4.1thus making them accessible for inspection and ref lect ion. Le t us consid erAlgebraLand, in particular i ts approac h to l inear equations.Search is not a central concept in algebra instruction today, but acentral insight of cognitive s cien ce is that learning problem-solving skillsin mathematics requires well-developed search procedures, that is ,knowledge about when to se lec t what subgoals and in what s eque nce . Inmost classroom instruction of algebra equation solving, the teacherselects the opera tor to be applied to an equation (e.g. , add-to-both-sides),and the student carr ies out the ar i thmetic. The pedagogical f law in thismethod is that the student does not learn when to select the varioussubgoals (Simon, 1980), but only how to execute them (e .g . , to do theari thmetic once th e divide operation has been selected) .Originally created several yea rs ago at Xerox PARC by J. S . B r o w n , K .Roach , and K . Va nLeh n, and c urrently being revised by C . Fos s for workwith middle school stud ents, AlgebraLand is an experimental sys tem forhelping students learn algebra by doing problems (Brown, 1984b). Figure4.1 i l lustrates som e of the features of the system to be discuss ed.The task in Fig. 4.1 for the student is to solve the equation for N(shown in the Solve for N window on the figure's right side). Algebraic


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4 . COGNITIVE TECHNOLOGIES FOR MAT HEMA TICS EDUCATION 113operators listed in the Basic Operations window on the bottom right-handside (such as Combine-Terms, Add-to-Both-Sides, Distribute) can beselected to apply to the whole equation or to one of i ts subexpressions.After selecting the operation and where to apply it , the student canexe cute i t . This creates a second algebraic expression.T h e Record window, upper right, records the steps taken by thestud ent to wa rds a solution. Its left-hand column lists all the interm ediateexpress ions; i ts right-hand column show s each operat ion used to t rans-form the recorded expression. The Search Space window represents allthe stud ent 's step s as a search tree; i t displays solution paths depicting allthe student 's moves, including backtracking. In the solution attemptdepicted here, the student took three different approach es to solving theequ ation . These app roaches are reflected in the three branches that issuefrom the original equation. Each intermediate expression that resultedfrom applying the do-arithmetic op erato r appears in boldface for clarity.Algeb raLand perform s all tactical, algebraic operations and arithmeticalcalcu lations , effectively eliminating e rro rs in arithmetic or in the applica-tion of operators. The student, whose work is l imited to selecting theop erato r and the scope of i ts operation, is free for the real mental work ofsear ch and op erato r evaluation.Operators are also provided for exploring solution paths. There is anU n d o op erato r that returns the equ ation to the sta te immediately preced-ing the current on e and a G o to operator, which is not on the menu, thatretu rns th e equat ion to any prior state . T he student can also back u p asolution path by applying the inverse of a forw ard op erator (e.g ., selectingdivid e just after applying multiply).Beca use the windows show every o perator used and every state intowhich the equation was transformed, students have valuable opportuni-ties to learn from their tracks and to play with possibilities. They canexp lore the search paths of their solution sp ac e, examining bran ch pointswh ere, on o ne s tem , an operat ion w as used that led dow n an unsuccessfulpath , and o n another stem , the operation chosen led down a path tow ardsth e solution. The n they can decide which features of the equation at thebran ch point could have led to the best ch oice . Transforming that decisioninto an hypothesis, they can test out that hypothesis in future equationsolving. These learning activities are not possible with traditionalme thod s for learning to solve equations. I nde ed. the cognitive technologyoffered by AlgebraLand affords opportunities for new and different typesof learning through problem solving than were available in static, non-com pute r-ba sed symbolic technologies.In summary, the computer environment AlgebraLand emphasizes apro ced ure diametrically opposed to th e traditional instructional m etho d.


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114 PEA

comp uter carries out the mechanical procedures tha t t ransform the equ a-t ion. Stud ents do face the problem of searching for and discovering a pathof operat ions leading from th e problem state to the solut ion. Th e graphicrepresentat ion p rovided by the se arch map al lows studen ts to reflect onthe means they used to solve the problem, and problem solving is nolonger an ephemera l process .AlgebraLand provides a very rich learning and research env ironm ent .But we sti l l know far too li t t le about the potential effectiveness of thesetypes of educat ional instruments, especial ly since they introduce newlearning problems: H ow d o students learn to "read" and use their t races?How do such learning-how-to-learn ski l ls develop, and what anci l laryfeatures of software programs such as games wil l help students under-stand how to make effect ive searches in the space of their " thinkingtracks" ?

