Presentation for Topic Group 19: Computer-Based Learning Environments
The Eighth International Congress on Mathematical Education (ICME-8)

SEVILLE, SPAIN, July 14-21, 1996

Back to TG 19 Panel Discussion

Constructing Multiplicative Operations with Fractions using Tools for Interactive Mathematical Activity (TIMA) Microworlds

John Olive
The University of Georgia

Introduction

Children's construction of operations necessary for building the rational numbers of arithmetic has been an important issue in the psychology of mathematics education for several decades (Behr et al., 1980; Kieren, 1988; Behr et al., 1992). Some attempts have been made to use computing technology to aid children's learning of fraction concepts but this has most often been in the form of tutorials or drill and practice software. Few research efforts have attempted to use the computer as an integral part of the shared learning environment for the teacher/researcher and students. The computer microworlds that were used in this investigation were developed as part of an NSF research project on children's construction of the rational numbers of arithmetic. They have been designed as tools for the children to develop and enact their operations on discreet and continuous quantities. But they are also tools for the teacher/researchers to construct situations in which they can test their emerging models of the children's mathematics. As such, they may offer the constructivist researcher a powerful, dynamic medium for investigating children's constructive itineraries.

The Computer Microworlds

TIMA: Toys
The Toys microworld is an environment in which instances of manipulable shapes (called toys) can be created simply by clicking the computer mouse. Five different shapes (triangle, square, pentagon, hexagon and heptagon) are available. The "toys" can be joined together in a string (like a string of beads) by clicking on them in succession. These strings of toys can then be moved as a whole (a composite unit), copied (to make multiples of composite units through iteration), joined together to make longer strings, cut apart to make shorter strings, and combined into a new two-dimensional composite unit called a chain of strings of toys (the horizontal strings are joined together vertically to form the two-dimensional chain). In the latest version of Toys the chains can also be combined into a three-dimensional unit called a stack.
Several operations are available for working with strings and chains. Strings can be Repeated to create chains, and chains can be repeated into stacks. Toys can be added to or removed from the end of a string or each string in a chain using the One More or One Less buttons. These buttons can also be used to add to or remove strings from a chain. Numerical information on the number of toys, strings and chains can be obtained via menu selections. Any object or group of objects can be covered so as to hide them from view.

Figure 1: Screen from TIMA: Toys

TIMA: Sticks
The Sticks microworld takes the user into the realm of continuous linear quantities. Horizontal sticks (line segments) of arbitrary length can be created simply by dragging the mouse cursor across the screen. The extent of the dragging motion determines the length of the stick. This link between motion and resulting length is an important aspect of this environment. Once created, sticks can be moved around the screen, copied, marked arbitrarily by clicking at a position on the stick with the mouse cursor, or partitioned into equal parts using a numerical counter. The different parts can be filled with different colors. Parts can be "pulled out" of a marked or partitioned stick without destroying the original stick. These "pulled out" parts then become new sticks, thus allowing comparisons of part to whole and whole to parts without destroying the whole. A marked or partitioned stick can be broken up into its constituent parts. Sticks can also be joined together to form longer sticks consisting of parts representing the joined sticks.
Any stick can be designated as a "ruler" for measuring purposes. The measure of other sticks relative to the designated unit stick can then be obtained. A fraction labeler is also available for labeling any stick or part of a stick with a fraction numeral. Covers are also available in this environment.

Figure 2: Screen from TIMA: Sticks

TIMA: Bars1
The manipulable objects in TIMA: Bars are rectangular regions (referred to as "candy bars") that the child can make simply by clicking and dragging the computer mouse. The candy bar created in this way can be moved around the screen, copied, marked both horizontally and vertically by line segments, partitioned both horizontally and vertically into equal sized parts (the orientation and number of the parts determined by the user). The pieces created by the MARKS operation, or the parts created by the PARTS operation can be filled with different colors and unfilled. The subdivided bar can also be broken apart into its sub-components (pieces or parts). These sub-components are now new candy bars which can be further subdivided. There is a SHADE operation that enables the user to shade any part of a bar horizontally or vertically. This is a dynamic operation, controlled by dragging the mouse. The intersection of horizontal and vertical shadings is discernible as a solid color (the individual shadings are translucent). A CUT operation allows the user to cut a bar apart either horizontally or vertically. Bars can be hidden under covers and operated on while hidden. Regions called MATS can also be created on which to place candy bars. Mats are not movable and cannot be operated on. They can, however, be covered. [Mats were often used in the teaching episodes to represent elements (or people) among which the candy bars were to be shared.]

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

A disembedding operation called PULLPARTS (similar to that in Sticks) was added to this microworld about half way through the first year of the teaching experiment. It was always possible to make images of parts or pieces of a bar, but these images were ephemeral in that they were not new objects that could be operated on. They were used to make indirect comparisons of one part with another. The PULLPARTS operation enables the user to copy any connected set of parts of a bar as a new candy bar object consisting of those parts only. This proved to be a very powerful operation for the children in our experiment.
Candy bars which have at least one dimension the same length (height or width) can be joined together to form a new candy bar. This JOIN operation also proved to be very useful for the children. Candy bars can also be rotated through 90 degrees. Any bar can be designated as the unit bar so that the measures of other bars can be obtained relative to the designated unit bar. Numerical information concerning the number of bars, number of parts in a bar, or number of bars or parts in a designated region of the screen can be obtained via menu selection.

