# Instructional decisions: helping students build links between representations.

Instructional decisions about the nature and extent of teacher input in mathematics classrooms typically lie on a continuum somewhere between teachers directly telling students how to carry out mathematical procedures and free exploration situations where teachers provide little or no guiding input. Making decisions about instructional input is an area of practical concern for many teachers. Some worry that by providing too much information they may deprive their students of opportunities to reason and build knowledge of mathematical relationships. Others struggle with concerns that their students who freely explore mathematical situations with little direction may fail to reach established curriculum goals in a timely manner.

The challenge of instructional decision-making has been the subject of much research and scholarly consideration (Cobb, Yackel, & Wood, 1991; Lampert, 1985). Several researchers have identified characteristics for appropriate teacher intervention in mathematics classrooms. Chazan and Ball (1995) pointed out that it is appropriate for a teacher to contribute to and shape a classroom discussion by inserting substantive mathematical comments aimed at moving students away from entrenched disagreement or to provoke useful disagreement. Rittenhouse (1998) suggested that teachers might explicitly teach vocabulary, rules and conversational norms associated with a developing mathematical discourse, and conventional notation for a distinction that students already have made. Brousseau (1997) proposed a learning theory based upon strategic problem situations selected by a teacher to engage learners and facilitate their development of culture-specific mathematics knowledge and in which learners reason without teacher inp ut to produce knowledge intended to be useful beyond the immediate situation. But many teachers remain uncertain about judging the appropriateness of their input in the development of students' mathematics knowledge.

How can teachers insert substantive mathematical comments and teach notational conventions without impairing opportunities for learners to develop meaning from their instructional experiences? How can they insure the continuation of learning when students appear unable to progress? To address these concerns, one must consider the questions of what understanding in mathematics means and how teachers' inputs influence students' understandings. According to Lesh (1979), the ability to translate from one mode of representation to another is an important way by which students make mathematics meaningful. He identified five nondistinct representational modes: real-world situations, manipulative models, pictures, spoken symbols, and written symbols. Linking mathematics knowledge constructed in one representational format to another is valued as an indication of students' understanding of mathematics (Lesh, Landau, & Hamilton, 1983). Kaput (1989) described representational systems in which learners use their knowledg e of mathematics developed in one representational format to give meaning and to extend their knowledge of mathematics in a different representational format. When teachers assist learners to develop links between representations, they help their students to learn mathematics. By insuring that those links connect a "known" representational format to a less familiar format, teachers maximize the opportunities for reasoning and building knowledge of mathematics relationships.

Tracing its roots to Vygotsky's theory of intellectual development (1962, 1978, 1983), the notion of scaffolding may potentially provide an instructional mechanism through which students might be enabled to create links between representations. Wood, Bruner, and Ross (cited in Wood, 1989) describe "scaffolding functions" through which a more knowledgeable individual helps novices extend their competence beyond levels of their individual capability. Scaffolding is a term used by many to refer to the language-based guidance provided to a novice learner by a more knowledgeable individual (e. g., Brown & Ferrara, 1987; Rogoff, 1990; Silliman & Wilkinson, 1991). The assistance enables the novice to engage in intellectual activity beyond that independently accessible to him or her, Through the social interaction afforded by scaffolding, the individual's independent problem solving capabilities will develop.

In this study, I investigated instructional decisions as part of a study of the processes used by middle school students to interpret dynamic physical models of functions and to link their interpretations to tables and equations. In this context, I found that certain instructional decisions illustrate how instruction can enable students to develop their knowledge by building links from known representational formats to less familiar formats. In this paper, I will give examples of such instructional decisions within the context of learning about functions using dynamic physical models. These examples illustrate the kind of instructional scaffolding that enables learners to develop understanding without compromising their opportunities for reasoning. While these examples contribute to an emerging picture of instructional decision-making, they cannot be interpreted as a complete or defining picture of this notion.

Explorations with dynamic physical models have been shown to provide a useful context in which learners can develop meaning for functions, For example, Piaget, Grize, Szeminska, and Bang (1977) documented developmental levels in students' knowledge of functions using a physical system consisting of a board with tracks through which metal pieces were pulled by rotating axles of various sizes. Greer (1992) suggested that investigations with a physical system consisting of a bucket attached to a rotational handle could promote a view of multiplication as an implicit function relationship. Monk and Nemirovsky (1994), who closely examined the processes by which a high school student developed expertise in interpreting graphs generated by activation of a physical model, claim that the immediacy of the use of the physical model in the learning context enabled the student to move beyond a purely visual interpretation of graphs. The use of dynamic physical models may also provide tools through which teachers can scaff old students' learning.

Methodology

One of the purposes of this investigation was to describe and analyze the instructional decisions I made during a series of instruction episodes aimed at examining the processes used by middle school students to interpret dynamic physical models of functions and to link their interpretations to tables and equations. Dynamic physical models are mechanical tools that students use to visualize functions. They feature a user-manipulated domain variable and separately generated range values.

Using a teaching experiment design (Cobb & Steffe, 1983; Menchinskaya, 1969; Von Glasersfeld, 1987), I conducted the investigations with seven eighth-grade students who had no previous experience with dynamic physical models in an instructional setting and limited instruction interpreting tables and equations of functions. I was specifically interested in the kinds of instructional decisions I made to help the students overcome constraints in their interpretations of functions originating in dynamic physical models, and of tables and equations representing functions originating in dynamic physical models.

Four of the students were enrolled in pre-algebra, and two were enrolled in general mathematics. One student was drawn from a first-year algebra class and one was from a first year geometry course. With the exception of one student who worked alone with the investigator, all the students worked in pairs to explore the dynamic physical models. I met with the students two or three times weekly for a period of eight weeks and asked them to explain their thinking as I investigated their understandings of functions embedded in the models.

Although this procedure did not exactly replicate classroom teaching, it does parallel the kind of interactions that might occur between an individual student and teacher or between two students in a classroom setting. Data were collected through videotaping of students' responses to my questions, students' written and drawn responses to my questions, and from field notes I generated during the course of the investigation.

