01. Fractional Commensurate Composition and Adding Schemes Learning Trajectories of Jason and Laura...

8/2/2019 01. Fractional Commensurate Composition and Adding Schemes Learning Trajectories of Jason and Laura Grade 5

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Journal of Mathematical Behavior

22 (2003) 237295

Fractional commensurate, composition, and adding schemesLearning trajectories of Jason and Laura: Grade 5

Leslie P. Steffe

Department of Mathematics Education, University of Georgia, 105 Aderhold Hall, Athens, GA 30602-7124, USA

Abstract

A case study of two 5th-Grade children, Jason and Laura, is presented who participated in the teaching experiment,

Childrens Construction of the Rational Numbers of Arithmetic. The case study begins on the 29th of November

of their 5th-Grade in school and ends on the 5th of April of the same school year. Two basic problems were of

interest in the case study. The first was to provide an analysis of the concepts and operations that are involved in the

construction of three fractional schemes: a commensurate fractional scheme, a fractional composition scheme, and

a fractional adding scheme. The second was to provide an analysis of the contribution of interactive mathematical

activity in the construction of these schemes. The phrase, commensurate factional scheme refers to the concepts

and operations that are involved in transforming a given fraction into another fraction that are both measures

of an identical quantity. Likewise, fractional composition scheme refers to the concepts and operations thatare involved in finding how much, say, 1/3 of 1/4 of a quantity is of the whole quantity, and fractional adding

scheme refers to the concepts and operations involved in finding how much, say, 1/3 of a quantity joined to 1/4

of a quantity is of the whole quantity. Critical protocols were abstracted from the teaching episodes with the two

children that illustrate what is meant by the schemes, changes in the childrens concepts and operations, and the

interactive mathematical activity that was involved. The body of the case study consists of an on-going analysis of the

childrens interactive mathematical activity and changes in that activity. The last section of the case study consists of

an analysis of the constitutive aspects of the childrens constructive activity, including the role of social interaction

and nonverbal interactions of the children with each other and with the computer software we used in teaching the

children.

2003 Elsevier Inc. All rights reserved.

Keywords: Fractions; Schemes; Teaching experiment; Learning trajectories

Two basic problems are of interest in this paper. The first concerns accounting for the operations that

are involved in the construction of three fractional schemes: a unit fractional multiplying scheme that Irefer to as the unit fractional composition scheme, a scheme for establishing equal fractions that I refer

Tel.: +1-706-542-4194; fax: +1-706-542-4551.

E-mail address: [email protected] (L.P. Steffe).

0732-3123/$ see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0732-3123(03)00022-1


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238 L.P. Steffe / Journal of Mathematical Behavior 22 (2003) 237295

to as a commensurate fractional scheme, and a scheme for adding unit fractions that I refer to as a unit

fractional adding scheme. The second problem concerns accounting for the contribution of the interactivemathematical activity that was involved in the construction of these three schemes. I investigate these

problems in a case study of two children, Jason and Laura, who participated in the teaching experiment,Childrens Construction of the Rational Numbers of Arithmetic (Steffe & Olive, 1990), during their5th-Grade in school. The case studies that are presented essentially constitute learning trajectories of the

two children across the teaching episodes that we held with them. Simon (1995b) introduced the conceptof a hypothetical learning trajectory to refer to the teachers prediction as to the path by which learningmight proceed. It is hypothetical because the actual learning trajectory is not knowable in advance. It

characterizes an expectedtendency (p. 135). In elaboration, he commented that, it is meant to underscorethe importance of having a goal and rationale for teaching decisions and the hypothetical nature of suchthinking (p. 136).

In a reaction to Simons paper, Beatriz DAmbrosio and I emphasized designing a learning space that

is based, at least in part, on a working knowledge of students mathematics in the construction of ahypothetical learning trajectory (Steffe & DAmbrosio, 1995). In a reciprocal reaction, Simon (1995a)amplified our emphasis in his comments that:

They have (appropriately, in my opinion) emphasized the teachers construction of models of thestudents mathematics as one of the most important foci of models of teaching based on constructivism

and agreed that the teachers knowledge is constantly being constructed as she interacts with studentsas they construct knowledge. (p. 162)

Simons emphasis in his paper was on learning trajectories as constructed by practicing teachers. Be-cause teaching was used as a method of scientific investigation in the teaching experiment, the practice

of research and the practice of teaching were interwoven in the exploration of student learning. As a

consequence, the major focus of attention in the teaching experiment was on the construction of mod-els of the mathematics of the involved children and how to bring that mathematics forth, sustain it,

and modify it. So, rather than consider the construction of learning trajectories as the responsibility ofpracticing teachers, I consider the construction of learning trajectories in the context of the idea ofworlds being constructed, or even autonomously invented, by inquirers who are simultaneously partic-

ipants in those same worlds (Steier, 1995, p. 71). When viewing learning trajectories as co-producedby children, it is possible to construct learning trajectories of children rather than hypothesize learn-

ing trajectories based solely on thought experiments concerning the paths by which learning mightproceed.

A learning trajectory that is abstracted from the experience of actually teaching children consists of

an explanation of childrens initial schemes as they enter the experiment, an explanation of the observedchanges in the entering schemes that the children produce as a result of interactive mathematical activityin situations of learning, and an analysis of the contribution of the interactive mathematical activity

involved in the changes. The teaching experiment was designed to produce such a learning trajectory asa result of actually teaching children (Steffe & Thompson, 2000b). In a related paper, I have given an

account of Jasons construction of a partitive fractional scheme and Lauras construction of a partwholefractional scheme while they were in 4th-Grade in school (Steffe, 2002). Both of these schemes areschemes for producing proper fractions and they have their genesis in partitioning schemes. I have also

given an account of Jasons unit fractional composition scheme and his construction of a commensuratefractional scheme during the first month of the teaching experiment that was held during his 5th-Grade


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L.P. Steffe / Journal of Mathematical Behavior 22 (2003) 237295 239

in school1 (Steffe, in press). During the same time period, Laura modified her multiplying scheme forwhole numbers in situations that were designed to bring forth a commensurate fractional scheme. But

she constructed neither a unit fractional composition scheme nor a commensurate fractional scheme as a

result of her interactive mathematical activity during that time period (Steffe, in press).

1. The entering partitioning and fractional schemes of Jason and Laura

The case study of the two children starts with the teaching episodes held on 19 November 1993 whereit was the goal of the teacher to bring forth the production of a sequence of fractions commensurate withone-third in the children. Jason could already produce a fraction commensurate with a given unit fraction,

and a plurality of fractions commensurate with one half, so it was the goal of the teacher to explore whetherJason could independently produce a sequence of fractions commensurate with one third. Laura could not

independently produce a fraction commensurate with a given unit fraction, but after Jason produced sucha fraction, she could explain why the two fractions were commensurate. Jason could operate in fractionalsituations in ways that Laura could not, and it was a major goal of the teacher to use Jasons ways of

operating to induce similar ways of operating in Laura.

1.1. Jasons entering partitioning and fractional schemes

1.1.1. The equi-partitioning scheme

Jasons construction of equi-partitioning occurred in a situation where we asked him to cut a piece of

candy off from a candy stick for one of four people (Steffe, 1999, 2002). He was using the computer toolTIMA: Sticks that we had designed for the teaching experiment.2 To make the share of one of four people,

Jason independently marked off one part of a segment he had drawn in the screen and pulled the part outof the marked segment. In a test to find if the part was one of four equal parts, he made three copies ofthe part and joined them together with the pulled part. Marking a segment once in estimating one of four

equal parts implies that he used his concept, four, in gauging where to make the mark, which means thathe used four as a partitioning template. The ability to iterate the pulled part in a test to find if the estimatewas one of four equal parts was inherited from the iterability of his units of one. He introduced these two

ways of using his concept of four, and they served him in constructing the equi-partitioning scheme. Thepurpose of the equi-partitioning scheme is to estimate one of several equal parts of some quantity and toiterate the part in a test to find whether a sufficient number of iterations produce a quantity equal to the

original. This scheme proved to be basic in Jasons production of the partitive fractional scheme.