#### A Brief Introduction to the Research Project

Six pairs of children worked with a teacher/researcher for 45 minutes a week for approximately 75 weeks (over a three year period). Each of the 75 teaching episodes were videotaped, using two cameras: one focused on the children and teacher/researcher, and the other on the computer screen. This paper focuses on the two most advanced children in our Project: Arthur and Nathan. Nathan participated in all three years of the Project. Arthur joined the Project in the final two years (grades 4 and 5) as Nathan's partner. Through ongoing analyses of the videotaped episodes a model of the children's development of their multiplicative operations on fractions, and their reasoning with ratios is being constructed.
The basic hypothesis of our teaching experiment was that children would construct their fraction schemes through modification of their whole number operations based on their abstract number sequences. Our models of Arthur's and Nathan's mathematical development indicated that they had both constructed what we refer to as a Generalized Number Sequence prior to their work together in the final two years of our Project.

#### Children's Abstract Number Sequences

Steffe and Cobb (1988) developed the notion of children's abstract number sequences from their teaching experiments with young children. They described the development of three successive number sequences: the Initial Number Sequence (INS), the Tacitly Nested Number Sequence (TNS) and the Explicitly Nested Number Sequence (ENS). The following key psychological aspects of number sequences have emerged as a result of our work in the current teaching experiment:
· Number Sequences are mental constructs: schemes.
· The possible operations associated with a number sequence emerge from the interiorization of activities that children engage in through applications of their prior number sequence. That is to say that children may do things in action first what they are not yet able to do mentally.
· Re-interiorization is the product of recursively applying the operations of a scheme to the results of a scheme.
· An Iterable unit is the result of reversible operations (One iterated five times produces one five, that can be partitioned into five ones). Reversible Operations are produced through recursive thinking.
· The operations of lower order Number Sequences reappear in higher order sequences but are applied to the more complex unit items. This may be similar to Kieren and Pirie's (1991) notion of Folding Back in their Recursive model of mathematical thinking.
Explanations of the INS, TNS and ENS can be found in Steffe & Cobb (1988). For the purposes of this paper it is important to realize that a Generalized Number Sequence (GNS) is the result of a re-interiorization of a child's ENS.

A Generalized Number Sequence (GNS) The re-interiorization of the ENS results in iterable composite units in much the same way as the re-interiorization of the TNS resulted in iterable "ones." With an iterable composite unit of four, say, a child can conceive of "four" six times as being the same as having a composite unit of six with a unit of four items in each of the six unit items that constitute the six "ones." The "ones" in the composite unit of six have become place holders for any type of unit (singletons or composites). We propose that this re-interiorization takes place through the recursive applications of a child's units-coordinating operations to the results of those operations: abstract composite units.

#### From Iterable Composite Units to Lowest Common Denominator

A composite unit is iterable for a child if the child can re-present and combine iterations of the composite unit prior to acting (Steffe, 1992). In his last teaching episode with the microworld Toys (11/15/91), Nathan revealed his ability to coordinate operations on two different iterable composite units. His coordinations involved keeping track of the iterations of the two composite parts which he had united into a new composite whole -- a strong indication of distributive reasoning.
[Note: This teaching episode is taken from the first year of the Project before Arthur began working with Nathan.]
Nathan and his partner Drew had created a set of strings of toys (linked toys) with from one to six toys in each string. The task was to use copies of the 3-string and the 4-string to make a set of 24 toys (see Figure 4). Nathan reasoned out loud as follows:
Three and four is seven; three sevens is 21, so three more to make 24. That's four threes and three fours!

Figure 4: Using a string of three and a string of four to make 24 toys

Nathan created an iterable seven consisting of an iterable three and an iterable four. Nathan saw seven as composed of a three and a four, and he was able to keep track of each part within the seven as he multiplied the seven by three to get 21: three sevens was also three threes and three fours. This result was then the input for further operating on the separate sub-parts to arrive at the solution of three strings of four and four strings of three. These coordinations could be thought of as figurative distributive reasoning. Nathan was able to distribute the multiplication of seven by three across the constituent addends of seven (three and four) in order to obtain three threes and three fours. This was an application of his generalized number sequence. He established two number sequences "side-by-side", as it were, one for threes and one for fours. Moreover, he could operate on these two sequences and combine them element for element.
As a result of his coordinations, Nathan had constructed 24 as a partitioned unit with two sub-partitions: three fours and four threes. He also used his decomposed seven later in the episode to work out seven times eight: "That's 32 and 24!" -- strong confirmation that distributive operations were available operations within his generalized number sequence.
Although Nathan's actions in the microworld were minimal in the above situations, his prior activity within the microworld of creating, joining, copying and breaking apart strings and chains of toys contributed to his ability to mentally represent the combinations of the different strings. Our goal in working with the children in the microworlds was to progress from actions on the objects to mental representations of those actions by the children. Such progress is critical for the children's construction of mathematical operations.

