Computer Tools for Interactive Mathematical Activity in the Elementary School
John Olive, Ph.D.
The University of Georgia
While much of the commercial software available for elementary school mathematics appears to be designed to support the "traditional" arithmetic curriculum through tutorial sequences followed by practice exercises, the mathematics education community needs to focus on ways in which computer-based environments can enhance childrens own construction of mathematics through interaction with other children and their teachers. Hoyles (this conference) has referred to this difference of use of computers in school mathematics teaching and learning as a "major fault line" and also advocates for "computational applications which point towards new, more learnable, more widely accessible mathematics; towards a redefinition of what school mathematics might become and who might be involved in it." Balacheff (this conference), in further describing such desirable computer applications notes that "The key feature of these environments, which share the common characteristics of being specific microworlds, is that they do not do the mathematics instead of the users but that they allow them to express their own mathematical ideas."
Over the past seven years a research team in the Department of Mathematics Education at The University of Georgia, headed by Dr. Les Steffe and myself, has been attempting to define "what school mathematics might become" for young children when they are given the opportunity to interact with other children and a teacher/researcher using computer tools "that do not do the mathematics instead of the users but that allow the users to express their own mathematical ideas." These computer tools were designed and developed as part of a research project on Childrens Construction of the Rational Numbers of Arithmetic (Steffe & Olive, 1990), supported in part by the National Science Foundation. These Tools for Interactive Mathematical Activity (TIMA) provide young children (grades K-8) with possibilities for enacting their mathematical operations with whole numbers and fractions. The software tools provide children with on-screen manipulatives analogous to counters or beads (regular geometrical shapes that we call "toys"), Sticks (line segments) and Fraction Bars (rectangular regions), together with possible actions that the children can perform on these manipulatives. These possible actions potentially involve the fundamental operations involved in the development of numerical schemes: uniting, unitizing, fragmenting, segmenting, partitioning, replicating, iterating and measuring (Piaget & Szeminska, 1965; von Glasersfeld, 1981; Steffe, 1994). These are the basic operations involved in developing numerical operations with whole numbers and fractions: addition, subtraction, multiplication and division of whole numbers and fractions, as well as the development of place-value concepts, families of equivalent fractions and simplification of fractions.
In this paper I shall briefly describe each of the TIMA environments, illustrating how they evolved out of our interactions with children, our emerging knowledge of childrens operations with fractional quantities and our knowledge of childrens operations with whole numbers. I shall illustrate how children have used these tools to build their own mathematical constructs within our teaching experiment. I shall also describe how we are using the tools to help pre-service elementary teachers better understand childrens mathematics, and how experienced teachers are using the tools within their classrooms. Each of these different scenarios provides evidence of the power of such computer-based learning environments when used in ways that are compatible with the constructivist philosophy that shaped their development.
Brief Descriptions of The Computer Microworlds
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 creation of an instance of a shape by the motoric action of a mouse click provides a link for the child between action (clicking the mouse) and the production of pluralities. The production of many toys, in snake-like configurations was an enjoyable entree into mathematical play on the part of the children.
Figure 1: Screen from TIMA: Toys
In designing the possible actions for the Toys microworld we wanted to engage children in units-coordinations that are the basis of multiplicative operations (Steffe, 1992). 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. The COVER action was included in each of the computer environments in order to encourage children to reprocess their perceptual collections to form figurative pluralities; that is, to re-present in their minds what is hidden under the cover. This reprocessing is critical to the construction of numerical concepts (Piaget & Szeminska, 1965; Steffe, von Glasersfeld, Richards, & Cobb, 1983).
