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Merging Physical Manipulatives and Digital Interface Facilitates the Transition to Abstract Knowledge Anna Zacchi, Nancy Amato Technical Report 99-023 Department of Computer Science Texas A&M University October 1999

ABSTRACT

During the history of geometry people gradually acquired physical knowledge of space, and later consolidated it in a more abstract form of knowledge, geometry. In this paper we describe how elementary school students used physical manipulatives in conjunction with the digital interface of educational software for geometry. The addition of physical manipulatives helped the students to consolidate their physical knowledge of the world and smooth the transition from the physical world to the abstract. The blending of physical manipulatives and digital interface also helped them to overcome the limits of the representation and interaction modalities of the digital interface. Keywords

Manipulatives, children, educational software, interface for education, geometry, learning. INTRODUCTION

For a long time geometry was intimately tied to physical space, beginning as a gradual accumulation of subconscious notions about physical space and about forms, content, and spatial relations of specific objects in that space. This early geometry of the Egyptians is called 'subconscious geometry' [1]. Later, the Greeks consolidated some of the early empirical knowledge into a collection of somewhat general geometrical laws or rules, developing what is called 'scientific geometry' [1]. In the classic period they again changed the nature of geometry creating what is called 'demonstrative geometry' [1]. The truths in 'demonstrative geometry' must be supported by (deductive) proofs rather than only by inductive or empirical evidence as in 'scientific geometry'. This axiomatic-deductive method is the cornerstone of modern mathematics. Demonstrative geometry is part of the western tradition and is the teaching objective in the schools. The ways in which to reach this objective are different. To guide the students through the same phases

experienced in the development of geometry seems to be the most natural strategy. First the student accumulates "subconscious" notions about physical space, forms, and spatial relations, then consolidates these notions in geometrical rules, and finally moves to an abstract level and works with concepts, axioms, and deductive rules. How to enable and structure these transitions between levels of knowledge is very important. National educational reform movements are advocating that students be actively engaged in "constructing" their own knowledge. How can educational software help in this process of facilitating the student in constructing his/her knowledge? In particular, how can software help students in geometry to transit from 'subconscious' geometry to scientific geometry, and lastly to demonstrative geometry? Software running on a computer is something abstract: when the student rotates a geometrical shape on the screen s/he is actually dealing with an abstract representation of the shape and an abstract representation of the interaction that causes the object to rotate. How does the representation of this reality impact the student's ability to accumulate notions about space and spatial relations? In this paper we describe a study on the use of physical manipulatives with educational software and their effect on the students' ability to acquire 'subconscious' notions of geometry and transition to scientific geometry. Our study used SuperTangrams [6], a program developed by the EGEMS group at the University of British Columbia. SuperTangrams is a computer-based mathematics learning environment aimed at helping middle-school children learn two-dimensional transformational geometry. We introduced manipulatives that students could use while playing with the software and studied their effect on learning. The idea of manipulatives for mathematics is not new, they were first introduced by Montessori [3]. Our novel idea is to use manipulatives in conjunction with the software in order to overcome the representational

problems of the interface and smooth the transition from the concrete to the abstract world. Manipulatives allow the students to experience what they are doing on the screen in the real world. At the same time, manipulatives help researchers understand the student's comprehension of the interface. Observing a student moving a manipulative indicates what she expects the software to do. Our study revealed several representation and interaction problems in the interface. We describe how students used the manipulatives and we suggest that interface problems can be overcome by simultaneously designing the interface for educational software and the manipulatives. This paper focuses on the observations of the interactions. Detailed analysis of empirical data will be presented in a companion paper. RELATED RESEARCH

A variety of research efforts have recently explored computationally augmented interfaces that emphasize human interaction with the physical word. The Tangible Media group at MIT is creating a repertoire of physical devices for Tangible User Interface (TUI)[8]. In particular the Bricks [2] "graspable user interface" project involves putting one or more bricks -abstract physical blocks - onto some screen-based virtual object. Bricks can then be used to physically rotate, translate, or scale and deform the attached virtual entities by manipulating the brick devices. TUI seeks to build upon the sophisticated skills that people have developed for sensing and manipulating our environment. How can these skills be used in a learning environment? The goal of TUIs is to facilitate the human-computer communication by taking advantage of the richness of multimodal human senses and skills. The goal of educational software for geometry is to gradually lead the student to learn the language -geometry- that describes the physical word. How can we take advantage of the student's knowledge of the physical environment? Our research focuses on how blending physical devices with the interface can facilitate the transition from physical knowledge of the environment to abstract knowledge. Another important research trend in education explores the enhancement of physical objects by adding computational power. Digital manipulatives [5] are computationally-enhanced versions of traditional children's toys. The goal is to provide children with tools that enable them to explore complex concepts (for example "systems concepts" such as feedback and emergence) through direct manipulation of physical objects. Digital manipulatives should prepare the student to "manipulate" and "physically" learn complex concepts before s/he is able to deal with them in a formal way. In our research we started with simple physical objects that are already common to the student's world. It would be interesting to expand the study by blending digital manipulatives with

