Computer Tools for Interactive Mathematical Activity in the Elementary School

JOHN OLIVE

COMPUTER TOOLS FOR INTERACTIVE MATHEMATICALACTIVITY IN THE ELEMENTARY SCHOOL

ABSTRACT. The computer tools for interactive mathematical activity (TIMA) weredesigned to provide children a medium in which they could enact their mathematical opera-tions of unitizing, uniting, fragmenting, segmenting, partitioning, replicating, iterating andmeasuring. As such, they are very different from the drill and practice or tutorial softwarethat are prevalent in many elementary schools. The TIMA were developed in the contextof a constructivist teaching experiment focused on children’s construction of fractions.They were used to promote cognitive play that could be transformed into independentmathematical activity. Teaching interventions were often critical in bringing forth mathe-matical activity. Students’ interactions were also important provocations for mathematicalreasoning with the TIMA. The TIMA do not, by themselves define a microworld. Rather itis the children’s activity and their interpretations of the results of that activity, while inter-acting with others, that bring forth a microworld of mathematical operations. Designersof computational environments for children need to take into account the contributionschildren need to make in order to build their own mathematical structures. For teachers tomake effective use of software such as the TIMA they need to understand (and share) theviews of learning that shaped the development of the software.

While much of the commercial software available for elementary schoolmathematics appears to be designed to support the ‘traditional’ arithmeticcurriculum through tutorial sequences followed by practice exercises,the mathematics education community needs to focus on ways in whichcomputer-based environments can enhance children’s own construction ofmathematics through interaction with other children and their teachers.Hoyles (1998) has referred to this difference of use of computers inschool mathematics teaching and learning as a “major fault line” and alsoadvocates for “computational applications which point towards new, morelearnable, more widely accessible mathematics; towards a redefinition ofwhat school mathematics might become and who might be involved in it”.Balacheff (1998), in further describing such desirable computer applica-tions notes that “The key feature of these environments, which share thecommon characteristics of being specific microworlds, is that they do notdo the mathematics instead of the users but that they allow them to expresstheir own mathematical ideas”.

International Journal of Computers for Mathematical Learning5: 241–262, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

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Over the past decade a research team in the Department of Mathe-matics Education at The University of Georgia, headed by Dr. LeslieSteffe and myself, has been attempting to define “what school mathematicsmight become” for young children when they are given the opportunityto interact with other children and a teacher/researcher using computertools “that do not do the mathematics instead of the users but that allowthe users to express their own mathematical ideas”. These computer toolswere designed and developed as part of a research project on Children’sConstruction of the Rational Numbers of Arithmetic (Steffe and Olive,1990), supported in part by the National Science Foundation. These Toolsfor Interactive Mathematical Activity (TIMA) provide young children(grades K–8) with possibilities for enacting their mathematical operationswith whole numbers and fractions. The software tools provide childrenwith on-screen manipulatives analogous to counters or beads (regulargeometrical shapes that we call ‘toys’), Sticks (line segments) and FractionBars (rectangular regions), together with possible actions that the childrencan perform on these manipulatives. These possible actions potentiallyinvolve the fundamental operations involved in the development of numer-ical schemes: uniting, unitizing, fragmenting, segmenting, partitioning,replicating, iterating and measuring (Piaget and Szeminska, 1965; vonGlasersfeld, 1981; Steffe, 1994). These are the basic operations involvedin developing numerical operations with whole numbers and fractions:addition, subtraction, multiplication and division of whole numbers andfractions, as well as the development of place-value concepts, families ofequivalent fractions and simplification of fractions.

In this paper I shall briefly describe each of the TIMA environments,illustrating how they evolved out of our interactions with children, ouremerging knowledge of children’s operations with fractional quantitiesand our knowledge of children’s operations with whole numbers. I shallillustrate how children have used these tools to build their own mathe-matical constructs within our teaching experiment. I shall also reportbriefly on how different teachers have approached the use of such toolswith children in their classrooms. Both scenarios provide evidence of theefficacy of such computer-based learning environments when used in waysthat are compatible with the constructivist philosophy that shaped theirdevelopment.

TOOLS FOR INTERACTIVE MATHEMATICAL ACTIVITY 243

Figure 1. Screen from TIMA: Toys.

BRIEF DESCRIPTIONS OF THE COMPUTER MICROWORLDS

TIMA: Toys

TheToysmicroworld is an environment in which instances of manipulableshapes (called toys) can be created simply by clicking the computer mouse.Five different shapes (triangle, square, pentagon, hexagon and heptagon)are available. The creation of an instance of a shape by the motoric actionof a mouse click provides a link for the child between action (clicking themouse) and the production of pluralities. The production of many toys, insnake-like configurations was an enjoyable entree into mathematical playon the part of the children.

In designing the possible actions for theToysmicroworld we wanted toengage children inunits-coordinationsthat are the basis of multiplicativeoperations (Steffe, 1992). The ‘toys’ can be joined together in astring(like a string of beads) by clicking on them in succession. These stringsof toys can then be moved as a whole (a composite unit), copied (to makemultiples of composite units through iteration), joined together to makelonger strings, cut apart to make shorter strings, and combined into a newtwo-dimensional composite unit called achain of strings of toys (the hori-zontal strings are joined together vertically to form the two-dimensionalchain). In the latest version ofToys the chains can also be combined intoa three-dimensional unit called astack.