The Sticks microworld takes the user into the realm of continuous linear quantities. Our goal in designing Sticks was to link the childs intuitive concept of "length" with their emerging concept of composite units in the creation of what we have called a "Connected Number Sequence" (Steffe & Wiegel, 1994). Horizontal sticks (line segments) of arbitrary length can be created (after selecting the DRAW button) 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 as the child, through the coordination of the motoric action with the visual result constructs a sensory experience of the "length" of a stick. Once created, sticks can be moved around the screen, copied (using COPY), marked arbitrarily by clicking at a position on the stick with the mouse cursor (after selecting the MARKS button), or partitioned into equal parts using a numerical counter (after selecting the PARTS button). The different parts can be filled with different colors (using the COLOR and FILL buttons). Parts can be "pulled out" of a marked or partitioned stick without destroying the original stick using PULLPARTS. 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 (using BREAK). Sticks can also be joined together (using JOIN) to form longer sticks consisting of parts representing the joined sticks.
Figure 2: Screen from TIMA: 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 using the MEASURE button. A fraction labeler is also available for labeling any stick or part of a stick with a fraction numeral. It is important to note that sticks are not labeled automatically. The child has to establish the numerical symbol that is meaningful for the child. Covers are also available in this environment.
The manipulable objects in TIMA: Bars are rectangular regions that the child can make simply by clicking and dragging the computer mouse. As in sticks, the coordination of the motoric action of dragging the mouse with the resulting visual display helps the child establish the size of the bar as a sensory experience. The 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 then new bars that 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 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 bars (representing candy or pizza) were to be shared.]
Figure 3: Screen from TIMA: Bars (finding 1/4 of 2/5)
A disembedding operation called PULLOUT (similar to PULLPARTS 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 PULLOUT operation enables the user to copy any connected set of parts of a bar as a new bar object consisting of those parts only. This proved to be a very powerful operation for the children in our experiment, as it provided a means of making comparisons between parts of a whole and the whole while not destroying the whole. The child could literally take the part out of the whole while still leaving it in the whole.
Bars which have at least one dimension the same length (height or width) can be joined together to form a new bar. This JOIN operation also proved to be very useful for the children as it provided a means of creating a referent whole from a part of the whole. 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.
Design and Use of the Microworlds in the Context of the Teaching Experiment
Our teaching experiment with 6 pairs of children spanned three years. We started working with the children in their third grade at school and continued through the end of their fifth grade year. We worked with a pair of children at a time, outside of the classroom for approximately 45 minutes each week for approximately 20 weeks each year. The six pairs of children were selected to provide us with a range of numerical development. All teaching episodes were videotaped using two cameras, one focused on the computer screen and one on the children and the teacher/researcher.
Our goal in using the TIMA tools in the teaching experiment was to provide the children with dynamic learning environments in which they were the primary actors. At the same time, these computer-based environments provided us (the teacher/researchers) with a medium in which the children can enact their mathematics in ways that we can interact with this mathematics, while trying to maintain the spontaneity of childrens actions.
Initial designs of the computer tools (Biddlecomb, 1994) were based on our understanding of childrens multiplying and dividing schemes (Steffe, 1992), and on information from research on childrens part-whole operations, pre-fraction and fraction schemes (Hunting, 1983; Kerslake, 1986; Kieren, 1988; Nik Pa, 1987; Ning, 1992; Saenz-Ludlow, 1994). These environments embody (for us, the designers) childrens conceptual operations. The possible actions programmed into the computer tools provided the children with ways to enact many of their conceptual operations. In particular, operations of unitizing, uniting, iterating, splitting, segmenting, partitioning, and disembedding with both discrete and continuous quantities were realizable through the possible actions with the computer tools.