software that helps the student to formally learn the complex concepts after s/he had learned them "physically". MANIPULATIVES

In 1907 Maria Montessori [3] developed special manipulative materials for the children of her "Children's House" to help them learn to read, write and do mathematics. Later Cuisenaire invented a set of colored wooden rods that quickly became famous all over the world. Since then manipulatives have been widely used in schools. The use of physical teaching aids gives rise to active manipulation and discoveries by the child herself. The operations done on a computer can be considered an extension of the use of manipulatives. However, since these operations are far more abstract than the operations on physical objects, the question is how well are the students able to relate them to the physical world? SUBJECTS AND SETTING

The study involved approximately 180 fifth grade students from a College Station, Texas, elementary school. These students were from six different classes (two teachers), each of approximately 30 students. None of the participants had used SuperTangrams before. The students had not been exposed to transformational geometry before the study and both teachers participating in the study agreed not to teach any topics related to geometry or transformational geometry during this time. All the students were very familiar with the Macintoshes in the school's computer lab. METHODS

All the students had access to the lab for a period of 50 minutes a day for 10 school days that covered approximately two weeks. The study was divided in two sessions, with three classes in each. To study the effect of the manipulatives we divided the students in each section in four groups and we provided each group with a different set of tools. The following table details the group subdivision: Groups

Session 1 Session 2 1 2 3 4 1 2 3 4 X X X X X X X X X X X X X X X X

SuperTangrams Notebook (scratch paper) Manipulatives set 1 Manipulatives set 2 Table 1: Division of Tools among groups.

The task of each student during the lab was to play with SuperTangrams to solve the tangram puzzles. The student was free to use the tools that we placed on the side of his computer according to the group division. During the lab hours, two or three researchers were always present in addition to the class teacher and the lab operator. In our research protocol researchers were free to observe the students and interact with them, to talk with them and to answer to their questions.

Data collection techniques included class observations, video-recordings, scratch paper used, pre-tests and posttests, questionnaires and log files of computer sessions. SUPERTANGRAMS

SuperTangrams is a computer-based learning environment for two-dimensional transformational geometry. Learning of this mathematical domain is situated in the context of the traditional tangram puzzles game activity. Game activity

The player is challenged to put together several geometric shapes to fill a given outline (fig 1.) To move the shapes on the screen the student selects and applys to them one of three transformations: translation, rotation, and reflection.

manipulation must be directed at the representation of those concepts – that is, interface elements that represent the structural and semantic properties of the concepts. For example, in the case of rotation the object is the shape and a representation of the concept of rotation can be the center of rotation and the angle. Sedighian's study [6] shows that DCM improves the learning experience of the students. We extended the study focusing on a different aspect: the representation chosen for the concepts. The representation and the chosen interactions, according to the DCM style, are the means through which the children come in contact with the concepts. They should represent a mirror of the actions performed on the objects in the real world. Through them the students should learn about the language of geometry. Our questions are: Are these representations appropriate? Can the use of manipulatives enhance the children's ability to acquire subconscious notions? Can the merging of physical manipulatives with the digital interface help the students' transition between the physical and the abstract world? Representation of geometric concepts

SuperTangrams implements three types of geometric transformations: translation, rotation and reflection. These concepts are directly represented in SuperTangrams with images upon which the students can interact. Translation Fig.1. Screen shot of SuperTangrams showing a translation[6]. At the first level the game displays the ghost image. The translation vector is not connected to the shape. Moving the tail or the head of the vector moves the ghost image.The shape is translated and goes to match its ghost image when the player presses the button "go". SuperTangrams interface style: DOM vs. DCM

Direct Object Manipulation (DOM) style refers to system interfaces which allow the user to manipulate the objects on the screen with some kind of pointing device [7, 4]. SuperTangrams implements a Direct Concept Manipulation (DCM) [6] interface style which on the other hand enables users to interact with and think in terms of the concepts (e.g. rotation) being learned rather than the objects that the concepts act upon (e.g. square). However, since, unlike objects, concepts are abstract entities, this