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Several operations are available for working with strings and chains.Strings can beRepeatedto create chains, and chains can be repeated intostacks. Our intention was that the result of a repetition might be construedby the children as a multiplicative structure (e.g. a chain of 4 strings with 5toys in each string produces 20 toys in the chain). Numerical informationon the number of toys, strings and chains can be obtained via menu selec-tions. This numerical information can be used to generate mathematicalrelations among the different unit types involved in the structures createdvia strings and chains. Toys can be added to or removed from the endof a string or each string in a chain using theOne More or One Lessbuttons. These buttons can also be used to add to or remove strings from achain. The quantitative comparisons of ‘one more’ and ‘one less’ could beinvestigated dynamically using these action buttons.

Any object or group of objects can be covered so as to hide themfrom view. TheCOVER action was included in each of the computerenvironments in order to encourage children to reprocess their percep-tual collections to form figurative pluralities; that is, to re-present in theirminds what is hidden under the cover. This reprocessing is critical to theconstruction of numerical concepts (Piaget and Szeminska, 1965; Steffe,von Glasersfeld, Richards and Cobb, 1983).

TIMA: Sticks

TheSticks microworld takes the user into the realm of continuous linearquantities. Our goal in designing Sticks was to link the child’s intuitiveconcept of ‘length’ with their emerging concept of composite units in thecreation of what we have called a “Connected Number Sequence” (Steffeand Wiegel, 1994). Horizontal sticks (line segments) of arbitrary lengthcan be created (after selecting theDRAW button) simply by dragging themouse cursor across the screen (see Figure 2). The extent of the drag-ging motion determines the length of the stick. This link between motionand resulting length is an important aspect of this environment as thechild, through the coordination of the motoric action with the visual resultconstructs a sensory experience of the ‘length’ of a stick. This sensoryexperience of length is missing when the child works with sets of physicalsticks that are pre-made.

Once created, sticks can be moved around the screen, copied (usingCOPY), marked arbitrarily by clicking at a position on the stick withthe mouse cursor (after selecting theMARKS button), or partitioned intoequal parts using a numerical counter (after selecting thePARTS button).The different parts can be filled with different colors (using theCOLORandFILL buttons). Parts can be ‘pulled out’ of a marked or partitioned


Figure 2. Screen from TIMA: Sticks.

stick without destroying the original stick usingPULLPARTS . These‘pulled out’ parts then become new sticks, thus allowing comparisons ofpart to whole and whole to parts without destroying the whole. A markedor partitioned stick can be broken up into its constituent parts (usingBREAK ). Sticks can be joined together (usingJOIN ) to form longer sticksconsisting of parts representing the joined sticks.

Any stick can be designated as a ‘ruler’ for measuring purposes bycopying it into theRuler portion of the screen. The measure of othersticks relative to the designated unit stick can then be obtained usingthe MEASURE button. The measure is given in the Number box as aratio-number. Notation such as 7/5 can thus be generated by measuring astick created by repeating a stick 7 times that was 1/5 of the stick in theruler. A fraction labeler is also available for labeling any stick or part ofa stick with a fraction numeral. It is important to note that sticks are notlabeled automatically. The child has to establish the numerical symbol thatis meaningful for the child. Covers are also available in this environment.

TIMA: Bars1

The manipulable objects in TIMA: Bars are rectangular regions that thechild can make simply by clicking and dragging the computer mouse. As inSticks, the coordination of the motoric action of dragging the mouse with

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Figure 3. Screen from TIMA: Bars (finding 1/4 of 2/5).

the resulting visual display helps the child establish the size of the bar as asensory experience. The bar created in this way can be moved around thescreen, copied, marked both horizontally and vertically by line segments,partitioned both horizontally and vertically into equal sized parts (theorientation and number of the parts determined by the user). The piecescreated by theMARKS operation, or the parts created by thePARTSoperation can be filled with different colors and unfilled. The subdividedbar can also be broken apart into its sub-components (pieces or parts).These sub-components are then new bars that can be further subdivided.

A disembedding operation calledPULLOUT (similar to PULL-PARTS in Sticks) was added to this microworld about half way through thefirst year of the teaching experiment. ThePULLOUT operation enablesthe user to copy any connected set of parts of a bar as a new bar objectconsisting of those parts only. This proved to be a very powerful opera-tion for the children in our experiment, as it provided a means of makingcomparisons between parts of a whole and the whole while not destroyingthe whole. The child could literally take the part out of the whole whilestill leaving it in the whole.

Bars which have at least one dimension the same length (height orwidth) can be joined together to form a new bar. ThisJOIN operationalso proved to be very useful for the children as it provided a means ofcreating a referent whole from a part of the whole. Any bar can be desig-


nated as theunit bar so that the measures of other bars (as ratio numbers)can be obtained relative to the designated unit bar. Numerical informationconcerning the number of bars, number of parts in a bar, or number ofbars or parts in a designated region of the screen can be obtained via menuselection. This numerical information was used to pose problems involvingmultiplication and division of whole numbers as well as connecting theseoperations with fractional results.