Dynamic Physical Models

I used dynamic physical models to introduce function concepts to students because of their potential to help students concretely experience variables related through systematic change. One dynamic physical model, the spool elevating system (Hines, 2002), consisted of an arrangement of spools of varying circumference, a cord with a weighted object attached at one end that could be connected to any of the spools, and a ruler to measure the position of the object (see Figure 1). The position of the object was controlled by lengthening or shortening the cord through turns of the handle attached to the spools' axle. Each turn of the handle produced a change in the position of the object. Starting with the object positioned at zero, and treating the number of handle-turns taken as the initial or input value to a function, the position of the weight varied and was dependent on the number of turns taken.

For example, on one particular spool called the 3-spool, a clockwise turn raised the object 3 inches. A second turn raised the object an additional 3 inches. Using the model, students experienced the variability of variables and later selected symbols to represent the variables. Students also experienced the function relationship in a nonsymbolic format.

Another dynamic physical model, the slack rope board, consisted of a 15-inch cord whose endpoints were tacked 10 inches apart horizontally, one beside the other on a small bulletin board (see Figure 2). The cord was pulled taut by using the eraser end of a pencil to produce two segments with a common end point. For identification purposes, the segments are referred to as AB and BC. The left segment length (measure of AB) was selected to represent the domain of a function; the right segment length (measure of BC) corresponded to the image or output value. Repositioning the pencil to various locations along the cord changed the length of AB, the input variable, and produced corresponding changes in the length of BC, the output or image variable. Despite changes in the pencil position and the lengths of AB and BC, the relationship between AB and BC remained the same, which is, measure of segment AB plus the measure of segment BC equals 15.

The slack rope board illuminated two roles of variables in functions: first, that a particular variable can assume more than one value, and second, that a particular value of one variable may be qualitatively and quantitatively related to that of a second. As with the spool system, students using this model encountered variables taking on several different values and later selected symbols to represent the variables.

Analysis of Data

The analysis of data involved several phases. In the initial phase that occurred while the data was being collected, I attempted to understand and characterize the students, conceptualizations of functions and to make decisions about how instruction should proceed. At the close of data collection, I conducted a two-phased analysis (Merriam, 1988). In the first phase, any data item I judged pertinent to making instructional decisions was identified and categories of these data were developed. Then, during a second phase of analysis, I examined the categories to delineate overlapping categories and to combine non-distinct categories. During this process, I realized that some of my decisions to provide or withhold instruction were based upon my interpretation of a student's immediate conceptualizations and of how students were developing links between representational formats.

The results of the investigation reported here Consist of two examples that illustrate how instructional decisions enabled students to link their interpretations of functions originating in dynamic physical models to symbolic representations without compromising their opportunities to reason and build knowledge of mathematics relationships. These examples are representative of a number of instructional decisions that I made during the investigation. They show how teacher input, specifically scaffolding instruction, can help students to anchor their knowledge for further reasoning, reflection, and communication.

Claudia and Gabe Write an Equation for the Spool Elevating System

In a series of instructional sessions, two students named Claudia and Gabe explored functions originating in the spool elevating system. Both Claudia and Gabe were enrolled in general mathematics at the time they participated in the investigation and had no previous experience with dynamic physical models in instructional settings nor of interpreting equations of functions. Their initial efforts focused on sorting out the variables so as to begin to develop an organized interpretation of the system. Claudia focused on the incremental amounts of elevation resulting from each rotation of the handle. Gabe considered the number of rotations needed for the object to reach the top of the frame and decided that when the object was on the 3-spool, a spool with circumference of 3 inches, the system was multiplying the number of rotations of the handle by three. The students explored variable features and systematic consistencies they observed as they operated the system. Both realized that on any particular spool, the number of rotations of the handle was changeable, as was the height attained by the object. The amount of elevation attained with each single rotation was unchanging. Without specific instruction, Claudia began to build a table to organize the information (see Figure 3), (Hines, 2002).

Claudia placed the object on each different spool and systematically recorded the amount it was elevated on one rotation of the handle for each spool. As she constructed her table, she explained, "OK, I just wanted to find out the difference. I just wanted to find out how many times and in what way they're [spools] different". Claudia confirmed that the amount of elevation using the so called 2-spool, was 2 inches from a single rotation of the handle and that when the next rotation was taken the object would be at 4 inches. She confirmed that on the 3-spool the object would first go to a height of 3 inches and then to 6 inches. She remarked, "So as far as we can tell it doubles all the time." But as she continued to elevate the object on the 3-spool, she developed a conflict in her doubling interpretation. "The first time it's doubling. Then after that, it's something different (not doubling), but they're the same [amounts] after that." Claudia realized that by doubling the successive output values in her tab le she was missing some values that occurred as she operated the spool system.

In a later session we returned to explore the spools system This time, both students viewed the relationship on the 3-spool as one of repeated addition regardless of which spool was being considered. With the 3-spool Claudia continued to view the change in height of the object from 3 inches to 6 inches as doubling, and subsequent changes as adding 3 inches. Gabe counted orally, "3, 6, 9, 12," for each crank." I encouraged him to complete a table of values for the 3-spooi to record his interpretations. Then the students created symbols for the variable features they observed as they operated the 3-spool. They each chose symbols to represent the number of rotations of the handle and the height of the object. I challenged Gabe to use his symbols and think about the numbers in his table to develop an equation. He responded, "Oh, I get it. The number of cranks times ... (pause) times the height equals, no wait, I got it now... (pause) by the height equals ... (pause) all right, 4 cranks, 1, 2, 3, 4. If the height' s 12, you'd take 4 cranks. You're doubling by 3, increasing each time by 3, 4 x 3 = 12."

Although Gabe did not use his symbols to write an equation, he was expanding the repeated addition interpretation to multiplication. English was a second Language for Gabe, and he appeared to struggle for language to express the concept of multiplication. Pointing to the spool, he continued to explain, "This right here is 3." Then pointing to the ruler, he exclaimed, "This is the height." Finally, pointing to the handle, he added, "This is the crank." So, 12 divided by the 3 is 4. I decided to record Gabe's words symbolically. Using h to stand for the height of the object and c for the number of rotations of the handle, I wrote as Gabe spoke. "The height could be divided by the spool size or the number of cranks." I wrote, h + 3 = c." Gabe replied, "The height could be divided by the number of cranks to equal 3, the spool size." I wrote, "h + c = 3."