#### Nathan's Strategy for finding the Lowest Common Denominator

Nathan was able to apply his units-coordination scheme for whole numbers to construct a procedure for finding the lowest common denominator (LCD) of two fractions. This construction also took place during Nathan's third grade year, one year prior to his work with Arthur.. Nathan did not have as his goal the generation of the LCD of two fractions. The LCD was never referred to in any of our teaching episodes with Nathan. I use it only as a short hand name for Nathan's procedure.
Nathan's goal was to find a partition for a candy bar (in the Microworld TIMA: Bars) that would allow him to pull out both one third of the bar and one fifth of the bar. His procedure was to count by threes and by fives until he found a common number in the two sequences. He would think to himself in the following way: 3, 6; 5, 10; 9, 12, 15; 15. It's 15. He would then put 15 parts in his bar, pull five out for one third and three out for one fifth (see Figure 5). He was able to coordinate and compare his two sequences of multiples until a common multiple was found, but he also knew how many of each multiple he had used to get to this common denominator; he had kept track of each sequence. Nathan was able to carry out these coordinations because "three" and "five" were available to him as iterable units.

Figure 5: Pulling out 1/3 and 1/5 from a 15-part bar.

The following year Nathan was able to use his knowledge of common multiples to obtain a similar result in a more efficient way: In a more complicated task (situated in TIMA: Sticks) requiring the children to make a fraction of a stick starting from a different fraction (e.g make a ninth of a unit stick using a twelfth of a unit stick), Nathan was eventually able to partition the 1/12 into three to make 36ths. He knew this would work because "both 9 and 12 add up to 36...four nines are 36 and three twelves are 36, so four of these will be 1/9." Nathan had begun a process that would lead to what we call a co-measurement unit for the two fractions.

#### Constructing Fractions of Fractions

The example given in the last section of making a fraction using a different fraction as a starting point was not easily achieved by the two boys (Arthur and Nathan). Even though we were convinced by their prior actions that both children had constructed a Generalized Number Sequence, it took Nathan four teaching episodes over a period of a month to construct a scheme that he knew he could use to solve such problems. For Arthur, the struggle took much longer. One stumbling block that they met was to name a fraction of a fraction as a new fraction. They could produce and anticipate a "fifth of a fifth" but did not know its value as a single fraction. In order to establish its value they had to iterate the result 25 times to reproduce the unit stick. They then knew that the result was 1/25. Their operations for making a fraction of a fraction, while recursive in nature, were not yet reversible. Eventually, they developed the ability to mentally project the partition of three equal parts of say one twelfth, into each of the 12 twelfths in the unit in order to establish the value of a third of a twelfth as 1/36.
Even after developing the ability to create and name a unit fraction of a unit fraction (as in the above example) the two boys had difficulties extending their strategies for finding, say 1/4 of 3/7 of a unit stick. One particular type of task, at first using the Sticks microworld and eventually transferring the task to the Bars microworld, proved to be engendering for this development. The different potential actions provided by both of these microworlds, we believe, were critical in the eventual development of a co-measurement scheme for fractions.

Modifications of Arthur's Schemes for Multiplying Fractions

The limitations of Arthur's fraction schemes for constructing a fraction of a non-unit fraction became very evident in the two consecutive teaching episodes in which he worked without Nathan, who was absent from school. The constraints Arthur encountered in these episodes and the modifications he made in the application of his unit fraction schemes in overcoming these constraints provide the bases for the development of his reversible partitioning operations. Arthur was working with Azita, one of the graduate research assistants on our project.

Protocol I (from 3/8/93)
Task 1: A Pizza Sharing Problem in the STICKS microworld.
In this task Sticks are used to represent pizzas.
Azita (T) poses the situation: Three people got 3/7 of a pizza.
Arthur makes a stick and uses PARTS to divide it into seven pieces; he pulls out three, unconnected pieces using PULLPARTS.
T: Then another person came in and we had to share it (the 3/7) fairly among the four people. How much did each person get?
After a false start on the problem in which Arthur used five instead of four as his new partitioning unit the teacher attempt to refocus on the beginning situation:
T: What are you trying to do with these three pieces (the 3/7)?
A: Cut it into four equal pieces.
T: What would be the easiest way of doing such cutting?
Arthur erases all marks on the 3/7, divides it into three parts again, and then divides each of these three parts into four parts. He then counts the 12 parts in four triplets: 1,2,3; 1,2,3; 1,2,3; 1,2,3 (indicated by the movement of the mouse).
T: Can you tell me what your are thinking?
A: I think every person gets three (of the 12 parts).
Azita asks him to fill in the share of one person and prove that it works.
Arthur fills three of the 12 parts, pulls these three out (see Figure 6), REPEATs them four times to make a stick the same length as the 3/7 stick.
[This action is an application of his Unit Fraction Scheme.]
T: How much of the whole pizza does one person get?
A: (To himself) Of the whole pizza, of seven pieces?
Three fourths of the seven.
T: Do you think so? Show me 3/4 of the whole pizza.

Figure 6: Four people share 3/7 of a pizza stick.
Arthur restates his answer: Three fourths of a seventh of the whole pizza. He then tries to use the MEASURE button to find out what fraction this would be but is prevented by his teacher. He takes the share (3/28) and repeats it, trying to replicate the whole stick, but it does not fill up the whole stick (because 3/28 is not reducible to a unit fraction and, therefore, does not segment the whole). He ends up with nine iterations of one share (27/28). (See Figure 7.)

Figure 7: Nine iterations of 3/4 of 1/7 of a pizza stick.