1.1.2. The partitive fractional scheme

Jason constructed the partitive fractional scheme while he was in his 4th Grade (Steffe, 2002). Thisscheme permitted him to establish meaning for unit fractional number words like one tenth as one out

of ten little pieces that could be iterated ten times to produce a partitioned segment of length equal tothe original. The phrase out of was a crucial indicator that he regarded one little piece as one unit part

1 These teaching episodes were held on the 25th of October and on the 1st, 8th, 15th, and 22nd of November of his 5th Grade.2 In TIMA: Sticks, a segment can be drawn using the mouse cursor, the segment can be marked using hash marks, marked

parts can be pulled out of the whole stick [an operation that left the marked stick intact], copies of the pulled part can be made,

and the copies can be joined together to make another stick.


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out of ten unit parts that constituted the partitioned fractional whole. He also knew that ten tenths was

the length of the partitioned fractional whole because he could iterate one-tenth ten times to produceten-tenths. So, one tenth was a fractional unit that he could use in iteration to produce identical copies in

the sense that I have just explained. The dominant purpose of the partitive fractional scheme is to partitiona unit into so many equal parts, take one out of those parts, and establish a one-to-many relation betweenthe part and the partitioned whole. The iterative aspect of the scheme serves in justifying or verifying

that a unit part of the partitioned unit is one of so many equal parts. By iterating a unit fractional part somany times, Jason produced fractional language such as two tenths, . . . , ten tenths as descriptionsof the results of iterating one tenth so many times. But it was a major surprise that he could not produce

improper fractions as a result of his productive activity while he was in his 4th Grade. That monumentalevent entailed the construction of the splitting operation, which I explain below.

1.1.3. The splitting operation

We asked both Jason and Laura to make a segment so that a given segment was five times longer (aswell as two times longer) than the segment they made toward the end of their 4th Grade (Steffe, 2002),and neither child could do it. Making such a segment entails positing a segment in thought that could be

iterated five times to produce a segment of length equal to the given segment, a process which produces aconceptual partition of the given segment. Realizing that the desired segment can be produced by simplysplitting the given segment into five parts implies a composition of the two operations of iterating and

partitioning, where the partitioning produces parts any of which can be iterated five times to producethe partitioned segment, and where any segment for which the given segment is five times longer can be

used to partition the given segment. The splitting operation is the basic operation that is involved in theconstruction of recursive partitioning.

1.1.4. Recursive partitioning

The ability to produce a partition of a partition in the service of a goal is fundamental in childrensprogress in the construction of fractional schemes beyond the partitive fractional scheme. When the resultof a partition is given, recursive partitioning occurs when children can produce the result using two other

partitions. For example, when asked to partition a stick into 12 parts under the constraint of not dialingthe counter of Parts to 12,3 Jason insightfully produced a partition of a partition to partition the stickinto twelve parts in a teaching episode held on the 8th of November 1993 (Steffe, in press). Recursive

partitioning is the inverse operation of first producing a composite unit, multiple copies of this compositeunit, and then uniting the copies into a unit of units of units. So, producing a recursive partitioning implies

that a child can engage in the operations that produce a unit of units of units, but in the reverse direction.

Recursive partitioning is fundamental in the production of the unit fractional composition scheme.

1.1.5. The unit fractional composition scheme

The initial observation that led to my construction of the unit fractional composition scheme occurredwhen Jason was at the beginning of his 5th Grade (Steffe, in press). He found how much three fourths ofone fourth of a segment was of the segment by reasoning, See, if we would have had it in that (points to

each one fourth part) four, four, four, and four sixteen. But you colored three, so it is three sixteenth!

3 In TIMA: Sticks, a child can use the computer action Parts to dial a counter to any one or two-digit number and then click

on a stick to mark the stick into the number of parts indicated on the dial.


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Recursive partitioning was essential in his reasoning, because it was his goal to find how much three

fourths of one fourth of a segment was of the segment, which is a non-partitioning goal. For a partialresult of a composition of two partitionings to be judged as recursive in the fractional context, there must

be good reason to believe that a child, given the partial result (three fourths of one fourth), can producethe numerosity of the full result. But this is not all, because the child must also use the second of the twopartitions (the one that is not fully implemented) in the service of a non-partitioning goal.4 The importance

of this latter judgment is that, to produce the numerosity of the full partition, the child must intentionally

choose to partition each part of the original partition using the second partition. In Jasons case, whatthis means is that he intentionally chose to partition each fourth of the original stick into fourths. This

amounts to embedding recursive partitioning in the reversible partitive fractional scheme in the processof achieving the goal. Not only is recursive partitioning essential in the production of a unit fractionalcomposition scheme, it is also essential in the production of a commensurate fractional scheme.

1.1.6. The commensurate fractional schemeWhile Jason was in his 4th Grade, we focused on using his multiplying schemes for whole numbers inthe production of a situation he could then use to abstract commensurate fractions.5 For example, Jason,along with Laura, chose to use segments from a collection of unmarked segments of length from one to

ten units to produce a segment that was 24 units in length. The children were to find which of the tensegments could be used and how many times. After using an unmarked segment of length six units four

times to produce a six-part segment of length 24 units, the children thought that the segment they usedwas six fourths of the segment of length 24 units. This kind of error recurred in similar situations in spiteof our attempts in teaching the children to overcome it. The errors that we encountered in the childrens

production of commensurate fractions were necessary errors because the operations that produce a unitof units of units, and their inverse, recursive partitioning, were yet to be constructed by these children.

When considering a segment of length twenty-four units, and a segment of length six units, the childrenwere yet to conceive of the segment of length six units as a measurement unit to measure the unmarkedsegment of length 24 units. That is, they were yet to conceive of the unmarked segment of length 24 units

as partitioned into subsegments of length six units prior to activity. They could repeat the segment oflength six units four times to produce a segment of length 24 units, and they were definitely aware of thatthey repeated the segment four times. But they operated with two levels of units rather than three levels of

units, and this produced what to us were conflations of the three units that we could see in the situations.When recursive partitioning emerged in the Jasons thinking at the beginning of his 5th Grade (Steffe,in press), the conflations that I have just noted disappeared. At this point, he became able to produce a

plurality of fractions commensurate with 1/2.

1.2. Lauras entering partitioning and fractional schemes

1.2.1. The simultaneous partitioning scheme

Laura could engage in simultaneous partitioning while she was in her 4th Grade (Steffe, 2002) in thatshe made uncannily accurate estimates of where to mark off one of up to ten parts of an unmarked segment.

4 In the example, Jasons goal was to find how many subsegments of length equal to one fourth of one fourth of the segment

would fit into the segment.5 I consider two fractions as commensurate if one of the two fractions is produced by a quantity preserving transformation of

the other.


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The main difference between her simultaneous partitioning scheme and Jasons equi-partitioning scheme

is that she did not explicitly disembed a part from the partitioned segment and use it in iteration in a testto find whether a sufficient number of iterations produced a segment in length equal to the original.

1.2.2. The partwhole fractional scheme

Using her simultaneous partitioning scheme, Laura produced a fractional scheme in the 4th Grade thatI refer to as the partwhole fractional scheme. This scheme was used by Laura to establish meaning forunit fractional number words like one tenth as one out of ten little pieces. Establishing meaning for

seven tenths entailed pulling seven distinct parts out of a stick partitioned into ten equal parts ratherthan iterate one-tenth seven times (Steffe, 2002). Iterating a unit fractional part is not a dominant part of

this scheme, so seven-tenths is yet to refer to the length of a stick that has been produced by iteratingone-tenth seven times. Each part of a partwhole fraction is a distinct part that is only equal in length tothe other parts rather than identical to them. When the unit fractional part one tenth is an iterable unit, the

copies produced are identical in the sense that they are implementations of the same conceptual unit item.Basically, a child who has constructed a partwhole fractional scheme is yet to construct unit fractions6

as iterable fractional units.