As we interacted with the children in these early environments, our understanding of the childrens conceptual operations was illuminated by the childrens actions in the microworlds that they created. In many situations we have referred to the computer environments as "microworlds" as if these microworlds existed independently of any action of the children using them. From the childrens frame of reference, a "microworld" is established by them when actually working in the TIMA environments. In this we agree with Kieren (1994) who stated that "A microworld is brought forth by the child, based on her/his structures, occasioned by both the computer environment and the interactions with adults and other children in it." (p. 134)
During the course of the teaching experiment the possible actions (of the microworlds) were modified to better fit how the children appeared to operate, and new actions and features were added to broaden the scope of the childrens activity and to provide supports and constraints that might engender modifications in their ways and means of operating. In particular, we made each environment configurable by the teacher (or students) so that different microworlds could be created for (and by) different children by making certain actions and information available and other actions and information unavailable. This cycle of "design-interaction-understanding-improved design" continues as we further develop our models of childrens operations that generate the rational numbers of arithmetic.
From Cognitive Play to Independent Mathematical Activity
Papert (1980) portrayed a microworld as a self-contained world in which children "learn to transfer habits of exploration from their personal lives to the formal domain of scientific construction" (p. 177). It was our intention to create computer tools that would provide children with opportunities for cognitive play and subsequent mathematical activity that can follow on from cognitive play under the careful guidance of a teacher. In their paper on "Cognitive Play" Steffe and Wiegel (1994) reported on the spontaneity of childrens actions within the microworlds. They described how the children used the possible actions of the microworlds for functional pleasure, and how those play activities were transformed into pleasurable mathematical activity. The dynamic nature of the computer tools was a key factor in promoting cognitive play activity. For example, the dynamic nature of the Toys microworld encouraged the production of sensory pluralities through the coordination and repetition of the motoric act of clicking the mouse and the appearance of a "toy" on the screen. This production can be regarded as a fundamental operation of intelligence on which the construction of numerical concepts, composite units, number sequences, and more general quantitative reasoning is based (Steffe, 1991). Thus the childrens play activities in Toys became the basis for enactments of these abstracted structures. Steffe and Wiegel (1994) illustrated how the ensuing mathematical activities were shaped by the possible assimilations and accommodations of the childrens existing counting schemes and the interactions among the children and the teacher. For instance, the children spontaneously created designs (using the repeated clicking action) such as faces, Jack-O-Lanterns or elaborate names. The simple question of "How many toys do you think are in your design?" turned this play activity into a mathematical activity that was enjoyable for the children. They wanted to see who could make the best estimates! I call this important component of these teaching episodes "Cycles of Cognitive Activity." Children can progress from cognitive play to teacher directed mathematical activity, and then to independent mathematical activity, which may become mathematical play, that is, self-initiated independent mathematical activity with a playful orientation. The cycle may then continue on a more abstract level, thus creating a learning spiral. Steffe and Wiegel (1994) pointed out that social interaction between the students of each pair and among the teacher and students was a vital component of all activities in these cycles.
Teaching Interventions that bring forth Mathematical Activity
It has been suggested by Kieren (1994), that the mathematical features of the play (in Sticks) became available to the children mainly when occasioned by the teachers interventions. Although we must admit that the children did tend to find the Toys microworld more appealing than Sticks, the situations of learning in Sticks gave rise to more reasoning activities that pertained to childrens construction of fraction schemes than did the Toys environment. (That was our main reason for moving the children into the Sticks microworld.) The following example, taken from Olive (1996), illustrates the powerful reasoning that one child (Arthur) was able to bring forth while working in the Sticks microworld. The teachers interventions in this episode were definitely critical to Arthurs successful mathematical activity.
Constructing fractions of fractions. The task (posed to Arthur and Nathan during the Spring semester of their fourth grade year) was to share part of a pizza (represented at first by a stick in the Sticks microworld) among a number of friends and to find out how much of a whole pizza each friend would get. For example: A pizza (stick) is cut into seven slices (pieces). Three friends each get one slice. A fourth friend joins them and they want to share their three slices equally among the four of them. How much of one whole pizza does each friend get? The typical approach that the two boys took to this type of problem was to partition the stick into seven parts, pull out three parts, partition each of those three parts into four parts and pull three of these parts out for the share of one friend (see Figure 4.) They would then iterate this share four times to check that it matched the part of the pizza (stick) that they were given.