DESIGN AND USE OF THE MICROWORLDS IN THECONTEXT OF THE TEACHING EXPERIMENT

Our teaching experiment with 6 pairs of children spanned three years.We started working with the children in their third grade at school andcontinued through the end of their fifth grade year. We worked with apair of children at a time, outside of the classroom for approximately45 minutes each week for approximately 20 weeks each year. The sixpairs of children were selected to provide us with a range of numericaldevelopment. All teaching episodes were videotaped using two cameras;one focused on the computer screen and one on the children and theteacher/researcher.

Our goal in using the TIMA tools in the teaching experiment was toprovide the children with dynamic learning environments in which theywere the primary actors. At the same time, these computer-based environ-ments provided us (the teacher/researchers) with a medium in which thechildren can enact their mathematics in ways that we can interact withthis mathematics, while trying to maintain the spontaneity of children’sactions.

Initial designs of the computer tools (Biddlecomb, 1994) were basedon our understanding of children’s multiplying and dividing schemes(Steffe, 1992), and on information from research on children’s part-wholeoperations, pre-fraction and fraction schemes (Hunting, 1983; Kerslake,1986; Kieren, 1988; Nik Pa, 1987; Ning, 1992; Saenz-Ludlow, 1994).These environments embody (for us, the designers) children’s concep-tual operations. The possible actions programmed into the computertools provided the children with ways to enact many of their conceptualoperations. In particular, operations of unitizing, uniting, iterating, split-ting, segmenting, partitioning, and disembedding with both discrete andcontinuous quantities were realizable through the possible actions with thecomputer tools.

As we interacted with the children in these early environments, ourunderstanding of the children’s conceptual operations was illuminated by

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the children’s actions in the microworlds that they created. In many situ-ations we have referred to the computer environments as ‘microworlds’ asif these microworlds existed independently of any action of the childrenusing them. From the children’s frame of reference, a ‘microworld’ isestablished by them when actually working in the TIMA environments. Inthis we agree with Kieren (1994) who stated that “A microworld is broughtforth by the child, based on her/his structures, occasioned by both thecomputer environment and the interactions with adults and other childrenin it” (p. 134).

During the course of the teaching experiment the possible actions (ofthe microworlds) were modified to better fit how the children appeared tooperate, and new actions and features were added to broaden the scope ofthe children’s activity and to provide supports and constraints that mightengender modifications in their ways and means of operating. In particular,we made each environment configurable by the teacher (or students) sothat different microworlds could be created for (and by) different childrenby making certain actions and information available and other actions andinformation unavailable. This cycle of “design-interaction-understanding-improved design” continues as we further develop our models of children’soperations that generate the rational numbers of arithmetic.

From Cognitive Play to Independent Mathematical Activity

Papert (1980) portrayed a microworld as a self-contained world in whichchildren “learn to transfer habits of exploration from their personal lives tothe formal domain of scientific construction” (p. 177). It was our intentionto create computer tools that would provide children with opportunitiesfor cognitive play and subsequent mathematical activity that can follow onfrom cognitive play under the careful guidance of a teacher. In their paperon ‘Cognitive Play’, Steffe and Wiegel (1994) reported on the sponta-neity of children’s actions within the microworlds. They described howthe children used the possible actions of the microworlds for functionalpleasure, and how those play activities were transformed into pleasur-able mathematical activity. The dynamic nature of the computer toolswas a key factor in promoting cognitive play activity. For example, thedynamic nature of the Toys microworld encouraged the production ofsensory pluralities through the coordination and repetition of the motoricact of clicking the mouse and the appearance of a ‘toy’ on the screen. Thisproduction can be regarded as a fundamental operation of intelligence onwhich the construction of numerical concepts, composite units, numbersequences, and more general quantitative reasoning is based (Steffe, 1991).Thus the children’s play activities in Toys became the basis for enactments


of these abstracted structures. Steffe and Wiegel (1994) illustrated how theensuing mathematical activities were shaped by the possible assimilationsand accommodations of the children’s existing counting schemes and theinteractions among the children and the teacher. For instance, the childrenspontaneously created designs (using the repeated clicking action) such asfaces, Jack-o-Lanterns or elaborate names. The simple question of “Howmany toys do you think are in your design?” turned this play activity intoa mathematical activity that was enjoyable for the children. They wantedto see who could make the best estimates! I call this important componentof these teaching episodes ‘Cycles of Cognitive Activity’. Children canprogress from cognitive play to teacher directed mathematical activity, andthen to independent mathematical activity, which may become mathema-tical play, that is, self-initiated independent mathematical activity with aplayful orientation. The cycle may then continue on a more abstract level,thus creating a learning spiral. Steffe and Wiegel (1994) pointed out thatsocial interactions between the students of each pair and among the teacherand students were a vital component of all activities in these cycles.