1.2.3. Lack of recursive partitioning

In the teaching episodes held on the 25th of October and on the 1st, 8th, 15th, and 22nd of Novemberof her 5th Grade (Steffe, in press), Laura did not construct recursive partitioning in spite of the best

attempts of the teacher to bring it forth nor did she independently produce a fraction commensuratewith a given fraction. Apparently, the two operations of partitioning and iterating that were available toJason opened possibilities for him that were not present for Laura. The difference in the two children

was striking throughout the four teaching episodes during the month of November. For example, in the

teaching episode held on the 8th of November, to partition a stick into twelve parts without using twelve,Laura partitioned the stick into eleven parts and then pulled one part from the stick and joined it to theeleven-part stick. In this case, Laura did engage in independent mathematical activity, but her way ofoperating constituted contraindication of recursive partitioning operations. It might be conjectured that

Laura simply didnt think to operate in the way Jason operated (first partition the stick into three partsand then each of these parts into four parts) because once she observed Jason operate, she did knowwhat to do and proceeded quite smoothly. However, it was characteristic for Laura to need to re-enact

an explanation made by Jason, or for there to be visual cues in her perceptual field, before she couldengage in the actions that were needed to be successful in explaining why a fraction such as one third was

commensurate to, say, four twelfths after she measured the 4/12-stick and 1/3 appeared in the Number

Box

of TIMA: Sticks. Jason could independently engage in the operations that were necessary to producesuch explanations or actions, and beyond that, he could independently produce a unit fraction that was

commensurate with, say, three fifteenths. I consider such independent productions as necessary in orderto judge that a child has constructed a commensurate fractional scheme.

It was the case that Laura did not provide any indication of the splitting operation, recursive partitioning,

or a commensurate fractional scheme. However, she did engage in mathematical interaction with Jason in

6 I use quotation marks to indicate that, say, 1/3 was a partitive unit fraction for Laura because she was yet to engage in

reciprocal reasoning of the kind explained by Thompson and Saldanha (in press). That is, she was yet to reason that if a segment

A is 1/3 of a segment B, then B is three times the length of A, and reciprocally. Reciprocal reasoning is based on the splitting

operation.


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those cases where her multiplying scheme for whole numbers was called forth. But this was not enough

for her to engage in making independent explanations for why a fraction was commensurate to anotherfraction, and she remained dependent on what Jason said or did, what the teacher said or did, or the

context of the situation in order to know what to do to be successful.

2. Producing a sequence of fractions commensurate with one third

Because independent explanations rely on the nature and use of the involved schemes, the teacherfocused on eliciting the production of a sequence of fractions commensurate with one third in the 29

November 1993 Teaching Episode. The goal of the teacher in Lauras case was to elicit independentexplanations for why, say, two sixths is commensurate to one third by encouraging interactive mathemat-ical activity with Jason. In Jasons case, the goal of the teacher was to explore whether his production

of a plurality of fractions commensurate with one half was limited to one half. The teacher established aconventional language during the first seven or eight minutes of the teaching episode to refer to partitionedsticks that were to be regarded as fraction sticks. Make a twelve twelfths stick was interpreted by the

children as partitioning a stick into twelve equal parts. They understood that each part was one twelfthof the 12/12-stick and that there were twelve such parts. In the protocols, T stands for Teacher,7 J

stands for Jason, and L stands for Laura.

Protocol I: Production of fractions commensurate with one third.

T: Now, please start with a three-thirds stick. Make me a three-thirds stick.L: Copy, copy, copy (while making a copy of the stick in the Ruler.8 She then partitions it into

three parts.)

T: (Asks the children to color the parts different colors.) You remember that last week we workedwith halves. Today we are working with thirds. I would like you to partition that stick in a

different way so you can pull out one third. I want another fraction, another fraction that willbe like one third.

J&L: (Sit quietly for approximately twelve seconds.)

J: (Jason takes the mouse and dials Parts to 5 and clicks on each of the three parts, partitioningeach into five equal parts. He then activates Pull Parts9 and pulls out the first three parts.

The teacher asked Can you pull out a third for me? while he was pulling the three parts,but that seemed irrelevant in Jasons activity.)

T: Is that piece one third of the whole stick (the stick in the Ruler)?

L: No, it is three fifteenths!

J: (Drags the 3/15-stick into the Trash. He then activates Pull Parts and sits quietly for aboutfifteen seconds, then pulls five parts out of the 15/15-stick he made.)

T: Is that one third?L: Because there are three whole pieces there (points three times in succession at the 15/15-stick

from the left to the right) and there is one there (points to the 5/15-stick Jason pulled out).

7 Dr. Ron Tzur served as the teacher of the children (Tzur, 1999).8 A stick could be copied into a space at the bottom of the screen and used as a ruler to measure another stick in the screen by

activating Measure and clicking on the stick.9 Pullparts can be used to pull one or more parts from a marked stick. The marked stick is left intact.


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T: Ok, can you give it another name? Thats one third, so can you give it another fractional

name?J: Five fifteenths. (He then uses Label to label the stick he pulled out 5/15.)

T: Five fifteenths. And before that you said one third. One third or five fifteenths of what?J&L: Of the whole cake. (Laura actually made a copy of the stick in the Ruler and partitioned it

into three equal parts using Parts. She then said, Thats a cake. when the teacher asked

her what it was.)

Both of Jasons and Lauras language and actions in Protocol I were remarkable, but for quite different

reasons. After the teacher said, I want another fraction, another fraction that will be like one third.Jason sat for approximately twelve seconds deep in thought. He then recursively partitioned the 3/3-stick

by partitioning each part into five parts. Interestingly enough, he then pulled out three rather than fiveparts to make one third. I consider pulling three rather than five parts out to make one third an indication

that his partitioning of each of the three parts into five parts was not an application of a general schemefor making fractions commensurate with a given unit fraction. The lack of a general scheme for makingcommensurate fractions, when coupled with the twelve seconds he sat deep in thought, indicates that hisrecursive partitioning act was indeed a creative act that might serve in the construction of a commensurate

fractional scheme. Recursive partitioning was the basic operation underpinning the act of creating a newpartition that could be used to make a fraction commensurate with one third. After the teacher asked, Is

that one third? and Laura answered, No, it is three fifteenths! after Jason made the 3/15-stick, Jasondid make a correction and pulled out a 5/15-stick after trashing the 3/15-stick. I consider this experienceof producing a 5/15-stick given a 1/3-stick as one that he had not experienced before. It was a novel

experience.When Laura said, No, it is three fifteenths! initially I considered this comment to indicate that she

understood that three fifteenths was not one third because it was not a fraction commensurate with onethird. In hindsight, however, her judgment that three fifteenths was not one thirdwas basedon the fact that itwas three out of fifteen parts rather than one out of three parts because she could not independently produce

a fraction commensurate with 3/15. Nevertheless, after Jason pulled the 5/15-stick from the 15/15-stick,she explained that it was one third Because there are three whole pieces there . . . and there is one there. . . . She obviously knew the 5/15-stick was called five fifteenths, but her explanation for why it could

be called one third as well indicates that she made a unit containing three composite units each ofwhich contained five elements. She appeared to be very confident in her explanation, and her additionalact of making a copy of the stick in the Ruler and partitioning it into three parts as a representative of

the whole cake, together with her confident attitude, corroborates the inference concerning making and

comparing composite units. For the first time it seemed as if she might be able to independently initiateher own partitioning activity. Being encouraged, the teacher continued on.

Protocol I: (First Cont.)

T: Can you now make another fraction? You already have one third, and you already have fivefifteenths. Can you make another part so that you can pull out one third?

J: (Shakes his head no and Laura sits quietly looking at the teacher.)T: Cant do another one?J: Uh Huh (no).

T: Last week you had a lot for halves. So I bet that you can have more number names for a third.


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J: (Sits in deep concentration for approximately ten seconds. He then drags the 3/3-stick Laura

made from the bottom of the screen upward, activates Parts and it is still dialed to 3, buthe does not use it because the stick is already partitioned into three parts. He then fills the

three parts different colors.)L: (Sits looking straight ahead with no apparent overt indications of mental activity.)T: Copy that one five or six times up here (the top of the screen) so we always have them filled.

J: (Makes several copies at the top of the screen. He then activates Parts and partitions eachpart of the 3/3-stick he colored into three parts. He then pulls the first three parts.) Threeninths.

L: Three ninths.T: (To Laura) you want to label it?L: (Labels it 3/9 using Label.)

T: Can you make another one?

J: (Takes the mouse and starts.)T: No, let Laura do the next one!L: (Drags a stick from the copies Jason made to the middle of the screen and partitions each of

the three parts into four parts.) four, four, four.

T: Before you pull it out, what will be the fraction that you are pulling out?L: Four, ah, four . . . .J: Twelfths (almost simultaneously with Laura).

T: How did you know that?J: Because . . . .T: Wait, wait, wait. Let Laura. We have to take turns because we cannot all talk at the same time.

You know we cannot all talk together.

L: Four, four, and four make twelve. Four and eight and twelve. (She then labels the part 4/12and pulls it out of the 12/12-stick she made.)