Figure 4: Four people share 3/7 of a pizza stick.
The difficulty for the two boys was in naming the share as a fraction of the original whole pizza. For instance, in the above situation Arthur named the share as three fourths of a seventh of the whole pizza. He then attempted to measure the original stick with the share of one person by repeating it to make a stick the same length as the unit stick. The repeated stick, of course, came up short (see Figure 5).
Figure 5: Nine iterations of 3/4 of 1/7 of a pizza stick.
In the very next teaching episode with Arthur he appeared to make the necessary modifications in his mathematical schemes and developed a powerful strategy for finding the product of two fractions. The teacher 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). The task was to share the four slices equally among five people so that each person gets a piece of each slice. Arthur 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 6).
Figure 6: Sharing 4/9 of a pizza stick among 5 people
The teacher asked Arthur to join the pieces that make the share for one person and to find out how much this was of the whole pizza. Arthur joined the four pieces in one row and compared this share to 1/9 of the original stick. He then thought for more than one minute, looking intently at the screen:
From Teaching Episode on 3/10/93
T: What do you think ?
A: I know it is 4/5 of a ninth of a pizza...
The teacher confirmed his response and asked if there was any way to find out how much that was 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)
T: Why is that?
Arthur explained: There are nine pieces (in the whole stick), five in each, so thats 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 thats 4.... and in the whole thing there are 45, so the share of one person is 4/45!
By focusing Arthurs attention on the unit of unit of units relation, the teacher helped to bring what I have termed (Olive, 1996) Arthurs 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. These actions were the building blocks for Arthurs reversible operations with fraction quantities. The teachers interactions with Arthur within the microworld that they had created together were instrumental in bringing forth Arthurs mathematical reasoning.
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.
From the Teaching Episode on 4/19/93
T (Olive): Lets have nine pieces in our pizza to start with. Arthur, how many pieces shall we use?
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: Ive 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?
Both children think for 30 seconds. Arthur stares intently at the screen, while Nathan stares off into space.
N: Its easier to do it when youve got it done. [Meaning: Its 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.
T: (To Arthur) Share this among seven people, please.
N: Ive no idea! [My] heads busted!
Arthur uses PARTS to partition the four-part piece HORIZONTALLY into seven rows of four.
N: Youve 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 7) 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 7: Filling 1/7 of 4/9 of a bar.
It is interesting to note that this was Arthurs 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! In this respect it was the interactions of the two children within the microworld that prompted a solution to the problem that had been created through the teachers intervention.
Use of the TIMA Environments in Pre-Service Elementary and Middle School Education
The TIMA software are used as tools for helping pre-service elementary and middle school teachers strengthen their own mathematical concepts as well as environments in which to work with young children. We have found that many of the students enrolled in our elementary and middle school education programs have (at best) only an instrumental understanding (Skemp, 1976) of the mathematical concepts underlying elementary school mathematics. For instance, before working with the TIMA, when given the following problem concerning the creation of a unit whole from a given fraction of the whole, over 90% of the class gave an understandable response similar to the one illustrated.
6. Given that the following bar is 3/5 of a whole candy bar, make the whole candy bar as accurately as possible.
Whereas less than 18% gave an acceptable response to the following similar problem:
12. The following segment is 7/5 of another segment. Explain how to make the other segment.
The same items were given to the class at the end of the course after the students had worked on similar problems using the TIMA environments themselves and with children at a local elementary school. The acceptable responses were the same for the 3/5 item but increased to 60% for the 7/5 item. There were, unfortunately, some students who could still not conceive of having 7/5 of something!
The college students in this class worked one-on-one with children at a local elementary school for approximately 40 minutes a week for seven weeks. During these intensive sessions with children, students were encouraged to use the TIMA to create microworlds in which they and their child could interact mathematically. Several did so and reported that they gained a lot of insight about the childrens mathematical thinking while working in these microworlds. They also reported that the children enjoyed working in the computer environment and posed challenging problems for one another. It is interesting to note that the one or two college students who found it difficult to create stimulating problem situations with the TIMA for their children were the same students who indicated a lack of relational understanding (Skemp, 1976) of their own mathematics.