One activity that proved successful in bringing about this progression tomathematical play was a game called ‘Baking and Selling’. In this activity,the shapes in TIMA: Toys represented cookies that were to be baked by thebaker and sold by the seller. The children decided how many cookies wouldbe baked at a time by placing that many shapes under a cover (the ‘cookietray’) in a designated part of the playground called the ‘oven’. The bakerthen baked one, two or three trays of cookies by copying the cover (alongwith the shapes under it) into the region of the playground designated asthe ‘Store’. The seller then decided how many trays of cookies would besold and moved that many covers (along with the shapes under them) overto the ‘sold’ portion of the playground (or erased the covers being sold). Ateach stage of the game, both players were to calculate how many cookieswere left in the store. No cookies were visible – only the covers. Theycould check their calculations by ‘peeking’ under the covers (an actionthat was added to the Toys environment specifically for this activity) orby using a menu item to find out ‘Number of Toys in a Rectangle’. Thefollowing protocol is an example of how the interactions generated throughthis activity between two girls during the first year of the project (thirdgrade), led to an abstraction of the mathematical structure of the game.

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Figure 4. Baking and selling trays of cookies in TIMA: Toys.

PROTOCOL I

The TIMA: Toys screen was set up with a cover in the ‘oven’ portion of the playgroundwith a thin cover used as a separator. The cookie store was below the separator. At thispoint in the game there were 4 covers (trays of cookies) in the store (see Figure 4).

The children had chosen to bake 4 cookies on a tray. Each cover therefore covered 4toys. Tanya (T) was the baker and could bake only 2 trays each round. Rebecca (R) wasto eat one tray of cookies each time (rather than sell them!). The children were to agreeon the number of cookies in the store at each stage of the game before checking by using‘Toys in Rectangle’ from the menu bar. Both children agree that there are now 16 cookiesin the store. They check and “16” appears in the Number Box on the screen. Tanya copies2 more trays and then changes the color of all the trays from red to purple.

R: (After counting sub-vocally) I know what it is! 24.T: (Says nothing)R: I don’t know . . .[Rebecca erases one tray (cover).]T: Twenty.R: I’m full of eating the cookies.[Tanya copies two more trays of cookies.][Teacher (Olive): How many, Rebecca, are left?]R: She added!O: Okay . . .T: (Turns to Rebecca) How many were left, not the total, how many were left?R: Twenty![Tanya turns to the teacher and looks at him as if puzzled.]O: That’s what you said . . .T: No, I’m telling the answer – how many were left, after five . . .R: Can I cook after her?T: I mean after 4.R: Can I be the cook after this one?O: You want to switch? Okay. How many, how many . . . ?[Both children count to themselves.]R: (Turning to Tanya) Is it 28?T: (Turns to teacher) Thirty-two.O: (After slight pause) You decide. You gotta convince one another before you check.[Tanya turns to Rebecca.]


R: I think it’s 28, what I said from 24–25, 26, 27, 28 (extends 4 fingers as she utters thenumber words).T: (Turns to teacher, then to Rebecca) but how many in the rectangle? (Covers the trays onthe screen with both hands) Did I put 1, 2, 3, 4 or 5 . . . ?R: You put 6! (Laughs).T: (With emphasis) No! I put two.R: I know. (She counts silently to herself.)T: What’s 3 times – let’s see, what’s 3 times . . . ?R: Thirty-two.O: What?R: Three times what is 32.O: Three times? Why 3 times, Tanya?T: I mean four.[Tanya sets out to check the number of cookies but Olive stops her because they have notyet agreed on the total. Both children state that they have agreed on 32.]O: (To Rebecca) Do you understand why?R: Yes, because I only counted up one. Count two.[They check “Number in Rectangle”. The Number Box shows “28”.]R: Twenty-eight? I was right before!T: Wait a minute . . .O: Why?T: But I put two in . . .R: Oh! (Laughs) We took one away and you have to add it back, 28, so all you ever haveto do is count up one.

In the first part of this protocol, Rebecca was keeping track of thenumber of cookies by adding eight after each baking, and then subtracting(counting down) four after she ate a tray of cookies. Rebecca correctlystated the total as 24 after Tanya added the first two trays, and Tanyacorrectly stated the total as 20 after Rebecca had erased one tray. At thispoint, Rebecca indicated her sense of fun with the game by stating thatshe was full from eating all those cookies. Tanya added two more trays ofcookies to the store before Rebecca had a chance to work out the totalleft after she erased (ate) the last tray. Tanya then took on the role ofteacher and tried to clarify the question by asking Rebecca “How manywere left, not the total, how many were left?” Rebecca interpreted Tanya’squestion as meaning how many were left after she ate the last tray andresponded correctly with “20”. Tanya may have been about to answer herown question or may have meant to ask how many were left before thecurrent round when only four trays were in the store. At this point, Rebeccawanted to switch roles with Tanya and may have lost track of how manycookies were currently under the trays. After I asked them to figure outhow many were currently in the store, Rebecca counted up from 24 to 28.This was correct but Tanya figured the total to be 32. In trying to come to anagreement, Rebecca was convinced that it should be 32 because she had notyet eaten a tray of cookies for this round (24 and 8 more makes 32). When

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the computer feedback verified Rebecca’s first answer of 28 she made theabstraction that after each round of baking and eating “all you ever have todo is count up one (set of four)”. In neutralizing the perturbation caused byher interactions with Tanya and the feedback from the computer, Rebeccahad abstracted the operational structure of the game situation.