T: Now this is interesting. You have three ninths, four twelfths, and five fifteenths (pointing atthe respective parts). Can you think of something that will be . . . even smaller than what we

have?J: Umm (yes).T: What would you do? Which one are you going to try?

L: I know one . . . Maybe not. Three seconds! Three twos!T: You go ahead (gesturing toward the computer).

J: I dont know what she is talking about.

L: (Partitions each part of a 3/3-stick into two parts. She then pulls the first two parts out of the6/6-stick she made.)

T: Go ahead. Pull it out and label it, please.L: Two, two . . . . (subvocally utters number words) six!T: So this is what she has to label (To Jason)?

J: Because she had, there are six pieces and there are two. They are kind of a part.L: (Labels the part 2/6).

The claim that Jasons act of partitioning each part of the 3/3-stick into five parts in Protocol I was a cre-

ative act is corroborated by the fact that initially he said that he could not do another one. After the teacher


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reminded him that he had done a lot for halves last week, and asserted that he could have more number

names for one third, that apparently reoriented Jason in such a way that he considered it possible to make apartition other than the one he had already made. That is, he had eliminated the perturbation that drove his

generation of five fifteenths, and the teachers provocation served in his reestablishment of a goal to gen-erate another number name. That he sat in deep concentration for approximately ten seconds does indicatethat partitioning each part into three parts did not immediately occur to him. I suggest that his experience

was more or less one of being in a state of perturbation in a search mode but with nothing appearingin his consciousness. But his recursive partitioning operations were activated, and to partition each of thethree parts into three parts appeared to him rather suddenly. This is indicated by the activity in which he

engaged that marked the end of the period of search. All at once he knew what to do. Moreover, afterthe teacher asked, Can you make another one from the same family? he immediately initiated activity,which indicates that he was now aware of how to proceed. That is, he had abstracted how he operated.

The teacher sensed that Laura could now initiate making another fraction commensurate with one third.

So, he stopped Jason from acting by saying, Let Laura! It might seem as if he should have made thissuggestion much before this time, but his suggestion was made on the basis of his interpretations of notonly Lauras language and actions, but on the basis of his interpretation of her body language as well.Because she now seemed confident when saying three ninths the teacher decided to ask her to make the

next fraction. She immediately partitioned each of the three parts of the 3/3-stick into four parts, and saidthat it was four twelfths of the whole stick. My interpretation is that she also knew that it was one third ofthe stick because, when the teacher asked the children to think of something so that the numbers would

be smaller, she generated three twos. What she meant was that she would partition each one of thethree parts into two equal parts. She obviously focused on the three partitioned 1/3-sticks (the 3/9-stick,the 4/12-stick, and the 5/15-stick) because she did not immediately complete the fractional number word

two sixths. Instead, she counted how many parts the 6/6-stick comprised before she could say two

sixths, and then she said six rather than sixths because she had just counted the six parts.Both children seemed to be poised to make the generation of fractions commensurate with one third

systematic. So, the teacher asked them to arrange the fractions that they had made in a systematic order.

Protocol I: (Second Cont.)T: Can you now arrange the screen so that we will have two sixths, then we will have three

ninths, then we will have, you know, like, a sequence.J: (Arranges the sticks so that the pair of sticks corresponding to 2/6 is on the bottom, then

3/9, then 4/12, then 5/15. Laura was active during this time, making several sugges-tions.)

T: Now comes the question. Without making the stick, can you think and tell me what will bethe next one in the sequence that will be one third? Which one will it be?

L: The next to highest or to lowest?T: You started with the two sixths, then three ninths . . . .

J: (Points to the 3/3-stick at the very top of the screen over all the others) that one will be thelowest.

T: Thats right. Thats the one first, so you want to put it down here (underneath the 2/6-stickand the 6/6-stick).

J: (Moves all the sticks upward to make room for the 3/3-stick and then drags it beneath all of

the others and labels it 1/3).


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T: (While Jason is arranging the sticks) While you are working, think what will be the next one,

the next one in the sequence upward.L: (After talking to herself) six thirty sixths!

T: Six thirty sixths? We will wait for Jason. Ok, she is saying the next one will be six thirtysixths. (To Jason) What do you say? (He didnt ask Laura to explain because he was waitingfor Jason to complete the rearrangement of the sticks.)

. . .

T: Ok, Laura said that the next one that you are going to put will be six thirty sixths.L: (Grabs the mouse) I know . . . (Plays with copies of the 3/3-stick remaining at the top of the

screen.)T: Wait, wait, wait. Jason, what do you say?J: Seven twenty oneths. Three times sevenis twenty-one,and twenty-one comes before thirty-six.

L: (Just as Jason is starting to say Seven twenty oneths) Oh, oh, I know!! (After Jason is done

explaining) six times three is. . .

six eighteenths. Six eighteenths (with confidence)!T: So, it will be six eighteenths now?J: Yeah, six eighteenths.T: And what will come next? After the six eighteenths, what will come?

J: (Puts his head down and thinks) I dont know . . . .T: (To Laura) do you want to do the six eighteenths?L: Yes.

T: (To Jason) Think of the next one while you work.J: (After about three seconds) I got it. I got the next one!

L: (Activates Parts, dials to 6 and clicks on each part of the 3/3-stick at the top of the screen.

She then pulls six parts and labels it 6/18).

T: Beautiful. (To Jason) you have another one? Tell us what it is.J: Seven twenty . . . .T: Seven twenty firsts! Before you do it, can you think of what will be the next one?L: I think, there will be eight in there, and eight in there, and eight in there (looking up and

pointing with her finger three times as if seeing a 3/3-stick in her visualized imagination).Eight twenty fourths!!

T: (To Jason) This is what you thought of? What will be the next one?

L: (Again looking upwards as she points three times) nine, nine, nine. Nine twenty sevenths!T: (To Jason) I wanted Jason to say it. Ok, what will be the next one after nine twenty sevenths?

J: Ten thirtieths.

L: (Again points three times in the air as Jason is answering Ten thirtieths) eleven, eleven,eleven. Eleven thirty three!

Laura went on in the same way, generating 12/36, 14/42, 15/45, 16/48, 17/51, 18/54, 19/57, and 20/60.

She used her multiplicative computational algorithm to calculate those products (e.g., 3 17) that shecould not quickly find by using mental addition. Jason could not keep pace with her fast calculations

and this is one of the first times that she appeared to be the more powerful of the two. Jason definitelywas aware of the sequence of fractions being calculated because he guessed 16/78 after Laura hadsaid 15/45. He was aware that Laura could produce fractions of the sequence faster than he could, and

appeared abashed that he could not keep up with her.


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It could be said that both Jason and Laura had established a scheme for producing a sequence of fractions

each commensurate with one third. The operations of the scheme for Jason included recursive partitioningand the activity of the scheme was in part to calculate the numerosity of the parts produced by using

the recursive partitioning operation. He did not use a standard multiplicative algorithm in calculation,but instead used progressive integration operations [e.g., 16 and 16 is 32, and 32 and 16 is 48]. Theactivity of Lauras scheme also was to calculate the numerosity of the three composite units produced

by partitioning, but in doing so, she used her standard multiplicative algorithm mentally. Whether sheengaged in recursive partitioning when partitioning the three parts is problematic because it was Jason whogenerated five fifteenths and three ninths to start the sequence. It was only then that Laura generated four

twelfths. Her production of this fraction was based on her assimilation of Jasons language and actionsusing her units-coordinating scheme10 and her scheme for producing proper fractions. For example,when Jason generated three ninths, she also said three ninths in recognition of Jasons results. This

recognition, when coupled with her production of four twelfths immediately afterwards, does indicate

that she distributed partitioning into three parts across each one third and produced nine as the totalnumber of parts (which is a units-coordination). But it does not necessarily indicate that she engaged inrecursive partitioning because making that inference requires an independence of operating that she didnot demonstrate. So, the teacher tested to find if she could independently find another fraction for two

thirds. If she could, then a case could be made that she engaged in recursive partitioning in doing so.

Protocol II: Production of fractions commensurate with two thirds.T: Copy the cake and put it into thirds. Make a three thirds stick. And fill it with different

toppings, please (for the children, this meant to color the parts different colors).

L: (Makes a copy of the stick in the Ruler, partitions it into thirds, and fills the two outer thirdswith different colors.)

. . .

J: (Fills the middle third with the same color as the first third and pulls the first two thirds fromthe stick as requested by the teacher.)