This action was again an application of his Unit Fraction Scheme, but this time the action was inappropriate as Arthur was trying to iterate a non-unit fraction to reproduce the whole. The 3-part stick, that was one of the four children's share, was a co-measure for the three fours and the four threes that Arthur had created, but it was not also a co-measure of the whole 28-part stick. The goal of finding 3/4 of 1/7 of a stick was not attainable with his current operations that were based on his strategies for finding a unit fraction of a unit fraction--a modification of these unit fraction schemes was required that would enable Arthur to make units-coordinations with three different levels of units. He needed to be able to decompose the 3/4 of 1/7 as 3 of 1/4 of 1/7. He could then use his recursive partitioning operations to find 1/4 of 1/7, and use his uniting and unitizing operations to take 3 of these 1/28 as one thing.

Protocol II (continuation of Protocol I)
Arthur seems to be frustrated, because he does not know why he did not fill out the whole stick with his nine iterations.
A: An unequal amount!
T: What do you mean?
A: One person is gonna be stuck with a little teeny bit (the extra 1/28 at the end of the original stick).
T: I want you to be thinking about it, how much of the whole pizza does one person get?
A: Less than what they were supposed to (get).
T: What do you mean?
A: They were supposed to get 1/7, now they have less than 1/7--they have 3/4 of a seventh.
[Arthur is referring back to the original context of the problem statement--three children share 3/7 of the pizza, then a fourth child joins them.]
T: You think so?
A: They have 3/4 of a seventh of a whole.
T: How much of the whole pizza is that?
Arthur was not able to answer this question because 3/4 of 1/7 was not an iterable unit for the whole--it could not be used to find a whole-number measure of the whole. This was a constraint of his Unit Fraction Scheme. Arthur was able to make coordinations with three levels of units as indicated by his response that the children would get 3/4 of 1/7 of a whole pizza. This was a strong indication that Arthur had constructed a Generalized Number Sequence. With this number sequence, a reversible partitioning scheme should be within the zone of Arthur's potential construction (Olive, 1994). In a prior episode with me (2/2/93) Arthur had eventually worked out 1/33 of 1/5 by multiplying 33x5, thus I was sure that 1/4 of 1/7 was a solvable problem for Arthur.

Two days later (3/10/93) Arthur appeared to make the necessary modifications in his schemes and developed a powerful strategy for finding the product of two fractions. Azita had posed the situation of a pizza stick with nine slices, each slice having a different topping (indicated by filling each of the nine parts in a different color). She asked Arthur to pull out four different slices, which he did (three of them attached and one separate piece). She then posed the following problem:

Protocol III (from 3/10/93)
T: We want to divide the four slices equally among five people so that each person gets a piece of each slice.
Arthur at first was confused and divided each of the four slices into four parts. Through his actions, however, he eventually came to understand the problem situation and started over by erasing the parts from the slices. He then divided each of the four slices into five parts and broke each slice up. He arranged the broken slices in five rows of four (one piece from each of the four slices in each row -- see Figure 8). The teacher asks Arthur to join the pieces that make the share for one person.

Figure 8: Sharing 4/9 of a pizza stick among 5 people
T: How much of the whole pizza did one person get?
Arthur joins the four pieces in one row and compares this share to 1/9 of the original stick. He then thinks for more than one minute, looking intently at the screen.
T: What do you think ?
A: I know it is 4/5 of a ninth of a pizza...
Azita confirms his response and asks if there is any way to find out how much that is of the whole stick.
A: Yes there is, but... (trails off and thinks some more).
T: How many of these small pieces do you have in the whole thing ? (pointing to one of the four parts of one share)
A: 45
T: Why is that?
Arthur explains: There are nine pieces (in the whole stick), five in each, so that's 45.
T: How much of the whole pizza is one share then?
A: 4/45. He explains: Because this is the share of one person..... and that's 4.... and in the whole thing there are 45, so the share of one person is 4/45!
By focussing Arthur's attention on the unit of unit of units relation, the teacher helped to bring Arthur's Recursive Partitioning Scheme into play, enabling him to work out the unit fraction size for the smallest part. It was then a simple matter of uniting the four unit fraction pieces to establish the share as 4/45 of the whole. He was now able to decompose 4/5 of 1/9 as 4 of 1/5 of 1/9. That Arthur had constructed this as an enactive strategy was confirmed in the next two tasks.

Protocol IV (continuation of Protocol III)

T: (Draws a stick) The stick that I am thinking of is 2/3 of 1/7 of that stick.
Arthur uses PARTS to divide the stick into seven parts, then the first part into three, then pulls two of these three pieces (see Figure 9).

Figure 9: 2/3 of 1/7 of a stick

T: How much of the whole stick is that?
A: There are 21 of that [the small piece] in the whole stick so its 2/21.
T: How can you make sure?
A: Use the measure button.
T: Without measuring!
Arthur colors the two pieces red, pulls a third part, joins it with the 2/21, then REPEATs the 3/21-stick seven times to form a stick the same length as the original stick (see Figure 10). He explains why it is 2/21:
A: The two that are filled in are the two that I started with, and in the whole thing there are 21 [referring to the sub-partitions].