The technology is seldom used in this way, and few developers areworking along these l ines. Nonetheless, when i t is used this way, i tprovides a qual i ta t ively new kind of tool for s tudents to learn moreeffectively from their problem-solving experience. It can also provide arich d ata source for analyzing studen t understanding of problem-solvingprocesses and m ethods .Providing explicit traces of a student 's problem solving activitiesduring an episode reveals possible entry points for tutorials in prob lem-solving skills and for intell igent coaching. In a variant on AlgebraLand,studen ts could be pro mpted to check the operat ion (such as factoring) thatthey have been using during equation-solving activities, for example, ifthey have used that operator in a way that has led to a more complexequat ion rather than a simpler one.Along the same l ines, the information avai lable through such t racesprovides the data for model ing what a student unde rstand s; these mo delscould be based on student interact ions with a computer in a specificmathem atics problem-solving do main . As experience with cu rrent proto-types such as the ari thmetic game West indicates (Burton & B r o w n ,1979), such systems al low for coaching dialogues that ar e sensi t ive t o astude nt 's d evelopmental level and qui te subt le in their coaching m ethod s.Unlike intell igent tutoring system s such a s the Geom etry T utor (Boyle &An derso n, 1984) and th e L isp Tutor (Anderson & Reiser, 1985), theydon ' t co rrec t the s tudent a f te r every subopt imal m ove.Th ere are two problems with using this type of tool for learning how tolear n. First , it i s not yet app arent h ow broadly modeling and co aching canbe appl ied to student misconcept ions. Second, al l the programs thatexemplify this category of funct ion run on minicomputers and have notbeen developed com mercial ly for schoo ls.

5 . Tools for learning problem-solving methods. This category of tools


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4. COGNITIVE TECHNOLOGIES FOR MATHEMATICS EDUCATION 115encourages reasoning strategies for mathematical problem solving. Re-cent work on the development of mathematical thinking highlights theimportance of reasoning s trategies . People who have expert ise in ap-proac hing problem-solving activities in m athem atics utilize, in addition t oknowledge of mathematical facts and algorithmic procedures, strategiesto guide their work on difficult problems that they cannot immediatelysolve. Such heuristics, well known from the work of P6lya (1957) andmodern studies of mathematical problem solving (e.g. , Silver, 1985),include drawing diagrams, annotating these diagrams, and exploitingrelated prob lems (Scho enfeld , 1985a). Sega l, Chip man , and G laser (1985)review related instructional programs t o teach thinking skills.Few of the existing examples of educational technologies aim to helpstudents develop problem-solving heuristics of this kind. However, oneprom inent exam ple should be briefly men tioned . W umpus (Yob, 1975) is afan tasy co m pu ter game in which the player m ust hunt and slay the viciousW um pus to avoid d eadly pitfalls. Goldstein & colleagues (e.g. , Go ldstein,1979) cre ated artificial intelligence program s t o help studen ts acquire thereason ing strategies in logic, probability, decision analysis, and g eom etrythat are needed for skil lful Wumpus hunting. The Wumpus coaches areminicomputer programs that have not been evaluated in educationalsettings, and the issue of transfer of the skills acquired by students tosett ings othe r than this game has not been s tudied.Sunburst Corporation (1985) has developed a problem-solving matrixtha t is index ed to th e software it sells for sch ools, so that purported ly th eteache r can kno w what problem-solving skil ls and s trategies (e.g. , binarysearch) the s tudent wil l learn by using the program. However, thesestrategies are not explicitly taught by the software. Furthermore, theschem e erroneously presupp oses that o ne ca n identify a priori the prob-lem-solving skills that all students will use at all times in working with aspecific program. But the problem-solving processes or comp onent think-ing skills a student will use in working with a software program changewith cognit ive development. The skil ls that are used also vary acrossindividuals bec aus e of cognitive style and oth er variables.These reservations notwithstanding, if sufficient attention were de-voted to th e effort, cu rrent tools in artificial intelligence could b e used t ocre ate learning environm ents in which t he application of a mathem aticsproblem-solving heuristic, or set of heuristics, would be exemplified formany problems. Students could explore these applications and then beoffered transfer problems that assess wh ether they have induced how t heheuristic works sufficiently to carry on independently (Wittgenstein,1956). If they hav e not, the system could offer various levels of coaching,o r different layers of hints, that would lead u p to a modeling exam ple for