T: Now comes the question. You gave me, like, twenty different thirds (referring to the children

making fractions commensurate with one third). Can you give me now a different two thirdsthan you have here? A different two thirds of the cake.

L: I know. (The teacher nods yes, so she continues. She partitions an extra copy of the stickshe made into three parts and pulls out the last two parts!)

T: Ok. Thats very good. But, can you give me another fraction. Can you give me a fraction

that will be two thirds out of the whole, but with a different partitioning, a different numberof pieces. (Both children sit quietly for approximately 20 seconds.) Can you find a way topartition the cake so that you will be able to pull out two thirds?

L: I know.J: (Following Laura) I know.

T: (Encourages Laura to carry on.)

10 To coordinate three and four, for example, the composite unit, four, is inserted into each of the three units of one contained

in the composite unit, three. This produces three composite units containing four units of one. The activity of the scheme is to

calculate the numerosity of the units of one (c.f. Steffe, 1992). Using this scheme in the context of segments involves using the

composite units as templates for partitioning.


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L: (Makes a copy using the stick in the Ruler. She partitions it into three parts and fills the

outer two parts with the same color. She then partitions the middle part into two equal parts.)T: I want you to pull out two thirds.

J: I know.T: (To Jason) Just wait a little. (To Laura) Thats a very good direction to start with.J: (Intervenes in spite of the teachers admonition to Just wait a little.) Put another one in

here (partition the first one third into two parts), and another one in here (the last one third),and then that would be two, two, and two and six in all (pointing appropriately to the screen.)

T: Now, can you pull out the two thirds?

L: Wait, wait, wait. (Partitions each of the two outer thirds into two equal parts, pulls the fourparts she just made, and joins them together.)

T: How much is it of the whole thing now that you joined it together? How much is it?

J&L: (Together) four sixths!

T: (To Jason) You want to make another one? Another one that will be two thirds? Start with afull cake and do it a different way.J: (Partitions an unmarked copy of the stick in the Ruler into twelve parts. He then colors the

first four parts.)

T: How many are you going to take?J: Eight.T: Why eight?

J: Because eight is two thirds of twelve!!

Although Laura partitioned each of the two outer thirds into two equal parts and pulled out the fourparts she just made after the teacher asked Now, can you pull out the two thirds? she did so only

after Jason directed her to partition the two outer thirds as well as the middle third. Why she partitionedonly the middle third of the stick into two parts can be explained when considering that her successfulmathematical activity in the two continuations of Protocol I was based on her first partitioning one of the

three parts. She simply repeated what worked for her in the case of finding another fraction for one third.Although in the continuations of Protocol I, her multiplying activity symbolizedpartitioning the other twoparts, she apparently did not anticipate partitioning the other two parts into two parts each in Protocol II.

Jason, on the other hand, first conceptually partitioned each of the three parts and only then pulled out theappropriate parts to establish a fraction commensurate with two thirds. His partitioning of an unmarked

copy of the stick in the Ruler into twelve parts when it was his intention to make another one that willbe two thirds warrants this claim. The teacher did encourage Jason to Start with a full cake and do it adifferent way. and this statement obviously oriented Jason to find a different way. However, the teachers

statement cannot be used to explain Jasons subsequent mathematical activity when he partitioned thestick into twelve pieces because the teacher did not mention twelve. Jasons partitioning act, along withhis explanation Because eight is two thirds of twelve. corroborates the claim that Jason could indeed

engage in recursive partitioning. However, I could not impute recursive partitioning to Laura.

3. Lauras explanations in the context of fractional compositions

Fortunately, Jason was absent from the teaching episode held on 10 January 1994 because it permitted

Laura to be the primary actor in solving tasks posed by the teacher. Even though Laura assimilated Jasons


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language and actions in Protocols I and II and thereafter acted as ifshe had constructed a commensuratefractional scheme, she did not solve commensurate fractional tasks independently of Jasons solutions.Jasons absence in this teaching episode forced the teacher to confront the lacuna in Lauras reasoning.

His strategy was to bring forth Lauras establishment of one half of one forth, and then measure the stickusing Measure. Of course, 1/8 appeared in the Number Box, and Laura was asked to explain whyone half of one fourth could be one eighth.

Protocol III: Lauras explanation for why one half of one fourth is one eighth.

T: Now we will start with a four-fourths stick. Do you want to prepare one, a four-fourthsstick?

L: (Dials Parts to 4 and clicks on a stick that she had drawn.)

T: First, can you pull out one fourth of the stick?L: (Pulls out the third one fourth of the stick, presumably because she liked its color, purple.)

T: Ok, now here comes the surprise. (Using Cover, covers the last three parts of the 4/4-stick,leaving the first part visible. He then establishes that Laura knew that the visible part wasone fourth of the stick as well as the part she pulled.) Now we have two children who have

to share the one-fourth pizza. Lets say that both of us have to share it, Ok? Show me yourshare and tell me how much it will be of the whole pizza.

L: Just that one piece (points to the 1/4-stick she pulled out)?

T: You can use whatever you want. You will have to show me, we will have to share one fourthof the pizza.

L: (Repeats the 1/4-stick to make a 4/4-stick. She then fills the first two parts) two fourths of

the Pizza will be one child.T: All right. Ill repeat the question because I can see it was my mistake. We can only share

the one fourth. Take this away (the 4/4-stick she just made) because we dont have a wholepizza to share. (After Laura trashes the 4/4-stick, he asks her to pull the visible part out ofthe partially covered 4/4-stick.) here is the question. Its only you and me, and we have only

the one fourth. We have to share this one (the 1/4-stick). Can you show me your part, andtell me how much will it be of the whole pizza?

L: (Dials Parts to 2 and clicks on the 1/4-stick.)

T: Now, what is your share?L: Umm, umm (fills the first part with a color different than the second).

T: What type of a pizza is that one?L: One half of a fourth.

T: So, how much is it of the whole pizza? That is very good.L: Umm (after about ten seconds) three and a half!!T: Three and a half what?L: Well, thats one half, and then theres the whole one (the three covered parts and the one

half of one fourth).T: I see, umm, how much is that piece (points to the 2/8-stick Laura made by partitioning the

1/4-stick into two parts) of the whole pizza?L: One fourth.T: One fourth. And you took one half of the one fourth. You said its one half of one fourth.

Can you think of a fractional name for that piece?


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L: Four and a half!

T: Four and a half?L: Cause there will be four of them and then one half of it.

T: I dont think I see what you are saying. That is why I am asking questions. You want to usethat one (the stick in the Ruler). Do you want to use that one (the stick in the Ruler) andshow me what you are saying?

L: I dont know.T: Can you use that one (the partially covered 4/4-stick) to show me what you mean?L: (Shakes her head no).

T: Can you pull out your part?L: (Pulls out one of the two parts of the stick she refers to as one half of one fourth.)T: Well, how much is this of the whole pizza?

L: It would be half of one fourth.

T: Can you think of a way to find out how much it is of the whole pizza?L: I can measure it!T: Go ahead!L: (Measures and 1/8 appears in the Number Box.)

T: Can you explain to me why?L: Yeah. Because if you would put half on all of them, on all of, umm, and then if youll half

them all, then they would be one eighth because there are eight pieces!

As this teaching episode was held on 10 January 1994, there was no indication whatsoever that Laura

had constructed recursive partitioning over Christmas vacation. In fact, Laura interpreted the teacherscomment, You will have to show me, we will have to share one fourth of the pizza, by repeating the

stick into a 4/4-stick and then pulling the first two parts. Her comment, two fourths of the pizza will beone child, when coupled with repeating the 1/4-stick four times to make a 4/4-stick, solidly indicatesthat she interpreted the teachers comments as ifshe had constructed a partitive fractional scheme. That

is, her goal was to make two out of four equal shares of the whole pizza for one of the children and twofourths for the other.

After the teacher attempted to clarify the question, Laura partitioned the 1/4-stick into two equal parts

and said that her share would be one half of a fourth. This production of fractional language indicatesthat she took one out of four parts of the stick as input for further partitioning. Moreover, her surprising

answer of Four and a half as a fractional name for one half of one fourth and her rationale, Causethere will be four of them, and then one half of it. indicates an awareness of all four parts of the stick.