Gabe invented a multiplication interpretation (i.e. "doubling by 3"), but faced a challenge when he attempted to write a symbolic equation for the relationship. At that point, I believed that he had constructed sufficient knowledge of the situation so that what I wrote in a different representational format would be meaningful to him. I assisted him as he spoke by writing symbolically the relationships he was expressing verbally. He arrived at multiplicative thinking by dividing the total height to be obtained by an object on a spool by either the number of rotations taken or by the number of inches of elevation resulting from one rotation. In this case, he created the symbols for the variables and the language of the relationship. I scaffolded his thinking of the relationship by assisting him with the creation of a symbolic equation. Since Claudia had not fully realized the underlying multiplicative nature of the relationship, I chose not to give the same assistance. I believed that to have done so might hav e compromised Claudia's opportunity to reason about the situation and build knowledge of the relationships linking activity with the physical system to the meaning of a representing symbolic equation. The symbolism of the equation potentially could have lost meaning for her. However, at the time of Gabe's experiences, Claudia expanded her view of the situation from one of repeated addition to one of multiplication, and using symbols she had created, she wrote "0 x 3 1," where 0 represented the number of rotations of the handle of the spool system and I represented the height attained by the object.

Jim Writes an Equation for the Slack Rope Board

In a different instructional session, a student named Jim explored the slack rope board. Jim was enrolled in pre-algebra at the time he participated in the investigation and had no previous experience with dynamic physical models in an instructional setting and limited instruction interpreting equations of functions. I held the slack rope taut in various positions and Jim measured and recorded the segments' lengths. Jim noticed that the sum of the lengths of the two changeable sides was always 15 inches and that the two changeable sides along with the edge of the slack rope board produced a triangle. I asked Jim to compare one of the triangles to a previous triangle and to notice what changed and what stayed the same. Jim indicated that two of the points and the 'base" stayed the same and that all the triangles would have the same perimeter. To explain how the two sides change together, Jim gave sets of specific numeric values for the lengths. Later he explained, "As one side gets bigger, the other side gets smaller." At this point, Jim did not appear to fully integrate his knowledge of the increasing/decreasing relationship of the sides, with his knowledge that the sum of the sides was always 15.

On a second visit I asked Jim to make a table with three columns to represent the relationship of the slack rope board (see Figure 4). He listed several whole number sets of values for the lengths of the left and right sides. He recognized the impossibility of listing all values in the tables, and agreed that tables with different specific sets of values could be used to represent the relationship of the slack rope board. Although these experiences were novel for Jim, I did not detect any cognitive challenge for him.

I proceeded to ask Jim to select symbols to stand for the lengths of the left and right sides. For no particular reason, Jim selected a tree-like symbol, hereafter referred to as t, which he labeled as the left-side length and a brick wall symbol, hereafter referred to as bw, to stand for the right-side length. Next, I asked Jim to use his symbols and write a mathematical sentence for the relationship between the sides. This task was challenging. I hinted that he should consider the values in his table and think about writing a mathematical sentence that incorporates both the ideas of one side increasing as the other decreases and that together the parts total 15. After listing several sets of specific numeric values for the variables, Jim wrote t + bw = 15. The equation corresponded to his thinking of the relationship in the physical context where two parts were joined to equal an unchanging total of 15. The strength of that interpretation became apparent when I asked Jim to reason about the equation.

I began a later session by presenting an equation: a+b = 15 and asking Jim about the meaning. Jim interpreted the equation as a representation for the slack rope board. He was aware that each variable could have more than one value and that the values change in opposite directions. His confidence in these interpretations arose from the consistencies he saw in the action of the variables in the physical model, table, and equation. He indicated that together the sides equaled 15, referring to the sides of the slack rope board.

Next, I asked Jim to think about how he could express a value for a alone. I presented him with the following equation: a =_____and requested that he fill in the blank Jim was puzzled and somewhat stunned by the question. Insisting that it would be the same as the earlier equation, he wrote, "a=a+b=15." I decided to change the language of my request Since Tim's interpretation of his equation was related to the action of the physical model, I asked, 'Think about the process you use to figure out a's value." He explained that a would have to be something within 15 and that it could not take on a value of 15. When I asked, "How would you explain to someone else what a is if a +b is 15?" he stated that I would need to tell them about the slack rope board and the string always 15 inches in length. When I pressed him to write an equation expressing the value for a, he insisted that he would need a symbol for within. He finally wrote, "a = 15 - a = b, stating that a was "an answer within an answer."

At this point Jim had blurred the distinction between a and b. a referred to one of the parts, not specifically the left part. But more importantly, the meaning that Jim ascribed to the original equation was rooted in the action of the physical model by which the equation was constructed. Writing an expression for a alone appeared to violate Jim's view of how a and b worked together.

Jim' s reluctance to rearrange the equation a + b = 15 and the strength of the equation's link to the action of the dynamic physical model, suggested to me that Jim was not ready to reason about and operate with symbolic equations in isolation from a real-world context. I considered telling Jim, "Just solve for a by subtracting b from both sides." But because Jim's thinking of the equation was so closely linked to the dynamics of the physical model, that would have compromised his opportunity for reasoning about the relationship that occurs in the physical system and the relationship as expressed in a symbolic equation. Jim did not think of the equation without linking it to the physical model to interpret it. I decided that one way to further scaffold Jim's knowledge of equations was to create a different but related action scenario. When I asked Jim directly. 'Tell me in words a general rule for how to figure out the length of a, if you have two segments together that equal 15, he arrived at subtraction on ly by considering specific numeric cases (e.g. 15 - 6 = 9). Then later, he reluctantly wrote a = 15 - a = b.