Teaching Interventions that bring forth Mathematical Activity

Hoyles and Noss (1992) have pointed out that, in computer environmentsdesigned to offer students “rich and diverse ways of exploring and solvinga problem . . . pupils are as likely to avoid encountering the mathematicalnuggets so carefully planted by their teachers. . . ” (pp. 45–46). Theyrefer to this possible outcome as theplay paradox. We encountered thisphenomenon in many situations and, like Hoyles and Noss, relied onthe interventions of the teacher to direct the play activity towards theintended mathematical activity. It has been suggested by Kieren (1994),that the mathematical features of the play (in Sticks) became availableto the children mainly when occasioned by the teacher’s interventions.Although we must admit that the children did tend to find the Toys micro-world more appealing than Sticks, the situations of learning in Sticks gaverise to more reasoning activities that pertained to children’s constructionof fraction schemes than did the Toys environment. (That was our mainreason for moving the children into the Sticks microworld.) The followingexample illustrates how one child (Joe) was able to use TIMA: Sticks asembodiments of his “connected numbers” (composite numbers composedof connected unit segments) to generate fractional quantities. Joe (J) wasworking with Azita (A), a graduate assistant in the Fractions Project. Thisteaching episode took place in the fall of Joe’s fourth grade year, the secondyear of the Project. A set of ‘Number-Sticks’ had been constructed at thetop of the playground area of the computer screen, separated by a longthin cover that stretched across the full width of the screen (see Figure 5).The length of each stick was a multiple of the smallest stick. The smalleststick was designated as the unit stick and each of the other sticks werenamed as an n-stick where n could be any number from 2 to 10 (e.g. a 5-stick). Sticks created by repeating or joining copies of sticks from this setof number-sticks were also named in the same manner (e.g. a stick createdfrom 4 repetitions of the 6-stick was a 24-stick).


Figure 5. A set of number-sticks in TIMA: Sticks.

PROTOCOL II

While Joe has his eyes closed, Azita makes a 24-stick below the cover using repetitions ofa copy of one of the sticks above the cover and erases the marks from her 24-stick.

A: The stick that I used was one third of the length of the stick I have right here (pointingto the unmarked 24-stick).

Joe measures the stick (24 appears in the number box). He then smiles to himself andcounts down the set of the number sticks ending on the 8-stick. He copies this stick andrepeats it 3 times to make a stick the same length as the 24-stick.

A: That is right!J: You said one third, so what will be . . . three times eight is 24.

Azita suggests doing more problems with the 24-stick. Joe wants to use a 21-stick but Azitaasks him to do one more with the 24-stick.

A: Think of a stick you could use to make the 24-stick and tell me what fractional part ofthe 24-stick it would be, and I will try to tell you what size stick it is and how many timesI should use it.J: Close your eyes.[Joe trashes the 3-part stick and looks at the set of number sticks.]J: O.K. I didn’t have to do nothing.J: It’s umm . . . It’s one sixth.A: The stick that you used is one sixth of the 24 stick?[Joe nods his head.A: So, I want something, I want a stick that when I repeat it six times would give me . . .J: No!A: Would give me the 24.J: One fourth! (at the same time as Azita is speaking).A: Oh! You used the one-fourth stick?[Joe nods yes.A: You used one-fourth, so I want a stick that when I repeat it 4 times will give me the 24,and I think that is the 6-stick! What do you think?[Joe nods yes.

Azita copies the 6-stick and repeats it 4 times to make a stick the same as the 24-stick.

Azita created a playful situation in Protocol II by asking Joe to closehis eyes. The situation became a game for Joe that he readily joined ascan be seen from his admonition to Azita to close her eyes when it was

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his turn to pose the problem. The game, however, was designed to bringforth reasoning with whole numbers that would generate meaning forunit fractions. Joe’s interpretation of ‘one third’ was something that whenmultiplied by 3 gave the total number. This was an iterative approach toestablishing unit fractions that appears to lead to inverse operations. Heused whole number division by the inverse of a fraction to find the appro-priate stick. It was our hypothesis that fractions could emerge from suchreorganizations of the children’s whole number schemes (Steffe and Olive,1990; Olive, 1999). Joe’s realization that he did not have to do anythingin order to pose the problem for Azita (as the unmarked 24-stick was stillvisible on the screen) indicates that he was able toimaginehimself actingwithin the microworld. It is important to encourage such imagined actionsas it is through re-presenting mentally their actions (within the microworld)that children construct their numerical operations (von Glasersfeld, 1981).When posing his problem for Azita, Joe hesitated in naming the fraction.He was trying to hold both the imagined stick in his head and the numberof times he would have to use it. He ended up using the stick size togenerate a name for the fraction rather than the number of times he wouldhave to use that stick. Joe realized his mistake as soon as Azita voiced herinterpretation of 1/6. He corrected his error rather than going with it andaccepting Azita’s actions. This indicates that Joe could generate his resultprior to action through both numerical calculation and visualized action.These are the necessary operations to reason inversely.