This awareness is corroborated when she explained why the 1/8 appeared in the Number Box aftershe measured the stick that was one half of one fourth of the whole stick Yeah. Because if you wouldput half on all of them, on all of, umm, and then if youll half them all, then they would be one eighthbecause there are eight pieces. She definitely distributed the operation of partitioning into one half across

the four parts of the 4/4-stick because three of the four parts were not visible. It would be necessary forher to operate on a re-presented 4/4-stick. Had she independently produced one eighth without first

measuring the stick she purported to be one half of one fourth and four and a half of the whole stick,then the inference that she made a recursive partition would be indeed strong. As it is, all that can besaid at this point in the teaching episode is that she distributed the operation of partitioning into halves

across the parts of a re-presented 4/4-stick when it was her goal to explain why one half of one fourth was


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one eighth of the whole stick. Her explanation was encouraging, so the teacher presented the situation of

Protocol IV to her.

Protocol IV: Lauras production of one twelfth as how much one of three shares of one fourth of a pizzais of the pizza.

T: (After Laura had erased the mark on the visible 1/4-stick) Lets say we were lucky the firsttime because we were only two. Now, we are not that lucky anymore. We are three people

all together; its you and me and Mr. Olive. He would also like to get some pizza.L: (Takes the mouse and starts to make another copy of the stick in the Ruler.)T: Oh, no, no, no. We have only one fourth of the pizza. All the rest is out. Can you show me

your share, my share, and Mr. Olives share, and tell me how much of the whole pizza isyour share?

L: Out of that one piece right there (points at the visible part of the partially covered stick)?T: Yeah. Only on that one fourth of the whole.L: (Dials Parts to 3 and clicks on the visible one fourth of the partially covered stick. She

then fills the left most part she made purple, which is her preferred color, and the middlepart green.) Ok (Looks at the teacher with confidence).

T: Which one is yours?

L: The purple.T: So, how much is your share, whatever type it is, out of the whole pizza?L: Ok. Umm, one twelfth!

T: One twelfth!! How come? Why?L: Because there are four (puts up four fingers) spots, and you put three in each one, and uh,

four times three is twelve!T: Did you say four spots? I was not sure I heard you right.L: Yeah. (Re-explains) well, there are four pieces of pizza, and then there are three pieces in

each, and then, and then three and four makes twelve.T: I see. Thats very nice, so what is my share?L: One twelfth.

T: And what about Mr. Olive?L: One twelfth.

T: All right. So, how much is the one fourth in terms of twelfths? For all the three of us together?L: Three twelfths.

T: Three twelfths. Thats very good. Can we go back to the one that we had before, when onlytwo of us shared, you and me? If I had one and you had one, how much was it?

L: Two twelfths!T: Can you tell me in terms of twelfths, how much is the part that is covered of the whole, how

much is it of the whole pizza?L: (After about ten seconds) twelve twelfths (looking uncertain)?

T: Is it twelve twelfths?L: Or four twelfths?T: Do you want to think about it a little bit more? Why twelve twelfths? Or why four twelfths?

L: Cause, if you put three in each one, that would make twelve, and its twelve twelfths.


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T: Ok, so you have twelve. But Im asking only of these that are covered. How many threes

can you put there?L: Nine.

T: Only nine? Why nine?L: Cause three fits in each one.

The explanation that Laura gave for why she said one twelfth Because there are four (simultaneouslyputs up four fingers) spots, and you put three in each one, and uh, four times three is twelve. indicates

that she may have recursively partitioned the 4/4-stick when it was her goal to find how much hershare of one fourth of the pizza was of the whole pizza. The comment, Because there are four spots,when coupled with simultaneously putting up four fingers, indicates that she visualized four spaces. Her

comment, you put three in each one further indicates that she inserted a unit of three into each ofthese spaces (one can also think of her inserting the operation of partitioning into three parts into each

of the spaces). In that these operations were carried out to serve the goal of finding how much of thewhole stick one third of one fourth constituted, this is the first indication that she engaged in recursivepartitioning operations (recursive partitioning always involves a units-coordination). The teacher was

encouraged, so he specifically asked her how much the one fourth is in terms of twelfths. Although thiswas a quite specific question, Lauras appropriate answer, three twelfths, indicates that she at leastmaintained an awareness of the partitioning she just made when she produced one twelfth. In that

she also answered two twelfths when asked about the one we had before, she apparently did notregenerate her experience of the immediately prior partitioning.

Her lack of regenerating the immediately prior partitioning when coupled with her difficulty in finding

how much the covered portion of the stick was of the whole pizza in terms of twelfths indicates that herway of operating to produce one twelfth was novel in the situation. Her answer twelve twelfths was

based on putting three in each one, that would make twelve, and its twelve twelfths indicates that shewas consumed by operating and did not make a distinction between putting three in each of the coveredparts and putting three in all four of the parts. One could claim that disembedding the complement of

one fourth in twelve twelfths was not an operation that she could currently perform, and hence a lacunaappeared in her reasoning. If disembedding three units of three from the four units of three produced bypartitioning was not available to her, this would be a contraindication of her construction of recursive

partitioning because it would indicate that she did not take a unit of units of units as a given in furtheroperating.

Laura definitely formed the goal of finding how much her part (one twelfth) was of the whole stick,but in doing so, finding one third of one fourth was only implicit in her activity. There was no indication

that she explicitly made one third of one fourth because she sharedone fourth of the stick among threepeople. This sharing activity would be sufficient to evoke sharing each of the remaining three coveredparts. In this case, she would not need to intentionally engage in the operations of finding one third ofone fourth and then ask herself how much thatpiece was of the whole stick. Rather, she could go directly

to finding how much her part was of the whole stick without explicit consideration of one third of onefourth. By-passing this step was productive for her as she did produce one twelfth and the operations

on which it was based.However, when she said three twelfths for how much the three of them together had of the whole

pizza, she had just said one twelfth to indicate how much each share was of the whole pizza, so to answer

three twelfths, all she needed to do was to unite the three parts together and use her partwhole fractional


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scheme to produce the answer. She could then use this way of operating to answer the question concerning

the share of three people in terms of twelfths. For his reason, I could not impute a commensurate fractionalscheme to her at this point in the teaching experiment even though she acted as if she had constructed

the operations necessary to produce three twelfths as commensurate with one fourth. Moreover, becauseshe shared one fourth of the stick among three people prior to saying her piece was one twelfth of thepizza, whether she engaged in recursive partitioning remains ambiguous because she had already formed

a partitioning goal.Laura independently decided to share the visible one fourth of the stick among four people after she

had completed the solving activity in Protocol IV. Her interactions with the teacher after partitioning

the visible one fourth of the stick constitute further contraindication that she had constructed recursivepartitioning. After using Parts to make the partition and Fill to fill the parts different colors, the teacherposed the opening question of Protocol V.

Protocol V: Lauras conflation of one sixteenth and four sixteenths.

T: Ok, and here comes the question. What is your share, or my share, or Mr. Olives share ofthe whole?

L: Four sixteenths.T: Is that my share?L: No, thats my share.

T: Umm, can you tell me why?L: Because four times four is sixteen.T: Ok. That tells me that you have sixteen of them over here (points at the whole stick). But do

you have four of them?L: No, but there are four pieces of pizza in all (points at the four visible parts of the stick).

T: So, how much do you get? Do you get all of the four sixteenths?L: No.T: How much do you have?

L: One piece of pizza.T: How much is your piece out of the whole pizza?L: One sixteenth.

T: When you said four sixteenths, what did you mean? Did you think of something else?L: Yes, of something else.T: Of what?

L: I dont know.T: How much is yours and mine?

L: Two sixteenths.T: How much is yours, mine, and Dr. Olives?L: Three sixteenths.

The way in which the teacher asked the question may have oriented Laura to interpret the question as

pertaining to all four people rather than to each individual.11 In any event, her answer four sixteenthsagain indicates that sharing one of the four parts of the stick evoked sharing the three covered parts. It

also indicates that she focused her attention on finding the fractional part that four shares of one fourth

11 The fourth person was left unnamed in the teachers question.


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of the stick was of the whole stick rather than on how much one fourth of one fourth of the stick was of

the whole stick. For this reason, I still could not impute the operation of recursive partitioning to her norcould I impute a commensurate fractional scheme to her.