Pre-service and inservice teacher education programs at other universities are using the TIMA software with their students and have reported similar results. Dr. Ron Tzur at Penn State University recently sent me the following:
The results of using TIMA microworlds in MTHED420 are very encouraging. Using the microworlds impact the prospective teachers shift from thinking about mathematics teaching and learning as development of skills of performing pre-given mathematical algorithms, to thinking about it as development of abstract conceptions resulting from reflecting on ones activities. . . Integrating the computer microworlds also contributed to the prospective teachers understanding of, and disposition toward implementing the teaching cycle, that is, to plan tasks that are likely to enhance childrens mathematical understandings, to interact with children toward this end, and to reflect on their plans and interactions in order to improve the mathematical learning experiences of their students. . . Experiencing first-hand interaction with children led the prospective teachers to change their view of the computer as being beyond the young childrens range of abilities, a view that was rooted in the teachers own fears and frustrations prior to participating in the course. The prospective teachers clearly said that working with the children was a valuable experience. Specifically, they pointed to the importance of experiencing the interactive nature of working with children in the microworld, how quickly the children learned to use the microworld, how deeply engaged the children were in exploring mathematical ideas, and how the computer can be a powerful tool beyond an entertaining machine for students who completed their routine seat work. They began to see the pitfalls of the "show and tell" approach to teaching mathematics, and to develop an alternative to this approach. This alternative stresses the active role of children in building up their mathematical understanding as a result of interacting with others and reflecting on their activities. Posing and solving mathematical problems in the microworld supports this kind of approach for both teachers and children. (personal communication)
Lessons learned from Classroom Use of the TIMA Software
The TIMA software have been used in elementary and middle grades classrooms over the past five years as part of an NSF supported Teacher Enhancement project: Leadership Infusion of Technology in Mathematics and its Uses in Society (Project LITMUS). Teachers were introduced to the software during summer inservice courses and supported during the school year as they attempted to use it with their students in their own classrooms. The ways in which the teachers made use of the software differed considerably among this group of approximately 100 teachers. Case studies of individual teachers (Pieper, 1995; Hanszek-Brill, 1997) suggest that the teachers own beliefs about mathematics and their own comfort with the use of technology as a teaching and learning tool greatly influenced their use of the technology with their students. Those teachers who believed that mathematics was a finished body of facts and procedures, and that the teacher should select, direct and control classroom activity based on the dictates of the curriculum guide tended to make minimal use of the TIMA software, preferring, instead, drill and practice computer "games" that reinforced the textbook curriculum. Teachers who held beliefs more compatible with a constructivist perspective on mathematics and learning (e.g. each persons mathematics must be constructed from experience in situations involving number and space, and exploration, discovery and verification are the essential processes of mathematics,) and who believed that the mathematics teacher should be receptive to student suggestions and ideas and should capitalize on them were more likely to use the TIMA software with their students in ways that encouraged students exploration and conceptual development (Hanszek-Brill, 1997).
Pieper (1995) related the following concerning a third grade teacher in her study: "in introducing mathematics topics, she focused upon conceptual understanding. She explained that to introduce her students to multiplication, she would engage students in a discussion about repeated addition, that in turn would lead her to introduce the term array. Then, students would build their own arrays using beans and a software program called Toys." (p. 63) This same teacher made extensive use of the computer lab with her students and developed a problem-solving task based on the childrens playful activities in the TIMA: Bars environment. The children had created figures out of different sized bars (head, body, arms, legs and feet). The teacher used the childrens figures to ask questions concerning the fractional relations among the components of the figure. For instance: If the "body" is the unit bar, what fraction of the body is each arm? She also followed up with challenges such as: "Can you make a figure in which the arms and legs are half the body, and the feet are one fourth of the body?"