The teaching intervention that was critical in this episode was Azita’svoicing of her interpretation of 1/6: “I want a stick that when I repeat it sixtimes would give me . . .” This created a perturbation for Joe as he knewthat the 6-stick (his choice) would only need to be repeated 4 times toproduce the 24-stick. He resolved this perturbation by restating the frac-tional size of his stick as one fourth (of the 24-stick). Azita then voicedher rationale for choosing an appropriate stick in terms of an iterative unitfraction, thus reinforcing Joe’s reasoning in the first part of the episode.

Students’ Interactions while Using the TIMA that Bring ForthMathematical Reasoning

While the teacher’s role can be critical in using the TIMA to provokemathematical activity on the part of the students, the students’ interac-tions while using the TIMA can also bring forth powerful mathematicalreasoning. The following example is taken also from the second year ofthe project. Arthur and Nathan were regarded as our two most advancedstudents. Nathan worked in the project during third grade with a differentpartner. Arthur was asked to join the project during his fourth grade year


as a more appropriate partner for Nathan. The children had been workingtogether in the Sticks environment on problems dealing with compositionof fractions (fractions of fractions). While both children could produce afraction of a non-unit fraction (say 1/7 of 4/9 of a pizza), they had problemsnaming the resulting part as a fraction of the original whole. The decisionwas made to use the TIMA: Bars environment in the belief that the avail-ability of the cross-partitioning action would help the children connect theresult of taking a fraction of a non-unit fraction (using a cross-partition)back to the original whole. It was my hypothesis that this potential ofmaking cross-partitions of a bar would provide a more explicit modelfor creating a fraction of a non-unit fraction of a pizza. In the followingprotocol O stands for Olive (myself, the teacher), A is Arthur and N isNathan.

PROTOCOL III

O: Let’s have nine pieces in our pizza to start with. Arthur, how many pieces shall we use?A: Four.O: O.K. Pull out the four pieces, Arthur. [Arthur does so.]O: You are going to share those four pieces among seven people.A and N: Seven?O: Seven. Before you do anything, do you think you can figure out how much of one pizzaeach person will get?N: I’ve got it! [Nathan reaches for the mouse.]O: Wait. How much of a pizza do we have here?A and N: 4/9.O: O.K. And how many people are sharing it?A: Seven.

Both children think for 30 seconds. Arthur stares intently at the screen, while Nathan staresoff into space.

N: It’s easier to do it when you’ve got it done. [Meaning: It’s easier to figure it out afteryou carry out the actions.]O: Tell me what you would do.A: If there are seven pieces in four then you have to think about how many in eight andthen how many would be in the remaining one to make nine.O: (To Arthur) Share this among seven people, please.A: Alert.N: I’ve no idea! [My] head’s busted!

Arthur uses PARTS to partition the four-part piece HORIZONTALLY into seven rows offour.

N: You’ve done it! Each person gets one of those strips. (pointing to a horizontal row offour)

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Figure 6. Filling 1/7 of 4/9 of a bar.

While Arthur is filling the share of one person (the top row – see Figure 6) Nathan worksout the number of small pieces in the whole bar and the fraction name for the share of oneperson:

N: Four times seven is 28, 28 and 28 is 56, and seven more makes 63. Each person gets4/63!

Nathan’s immediate reaction was to act on the bar on the screen. Istopped him with the intention of having the children reflect on what theywould do before actually carrying out any actions. This created a blockagefor Nathan. He wanted to use his actions in the microworld as tools forhis thinking. Arthur was able to verbalize the situation in terms of sharingthe 4 pieces among 7 people: “If there are seven pieces in four then youhave to think about how many in eight and then how many would be in theremaining one to make nine”. I used Arthur’s description as a springboardfor his activity by suggesting that he share the pulled out 4/9 among 7people. He made the decision to use a horizontal partition of the 4/9-barinto 7 horizontal strips. Nathan immediately saw this cross-partition as asolution to the problem. In this respect it was the interactions of the twochildren within the microworld that prompted a solution to the problemthat had been created through the teacher’s intervention. A more detailedanalysis of the teaching episodes with these two students can be found inOlive, 1999.