Translation (fig. 2) is represented with an arrow with three mouse-sensitive handles at the tail, midpoint and head. The length and direction of the vector can be changed by dragging either the head or the tail. Rotation

Rotation (fig. 2) is represented by the center of rotation, a line that goes from the center of rotation to one vertex of the shape, and an arc whose direction and magnitude represent the direction and magnitude of the rotation. The configuration has two sensitive handles: the center of rotation and the head of the arc. Students select the direction, clockwise or counterclockwise and the amplitude of rotation by dragging the head of the arc. Reflection

Reflection (fig. 2) is described by the symmetry line, or line of reflection. In SuperTangrams reflection is

Figure 2: representation of the transformations in SuperTangrams: translation, rotation and reflection

represented by three lines: the reflection line, a horizontal line and bisector. The bisector goes from one vertex of the shape to the position the vertex will be after the reflection operation. Additionally an arc between the horizontal line and the line of reflection shows the angle between them. There are two handles in this configuration. The first handle is at the intersection of the reflection line and the horizontal line. It can be dragged all over the screen and it moves both lines at the same time. Using this handle does not change the orientation of the reflection line. The other handle is along the reflection line at the head of the arc. It changes the orientation of the reflection line. INTERACTION WITH THE DIGITAL INTERFACE

All the observations collected in this section come from direct observation of the students, interactions with them when answering their questions, video recordings, and the analysis of the scratch paper. The observations collected here attempt to shed some light on the students' conceptual model and on what they perceive from the software. Translation

At the beginning of the game the students did not have problems with translation. The ghost image was present and they quickly learned that moving the head of the arrow allows them to position the ghost image in the desired position. This however, resulted in them learning the physical use of the handle without requiring them to pay attention to the concepts involved. Rotation

Rotation was more difficult since they had to move both the center of rotation and the handle of the angle. Their approach was generally to bring the ghost image as close as possible to the target position, by alternatively using the two handles. The angle was used mainly as a way to move the shape in space and not to orient it. When the shape was close enough they fine tuned the position by moving the two handles. Thus, the use the students made of the handles was not coherent with the concepts involved. Rotation became very challenging when the difficulty level of the game increased and the ghost image was eliminated. The major problem was to deal with both the orientation and the position simultaneously. After several trials some students learned to place the head of the arc on the desired position. In this way they were able to place one corner of the shape on one corner of the target. Choosing the right corner to match was much more difficult since they had to figure out how the shape was going to rotate during the transformation. Orientation was the last big challenge. Several students learned to match the corners. Subsequently, for every rotation they first matched the corner, and then arbitrarily selected an angle. They repeated the transformation over and over until they guessed the right angle. In general, orientation was not well understood.

Reflection

Reflection was a challenging transformation. Many students tried to move the handle that positions the reflection line along the reflection line itself. This operation moved the horizontal line but did not change the ghost image. This evoked complaints about the software. A common comment was "The game does not work, I move this point and the ghost image does not move." The handles in rotation and reflection were the same, a red and a green dot. One of the two handles had the same effect on both transformations, i.e. the rotation of the ghost image. So it was natural to expect a similar effect for the second handle too, but this was not the case. The students did not really pay attention to the reflection line: most of them were just trying to physically learn the effect of the interaction with the handles. The dot for positioning the reflection line was treated as a means of translation. As for rotation, the purpose of the two handles for reflection was confusing. They must be used in conjunction to position the reflection line. The students did not always understand this. Instead, many saw the two handles simply as two handles to move the shape, each one in a different way, not two handles to represent the axis of reflection, and they were not able to figure out which transformation they were performing. Reflection was also confusing for many children (and teachers too) because they did not know how to deal with the horizontal line. The purpose of the horizontal line was only to calculate the angle of the reflection line with the horizon. But the children did not really need this angle. The angle between the shape and the target was more important and could not be measured directly with the angle shown. Moreover, dragging the handle caused the horizontal line to move too, and this was confusing since the students could not really tell what was the direct effect of their move and what was a side effect. Some students correctly concentrated on the reflection line, very few on the relation of the reflection line with the horizontal one, and the majority focused on the bisector, since using it they could easily determine the final position of one corner, similarly to the case of rotation. Student reactions