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116 PEASYNTHESIS OF FUNCTIONS: THE NEED FORINTERDISCIPLINARY RESEARCH ON COGNITIVETECHNOLOGIES FOR MATHEMATICS EDUCATION

I would like to close this discussion of functions for cognitive technolo-gies in mathematics education by making a plea for interdisciplinary.classroom-based research, involving mathematics educators, cognitivescientists, software makers, and mathematicians. We know very littleabout the educational impact. actual or possible, of different technologyapplications in mathematics education. We cannot predict how the role ofthe human teacher may change as the use of such technologies increasesor in what new ways teachers need to be trained to help students use thetechnologies to gain control of their own mathematical thinking andlearning. As these new technologies reduce the focus on teaching routinecomputation algorithms. teachers' jobs will become much more intellec-tually challenging; their activities, more like those of mathematicians.Teacher-training institutions will need to change in as yet unspecifiedways. We face a plethora of unknowns. Yet, mathematics educators willhave to understand these unknowns in order to use cognitive technologieseffectively. Empirical testing of exploratory new instructional curriculathat embody the various functions I have described is necessary, forwithout it, we will have little idea whether the functions actually promotemathematical thinking.

CodaThese are very exciting times for learning mathematics and for using newtechnologies to shape the futures of mathematical thinking. Mathematicswas a dreary subject for many of us in the past. but mathematical thinkingis now often learned through problem-solving activities that bury themechanical aspects of mathematics in interesting ideas. And the puritani-cal attitude that the mind is a muscle to be exercised through mechanicalrepetition is giving way to a richer view of the creative, exploring mind,which can be nurtured and guided to discover and learn through meaning-ful problem-solving activities. By infusing life into the learning tools formathematics, by integrating supports for the personal side of mathemati-cal thinking with support s for knowledge, we can perhaps help each childrealize how the powerful abstractions of mathematics confer personalpower. In such a utopia, learning mathematics is but one more way oflearning how to think and how to define one's personal voice in the world.It is hoped that the transcendent fi~nctionsproposed for cognitivetechnologies in mathematical education will be useful for crafting newgenerations of cognitively support ive and personally meaningful learning


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4. COGNITIVE TEC HNOLOGIES FOR MATHEMATICS EDUCATION 117and teaching tools for mathematical thinking. They may also providepractitioners with new ways of evaluating these educational tools. Otherresearchers are bound to offer a different set of transcendent functionsthan those I have proposed; debate should clarify the issues underdiscussion and contribute to the fundamental goal of using cognitivetechnologies in mathem atics education to pre pare s tudents effectively forthe comp lexities of mathem atical thinking.Although we ca nno t predict w hat sh ape the cognitive technologies formathematics education will take, we can certainly monitor whether theyare congruent with emerging concepts of mathematical thinking and thenature of the learner, and assure that the science and tools of educationare never fa r apar t .

ACKNOWLEDGMENTSI have had opportunity to discuss some of these issues with WallyFeurzeig, Ron M awby , Andrea Pet t i to , Al Schoenfeld, Judah Sch wartz ,Kurt VanLehn, Ron Wenger, Jim Wilson, and members of the Sloanworkgroup on Cognit ive Science and Math Education. Jim Kaput , inparticular, has greatly influenced this account of how drastically informa-tion techn ologies a re changing and will con tinue to change our curriculumtopics and educational goals in mathematics. None of those mentionednecessarily s ubscribes to the taxonomy and arguments presented here .