LESSONS LEARNED FROM CLASSROOM USE OF THE TIMASOFTWARE

The TIMA software were used in elementary and middle grades class-rooms during 1991 through 1998 as part of an NSF supported Teacher


Enhancement project:Leadership Infusion of Technology in Mathematicsand its Uses in Society(Project LITMUS). Teachers were introduced to thesoftware during summer inservice courses and supported during the schoolyear as they attempted to use it with their students in their own classrooms.The ways in which the teachers made use of the software differed consid-erably among this group of approximately 100 teachers. Case studies ofindividual teachers (Pieper, 1995; Hanszek-Brill, 1997) suggest that theteacher’s own beliefs about mathematics and their own comfort with theuse of technology as a teaching and learning tool greatly influenced theiruse of the technology with their students. Those teachers who believedthat mathematics was a finished body of facts and procedures, and thatthe teacher should select, direct and control classroom activity based onthe dictates of the curriculum guide, tended to make minimal use of theTIMA software, preferring, instead, drill and practice computer ‘games’that reinforced the textbook curriculum. Teachers who held beliefs morecompatible with a constructivist perspective on mathematics and learning(e.g.each person’s mathematics must be constructed from experience insituations involving number and space, and exploration, discovery andverification are the essential processes of mathematics), and who believedthat the mathematics teacher should be receptive to student suggestionsand ideas and should capitalize on themwere more likely to use the TIMAsoftware with their students in ways that encouraged student’s explorationand conceptual development (Hanszek-Brill, 1997).

Pieper (1995) related the following concerning a third grade teacherin her study: “in introducing mathematics topics, she focused uponconceptual understanding. She explained that to introduce her students tomultiplication, she would engage students in a discussion about repeatedaddition, which in turn would lead her to introduce the term ‘array.’Then, students would build their own arrays using beans and a softwareprogram called ‘Toys’ ” (p. 63). This same teacher made extensive useof the computer lab with her students and developed a problem-solvingtask based on the children’s playful activities in the TIMA: Bars environ-ment. The children had created figures out of different sized bars (head,body, arms, legs and feet). The teacher used the children’s figures to askquestions concerning the fractional relations among the components of thefigure. For instance: If the “body” is the unit bar, what fraction of the bodyis each arm? She also followed up with challenges such as: “Can you makea figure in which the arms and legs are half the body, and the feet are onefourth of the body?”

In a companion study to the research project in which the TIMA soft-ware were developed, D’Ambrosio and Mewborn (1994) worked with

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children from a fourth grade classroom in the same school as the researchproject. During the three weeks that they worked with groups of childrenfrom this classroom they found ways to implement the constructivepedagogy espoused by the Project. Although they found both advantagesand disadvantages of using manipulatives such as Pattern Blocks and paperfolding, they found the dynamic visualization capabilities of the micro-worlds to be empowering for the children. The microworlds, however,could not be used without the presence of a teacher to pose problems andguide the children in their explorations. A major conclusion from theirexperience was the need for children to verbalize their thinking rather thanus ‘filling in the blanks’ for them and assuming that we know what thechildren really mean.

The most striking difference between the teaching experiment andthe classroom setting was the role of the teacher. D’Ambrosio’s andMewborn’s intention in their activities in the classroom was to create ateaching-learning situation in which they could seek to understand thestudents’ meaning making from the experiences they invited them toengage in, and find ways of modifying those meanings when necessary.To this end, the focus of their attention was on the Children’s Mathe-matics, whereas, in contrast, the focus of the classroom teacher was onthe mathematics of the textbook, and her interactions with the childrenwere to determine whether she could move on to the next topic or whetherfurther practice or instruction was necessary.

INTENTIONS, ACTIONS AND INTERPRETATIONS:DEFINING A MICROWORLD

Kieren (1994) has strongly suggested that it is the subsequent action ofthe child that determines the effectiveness of an intervention rather thanthe intent of the teacher. Thus, in considering the design of a computerenvironment or the nature of teacher or researcher interventions, we shouldlook at what the child does rather than simply characterize the interventionby its intent. Kaput (1994) also suggests that “whether or not something isa feature for a student depends very much on what the student is capableof seeing at a particular time” (p. 144). We would go further and say thatthe features (of a microworld) are products of the student’s actions withinthe microworld and not independent of those actions. Hoyles and Noss(1992) would appear to agree with this sentiment when they state that “amicroworld is best thought of as considerably more than software. . . ”(p. 31). They go on to say that:


Pieces of knowledge are appropriated (or not) depending upon the pupils’ own agendas,how they feel about participation, teacher intervention, and above all, the setting in whichthe activities are undertaken. Thus it is misguided to argue that simply by interacting withthe computer, children are likely to ‘acquire’ specified mathematical ideas (although theymight, of course, acquire others). (p. 31)

These concerns, regarding whether intended features of a softwareenvironment are constructed as such by the children using the software,were brought home to us during the use of TIMA: Sticks by Joe andPatricia, two fourth grade children in the Fractions Project. Sticks werebeing used to represent pizza. The situation was that there were two parties.At one party there were 14 people, at the other there were only 5 people.Both parties ordered one pizza. Before representing the pizza for eachparty using sticks, Azita, the teacher in this episode, asked the question“Which is larger, the share of 2 people from the party of 5 or the share of 3people from the party of 14?” Joe thought the share of 2 from 5 was bigger.Patricia thought they might both be the same. The children then copied twosticks of the same size on the screen, partitioned one into 5 parts and theother into 14 parts. The 14-part stick was lined up underneath the 5-partstick. Patricia made a visual comparison between the first part of the 5-part stick and the first 3 parts of the 14-part stick, remarking that theywere almost the same. She forgot that the original question asked her tocompare two shares from the party of 5 with three shares from the party of14. Her new intention was to check that one share from 5 was the same as 3shares from 14. After pulling out 3 parts from the 14-part stick and one partfrom the 5-part stick she attempted tomeasurethem using the MEASUREbutton. As no stick was currently in the Ruler section of the screen, nomeasurement was given. Azita told Patricia that she needed to copy a unitstick into the ruler in order to measure something. Patricia immediatelycopied the 3/14-stick into the Ruler and used MEASURE to click on theoriginal 3/14-stick that she had just copied. The numeral ‘1’ appeared inthe Number Box. Patricia then copied the 1/5-stick into the Ruler and usedthe MEASURE button to click on the original1/5-stick. The numeral ‘1’again appeared in the Number Box. Patricia immediately exclaimed “One.They’re both the same! They are both one”.