Figure 10: Iterating 3/21 to make a whole stick.
Arthur's actions of re-forming a seventh by adding the third piece to his 2/21 and then repeating this partitioned seventh seven times to make the whole were in contrast to his application of his Unit Fraction Scheme in the previous teaching episode (3/8/93). He was able (this time) to reverse his actions to form a measurement unit (3/21 or 1/7) for the whole stick instead of trying to use the 2/21 as a measure for the stick. This was the first indication that Arthur was possibly in the process of constructing a reversible partitioning scheme as is indicated by his actions in the final two tasks.

Protocol V (continuation of Protocols III and IV)

Task 3: Sharing 5/11 among 7 people.
The teacher instructs Arthur to draw a stick, divide it into 11 parts, pull five parts out, and share those five parts among seven people.
T: How much of the whole stick does each person get?
Arthur looks at the screen, thinks for a few seconds and then divides one of the five parts into seven; he pulls a small piece out of this partitioned part. He then erases the marks and thinks some more.
T: What were you going to do?
A: I don't know...every body would get... If you divide each piece into seven pieces then everybody would get a piece from each piece, so five pieces. If there are 7 in 1/11 then there would be 77 in the whole and that would be 5/77.
Arthur then repartitioned the 1/11 part by seven and pulled five of the 77ths out of this piece, even though he had said "everybody would get a piece from each piece." He must have seen the equivalence of the 5/7 of 1/11 with 5 of 1/7 of 1/11 and used the former as the more efficient way of operating.
The observer then posed a question to Arthur: What's 1/7 of 5/11?
A: I don't know.
The observer repeats the question. Arthur thinks for several seconds, looking at the screen, and then responds:
A: I think I have--It would be the same piece, I think. [He drives the 5/77 piece around on the screen.]

In the last task in this teaching episode Arthur illustrated the power of his generalized number sequence for computing 1/8 of 1/12, and also showed us that the result of his newly formed reversible partitioning scheme was a unit that he could use in iteration.

Protocol VI (continuation of Protocols III, IV and V)

The teacher instructs Arthur to draw a stick, divide it into 12 parts, pull out five, and share these five among eight people. She then makes the following request:
T: Show me the share of two people and how much that is of the whole stick.
Arthur wasn't sure of the problem so she repeats the question.
A: (After several seconds of thought) Maybe 5/8 of 1/12 of a whole for one person. [Again indicating the reversibility of his order of operations.]
T: How much of the whole stick?
Arthur divides 1/12 into 8 parts and pulls out five.
A: If there are 8 in 1/12 then there would be....16 in 2, 24 in 3, 32 in 4, 62 in 8 and...wait 4x8 is 32 and 4x32 would be....
T: What are you trying to do?
A: I'm trying to find out how many in the whole, I think. If there are 8 in one piece, I'm finding out the number in 4 pieces, and since there are three 4's in 12, I'm adding the amount in 4 three times....So 32 to 64 to 96.....[An illustration of the operations of his generalized number sequence.]
T: So how much of the whole pizza would that be? -- The share of one person?
A: 5/96 and two people would be 10/96.

The contrast in Arthur's capabilities at the end of these two teaching episodes (3/8/93 and 3/10/93) is quite remarkable. In the first he was constrained by the use of his unit fraction scheme, which he inappropriately applied to a non-unit fraction; in the second episode, his recursive partitioning scheme was brought to the fore through the attention of his teacher on the relation of the result of his recursive partitioning action to the original unit whole. Constructing this relation appeared to allow Arthur to unpack the composite unit fractions (to decompose) and recompose the order of operations. These actions were the building blocks for Arthur's reversible operations with fraction quantities.

#### The Pizza Task using TIMA: Bars

Nathan had similar problems with the Pizza Task as Arthur, especially in naming the resulting share of one person when sharing 4/9 (say) of a pizza among seven people. I decided to introduce the Bars microworld as an environment in which they could model this type of problem. While the Bars microworld did not provide actions for easily enacting recursive partitioning operations in the same way that the Sticks microworld did, it did provide a two-dimensional environment with the possibility of partitioning bars both vertically and horizontally. It was my hypothesis that this potential of making cross-partitions of a bar would provide a more explicit model for creating a fraction of a non-unit fraction of a pizza.

Protocol VII (4/19/93)

T (Olive): Let's have nine pieces in our pizza to start with. Arthur, how many pieces shall we use?
A: Four.
T: O.K. Pull out the four pieces, Arthur. [Arthur does so.]
T: You are going to share those four pieces among seven people.
A & N: Seven?
T: Seven. Before you do anything, do you think you can figure out how much of one pizza each person will get?
N: I've got it! [Nathan reaches for the mouse.]
T: Wait. How much of a pizza do we have here?
A & N: 4/9.
T: 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 stares off into space.
N: It's easier to do it when you've got it done. [Meaning: It's easier to figure it out after you carry out the actions.]
T: Tell me what you would do.
A: If there are seven pieces in four then you have to think about how many in eight and then how many would be in the remaining one to make nine.
N: I've no idea! [My] head's busted!
Arthur uses PARTS to partition the four-part piece HORIZONTALLY into seven rows of four.
N: You've done it! Each person gets one of those strips. (pointing to a horizontal row of four)
While Arthur is filling the share of one person (the top row -- see Figure 11) Nathan works out the number of small pieces in the whole bar and the fraction name for the share of one person:
N: Four times seven is 28, 28 and 28 is 56, and seven more makes 63. Each person gets 4/63!