There was good reason to believe that her partwhole fractional scheme was a reversible schemebecause she formed the goal of finding how much the four parts of one fourth of the stick was of thewhole stick.12 She also embedded the activity of units-coordinating within this reversible scheme in the

sense that sharing one of four parts of a stick into four parts evoked sharing the three other parts ofthe stick into four parts. Lauras activity in the very next situation of learning posed following ProtocolV contains corroboration that her partwhole fractional scheme was a reversible scheme and that she

had embedded units-coordinating in that scheme. The teacher told her that her share was one thirtyseconds, and asked her to figure out how many people would have to share the visible one fourth ofthe partially covered stick in order that she could get one thirty second. She immediately dialed Partsto 8 and clicked on the visible part. In answering the teachers question concerning why she knew

how to do that, she said, Because eight times four is thirty-two. In other words, she could produce thepartitioningoperation, eight, given a result, one thirty seconds,of the units coordination. The corroborationof embeddedness is found in her explanation because it indicated that she used the inverse of herunits-coordinating scheme to produce 8. Reversibility was involved in that one thirty seconds meant

that the stick was partitioned into thirty-two parts. After she produced eight as the partitioning operation,the most significant event of the teaching episode occurred. The teacher asked her how much three fourthsis in terms of thirty seconds, and she answered, twenty four thirty seconds because eight times three is

twenty-four. The consequences of this unexpected answer remain to be explored further in subsequentteaching episodes.

4. Lauras apparent construction of a fractional composition scheme

Of interest in the teaching episode held on 8 February 1994 is whether Lauras production that oneof three equal shares of one fourth of a stick is one twelfth of the stick meant that she could engage in

the productive thinking that would be an indication of a fractional composition scheme. That is, was afractional composition scheme in her zone of potential construction? To begin the investigation, afterLaura had drawn a stick the length of the screen, the teacher asked Jason to make two halves in the stick

and Jason partitioned the stick into two parts using Parts.

Protocol VI: Lauras enactment ofone half of one half.T: Lets say this is Jasons part (pointing to the left most one half of the stick), you see that

this one (the whole stick) is the whole stick, and you took half of it. Now, you are going to

take half of Jasons part.L: Right now? What do I do?

T: Yeah, right now. Jason, pull that one out (points to the right most one half) and label it,please.

J: (Pulls the right most one half out of the 2/2-stick and labels it 1/2.)

12 Laura did form the goal of finding how much one half of one fourth of a stick was of the whole stick in Protocol III, which

is also an indication of reversibility of her scheme for making proper fractions.


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T: Laura, now you have to take one half of this one half (points to the left most one half of the

stick.)L: Half of this half (picks the mouse up). Half of that half, half of that half . . . . Ok. (Clicks

on Parts and 2 is already showing, so she clicks on the left most half of the stick. Aftercoloring each of the two one fourth parts she made, pulls out one of the two parts.)

T: (Points to the part Laura pulled out) You could label it in terms of the w-h-o-l-e stick (runs

his finger along the whole stick)?J: (After a few seconds) I know (smiling).

L: (After approximately tens seconds) a half of one half.

T: Thats right. Thats a good one. So how much is it of the whole?J: I know it!

L: I dont know!

T: Can you use the computer to tell you?

L: Measure it. (Clicks on Measure

and then on the 1/4-stick, and 1/4 appears in the numberbox).T: Why is it (one fourth)?L: Because if you had all, if you had halved this one (points to the right most one half of

the original 2/2-stick), this one would be one fourth (pointing to the 1/4-stick she pulledout). Half of that half, half of that half . . . . Ok. (Clicks on Parts and 2 is already show-ing, so she again clicks on the left most half of the stick and then drags the 1/4-stick

underneath the left most one fourth of the original stick and then underneath the next onefourth.)

After reenacting the teachers language Half of that half, half of that half . . . , it is not surprising

that Laura partitioned the left most part of the 2/2-stick into two equal parts. Laura could obviously givemeaning to Half of that half, at least enactively. However, she could not say how much it was of the

whole stick. Nevertheless, she could explain (but not produce) why one half of one half is one fourthafter she measured the stick that was one half of a 1/2-stick: Because if you had all, if you had halvedthis one (the right most one half of the original 2/2-stick), this one would be one fourth (pointing to the

1/4-stick she pulled out). This comment seems to indicate that she at least completed partitioning the2/2-stick in her visual field into a 4/4-stick.

There is a crucial difference between explaining why one half of one half is one fourth after measuring

the stick that was created by taking one half of a 1/2-stick, and in producing one fourth as referringto the measure of that stick prior to measuring it as did Jason when he said he knew the answer prior

to measuring. After she knew that one half of one half is one fourth, only then did Laura partition theoriginal stick into four parts. The difference seems to reside in the availability for Jason of the operationsof recursive partitioning. When it was Lauras goal to explain why one half of one half is one fourth, she

used her units-coordinating scheme in the context of connected numbers, which to the observer produceda composite unit containing two composite units of two. But whether she structured the results of using herscheme in this way is equivocal even though she did seem to be explicitly aware of two units of two after

she measured one half of one half of the stick and found that it was one fourth of the stick. In any event,making one half of one half did not evoke her units-coordinating scheme, which is contraindication that

she had constructed recursive partitioning. But it cannot be said that Laura was incapable of abstractingthe operations she uses as indicated in Protocol VII.


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Protocol VII: Laura finding how much one half of one fourth is of the original stick: A contextual solution.T: Take half of this piece (points to the left most one fourth of the original 2/2-stick. By this

time, the left most one half of the original 2/2-stick was marked into two parts and the right

most one half was blank.)L: Lets see, that would be . . . one, two (in synchrony with moving her left hand and then

her right hand.) And that one will be (eyes upward, mumbling to herself. She then turns

and points to the teacher) I know what yours would be. (Sits quietly for approximately 80seconds while the teacher and Jason interact concerning the situation.)

T: (To Laura) Do you know? Do you want to say?L: (Puts two fingers up and moves them in synchrony with uttering) two, four, six, eight

(gesturing toward the stick with her two fingers.) It would be one eighth.

T: (Again, after interacting with Jason for approximately 50 seconds) Laura, how did youcome to know it will be one eighth?

L: (Leans toward the screen enthusiastically and refers to an 8/8-sick Jason had made duringthe approximately 50 seconds she sat idly) Because if you have, if you had just that onewhole piece (points to the left most 2/8 of the 8/8-stick placing her right index finger on

the left endpoint of the stick and her right thumb on the mark at the end of the secondone eighth of the stick) you can just copy it (moves her extended right index finger andthumb along the 8/8-stick as if she is making copies of initial 2/8-stick. When moving, she

places her right index finger and thumb so that they span each successive 2/8-stick.). I meanfour pieces, and you halved it, so you have two in each, and two times, so two and fourare . . . .

Lauras goal was to find how much one half of one fourth was of the original stick. Her comment, Because

if you have, if you had just that one whole piece. . .

you can just copy it. I mean four pieces, and youhalved it, so you have two in each, and two times . . . does indicate that she viewed the whole stick on

which she was operating as a 4/4-stick, and that she halved each fourth. It also indicates that she iteratedthe 2/8-stick to complete an 8/8-stick. Iteration was emerging in her partwhole fractional scheme, andshe seemed to be in the process of transforming it into a partitive fractional scheme. Because she iterated

the unit of two four times, the iteration emerged as an enactment of coordinating the units of four andtwo. Putting up two fingers up and moving them in synchrony with uttering 2, 4, 6, 8 while gesturing

toward the stick with her two fingers and moving her extended right index finger and thumb along the8/8-stick as if she is making copies of initial 2/8-stick, are both enactments of a units-coordination of fourand two. Both completed the production of a connected number consisting of four units of two.13 But, to

transform her partwhole scheme into a partitive fractional scheme, her iterative operation would needto emerge for unit fractions.

Lauras operating occurred after she had explained why one half of one half is one fourth in Protocol

VI, so the operations she used to partition each fourth into halves were still in a state of activation.Nevertheless, it is quite possible that such a re-enactment of immediate past operating in a new but highly

related task would engender the construction of recursive partitioning in the context of finding a fractionof a fraction. So, the teacher continued on, exploring whether she could find one half of one eighth.

13 In the case of counting by two, the results of the counting activity symbolized such a connected number as indicated by her

moving her right index finger and thumb along the 8/8-stick.


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Protocol VII: (Cont.)

T: That is so nice. So, what do you think will be your next step, after he labels it (Jason wastrying to use Label to label the piece 1/8).