These first observations suggest the importance of differentiating the widgets that correspond to different concepts both visually and also by the method of manipulation. Even if initially the students were not expected to really interact with the concepts involved it is important that the type of interaction they have with them is correct. For example they should use angles for rotation and not for translation. It is interesting to note that the vast majority of the students did not consider the possibility of moving a shape in two or more steps: they always tried to move the shape

as close as possible to the target first, and then fine tune the position. The form of representation chosen for rotation is the general form where the center of rotation is outside the shape. This forced the students to deal with two different issues at the same time: selecting the angle and the position of the center. Since these are both challenging decisions, a different kind of scaffolding is needed that allows the students to deal with the two concepts separately. Not all students found this difficult, and one student discovered a 'trick' to do rotation without the ghost image that is worth mentioning. He explained the trick to us as well as to most of the class. For a few days he was a celebrity. The trick: 1.

Rotate the shape using many small steps until it reaches an orientation that is opposite to the target (mirror image). The position is not important, but one close to the target is more convenient.

2.

Set the rotation angle to 180° and then drag the center of rotation until the arrow on the rotation matches the right corner of the target position.

This discovery is very important for several reasons: a)

The student developed a procedure involving more than one step. Thus, he added a level of abstraction to the simple movement of the shapes on the screen.

b) The first part of the procedure deals only with the orientation of the shape, i.e., with the angle of rotation. This shows that it was easier for him to deal separately with the orientation and center concepts. c)

He did not calculate the angle directly but just tried small successive rotations until it worked. Nevertheless, this was a methodical approach, as opposed to the casual guessing of most other students.

d) He discovered the concept of opposite orientation which is connected with 180° degrees. Basically, he discovered a property of rotation, transformed it into a rule and was able to describe it correctly and to see the different situations where it could be applied, even if he could not formalize it using some mathematical language. The last point is very important since students are supposed to learn this kind of knowledge. It also shows a success of the game for providing an opportunity to discover knowledge. But it is interesting to note that very few students reached this confidence. After he explained his trick to others, a few students learned the trick very quickly, while many others were not able to apply it correctly and asked us for help. We believe this is due to the difficulty of the interface or representation chosen. While some students could understand it, many could not.

Suggestions

In summary, the representation chosen were too difficult for most children since it forced them to deal with different challenges at the same time and also it was not clear to many students what the effect of the handles was. The first issue leads us to propose additional, or different, scaffolding that will allow the children to deal with different concepts separately. In this case, rotation should first be presented with the center of rotation on one corner of the shape, so the children can practice with orientation and angles before dealing with the more difficult problem of the center of rotation. The second issue prompts us to guide the students gradually from the physical word to the computer screen through the use of manipulatives. Another issue is the overloading of information on the screen. This information is meant to help the student giving her several hints, but these actually confuse her since she must distinguish the basic ones that define the concepts from the secondary one. Also in this case manipulatives can be used to reduce the information on the screen in favor of physical practice. USE OF THE NOTEBOOK

In addition to the manipulatives, we provided some of the students with scratch paper that we called a notebook. The purpose of the notebook was different than the manipulatives. Many people use scratch paper to help them solve problems, especially when they are reasoning about something complex. It can help them pinpoint some intermediate state or simply visualize all the aspects of the problem while solving it. In addition to providing students scratch paper, the notebooks allowed us to see what kind of representation or "mental model" they were using and which points were giving them the most difficulty. Encouraging them to draw also forced them to chose a representation, i.e., make another step toward the abstract world of demonstrative geometry. The notebooks were used mainly as scratch paper to calculate sums of angles. Few students used it to draw pictures of rotation and reflection, and only one used it for translation. All the pictures of rotation focus on the arc. For example, the notebook entry in fig. 3 shows the child's concern for the orientation of the shape that results from the movement of one corner along the arc, or Fig. 3: notebook drawing his concern in finding the right arc. The arc seemed to catch the full attention of students, as if it were the main object involved with

Figure 4: Manipulatives: a) grid board with shapes; b) reflection mirror over board; c) popsicle sticks.

translation. They were trying to reproduce the animation of the shape moving along the arc. The animation of course was what they were seeing on the screen when pushing the 'go' button. And actually animation helped many of children understand. One wrote:

they also used the mirror on the screen. They would stand up or contort themselves in their chairs to observe the image displayed on the screen reflected in the mirror. They did not enjoy working far from the screen.