In an effort to further probe the strength of Jim's view of equations of the format a + b = 15 that arises from the slack rope board situation I introduced a new problem situation. An equation algebraically equivalent to a + b = 15 was generated from an entirely different action than that of the length of two segments together totaling 15 inches. I presented Jim with what became known as the pasta problem. In the pasta problem, several paper bags were filled with different combinations of two types of uncooked pasta. The two types of pasta, pine trees and wagon wheels, were easily distinguishable by touch. The intent of the pasta problem was to observe Jims' interpretation of the relationship between the number of wheels and the number of pine trees. In all cases, the total number of pieces of pasta in a bag was 12. I instructed Jim to determine the number of pine-tree pasta in each bag by removing the wagon-wheel pasta without looking inside the bag. I was interested to know if the separating action associa ted with this activity would suggest an equation of the format, 12 - a = b, which had been especially difficult to consider from the situation of the slack rope board.

After Jim had completed the activity with several different paper bags, I asked him to identify the variable and nonvariable features within the situation. He replied that the number of wheels and the number of trees were changeable and that the total amount of pasta in each bag was nonchangeable. I asked Jim to pick symbols for the changeable things. He drew a wagon wheel and a pine tree, hereafter indicated respectively as w and t. Then I asked him to describe in his own words how the process worked. Jim replied, "I stuck in my hand and pulled out the wheels and left the trees in there. I counted them [wheels] and I figured out how many wheels we had, which told us how many trees we had." He wrote, "12 = # of w or # oft and w + t = 12, explaining, "12 is the total in there. w is a certain number of wheels. We don't know how many unless you've already taken the stuff out." When I asked Jim to tell me again how he figured out how many trees, he explained, "Say it is 6; then the other is 6 'cause the total is 12." He understood a general process in the physical context. Using that as a known representational format from which Jim might build knowledge of symbolic equations, I decided to show Jim the equation, 12- w = t. He remarked, "it's kind'a what I meant. If there's 12 in there and you take one kind out, it tells you how many are left of the other kind." Then I pressed, "Do you remember the equation for the slack rope board? What if I said, "15 - bw = t?" Jim replied, 'That's the same thing as that [points to 12 - w = t], if the total number of noodles (pasta) was 15. But he further explained that the equations, 15 - bw = t and bw + t = 15, were not "the same." In order to make the equation, 15 - bw = t, sensible for the relationship of the slack rope board, Jim stated that he would need to change his thinking of how the slack rope board worked. He explained, "Move it the left segment] one way, measure one end, and it will tell you the other end [length of the right segment].

In the pasta problem Jim accepted the equation 12 - w = t, although he did not generate it. He argued that in the slack rope board situation we were not taking one number away from the total to equal the other. But after exploration of this situation, Jim was able to modify his thinking of the action on the slack rope board in a way which would justify an equation such as 15 - bw = t. In order to modify the equation format he had to change his view of the action on the physical model from which the equation was generated.

In the pasta problem, the action of the task was different and Jim was comfortable with an equation indicating an expression of the value of one variable in terms of the other. He was even able to consider changing the format of equations if the actions of the physical situations they represented were considered differently. The instructional decisions to encourage Jim to consider and compare the different physical situations, and their representing equations allowed Jim to encounter and reason about mathematically equivalent symbolic equations. By maintaining his connection of equations to the physical contexts through which they were created, Jim's meanings for the equations were not compromised, yet he expanded his knowledge of symbolic equations.

Conclusions

Scaffolding is the guiding language provided by a teacher to capture and signify students' emerging conceptualizations and to develop a structure through which students can reason further about, reflect upon, and communicate their conceptualizations. As a teacher initially becomes aware of students' conceptualizations, he/she can begin the scaffolding process by using the students' language to validate their thinking and to reflect back to the students how he/she has interpreted their' conceptualizations. For example, in Gabe's situation, I used his language, "doubling by 3", to acknowledge my understanding of his description of the function originating on the 3-spool as multiplication. I then showed him how to record symbolically the relationship he described verbally. Similarly, since Jim interpreted the function of the slack rope board in terms of the actions that occurred with the lengths of the sides, I began to use the action-based language that he created. This was helpful in enabling him to maintain h is interpretation of the function from the slack rope board as he reasoned about it in a table and an equation. By using students' language, the teacher acknowledges both for her or himself and for the students, an awareness of the initial development of their knowledge. From this point further scaffolding can be initiated to help the students extend their knowledge.

In this paper I described two ways in which instructional decisions scaffolded learners to link their knowledge created in a physical system to symbolic equations. In these examples, the investigator's interventions enabled the students to move beyond constraints in their creation and/or use of symbolic equations. With Gabe, I showed how to translate knowledge of functions generated in one representation, a verbal description of operation of the spool system, to another representation, a symbolic equation. Because be had developed an interpretation of the function from the physical model, instruction that showed him how to write an equation did not compromise his opportunity for reasoning about the function, but helped symbolically anchor his knowledge. With Jim, I enabled comparing and reasoning about the equivalence of two equations by encouraging him to modify his original interpretation of the action of the slack rope board through which one of the equations was generated. Again, because the instructional intervention was based upon Jim's interpretation of functions in a physical situation, his opportunity for reasoning and building knowledge of mathematical relationships was preserved.

These examples illustrate two ways to continue scaffolding after the initial use of students' language to validate their thinking and reflect back to the students how their conceptualizations have been interpreted. But language was not the only scaffolding tool. In Gabe's case, the symbolism of an equation anchored his knowledge of the function of the spool system to enable his understanding of the equation intended to represent it. In Jim's case, the reinterpretation of the action of the slack rope board scaffolded greater understanding of equivalent symbolic equations.

There are practical challenges associated with instructional decision-making. An interpretation of understanding as the building of links between different representations as described by Lesh et al, (1983) provides a useful framework in which to develop instructional decisions. In this investigation learners developed an interpretation of function concepts through explorations of physical systems. These explorations and interpretations served as anchor points supporting instruction intended to promote understanding of functions represented in equations. Viewed in isolation, the instruction provided in these examples may appear too directive to encourage student reasoning and building knowledge of mathematics relationships. When the larger picture of students' understanding is considered as the liking of representational formats, these instructional decisions illustrate how a teacher might build on students' existing knowledge and show students how to extend their knowledge to unfamiliar representational for mats. This kind of scaffolding presents no cause for concern about compromising learners' opportunities to reason and build knowledge of mathematics relationships. It suggests an important step in addressing practical concerns regarding teacher input in instruction.