Teachers from other parts of the US have reported similar excitement in their use of the TIMA software in their classrooms. A sixth grade teacher from Pennsylvania recently wrote the following: "I have found the microworld to be extremely helpful in exploring many different mathematical concepts with my students. It has served as an introduction, an extension, an enrichment and sometimes a complete curriculum for concepts we work with in our sixth grade mathematics curriculum. Currently the program is installed in our computer lab and my students each have access to a computer on which they can discover mathematics by engaging in such activities as manipulating shapes and drawing sticks that can easily be divided in parts and moved around the microworld." (Coon-Kitt, personal communication)
In a companion study to the research project in which the TIMA software were developed, DAmbrosio and Mewborn (1994) worked with children from a fourth grade classroom in the same school as the research project. During the three weeks that they worked with groups of children from this classroom they found ways to implement the constructive pedagogy espoused by the Project. Although they found both advantages and disadvantages of using manipulatives such as Pattern Blocks and paper folding, they found the dynamic visualization capabilities of the microworlds to be empowering for the children. The microworlds, however, could not be used without the presence of a teacher to pose problems and guide the children in their explorations. A major conclusion from their experience was the need for children to verbalize their thinking rather than us "filling in the blanks" for them and assuming that we know what the children really mean.
The most striking difference between the teaching experiment and the classroom setting was the role of the teacher. DAmbrosios and Mewborns intention in their activities in the classroom was to create a teaching-learning situation in which they could seek to understand the students meaning making from the experiences they invited them to engage in and find ways of modifying those meanings when necessary. To this end, the focus of their attention was on the Childrens Mathematics, whereas, in contrast, the focus of the classroom teacher was on the mathematics of the textbook, and her interactions with the children were to determine whether she could move on to the next topic or whether further practice or instruction was necessary.
Conclusions and Recommendations
The Design and Intent of Computer Environments
Kieren (1994) has strongly suggested that it is the subsequent action of the child that determines the effectiveness of an intervention rather than the intent of the teacher. Thus, in considering the design of a computer environment or the nature of teacher or researcher interventions, we should look at what the child does rather than simply characterize the intervention by its intent. Kaput (1994) also suggests that "whether or not something is a feature for a student depends very much on what the student is capable of seeing at a particular time." (p. 144) We would go further and say that the features (of a microworld) are products of the students actions within the microworld and do not exist independent of those actions. Bowers (1997) comes to a similar conclusion in her work on place value with two children using a computer environment designed to link actions on graphical icons and numerical symbolic representations of the results of those action in an inventory form:
It appears that Martine did not view the dynamic inventory form as a way to enact changes on the graphics because such a device is inconsistent with his interpretation of the microworld as an empirical simulation world in which one acts as one would in real life. That is, in real life, it is not possible to act on numbers to change objects; one acts on objects and then records the results with numbers. (p. 615)
Bowers continues that:
for Carolyn, the linked representation system was self-evident, and she was comfortable enacting transactions by clicking on the numbers on the inventory form or clicking on the graphics. The critical distinction between Carolyns interpretation and that of Martine is that her efforts to anticipate the results of her acting were implicit in her decisions. In contrast, for Martine, who had not generalized his situated activities, the link between the notation systems was not apparent, nor was it an integral part of his activities. (p. 616)
Bowers concludes that she would recommend a revision to the computer software that eliminated the linked representation system. "Activities with the revised microworld would involve having students conduct transactions and transformations with the graphics, but develop their own ways to record their actions. . . In this way, the use of numbers arises as a natural way to expedite the symbolizing process." (p. 616)
We actually witnessed this natural production of a numerical symbol system when two of our children in the teaching experiment were working with the TIMA: Bars microworld on recursive fractionizing tasks (Steffe and Olive, 1996). Joe and Melissa, two 10-year-old children in our teaching experiment, were asked to copy a unit bar and then "fourth" the copy, break it and fill one of the fourths; then "third" the filled fourth, break it and fill one of the thirds; etc. Joe was to make and fill fourths and Melissa, thirds. We pick up their mathematical activity after Joe had made and filled a fourth, Melissa had made and filled a third of the fourth, and Joe had made and filled a fourth of the third of the fourth (see Figure 8, from Steffe and Olive, p. 122).