" I enjoyed seeing the shape move on the computer screen because it helps me understand better"

One common use of the shapes was to put them on the screen and animate them. A video recording shows a girl using them for rotation. She started superimposing one plastic shape on the digital shape on the screen, and then she moved the shape along the arc trying to rotate it at the same time. While the operation of following the arc was easy, the rotation involved was arbitrary. If the arc was short the shape did not rotate much, while if it was long the shape should rotate more. This illustrates once again that she was not considering the orientation of the edges due to the angle. Even if the amplitude of the arc and the angle are equivalent, just inspecting the arc cannot give information as precise as when looking at the orientation of the segments that define the angle.

But while aiding comprehension, animation caused the children to try to reproduce animation or to concentrate on the main aspect of the animation (the arc). It did not help them to find how to move or to focus on the aspects that would have helped them with rotation, mainly angles and orientation of the edges. USE OF PHYSICAL MANIPULATIVES First set of manipulatives

During the first study session (the first two weeks), we provided the students with the following manipulatives: -a set of tangram plastic shapes (fig. 4a) -a reflection mirror (fig. 4b) -a colored grid board with a rotating stick (fig. 4a) The board had two different drawings on it: a line for reflection and a circle divided in 16 sectors of equal size for rotation. This is consistent with the software that allows the minimal angle to be 1/16th of a circle. A stick is connected to a hub at the center of the circle and can rotate 360o. The idea was that the students would place a shape on the circle and then rotate it by pushing against it with the stick. We intended for them to use the reflection mirror with the line drawn on the surface to find the reflection line. To do this they would place a shape on the surface in front of the mirror and rotate the mirror on the surface until the image of the shape was in the desired position. We expected the students to use the manipulatives on the desk next to the computer, and reproduce what they were supposed to do on the screen. That's not exactly what happened. A few students did actually use the manipulatives as we expected. However a majority of them placed the shapes directly on the screen. To our surprise

Rotation

The students who used the grid positioned the plastic shape on the circular drawing, trying to position the shape in the same orientation as the one on the screen and with the stick in the same orientation as the radius of rotation on the screen. Then they would push the shape with the stick until the shape could reach the orientation of the target. They would count the number of sectors the stick moved and move the digital handle for the arc of rotation for the same number of steps as the number of sectors (they both correspond to 1/16th of a circle.) Some students felt comfortable with this method, even if it was not always easy to push the shape keeping the same contact with the stick, while some others preferred to bring the shapes directly on the screen and rely on the arc of rotation. Reflection

Students didn't use the mirror on the grid board with the plastic shapes as we had expected. Instead they placed the mirror directly on the screen and looked for the reflected image of the digital shape. The first operation involved

was to position the mirror until they could see the reflected image superimposing the target. Once the position was found, they had to play with the digital handles to position the reflection line along the position found for the mirror. For this operation the only concern was to position the line. They didn't need to be concerned about the horizontal line, the bisector, or the different effect of the two handles on the shape position. The operation was much easier, more concrete, and they could just deal with the reflection line, which is the fundamental concept at this level of knowledge. At this stage is important to know how reflection works, they do not have to deal which properties such as the same distance from the reflection line, the symmetry between the angles and so on. They first had to physically experience the operation with their hand and only focus on the reflection line.

When the plastic shape reached the target, the opening of the two popsicle sticks represented the sought after angle. In this case they could hold the right "angle" in their hand. Finally, they would recreate the digital angle playing with handles, holding the physical one as a model. Finding the center of rotation required more skillful operations, since they should move the joined ends of the sticks leaving one free end in the original position. Also in this case the holding of the "physical" angle helped them in understanding the operations involved. This both gave them the opportunity to increase their physical knowledge, or improving their subconscious knowledge of geometry, and at the same time helped them to overcome the difficulties of using the two digital handles as separate effectors, and of dealing with the quantity of information used in the digital representation.

New Manipulatives

Student reactions

The use the students made of the manipulatives surprised us, since they preferred to use them on the screen, merging the physical operations with the digital ones. Thus, for the second session we modified the manipulatives to make it easier for the students to use them on the screen.

The second set of tools had more success than the first. It was more natural for the children to bring the tools to the screen, and they could physically experiment with the transformations.