Acknowledgements

This research was conducted as part of the author's doctoral dissertation completed at Northern Illinois University under the co-direction of Frank Bazeli and Helen Khoury. The author gratefully acknowledges their support and guidance.

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The challenge of instructional decision-making has been the subject of much research and scholarly consideration (Cobb, Yackel, & Wood, 1991; Lampert, 1985). Several researchers have identified characteristics for appropriate teacher intervention in mathematics classrooms. Chazan and Ball (1995) pointed out that it is appropriate for a teacher to contribute to and shape a classroom discussion by inserting substantive mathematical comments aimed at moving students away from entrenched disagreement or to provoke useful disagreement. Rittenhouse (1998) suggested that teachers might explicitly teach vocabulary, rules and conversational norms associated with a developing mathematical discourse, and conventional notation for a distinction that students already have made. Brousseau (1997) proposed a learning theory based upon strategic problem situations selected by a teacher to engage learners and facilitate their development of culture-specific mathematics knowledge and in which learners reason without teacher inp ut to produce knowledge intended to be useful beyond the immediate situation. But many teachers remain uncertain about judging the appropriateness of their input in the development of students' mathematics knowledge.

How can teachers insert substantive mathematical comments and teach notational conventions without impairing opportunities for learners to develop meaning from their instructional experiences? How can they insure the continuation of learning when students appear unable to progress? To address these concerns, one must consider the questions of what understanding in mathematics means and how teachers' inputs influence students' understandings. According to Lesh (1979), the ability to translate from one mode of representation to another is an important way by which students make mathematics meaningful. He identified five nondistinct representational modes: real-world situations, manipulative models, pictures, spoken symbols, and written symbols. Linking mathematics knowledge constructed in one representational format to another is valued as an indication of students' understanding of mathematics (Lesh, Landau, & Hamilton, 1983). Kaput (1989) described representational systems in which learners use their knowledg e of mathematics developed in one representational format to give meaning and to extend their knowledge of mathematics in a different representational format. When teachers assist learners to develop links between representations, they help their students to learn mathematics. By insuring that those links connect a "known" representational format to a less familiar format, teachers maximize the opportunities for reasoning and building knowledge of mathematics relationships.

Tracing its roots to Vygotsky's theory of intellectual development (1962, 1978, 1983), the notion of scaffolding may potentially provide an instructional mechanism through which students might be enabled to create links between representations. Wood, Bruner, and Ross (cited in Wood, 1989) describe "scaffolding functions" through which a more knowledgeable individual helps novices extend their competence beyond levels of their individual capability. Scaffolding is a term used by many to refer to the language-based guidance provided to a novice learner by a more knowledgeable individual (e. g., Brown & Ferrara, 1987; Rogoff, 1990; Silliman & Wilkinson, 1991). The assistance enables the novice to engage in intellectual activity beyond that independently accessible to him or her, Through the social interaction afforded by scaffolding, the individual's independent problem solving capabilities will develop.

In this study, I investigated instructional decisions as part of a study of the processes used by middle school students to interpret dynamic physical models of functions and to link their interpretations to tables and equations. In this context, I found that certain instructional decisions illustrate how instruction can enable students to develop their knowledge by building links from known representational formats to less familiar formats. In this paper, I will give examples of such instructional decisions within the context of learning about functions using dynamic physical models. These examples illustrate the kind of instructional scaffolding that enables learners to develop understanding without compromising their opportunities for reasoning. While these examples contribute to an emerging picture of instructional decision-making, they cannot be interpreted as a complete or defining picture of this notion.

Explorations with dynamic physical models have been shown to provide a useful context in which learners can develop meaning for functions, For example, Piaget, Grize, Szeminska, and Bang (1977) documented developmental levels in students' knowledge of functions using a physical system consisting of a board with tracks through which metal pieces were pulled by rotating axles of various sizes. Greer (1992) suggested that investigations with a physical system consisting of a bucket attached to a rotational handle could promote a view of multiplication as an implicit function relationship. Monk and Nemirovsky (1994), who closely examined the processes by which a high school student developed expertise in interpreting graphs generated by activation of a physical model, claim that the immediacy of the use of the physical model in the learning context enabled the student to move beyond a purely visual interpretation of graphs. The use of dynamic physical models may also provide tools through which teachers can scaff old students' learning.

Methodology

One of the purposes of this investigation was to describe and analyze the instructional decisions I made during a series of instruction episodes aimed at examining the processes used by middle school students to interpret dynamic physical models of functions and to link their interpretations to tables and equations. Dynamic physical models are mechanical tools that students use to visualize functions. They feature a user-manipulated domain variable and separately generated range values.

Using a teaching experiment design (Cobb & Steffe, 1983; Menchinskaya, 1969; Von Glasersfeld, 1987), I conducted the investigations with seven eighth-grade students who had no previous experience with dynamic physical models in an instructional setting and limited instruction interpreting tables and equations of functions. I was specifically interested in the kinds of instructional decisions I made to help the students overcome constraints in their interpretations of functions originating in dynamic physical models, and of tables and equations representing functions originating in dynamic physical models.

Four of the students were enrolled in pre-algebra, and two were enrolled in general mathematics. One student was drawn from a first-year algebra class and one was from a first year geometry course. With the exception of one student who worked alone with the investigator, all the students worked in pairs to explore the dynamic physical models. I met with the students two or three times weekly for a period of eight weeks and asked them to explain their thinking as I investigated their understandings of functions embedded in the models.

Although this procedure did not exactly replicate classroom teaching, it does parallel the kind of interactions that might occur between an individual student and teacher or between two students in a classroom setting. Data were collected through videotaping of students' responses to my questions, students' written and drawn responses to my questions, and from field notes I generated during the course of the investigation.

Dynamic Physical Models

I used dynamic physical models to introduce function concepts to students because of their potential to help students concretely experience variables related through systematic change. One dynamic physical model, the spool elevating system (Hines, 2002), consisted of an arrangement of spools of varying circumference, a cord with a weighted object attached at one end that could be connected to any of the spools, and a ruler to measure the position of the object (see Figure 1). The position of the object was controlled by lengthening or shortening the cord through turns of the handle attached to the spools' axle. Each turn of the handle produced a change in the position of the object. Starting with the object positioned at zero, and treating the number of handle-turns taken as the initial or input value to a function, the position of the weight varied and was dependent on the number of turns taken.