REFERENCESAbelson, H . , & diSessa, A. A. (1981). Turtle geometry: The computer as medium for

exploring rnarhematics. Cambridge, MA: MIT Press.Anderson, J . R ., & Reiser, B. (1985). The LISP tutor. Byte, 10, 159-175.Bereiter, C., & Scardam alia, M. (1986). The psychology of written composition. Hillsdale,NJ: Lawrence Erlbaum Associates.Boyle, C. F., & A n d e r s o n , J . R. (1984). Acquisition and automated instruction ofgeometryproof skills. Paper presented at the meeting of the American Educational ResearchAssocia t ion, New Orleans , LA.Brown, J . S . (1984a). Idea ampl$ers: New kinds of electronic learning environments. PaloAlto, C A: Xerox Palo Alto Research Center, Intell igent Systems Laboratory.Brown, J . S. (1984b). Process versus product: A perspective on tools for communal andinformal electronic learning. In S . Newrnan & E. Poor (Eds . ) , Report from the learninglab: Education in the electronic age (pp. 41-57). Ne w York: Educational Bro adca stingCorpora t ion, WN ET.Brune r, J . S . (1966). On cognitive growth 11. In J . S . Bru ner, R . R. Olver, & P. M . Greenfield( E d s . ) , Studies in cognitive growth (pp . 30-67). Ne w Y ork: Wiley.


30/34

118 PEA

Burton, R. R. , & Brown, J . S . (1979). An investigation of c om pute r coaching for informallearning activities. Internatiotlul Journal of Man-Machirte Studies. 11, 5-24.Cajori, F. (1929a). A history of mathematical notations. Vol. I: Notations in elernerltarymathematics. La Sa l le , IL : Open Cour t .Cajori, F. (1929b). A history of muthetnatical notations, Vol. 2: Notations mainly in highermathernatics. La Sa l le , IL : Open Cour t .Campione, J . C . . & Brown, A. L . (1978). Tram ng general metacognitive skills in retardedchi ldren. In M. Grunberg, P. E. Morr is , & R . N . S y k es ( E d s . ) , Practical aspects ofmemory (pp. 410-417). New York: Academic Press.Cassirer, E . (1910). Substance and function and Einstein's theory of relativity. Mineola. NY:Dover .Cassirer. E . (1944). An essay on man: An introduction to a philosophy of human culture.New Ha ven , C T Yale Universi ty Press .Cassirer, E. (1957). The philosophy of symbolic forms. Vol. 3 . The phenomenology ofknowledge. New Hav en, CT: Yale University Press.Champagne, A. B. , & Chaiklin. S . (1985). Interdisciplinary research in science learning:Great prospects and dim future. Paper presented at the rneetlng of the AmericanEducational Research Association, Chicago. IL.Char , C. , Hawkins , J . , Woot ten, J . , Sheingold, K . , & Rober ts , T. (1983). "Voyage of theMimi": Classroom case studies of software, video, andprlnt materials. New Y ork: BankStreet College of Edu cation , Center for Ch ~ld ren nd Technology.Cole , M. (1985). Non-cognitive factors in education: Subcommittee report. Washington,DC: National Research Council Commission on Behavioral and Social Scien ces andEducation.Cole. M., & Griffin, P. (1980). Cultural amplifiers reconsidered. In D. R. Olson (Ed .) . Thesocial foundations of language and thought: Essays in honor of Jerome S . Bruner (pp.343-364). New York: No rton .Conference Board of the M athematical Sciences (1983).The mathematical sciences curricu-lum K-12: What is still fundamental and what is not . Washington. DC: CBMS.

Dewey. J . (1933). How we think. Chicago: Henry Regnery.Dewey , J . (1938). LOGIC: The theory of inqurry. New York: Henry Holt & C o .Dickson, P. (1985). Thought-provoking software: Juxtaposing symbo l system s. EducationalResearcher, 14, 30-38.Dugdale, S. (1982. March). G reen Glob s: A microcom puter application for the graphing ofequations. Mathetnatics Teacher. 208-214.Dweck , C . S . , & Elliot, E . S . (1983). Achievement mo t ivat~ on. n P. H. Mussen ( Ed .) ,Handbook of child psychology , Vol. IV (pp . 643-691). New York: Wiley.E lgar ten , G . H . , Posament ie r , A . S . , & Moresh. S. E . (1983). Using computers inmathematics. Menlo Park , CA : Addison-WesleyFeldman . D. H . (1980). Beyond universals in cognitive development. Norwood. NJ : Ablex .Feurzeig, W .,& White, B. (1984). An articulate instruclional sy stem for teaching arithmeticproblems. Cambridge, MA : Bol t , Beranek. & Newman.F ey , J . T. (1984). (Ed.). Computing and mathematics: The impact on secondary schoolcurricula. College Park, MD: National Council of Teachers of Mathematics. TheUniversity of Maryland.Goldstein. I . (1979). The genetic epistemology of rule systems. International Journal ofMan-Machine Stud ies, 11, 51-57.Goodman, N . (1976). Latlguages of art. (Rev. ed .) Amhers t . MA: Universi ty of Massachu-setts Press.