We had assumed that these two children both knew how the Rulerand MEASURE function worked because they had used these featuresseveral times in previous episodes. In these past experiences with Rulerand MEASURE the computer readout had confirmed their predictions(e.g. with an arbitrary unit stick in the Ruler, one of 5 parts from thissame unit stick measured 1/5). In the above situation, Patricia seemed tothink that whatever she wanted to measure had to be in the Ruler, and thatMEASURE gave a value of the length of the stick in the Ruler in terms of

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somestandardunit. This interpretation fit with her experiences measuringobjects in other contexts. The measure would be so manyinchesor centi-meters, not an amount of some arbitrary unit length. Her interpretation andparticular use of these features also provided confirmation of her initialconjecture (that the shares could be the same). Thus her interpretationof these features was reinforced rather than discouraged by the results!This episode led to our adding a new feature to the Sticks microworld:Measure with images. This was a feature added to theStatusmenu andcould be turned on or off. WhenMeasure with Imageswas active, imagesof the stick in the ruler were superimposed, end to end, over the stick beingmeasured. This was only effective (as a visual comparison), however, whenthe measuring stick was a unit fraction of the stick being measured.

CONCLUSIONS

Our experiences in using the TIMA in the context of our research onchildren’s mathematical thinking (illustrated by the protocols in thisarticle), as well as our observations of how the TIMA have been used inclassrooms by different teachers, have led us to the following conclusions:

1. As designers of computational environments for children, we needto think carefully about the contributions that the children need tomake to the situation in order to build their own mathematical struc-tures. The possible microworld actions and representations do not(by themselves) determine the structure of the children’s actionsand representations. Rather, the children’s intentions, as well as theconceptual operations available to them, are also involved.

2. Even the most carefully designed tools can be used in ways that wereunintended by the designers of the tools. For teachers to make effectiveuse of software tools such as the TIMA, they need to share the epistem-ological and pedagogical goals of the designers, and be comfortableusing the tools themselves. In the case of the TIMA, these goalswould include building an understanding of the children’s own waysand means of operating mathematically, and a willingness to interactconstructively with the children rather than following a pre-determinedcurriculum.

If these conclusions are taken seriously, the effective use of computertools such as the TIMA could help bridge that “major fault line” (Hoyles,1998) and set us on a path that might well redefine school mathematicsin terms of a “mathematics for children” (Kieren, 1990) based on ourunderstanding of children’s mathematics.


NOTE

1 TIMA: Bars is published by William K. Bradford Publishing Company, 16 CraigRoad, Acton, MA 01720. 1-800-421-2009. A new version of Bars is being developedthat is a platform-independent Java application and is available from the author athttp://jwilson.coe.uga.edu/olive/welcome.html.

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D’Ambrosio, B. S. and Mewborn, D. S. (1994). Children’s construction of fractions andtheir implications for classroom instruction.Journal of Research in Childhood Education8: 150–161.

Hanszek-Brill, M. B. (1997).The Relationships Among Components of ElementaryTeachers’ Mathematics Education Knowledge and Their Uses of Technology in theMathematics Classroom. Doctoral dissertation. The University of Georgia, Athens, GA.

Hoyles, C. (1998). Lessons we have learnt: Effective technologies for effective mathe-matics. Paper Prepared for the Conference on Technology in the Standards 2000.Washington, D.C.

Hoyles, C. and Noss, R. (1992). A pedagogy for mathematical microworlds.EducationalStudies in Mathematics23: 31–57.

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Kieren, T. E. (1990). Children’s mathematics/mathematics for children. In L. P. Steffeand T. Wood (Eds),Transforming Children’s Mathematics Education. Hillsdale, NJ:Lawrence Erlbaum Associates.

Kieren, T. E. (1988). Personal knowledge of rational number: It’s intuitive and formaldevelopment. In J. Hiebert and M. Behr (Eds),Number Concepts and Operations in theMiddle Grades(pp. 163–181). Hillsdale: Lawrence Erlbaum.

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Papert, S. (1980).Mindstorms. New York: Basic Books.Piaget, J. and Szeminska, A. (1965).The Child’s Conception of Number. New York:

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The University of GeorgiaAthens, Georgia 30602-7124USA

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Computer Tools for Interactive Mathematical Activity in the Elementary School