For example, on one particular spool called the 3-spool, a clockwise turn raised the object 3 inches. A second turn raised the object an additional 3 inches. Using the model, students experienced the variability of variables and later selected symbols to represent the variables. Students also experienced the function relationship in a nonsymbolic format.

Another dynamic physical model, the slack rope board, consisted of a 15-inch cord whose endpoints were tacked 10 inches apart horizontally, one beside the other on a small bulletin board (see Figure 2). The cord was pulled taut by using the eraser end of a pencil to produce two segments with a common end point. For identification purposes, the segments are referred to as AB and BC. The left segment length (measure of AB) was selected to represent the domain of a function; the right segment length (measure of BC) corresponded to the image or output value. Repositioning the pencil to various locations along the cord changed the length of AB, the input variable, and produced corresponding changes in the length of BC, the output or image variable. Despite changes in the pencil position and the lengths of AB and BC, the relationship between AB and BC remained the same, which is, measure of segment AB plus the measure of segment BC equals 15.

The slack rope board illuminated two roles of variables in functions: first, that a particular variable can assume more than one value, and second, that a particular value of one variable may be qualitatively and quantitatively related to that of a second. As with the spool system, students using this model encountered variables taking on several different values and later selected symbols to represent the variables.

Analysis of Data

The analysis of data involved several phases. In the initial phase that occurred while the data was being collected, I attempted to understand and characterize the students, conceptualizations of functions and to make decisions about how instruction should proceed. At the close of data collection, I conducted a two-phased analysis (Merriam, 1988). In the first phase, any data item I judged pertinent to making instructional decisions was identified and categories of these data were developed. Then, during a second phase of analysis, I examined the categories to delineate overlapping categories and to combine non-distinct categories. During this process, I realized that some of my decisions to provide or withhold instruction were based upon my interpretation of a student's immediate conceptualizations and of how students were developing links between representational formats.

The results of the investigation reported here Consist of two examples that illustrate how instructional decisions enabled students to link their interpretations of functions originating in dynamic physical models to symbolic representations without compromising their opportunities to reason and build knowledge of mathematics relationships. These examples are representative of a number of instructional decisions that I made during the investigation. They show how teacher input, specifically scaffolding instruction, can help students to anchor their knowledge for further reasoning, reflection, and communication.

Claudia and Gabe Write an Equation for the Spool Elevating System

In a series of instructional sessions, two students named Claudia and Gabe explored functions originating in the spool elevating system. Both Claudia and Gabe were enrolled in general mathematics at the time they participated in the investigation and had no previous experience with dynamic physical models in instructional settings nor of interpreting equations of functions. Their initial efforts focused on sorting out the variables so as to begin to develop an organized interpretation of the system. Claudia focused on the incremental amounts of elevation resulting from each rotation of the handle. Gabe considered the number of rotations needed for the object to reach the top of the frame and decided that when the object was on the 3-spool, a spool with circumference of 3 inches, the system was multiplying the number of rotations of the handle by three. The students explored variable features and systematic consistencies they observed as they operated the system. Both realized that on any particular spool, the number of rotations of the handle was changeable, as was the height attained by the object. The amount of elevation attained with each single rotation was unchanging. Without specific instruction, Claudia began to build a table to organize the information (see Figure 3), (Hines, 2002).

Claudia placed the object on each different spool and systematically recorded the amount it was elevated on one rotation of the handle for each spool. As she constructed her table, she explained, "OK, I just wanted to find out the difference. I just wanted to find out how many times and in what way they're [spools] different". Claudia confirmed that the amount of elevation using the so called 2-spool, was 2 inches from a single rotation of the handle and that when the next rotation was taken the object would be at 4 inches. She confirmed that on the 3-spool the object would first go to a height of 3 inches and then to 6 inches. She remarked, "So as far as we can tell it doubles all the time." But as she continued to elevate the object on the 3-spool, she developed a conflict in her doubling interpretation. "The first time it's doubling. Then after that, it's something different (not doubling), but they're the same [amounts] after that." Claudia realized that by doubling the successive output values in her tab le she was missing some values that occurred as she operated the spool system.

In a later session we returned to explore the spools system This time, both students viewed the relationship on the 3-spool as one of repeated addition regardless of which spool was being considered. With the 3-spool Claudia continued to view the change in height of the object from 3 inches to 6 inches as doubling, and subsequent changes as adding 3 inches. Gabe counted orally, "3, 6, 9, 12," for each crank." I encouraged him to complete a table of values for the 3-spooi to record his interpretations. Then the students created symbols for the variable features they observed as they operated the 3-spool. They each chose symbols to represent the number of rotations of the handle and the height of the object. I challenged Gabe to use his symbols and think about the numbers in his table to develop an equation. He responded, "Oh, I get it. The number of cranks times ... (pause) times the height equals, no wait, I got it now... (pause) by the height equals ... (pause) all right, 4 cranks, 1, 2, 3, 4. If the height' s 12, you'd take 4 cranks. You're doubling by 3, increasing each time by 3, 4 x 3 = 12."

Although Gabe did not use his symbols to write an equation, he was expanding the repeated addition interpretation to multiplication. English was a second Language for Gabe, and he appeared to struggle for language to express the concept of multiplication. Pointing to the spool, he continued to explain, "This right here is 3." Then pointing to the ruler, he exclaimed, "This is the height." Finally, pointing to the handle, he added, "This is the crank." So, 12 divided by the 3 is 4. I decided to record Gabe's words symbolically. Using h to stand for the height of the object and c for the number of rotations of the handle, I wrote as Gabe spoke. "The height could be divided by the spool size or the number of cranks." I wrote, h + 3 = c." Gabe replied, "The height could be divided by the number of cranks to equal 3, the spool size." I wrote, "h + c = 3."