Figure 11: Filling 1/7 of 4/9 of a bar.

It is interesting to note that this was Arthur's first session using the Bars microworld, whereas Nathan had used it for more than half of his first year in the project. And yet it was Arthur who eventually came up with the cross partitioning that provided Nathan with a solution to the problem! The engendering power of this cross partitioning action in TIMA: Bars was evidenced the following year (2/22/94) in a similar problem dealing with investment shares of a gold bar.
Problem situation: Five people buy equal shares in a bar of gold. Three of the people pull out of this corporation, taking their shares with them. A new person invests in this group. What amount of the original gold bar does each of these four people now own?
The two boys solved the problem by pulling 3/5 out of the 5-part gold bar (three vertical parts), partitioning this 3/5 bar horizontally into four parts, pulling out one horizontal row and comparing this amount to the original gold bar (see Figure 12).

Figure 12: 1/4 of 3/5 of a gold bar.

They then superimposed the partitioned 3/5 over the original bar and mentally extrapolated the horizontal partition to arrive at 20 small pieces in the original bar, giving them 3/20 as the new share of the gold bar for each of the four people in the new corporation. The teacher then posed a follow-up question:

Protocol VIII (2/22/94)

T: What if two people leave the new corporation (selling their shares to the remaining two)?
N: That's easy! Half of 3/5 of a whole is 3/10.
Nathan then explained how this problem was related to the previous problem:
N: You just divide it by two. One sixth becomes 1/12, and 3/10 become 3/20. Two of these 20ths make one of these tenths.

In the next task (seven people share a gold bar, four pull out and form a new corporation with two new people) the boys are asked to solve the problem without pulling out the share of the people who leave the corporation. They quickly solve the task by filling four of the seven vertical parts and then partitioning the whole bar horizontally into six parts (see Figure 13). Nathan calculates the share of one person in the new corporation by multiplying 7 times 6 to get 42 and then taking 4 of these.

Figure 13: Finding 1/6 of 4/7 of a gold bar
N: That's 4/42!
A: 4/24.
N: 4/24 of the new corporation and 4/42 of the original corporation.
T: Can you measure one person's share?
Arthur designates the bar as a Unit Bar and pulls out one person's share from the shaded part of the bar.
T: What will it measure?
A & N: 4/42.
N: Wait! You can divide that by two! It will be 2/21.
They measure the pulled out part and verify it as 2/21 of the original bar.
In the next task, a situation that could be regarded as division of fractions was introduced.
T: I am in a corporation of four people sharing a gold bar. Three of us pull out, so we have 3/4 of the bar, but I can only afford to own 1/8 of a gold bar. How many people need to join the new corporation so that each person only owns 1/8 of a bar?
N: Three fourths. Oh, oh! We need three! Yes, three!
A: If we cut it in half we'll have eight pieces.
N: No, six (pointing to the last three parts in the bar).
A: Of the whole bar she said. That's eight (pointing to the whole of the bar).
N: Oh yeah!
Arthur partitions the bar into halves horizontally.
A: Everyone will have to sell half their share.
T: So how many people do we need in the new corporation?
N: Six!
The teacher then asked Nathan to set Arthur a similar problem.

Nathan's Problem: Four people buy a gold bar; two people pull out. They want to form a big corporation so that each only owns 1/16 of the original bar (they are very poor!). Arthur solves the problem by first modeling the situation (pulling 2/4 out of a 4/4 bar), and then partitioning the original bar into four parts horizontally to make 16 parts. At Nathan's suggestion he also partitions the 2/4 horizontally to get 8/16. He then says that 8 people need to be in the new corporation.
The above sequence of tasks with gold bars illustrates how useful the cross partitioning facility had become for these two children. Their reasoning also suggests that the multiplicative relations among the fraction quantities involved were becoming explicit. The fact that Nathan could quickly assimilate the "division" situation in order to pose a similar problem for Arthur suggests that these multiplicative relations were reversible for Nathan.

#### The Classroom Algorithm for Multiplying Fractions

Later in the same teaching episode (2/22/94), the teacher asked the two children to show her 2/3 times 4/5 using the microworld. Nathan says out loud "2 times 4 is 8 and 3 times 5 is 15, so it has to be 8/15." When asked to show that using TIMA: Bars, Nathan created a bar, copied it, partitioned one bar into thirds and one into fifths. He pulled out 2/3 from the thirds bar and 4/5 from the fifths bar. He lined these two pieces up at the bottom of the screen and said that he would put an X between them if he could! He appeared to make no connection between the learned classroom algorithm for multiplying fractions and the actions they had been carrying out in the Microworld. Arthur, on the other hand, when pressed by the teacher to verify that the result would, indeed, be 8/15 partitions the 4/5 piece horizontally into thirds and pulls out two of these. He then superimposed this cross-partitioned piece on the original fifths bar and explained how there were eight small pieces in his piece and there would be 15 of them in the whole bar.
In the next problem posed by the teacher (show me 3/4 of 9/16) Nathan again uses his mental algorithm to obtain 27/64. Arthur agrees with Nathan's answer and both children appear stumped in finding a way to show the result in the microworld. They create a bar partitioned into 16 parts vertically and pull out 12 of these parts saying "12/16 is the same as 3/4." It is only after the teacher reminds them that she wants 3/4 of 9/16 that Arthur again makes the breakthrough:

Protocol IX (continuation of 2/22/94)

A: Deah! Just like we did the other time! We cut it into four the other way and take out three lines. [Arthur carries out his actions, creating a piece with 27 parts.]
N: See! 27/64. I told you!
T: I see "27" but I don't see "64."
Nathan then partitions the 16-part unit bar horizontally into 4, pulls one piece out and measures it obtaining 1/64 in the NUMBER box.
N: There!
A: You can just do 4 times 16 gives you 64 (pointing to the 4 rows and 16 columns on the unit bar).
I was observing this episode at the time and asked the two boys the following question concerning the rule Nathan was using with the numbers to find the answer:
O: What makes more sense to you, the numbers or the model?
N: The numbers!
A: The model!
N: The numbers are easier for me. All you have to do is multiply them.
O: I didn't ask which was easier, I asked which makes more sense.
A & N: The model!
O: Why?
A: Well , it makes more sense with the model because you can show anybody on the model, but people who can't figure it with numbers have no idea!

Nathan and Arthur had been solving situated problems (that from our point of view involved multiplication of fraction quantities) using the microworlds TIMA: Sticks and TIMA: Bars for almost two years. These situated problems appeared to be separated in their minds from the algorithms they were learning in their classroom work. Their actions in the microworlds indicated that they could interpret fractions as both quantities and operations (9/16 was a quantity, taking 3/4 of it was an operation to be carried out), but for Nathan, when presented with two fractions to be multiplied, both were numerical quantities (as evidenced by him showing both fractions as parts of the same whole) and multiplication was an extension of his whole number multiplication procedure. Even though he stated at one point that "the X means 'of' with fractions" he did not easily connect this with his operational scheme for "taking a fraction of something."

#### Fractions as Measures: A Path to Rational Numbers

While fractions had been constructed by Arthur and Nathan as both quantities and operations at this point in the teaching experiment, we did not think that they had constructed what modern mathematicians would characterize as "rational numbers." From our point of view a fraction scheme would need to include fractions as measurement units to be regarded as a scheme for generating the rational numbers of arithmetic. With fractions as measurement units, division of fractions becomes meaningful. For example, the questions "How much of 3/4 is 1/8?" or "How many eighths in 3/4?" require finding the measure of 3/4 in terms of 1/8 as a measurement unit. When children have constructed fractions as measurement units they have the possibility of constructing any fraction from any other by finding a co-measurement unit for the two fractions. From a mathematician's point of view it is just such a possibility that generates closure on the field of rational numbers.
All measurement involves the ratio comparison of two quantities. An indication that Arthur and Nathan had constructed fractions as measurement units would be their ability to make ratio comparisons among fractions. We developed several tasks in the TIMA microworlds that we believed would engender the development of this ability. One such task was modeled after the paper folding activities described by Kieren (1990) in which children compared the results of successive halving actions by folding a sheet of paper several times. We simulated a more generalized, successive fractioning activity using TIMA: Bars by successively partitioning and breaking a bar, and then continuing the process on just one of the resulting sub-bars. In the following episode (3/8/94) Nathan was to make thirds of the selected sub-bar on each of his turns and Arthur was to make fourths of the subsequent sub-bar. Protocol X begins just before Nathan's second turn as the "thirder."

Protocol X (3/8/94)

T: Before you do that, predict how many, what size piece do you think you will get?
N: Thirty six. One thirty sixth is right there! (as he carries out his thirding activity -- see Figure 14).
T: Why 36?
N: Because you are timesing it by three. You are breaking up all the 12 pieces into three.
[Note that there are not 12 twelfths visible as a result of the first two steps! (see Figure 14) Nathan is imagining the results of "symbolic action." (Steffe and Olive, 1996)]

Figure 14: "Thirding" 1/4 of 1/3 of a unit bar.

The episode continues in this way until the resulting bar is so tiny that it is hardly visible on the screen. Arthur correctly identifies his resulting bar as 1/1728 of the original unit bar (see Figure 15).

Figure 15: Measuring 1/1728 of a unit bar
T: How many of those 1/1728 does it take to make 1/144?
A: Wait, which one of these was the 1/144? This one? Nathan nods "Yes."
A: Then it would be four times three, 12!
T: How many 1/1728 does it take to make 1/12?
A: (After correctly identifying a piece that is 1/12 of the original bar) O.K. That is 12 (pointing to the 1/144 piece that he had just worked with) and four times 12 is 48 and 48 times 3 is...
T: Do you need some paper? (Hands a sheet of paper to Arthur) The question is how many 1/1728 do we need to make 1/12?
A: (After writing the sum of 96 and 48 on his paper) 144!
N: Yeah, 144!
In both of the preceding problems Arthur reversed their prior actions in the microworld to produce his result. He was able to do this because the result of each action was represented on the screen. The following question asks the boys to make a comparison using a fraction that was not a result of their actions in the microworld.
T: How about if we want to use the 1/144 piece and we want to find out how to make 1/6?
A: First we need to make a sixth.
T: Use what you have there.
A: (Pointing to two of the twelfths) These two combined will make a sixth. (Arthur moves one of the 1/144 pieces.) Four, eight, twelve, 24. Twenty four 1/144 to make 1/6.
T: Why is that?
N: Because there are four in 1/36, and 3/36 make 1/12,
A: and three times four is 12,
N: then you have to double that to make 1/6,
A & N: So 24!
In the next task, the teacher used COVER to hide the fractured unit bar with all the resulting sub-bars after moving 1/432 piece to the bottom of the screen.
T: I want you to write down on your paper how you made this piece.
Nathan writes 3x144 on his paper.
T: I want you to write down all the steps starting with the unit bar.
Nathan writes: Unit ÷ 3 -> one third of that ÷ 4 -> one 12 ÷ 3 -> one 36th ÷ 4 -> one 1/144 ÷ 3 -> one 432 is the result.
Arthur writes: Took a unit bar, divided it by 3, took a third of that, divided it by 4, took a fourth of that divided it by 3, took a third of that divided it by 4, took a fourth divided by 3.
N: I was using it as 1/3 of the whole bar, then 1/12 of the whole bar and 1/36 of the whole bar, not just of the new fraction.
T: Then how is your's different from Arthur's?
N: He's using the new fraction as a measurement.
Nathan's last statement indicated to us that he understood the role of a fraction as a measurement unit. The teacher asked a clarifying question of both children at this point.
T: Do you mean Arthur changed the unit each time?
A & N: Yes.