Reflection

For the reflection manipulative we retained the reflecting mirror but added tracing paper. We instructed the students to position the tracing paper on the screen and trace the shape in its initial and desired final position. Then they should fold the tracing paper so that the two drawings would overlap. The folding line corresponded to the reflection line. We decided to add tracing paper since this made reflection more "physical", the shape drawn on the paper moves in space while folding the paper. It is easier to see the trajectory of the shape in space, as opposed to the mirror which rely on the light's reflection. This adds a level of indirection to the concept of reflection, since to fully understand it the student must understand optics. The major problem with the tracing paper was to deal with the cases in which the shape was crossed by the reflection line. Some children felt more comfortable with the mirror, some with the tracing paper. Rotation

We eliminated the grid board and built a rotation tool consisting of two popsicle sticks (fig. 4c) joined together at one end in such a way that they could rotate, and placed some sticky paste on the other ends. In this way the students could stick a shape on the edge of the stick and rotate it. Moreover they could overlay this tool on the screen. The popsicle stick had more success than the grid. The students would stick the chosen shape at the end of one stick, and position the two overlying sticks directly on the digital radius of rotation, with the plastic shape matching the digital shape. Then they would rotate the stick holding the shape and keep the empty stick in the initial position.

In the questionnaires we asked children if it was good to have manipulatives, which ones they used, and if they helped them learn. Here are some of the answers: " They helped because they were 3D." " they helped you to learn the movement." " all tools because they helped you think ahead." " you need some reference to look at." " because it is important to have stuff you can work with in your hand." "it helps understand" "all because I couldn’t visualize it." This was the effect we were expecting from the manipulatives. However, some students said they did not use them because they did not know how. Suggestions

Some children felt more comfortable with one tool, and some with another. Some did not need them and preferred to figure it out by looking at the screen. Therefore, it is important to give the children multiple alternatives, so they can choose the ones that fit them best. Our findings indicate that physical manipulatives can be a valuable addition to educational software. However, they must be designed so that they can be used with the digital interface. Therefore, development of the software and the manipulatives must be carried on in parallel and both must support similar types of operations. CONCLUSIONS

Software running on a computer is something abstract: when the student drags a handle with the mouse something

happens on the screen. In educational software for geometry (and other abstract theories) it is important that the student learns to relate what is happens on the screen to the real word. It is actually important that s/he learns physical properties of the world and then gradually transfers them to the formal world. In this paper we presented a study performed on 180 children using educational software for geometry together with physical manipulatives. We developed some manipulatives and let the students use them while playing with the software. Surprisingly the students used the manipulatives in a way we did not expect. They used them directly on the screen merging the operations in the real world with those in the digital world, blurring the distinction between the physical and the abstract world. We noticed how the use of manipulatives helped students overcome limits of the representation and of the types of interaction allowed by the interface. For example, additional information on the screen whose purpose was to help the students, such as additional lines, actually confused them, leading them to miss the important concepts they were supposed to learn. Using a physical representation of those concepts, such as two popsicle sticks joined together for an angle, or the base of a mirror for a reflection line, helped them to focus on the main aspects of the transformation without being lost in representation details. The physical object also eliminated the problem on exploring the effect of the handles. Once the transformation was clear, the use of the handles came more naturally. Physical manipulatives helped them to consolidate the physical knowledge of the world. Moreover, using them on the screen, and blending the physical operations with the abstract ones, helped them to transition smoothly from the physical world to the formal world. It is important that the design of physical manipulatives be carried out in parallel with the design of the interface, that the representations used for the interface be reproducible physically with the manipulatives, and vice versa. Otherwise students may be led to use some tools following a misleading representation on the screen. For example, students moved shapes along the arc of rotation on the screen. The use of the popsicle rotator instead of shapes alone could avoid this problem. Different students felt comfortable with different manipulatives, so it is important to have multiple options. Manipulatives can also be used as a way of scaffolding. Instead of using additional information on the screen at the beginning, and withdraw them later, only the basic information can be used and the additional information can be derived by the student through performing the operation in the real world. We also noticed how it is important to deal with single concepts separately, at least

initially, instead of using a general representation that incorporates multiple concepts. According to the constructivist theory of learning [9], the student builds his own knowledge. This process is facilitated by the use of physical manipulatives, since the student can experiment on the world and gradually transfer his knowledge to the software and the abstract world. This is somehow different from the process that leads the students to experiment the software, a computational representation of abstract concepts, in order to discover the hidden mechanism of the software. It may be difficult to relate it to the real world. This last process consists in giving the student a computational model of a formal language and asking her to discover how it works instead of leading him to discover the physical properties that in the past lead to the creation of that language. ACKNOWLEDGMENTS

We thank Maria Klave and the EGEMS group for the support with SuperTangrams, Linda Stewart and Carolyn Melick for helping in the research, the teachers and students of Oakwood for their enthusiastic participation, and Randy Smith for his valuable suggestions. REFERENCES

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