L: Ah, half on that one (points at the first part of the 8/8-stick)?T: Good, very good (raises thumbs up. Jason engages in extraneous activity for approxi-

mately 50 seconds in an attempt to label a one eighth part of the original 2/2-stick. At the

completion of this activity, the teacher returns to Laura.) Can you tell me what will be thelabel, how much will it be out of the whole stick (one half of one eighth)?

L: It will be, hold on, hold on. Lets see (after about ten seconds during which she touches

each part of the 8/8-stick with the cursor) one sixteenth.T: (To Jason) What do you think?L: I know how to explain it.

T: (To Laura) You just wait thats good (laughing).

J: One eighteenth.T: One eighteenth? Ok, Laura, you explain, and then Jason, because you have a disagreement,. . . (intends for the children to work it out between the two of them).

L: Because, that will be one eighth, ah, the pieces. All right, right there, since you halved it

in there (points to the first part of the 8/8-stick), theres eight pieces and you halved themso theres two, and two times eight is sixteen.

T: (To Jason) How did you get eighteen?

J: (While Laura is explaining, he pulls out a 1/8-stick and partitions it into two parts. Hethen repeats the 2/16-stick he made to produce a 16/16-stick) two, four, six, eight, ten,twelve, fourteen, sixteen. (He then counts each part of the 16/16-stick) I miscounted them

(in consternation).

Laura independently enacted distributing partitioning into two parts across each part of the 8/8-stick bytouching each part with the cursor. Afterwards, her comment I know how to explain it indicated that

the result of her activity was meaningful. So, her goal of finding how much one half of one eighth was ofthe original stick evoked the operation of distributing partitioning into two parts across the parts of the8/8-stick, which is what units-coordinating means in the context of connected numbers. Coordinating the

operations of partitioning into eight parts and partitioning into two parts was at least partially achievedin that, given the results of the first partitioning, she could carry out the second partitioning to achieve

the goal of finding one half of one eighth. This constitutes progress in that she now seemed more awareof the operation of units coordinating than previously. She had definitely abstracted a way of operating

in the case of finding one half of one fourth, one eighth, or one sixteenth.Jason was caught up in the operations of TIMA: Sticks and, as a result, he had to extract himself

from the experience of labeling, attempting to separate a label from the stick to which it was affixed,etc., in order to produce one sixteenth as the result of taking one half of one eighth. In fact, he actually

partitioned a 1/8-stick into two parts and iterated it to produce a 16/16-stick while counting by twos whenhe reinitialized his solving activity to check his answer. This can be considered as an enactment of the

mental operation of recursive partitioning that led to his answer, one eighteenth. Although this is one ofthose cases where his fascination with operating in TIMA: Sticks substantially interfered with using hisfractional composition scheme at the level of re-presentation, when he found out that one eighteenth

was not correct, he independently reinitiated his solving activity to eliminate his error rather than simply


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accept one sixteenth as the answer. His confidence that his way of operating would produce the correct

results and his engagement in independent self-corrections and verifications was characteristic of hismathematical activity.

In the very next task following the continuation of Protocol VII, the teacher intended to asked Laura tofind what one half of one sixteenth would be, but instead, inadvertently ask her to make half of a 1/8-stick.This unintended question opens the way for making an evaluation of Lauras ways of operating when

finding one half of one eighth.

Protocol VIII: Laura finding one half of one eighth for the second time .

T: All right, its your turn to make it. Half of this one sixteenth, and then make it and labelit (points to the first one eighth part of the 8/8-stick. By this time, the 16/16-stick Jason

made was moved to the Trash, but the 8/8-stick was directly below the original 2/2-stick,so that the first parts of the two sticks were in almost perfect alignment).

L: (Makes a copy of that part of the 8/8-stick to which the teacher pointed. She then partitions

it into two parts using Parts and pulls out one of the two parts. She then repeats this stickinto a 16/16-stick. She then labels the last part of the 16/16-stick 1/16 using Label.)

In that Laura proceeded without comment, it is not certain that she knew that one half of one eighth wasone sixteenth prior to iterating the part she pulled out sixteen times. If she did engage in iteration of the

part to find what fraction it was of the whole stick, this corroborates the inference that she had constructeda partitive fractional scheme as a result of her activity in Protocol VII and its continuation. But it wouldindicate more because it would imply that iteration had become part of the activity of a fractional compo-

sition scheme. If she already knew that the stick that was one half of the 1/8-stick was one sixteenth of thewhole stick prior to iterating, then the iterative activity would be carried out in verification of the result of

partitioning each of the eight parts into two parts. The latter seemed to be the case, because Jason pulled

one part out of the 16/16-stick that Laura made in Protocol VIII and partitioned it into halves. After onlyfive seconds Laura repeatedly said, I know, I know, . . . . Moreover, she quickly calculated that one half

of one thirty seconds would be one sixty fourths. In fact, she abstracted a pattern in a review of her activity.In explanation of how she found one sixty fourths, she commented, Because, right here all you had wastwo (the 2/2-stick) and you added two more and thats four. And four more would be eight, and eight

more would be sixteen, and sixteen more would be thirty-two, and thirty-two more would be sixty-four.This pattern had to be constructed, and it was the result of using the results of prior operating in further

operating. Abstracting this pattern establishes Lauras abstractive potential, but it occurred in the contextof a regularity of operating using her units-coordinating scheme. Whether Laura had constructed recursivepartitioning operations turns on whether she had constructed partitioning as a symbolic operation.

5. On the symbolic nature of Lauras partitioning operations

A primary issue in attributing a fractional composition scheme to a child is whether partitioning onefourth of a stick into two equal parts symbolizes partitioning each of the other three fourths into two

parts. That is, a key to the establishment of partitioning operations as recursive operations is to establishthem as symbolic operations. Based on the above analysis of Lauras attempts to find one half of onefourth, one eight, etc., was of the whole stick, it was apparent that her units-coordinating operations were

symbolic operations. But this is not to say that taking a half of, say, one third would symbolize taking a


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half of each of three thirds. We have seen that Laura abstracted how to operate when partitioning each

fourth, each eighth, etc., into halves. Whether similar anticipatory operating would emerge in the absenceof immediate past experience was tested in a teaching episode that was held on 22 February 1994. The

teacher introduced TIMA: Bars to the children in a teaching episode held on 15 February 1994, so in thecurrent teaching episode, Laura was facile with the operations of this computer tool. Jason was absent,so it is possible to focus solely on Laura. The teacher started by drawing a bar and then copying the bar

so that the two bars were side-by-side.

Protocol IX: Laura attempting to find a half of a half of a half.T: All right. Lets say we take turns like we did last week. Jason is not here, but lets say that

he was here and he would have the first turn of halving. Then you do the second one, andthen Ill be the third one. After I finish mine, how much would you have here? How muchwould you have out of the whole (Laura was to use the copied bar)?

L: (Sits silently for approximately 33 seconds) um, I am not sure.

T: What were you thinking of. Before you do it, what were you thinking of?L: Can I do it? Well, and halving, and . . . one tiny little piece.

T: Do you have a guess for how much it would be out of the whole?L: (Shakes her head no.)T: Ok, go ahead and see what happens.

L: (Dials Parts to horizontal 2 and clicks on the bar. She then breaks the bar after the teachermade Break available and colors the top one half purple. She then interchanges the top and

the bottom parts.)T: Now, that was Jasons turn. Now it will be your turn and then it will be mine. Will you be

able to say now what will be my share?

L: (Activates vertical Parts and moves the cursor to the lower half. Parts is still dialed to2). Half will be there (points the cursor at the lower one half). Will it be one eighth?T: Wow! How did you know that?

L: (Takes the mouse, points the cursor to the purple half) because if you halved that one, andthen you would halve that one (the top one half), that would be four pieces. And then if you

halved that one, then that will be eight!

As Laura sat for approximately 33 seconds, the inference that she was in a state of perturbation is solidbecause when the teacher asked her what she was thinking of, she said, Can I do it? Well, and halving,

and . . . one tiny little piece. This comment indicates that she was aware of a result of halving whichwas one tiny little piece. So, the inference that she was aware of a discrepancy between a situation that

she visually experienced and this one tiny little piece that she anticipated making is corroborated byher saying, Well, and halving. That is, she knew that she needed engage in halving to produce this onetiny little piece, but she seemed unable to imagine herself engaging in successive halving operations insuch a way that she could take

01. Fractional Commensurate Composition and Adding Schemes Learning Trajectories of Jason and Laura...

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