Gabe invented a multiplication interpretation (i.e. "doubling by 3"), but faced a challenge when he attempted to write a symbolic equation for the relationship. At that point, I believed that he had constructed sufficient knowledge of the situation so that what I wrote in a different representational format would be meaningful to him. I assisted him as he spoke by writing symbolically the relationships he was expressing verbally. He arrived at multiplicative thinking by dividing the total height to be obtained by an object on a spool by either the number of rotations taken or by the number of inches of elevation resulting from one rotation. In this case, he created the symbols for the variables and the language of the relationship. I scaffolded his thinking of the relationship by assisting him with the creation of a symbolic equation. Since Claudia had not fully realized the underlying multiplicative nature of the relationship, I chose not to give the same assistance. I believed that to have done so might hav e compromised Claudia's opportunity to reason about the situation and build knowledge of the relationships linking activity with the physical system to the meaning of a representing symbolic equation. The symbolism of the equation potentially could have lost meaning for her. However, at the time of Gabe's experiences, Claudia expanded her view of the situation from one of repeated addition to one of multiplication, and using symbols she had created, she wrote "0 x 3 1," where 0 represented the number of rotations of the handle of the spool system and I represented the height attained by the object.

Jim Writes an Equation for the Slack Rope Board

In a different instructional session, a student named Jim explored the slack rope board. Jim was enrolled in pre-algebra at the time he participated in the investigation and had no previous experience with dynamic physical models in an instructional setting and limited instruction interpreting equations of functions. I held the slack rope taut in various positions and Jim measured and recorded the segments' lengths. Jim noticed that the sum of the lengths of the two changeable sides was always 15 inches and that the two changeable sides along with the edge of the slack rope board produced a triangle. I asked Jim to compare one of the triangles to a previous triangle and to notice what changed and what stayed the same. Jim indicated that two of the points and the 'base" stayed the same and that all the triangles would have the same perimeter. To explain how the two sides change together, Jim gave sets of specific numeric values for the lengths. Later he explained, "As one side gets bigger, the other side gets smaller." At this point, Jim did not appear to fully integrate his knowledge of the increasing/decreasing relationship of the sides, with his knowledge that the sum of the sides was always 15.

On a second visit I asked Jim to make a table with three columns to represent the relationship of the slack rope board (see Figure 4). He listed several whole number sets of values for the lengths of the left and right sides. He recognized the impossibility of listing all values in the tables, and agreed that tables with different specific sets of values could be used to represent the relationship of the slack rope board. Although these experiences were novel for Jim, I did not detect any cognitive challenge for him.

I proceeded to ask Jim to select symbols to stand for the lengths of the left and right sides. For no particular reason, Jim selected a tree-like symbol, hereafter referred to as t, which he labeled as the left-side length and a brick wall symbol, hereafter referred to as bw, to stand for the right-side length. Next, I asked Jim to use his symbols and write a mathematical sentence for the relationship between the sides. This task was challenging. I hinted that he should consider the values in his table and think about writing a mathematical sentence that incorporates both the ideas of one side increasing as the other decreases and that together the parts total 15. After listing several sets of specific numeric values for the variables, Jim wrote t + bw = 15. The equation corresponded to his thinking of the relationship in the physical context where two parts were joined to equal an unchanging total of 15. The strength of that interpretation became apparent when I asked Jim to reason about the equation.

I began a later session by presenting an equation: a+b = 15 and asking Jim about the meaning. Jim interpreted the equation as a representation for the slack rope board. He was aware that each variable could have more than one value and that the values change in opposite directions. His confidence in these interpretations arose from the consistencies he saw in the action of the variables in the physical model, table, and equation. He indicated that together the sides equaled 15, referring to the sides of the slack rope board.

Next, I asked Jim to think about how he could express a value for a alone. I presented him with the following equation: a =_____and requested that he fill in the blank Jim was puzzled and somewhat stunned by the question. Insisting that it would be the same as the earlier equation, he wrote, "a=a+b=15." I decided to change the language of my request Since Tim's interpretation of his equation was related to the action of the physical model, I asked, 'Think about the process you use to figure out a's value." He explained that a would have to be something within 15 and that it could not take on a value of 15. When I asked, "How would you explain to someone else what a is if a +b is 15?" he stated that I would need to tell them about the slack rope board and the string always 15 inches in length. When I pressed him to write an equation expressing the value for a, he insisted that he would need a symbol for within. He finally wrote, "a = 15 - a = b, stating that a was "an answer within an answer."

At this point Jim had blurred the distinction between a and b. a referred to one of the parts, not specifically the left part. But more importantly, the meaning that Jim ascribed to the original equation was rooted in the action of the physical model by which the equation was constructed. Writing an expression for a alone appeared to violate Jim's view of how a and b worked together.

Jim' s reluctance to rearrange the equation a + b = 15 and the strength of the equation's link to the action of the dynamic physical model, suggested to me that Jim was not ready to reason about and operate with symbolic equations in isolation from a real-world context. I considered telling Jim, "Just solve for a by subtracting b from both sides." But because Jim's thinking of the equation was so closely linked to the dynamics of the physical model, that would have compromised his opportunity for reasoning about the relationship that occurs in the physical system and the relationship as expressed in a symbolic equation. Jim did not think of the equation without linking it to the physical model to interpret it. I decided that one way to further scaffold Jim's knowledge of equations was to create a different but related action scenario. When I asked Jim directly. 'Tell me in words a general rule for how to figure out the length of a, if you have two segments together that equal 15, he arrived at subtraction on ly by considering specific numeric cases (e.g. 15 - 6 = 9). Then later, he reluctantly wrote a = 15 - a = b.

In an effort to further probe the strength of Jim's view of equations of the format a + b = 15 that arises from the slack rope board situation I introduced a new problem situation. An equation algebraically equivalent to a + b = 15 was generated from an entirely different action than that of the length of two segments together totaling 15 inches. I presented Jim with what became known as the pasta problem. In the pasta problem, several paper bags were filled with different combinations of two types of uncooked pasta. The two types of pasta, pine trees and wagon wheels, were easily distinguishable by touch. The intent of the pasta problem was to observe Jims' interpretation of the relationship between the number of wheels and the number of pine trees. In all cases, the total number of pieces of pasta in a bag was 12. I instructed Jim to determine the number of pine-tree pasta in each bag by removing the wagon-wheel pasta without looking inside the bag. I was interested to know if the separating action associa ted with this activity would suggest an equation of the format, 12 - a = b, which had been especially difficult to consider from the situation of the slack rope board.