Figure 8: Fouthing and Thirding in TIMA: Bars
Joe had figured out from the pieces on the screen that his brown piece would be 1/48 of the unit bar and had verified this using the MEASURE feature of the microworld. Melissa had said that she was going to do "15 times four" to find the size of Joes piece. At this point the teacher handed Melissa a piece of paper and a pencil and asked where she got the fifteen times four from. Melissa wrote on her paper for approximately 45 seconds, creating a sequence of superscript whole numbers followed by a resulting fraction, along with a picture of the result of their activity in the Bars microworld (see Figure 9, taken from Steffe & Olive, p 124). In the following protocol (also taken from p. 124) T stands for teacher, J stands for Joe, and M for Melissa:
T: (After about 15 seconds) so youre starting to make a sequence of pieces here. (After about 30 seconds) now youre drawing a picture of the pieces. (Indicates to Joe that he should show what he did using his paper and pencil.)
How did you get your answer, Joe, while shes working on hers?
J: I did 12 times 4.
T: 12 times 4 -- O.K.
M: Yeah, thats 1/12. And the next one was one forty-eighth. First, he did his fourths, then I did my thirds, then he did his fourths.
Figure 9: Melissas numeric sequence and picture
We regard Melissas sequence of superscript numerals "4", "3", "4" as symbolizing using PARTS and BREAK. The fractional numerals symbolized for Melissa the results of each symbolic recursive partitioning operation because she wrote the results of "fourthing", "thirding", and then "fourthing" as "1/4", "1/12", and "1/48" rather than as !/4, 1/3, and 1/4. As we stated in our article:
There would be no possibility that she could write, say, "1/48" as the result of fourthing 1/12 unless she believed that each twelfth, which she had not made in the microworld, was partitioned into fourths. Moreover, when she drew the picture, she concentrated on her drawing and glanced only momentarily at the screen, which indicates to us that her drawing was produced using her symbolic partitioning operations. (Steffe & Olive, 1996, p. 125)
We concur with Bowers that a linked representational system in this situation, that might have automatically generated the symbolic representation of the results of the childrens partitioning acts in the microworld, would have robbed the two children of their own generative symbolic activity. As designers of computational environments for children, we need to think carefully about the contributions that the children need to make to the situation in order to build their own mathematical structures. The possible microworld actions and representations do not (by themselves) determine the structure of the childrens actions and representations. Rather, the childrens intentions, as well as the conceptual operations available to them, are also involved.
Implementing Interactive Software Tools in the Classroom
We believe that the intentions of our teaching experiment are realizable in classrooms where teachers share those intentions. The obstacle is a lack of understanding of those intentions and a lack of the knowledge and skills needed to make sense of and build on childrens meaning making. The challenge for us as mathematics educators and for teacher education in general is to find effective ways to help pre-service and inservice teachers construct these understandings, knowledge and skills. The use of software such as our TIMA in pre-service and inservice education, together with intensive interactions with children using these tools has helped students to construct their own understandings. The use of videotaped teaching episodes from such teaching experiments as well as from classroom settings could also be of great help. The new theory of mathematics learning that we are attempting to build (Olive, 1994; Steffe & Tzur 1994; Steffe & Wiegel, 1996) could also provide the necessary orientation for effective use of these resources and tools. The recommendations for use of technology in elementary mathematics in the Standards 2000 documents need to address this critical issue of teacher preparation for effective use of the technology as well as the issues of design and intent.
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