Goody, J . (1977). The domestication of the savage mind. New York: Cambridge Un ~ve rs i tyPress.Gould. S. J . (1980). Shade of L am arc k. In The panda's thumb. More reflections in natural'history (pp. 76-84). New Y ork: Norton .


31/34

4. COGNITIVE TECHNO LOG IES FOR MATHEMATICS EDUCATION 119Grabiner. J . V. (1974). Is mathematical truth time-dependent? Amer i can Ma the tna t i ca l

M o n t h l y , 81. 354-365.G r a v e s , D . H . , & Stua r t , V. (1985). Wri te rom the s ta r t : Tapp ing your ch i ld 's na tura l w r i t ingub i l i t y . New York: Dutton.Greenfield, P. (1972). Oral and written language: The consequences for cognitive develop-ment in Africa, the United States, and England. Language and Speech, 15, 169-178.Greenfield, P. M ., & Bruner, J . S. (1969). Culture and cognitive growth (revised version). InD. A . Gos l in (Ed . ) , Ha ndb ook of soc ial iza t ion theory and research (pp. 633-657).Chicago: Rand McNally.Ha rter , S . (1985). Competence as a dimension of self-evaluation: Toward a comprehensivemodel of self-worth. In R . Leahy (Ed . ) , The develop me nt of the se lf. New York:Academic Press.Hawkins , J . , Hamolsky. M. , & He ide, P . (1983). Paired problem solving in a comp utercontext (Tech. Rep . N o . 33), New York: Bank Street College, Center for Children andTechnology.Hawkins , J . , Sheingold, K . . Gearhart , M . , & Berger, C . (1982). Microcomputers in sc hools:Impact on the social life of elementary classrooms. Jou rna l of A ppl ied Deve lopme ntalPsycho logy, 3 , 361-373.Hillel, J . , & Sam urcay, R. (1985). An alys is of a Lo go environm ent for learning the conceptof procedures w ith varia ble. (Technical R epo rt). Montreal: Concordia University. Math-ematics Department

Kapu t . J . J . (1985). In format ion technology an d muthemur ics: Opening new represenra-t iona l w indows. Paper presented a t Na t~on alConference on Using Computers to TeachComplex T hinking, National Academ y of Scienc es, Washington, DC.Kaput . J . J . (in press). Towards a theory of symbol use in mathem atics. In C . Ja n v~ er E d . ) .Problem s of representat ion and t ranslat ion in tnathem ot ics. Hillsdale, NJ: LawrenceEr lbaum Assoc~a te s .Kelman. P . . Bardige . A. . Choate , J . , Hanify , G . . Richards . J . , Roberts . N. . Wal ters . J . . &Tornrose . M. K . (1983). Computers in teach ing mathemat ics . Reading, MA : Addison-Wesley.Kesse l , F. S . , & Siegel, A. W. ( Ed s.) . (1983). The chi ld and other cul tura l inve nt ions. NewYork: Praeger.Ke ssen . W. (1979). The A merican child and ot her cultura l inventions. Amer icctn Psyc holo-g i s t , 34, 815-820.Kline, M. (1972). Math ema t ica l thought f rom unc ien t to modern t imes. Oxford: OxfordUniversity Press.

Kunkle . D . , & Burch. C. I . Jr. (1984). Sym bolic com put er algebra: The classroom com putertakes a quantum leap. Mathemat ics Teacher , 209-214.Kurland. D . M . , C le m e n t, C . , M a w by . R . , & Pea. R. D . (In press). Mapping the cognitivede

cognitive tech maths edu

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