After Jim had completed the activity with several different paper bags, I asked him to identify the variable and nonvariable features within the situation. He replied that the number of wheels and the number of trees were changeable and that the total amount of pasta in each bag was nonchangeable. I asked Jim to pick symbols for the changeable things. He drew a wagon wheel and a pine tree, hereafter indicated respectively as w and t. Then I asked him to describe in his own words how the process worked. Jim replied, "I stuck in my hand and pulled out the wheels and left the trees in there. I counted them [wheels] and I figured out how many wheels we had, which told us how many trees we had." He wrote, "12 = # of w or # oft and w + t = 12, explaining, "12 is the total in there. w is a certain number of wheels. We don't know how many unless you've already taken the stuff out." When I asked Jim to tell me again how he figured out how many trees, he explained, "Say it is 6; then the other is 6 'cause the total is 12." He understood a general process in the physical context. Using that as a known representational format from which Jim might build knowledge of symbolic equations, I decided to show Jim the equation, 12- w = t. He remarked, "it's kind'a what I meant. If there's 12 in there and you take one kind out, it tells you how many are left of the other kind." Then I pressed, "Do you remember the equation for the slack rope board? What if I said, "15 - bw = t?" Jim replied, 'That's the same thing as that [points to 12 - w = t], if the total number of noodles (pasta) was 15. But he further explained that the equations, 15 - bw = t and bw + t = 15, were not "the same." In order to make the equation, 15 - bw = t, sensible for the relationship of the slack rope board, Jim stated that he would need to change his thinking of how the slack rope board worked. He explained, "Move it the left segment] one way, measure one end, and it will tell you the other end [length of the right segment].

In the pasta problem Jim accepted the equation 12 - w = t, although he did not generate it. He argued that in the slack rope board situation we were not taking one number away from the total to equal the other. But after exploration of this situation, Jim was able to modify his thinking of the action on the slack rope board in a way which would justify an equation such as 15 - bw = t. In order to modify the equation format he had to change his view of the action on the physical model from which the equation was generated.

In the pasta problem, the action of the task was different and Jim was comfortable with an equation indicating an expression of the value of one variable in terms of the other. He was even able to consider changing the format of equations if the actions of the physical situations they represented were considered differently. The instructional decisions to encourage Jim to consider and compare the different physical situations, and their representing equations allowed Jim to encounter and reason about mathematically equivalent symbolic equations. By maintaining his connection of equations to the physical contexts through which they were created, Jim's meanings for the equations were not compromised, yet he expanded his knowledge of symbolic equations.

Conclusions

Scaffolding is the guiding language provided by a teacher to capture and signify students' emerging conceptualizations and to develop a structure through which students can reason further about, reflect upon, and communicate their conceptualizations. As a teacher initially becomes aware of students' conceptualizations, he/she can begin the scaffolding process by using the students' language to validate their thinking and to reflect back to the students how he/she has interpreted their' conceptualizations. For example, in Gabe's situation, I used his language, "doubling by 3", to acknowledge my understanding of his description of the function originating on the 3-spool as multiplication. I then showed him how to record symbolically the relationship he described verbally. Similarly, since Jim interpreted the function of the slack rope board in terms of the actions that occurred with the lengths of the sides, I began to use the action-based language that he created. This was helpful in enabling him to maintain h is interpretation of the function from the slack rope board as he reasoned about it in a table and an equation. By using students' language, the teacher acknowledges both for her or himself and for the students, an awareness of the initial development of their knowledge. From this point further scaffolding can be initiated to help the students extend their knowledge.

In this paper I described two ways in which instructional decisions scaffolded learners to link their knowledge created in a physical system to symbolic equations. In these examples, the investigator's interventions enabled the students to move beyond constraints in their creation and/or use of symbolic equations. With Gabe, I showed how to translate knowledge of functions generated in one representation, a verbal description of operation of the spool system, to another representation, a symbolic equation. Because be had developed an interpretation of the function from the physical model, instruction that showed him how to write an equation did not compromise his opportunity for reasoning about the function, but helped symbolically anchor his knowledge. With Jim, I enabled comparing and reasoning about the equivalence of two equations by encouraging him to modify his original interpretation of the action of the slack rope board through which one of the equations was generated. Again, because the instructional intervention was based upon Jim's interpretation of functions in a physical situation, his opportunity for reasoning and building knowledge of mathematical relationships was preserved.

These examples illustrate two ways to continue scaffolding after the initial use of students' language to validate their thinking and reflect back to the students how their conceptualizations have been interpreted. But language was not the only scaffolding tool. In Gabe's case, the symbolism of an equation anchored his knowledge of the function of the spool system to enable his understanding of the equation intended to represent it. In Jim's case, the reinterpretation of the action of the slack rope board scaffolded greater understanding of equivalent symbolic equations.

There are practical challenges associated with instructional decision-making. An interpretation of understanding as the building of links between different representations as described by Lesh et al, (1983) provides a useful framework in which to develop instructional decisions. In this investigation learners developed an interpretation of function concepts through explorations of physical systems. These explorations and interpretations served as anchor points supporting instruction intended to promote understanding of functions represented in equations. Viewed in isolation, the instruction provided in these examples may appear too directive to encourage student reasoning and building knowledge of mathematics relationships. When the larger picture of students' understanding is considered as the liking of representational formats, these instructional decisions illustrate how a teacher might build on students' existing knowledge and show students how to extend their knowledge to unfamiliar representational for mats. This kind of scaffolding presents no cause for concern about compromising learners' opportunities to reason and build knowledge of mathematics relationships. It suggests an important step in addressing practical concerns regarding teacher input in instruction.

Acknowledgements

This research was conducted as part of the author's doctoral dissertation completed at Northern Illinois University under the co-direction of Frank Bazeli and Helen Khoury. The author gratefully acknowledges their support and guidance.

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Author: | Hines, Ellen |
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Publication: | Focus on Learning Problems in Mathematics |

Geographic Code: | 1USA |

Date: | Jan 1, 2003 |

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