Subsequent activities in this same episode also indicated that both children realized that changing the order of their operations would produce some different fraction pieces but the end result would be the same if they took the same number of turns. They always represented their action in the microworld as whole number division and the result of their action as a fraction. In verifying that the resulting fraction would be the same if Arthur took all his turns as "fourther" before Nathan took his turns as "thirder" Arthur wrote down the following:
1÷4=1/4÷4=1/16÷4=1/64÷3=1/192÷3=1/576÷3=1/1728.
At this point the teacher asked Nathan if he could write the same thing using fractions and multiplication. He was not to use division. Nathan wrote the following:
1 x 1/4=1/4 x 1/4=1/16 x 1/4=1/64 x 1/3=1/192 x 1/3=1/576 x 1/3=1/1728.
He commented that this was just using the reciprocal. The teacher asked him to explain.
N: One third is the reciprocal of 3 and 1/4 is the reciprocal of 4. That's how I do division on paper.

The above teaching episode indicated to us that Nathan and Arthur had constructed an isomorphism between the operation of taking a unit fraction of something and dividing by the whole number reciprocal of that fraction. Coupled with the notion of a fraction as a measurement unit these were strong indications that their fraction schemes were undergoing modifications that would result in a scheme for the rational numbers of arithmetic.
The modification of the children's fraction schemes to include whole number division as a reciprocal operation to fraction multiplication could also constitute a re-interiorization of their Generalized Number Sequence (GNS) inwards that produces recursive divisions of the number sequence rather than the outward direction of their GNS that provides for exponential structures. That is, instead of producing units of units of units, they could now produce units within units within a unit. And we believe that this re-interiorization inward is the necessary accommodation of a GNS that will generate Rational Numbers of Arithmetic.

#### References

Behr, M., Post, T., Silver, E., & Mierkiewicz, D. (1980). Theoretical foundations for instructional research on rational numbers. Proceedings of the Fourth Annual Conference of the International Group for the Psychology of Mathematics Education, Berkeley, CA.

Behr, M., Harel, G., Post, T. & Lesh, R. (1992). Rational number, ratio, and proportion. In D. Grouws (Ed.), Handbook of Research on the Teaching of Mathematics. Macmillan Publishing Company.

Kieren, T. E. (1988). Personal knowledge of rational number: It's intuitive and formal development. In J. Hiebert & M. Behr (Eds.), Number Concepts and Operations in the Middle Grades, 163-181. Hillsdale, NJ: Lawrence Erlbaum Associates.

Kieren, T. E. (1990). Children's mathematics/mathematics for children, in L. P. Steffe & T. Woods (Eds.) Transforming Children's Mathematics Education: International Perspectives, pp. 323-333. Hillsdale, NJ: Lawrence Erlbaum Associates.

Kieren, T. E. & Pirie, S. E. B. (1991). Recursion and the mathematical experience. In L. P. Steffe (Ed.) Epistemological foundations of mathematical experience. New York: Springer-Verlag, 78-101.

Olive, J. (1994). Building a new model of mathematics learning. Journal of Research in Childhood Education, 8(2).

Olive, J. (1993). Children's Construction of Fractional Schemes in Computer Microworlds, Part II: Constructing Multiplicative Operations with Fractions. A paper presented at the research presession of the Special Interest Group for Research in Mathematics Education (SIG-RME), National Council of Teachers of Mathematics, Seattle, WA.

Saenz-Ludlow, A. (1993). Ann's fraction schemes. Paper presented at the First Conference of the International Working Group on Rational Number. University of Georgia, Athens, GA.

Steffe, L. P. (1992). Schemes of action and operation involving composite units. Learning and Individual Differences, 4(3), 259-309.

Steffe, L. P. (1993). Interaction and children's mathematics. Paper presented at the Annual Meeting of the American Educational Research Association, Atlanta, GA.

Steffe, L. P. & Cobb, P. (1988). Construction of arithmetical meanings and strategies. New York: Springer-Verlag.

Steffe, L. P. & Olive, J. (1996). Symbolizing as a constructive activity in a computer microworld. Journal of Educational Computing Research, 14(2), 103-128.