Math in motion: using CBRs to enact functions.

This article reports on results of an exploratory study on undergraduate pre-service teachers' understanding of graphical representations of motion functions. The study described pre-service teachers' explorations using a CBR device. Pre-service teachers' growth was studied in two dimensions: (a) in their learning of the mathematics involved and (b) in their learning of the pedagogy related to the mathematics and the technology used. Through their interaction with the device, pre-service teachers were able to overcome common misconceptions with respect to the mathematics and also to develop pedagogical insights regarding the teaching of the concepts.

MATH IN MOTION: USING CBRS TO ENACT FUNCTIONS

Pre-service teachers enrolled in mathematics education courses bear a dual role. As Bowers and Doerr (2001) note, pre-service teachers are "simultaneously learners and teachers in transition" (p. 115). This is especially true when pre-service teachers are introduced to seemingly "non-traditional" topics such as the development of algebraic reasoning at the elementary school level. Elementary school pre-service teachers are expected to construct knowledge of new content that they may not have had the opportunity to learn or of which they have little or fragmented knowledge (Kilpatrick & Swafford, 2001). Furthermore, pre-service teachers are expected to develop pedagogical insights and models for teaching these unfamiliar concepts and skills to their own future students while they, themselves, have had little (if any) experience observing other teachers teaching these topics.

Teacher preparation institutions have traditionally drawn a line between content and pedagogy. This often has resulted in pre-service teachers being required to enroll first in mathematics courses (often taught separately in the mathematics department) and subsequently in "methods" courses that focus primarily on the development of pedagogical knowledge. However, research has shown that this scheme is not necessarily successful in helping preservice teachers construct interconnected knowledge of mathematics and pedagogy (e.g., Brown & Borko, 1992; Lappan & Theule-Lubienski, 1994). Drawing from this research, the Conference Board of the Mathematical Sciences [CBMS], in a report on its vision of the education of mathematics teachers, argues for a close coordination of content and pedagogical courses in order to facilitate the development of strong knowledge of the mathematics that pre-service teachers will teach (CBMS, 2001). Furthermore, pre-service teachers themselves have indicated that they would be more willing to take additional mathematics courses if they found more connections between the content of the coursework and the mathematics that they would be expected to teach (Smith, 2000).

One way to address this call for coordination of content and pedagogy is to embrace the pre-service teachers' dual role as learners and teachers. This can be accomplished by accommodating their needs in education courses that aim to teach both the new mathematical content and the pedagogy related to this content. A growing number of teacher educators advocate this approach to mathematics education courses. Teacher educators in this group believe that pre-service teachers' understanding of mathematics will be more directly strengthened when they learn by engaging in mathematical tasks similar to those their students will be asked to complete as well as in ways that reflect the style of teaching and learning that they are expected to practice and promote (e.g., Ball and Cohen, 1999; Simon and Schifter, 1991). Smith (2001) argues that the education and professional development of teachers should be "situated in practice." In this view, the everyday tasks and tools that are central to the work of teaching (including the curricular materials that are used in classrooms) should provide the focus and object of teacher education. Allowing pre-service teachers to engage in mathematics learning while reflecting on the practice of teaching affords them the opportunity to build strong networks of interconnected knowledge.

The mathematics of motion, and particularly graphical representations of motion functions, is one of the topics of which pre-service teachers have limited or fragmented knowledge. While the development of algebraic thinking, including the mathematics of motion, is one of the major goals in the K-12 mathematics (NCTM, 2000), the majority of elementary school teachers report that they feel inadequately prepared or unqualified to teach algebra (e.g., Weiss, 1995) and that they spend little or no time teaching the topic (Grouws & Smith, 2000).

In this article we describe a study designed to understand how engaging pre-service teachers in mathematical tasks related to mathematics in motion may impact their understanding of the mathematics involved in the task as well as how they might teach this topic to their own students. Given the limited or fragmented understanding of functions and, in particular, motion functions, of the pre-service teachers involved in this study, their participation allowed us to study them in their dual role of learners of mathematics and teachers in transition. We chose to advance our pre-service teachers' understanding of the mathematics of motion and its pedagogy by engaging them in technology-based mathematical tasks. It was hypothesized that novel technology environments would provide a variety of opportunities for advancement of these pre-service teachers' mathematical understanding of the topic and for their reflection on pedagogical issues related to the mathematics of motion. However, little is known about how technology can be used by pre-service teachers in their dual roles. Our goal was to investigate the use of specific types of technology in each of these two aspects of pre-service teachers' education.

Graphical Representations of Functions--A Brief Review of Education Research

Students' understanding of the various representations of functions, and, in particular, their understanding of graphical representations of functions has been the object of several research studies (for a detailed review of this literature see Leinhardt, Zaslavsky, & Stein, 1990). This literature focuses largely on the misconceptions and misunderstandings that characterize students' responses to the visual qualities of graphs. Inexperienced users of graphs often interpret graphs "iconically." With respect to graphs of motion (both distance versus time and velocity versus time graphs), inexperienced users often interpret such graphs as actual pictures of the path traveled by an object making an overtly direct connection between the visual features of the graph and the situation it represents (e.g., Janvier, 1978; Kaput, 1987). In addition, the literature points to student difficulties in making connections between graphical and other representations of a function--primarily symbolic and numerical (Leinhardt et al., 1990). Our collective experiences with student understanding of graphical representations suggest that graphs do not necessarily aid students to further their understanding of a novel (to them) concept. This difficulty of understanding graphical representations found in novice learners is contrasted by evidence that individuals who are skilled in mathematical problemsolving tend to rely on visual representations (including graphs and diagrams) as tools that add information in the problem-solving process (e.g., Leikin, Stylianou & Silver, in press; Stylianou, 2002). Similar results are described in the science education literature (e.g., Ochs, Jacoby & Gonzales, 1994).

The topic of graphical representations and, particularly, the issues surrounding the learning and teaching of graphical representations gained further importance with the advancement of technology and its new role in classrooms. Modern computer technology and software offered students of mathematics and professionals engaged in mathematical work unprecedented access to easily generated and manipulated types of representations. Educators were quick to realize that these new technological learning environments and tools could lift some of the obstacles interfering with the understanding of graphical representation. Hence, calls for instructional reform in mathematics have explicitly focused on the importance of multiple representations (including numeric, graphic, and symbolic) in the learning of core concepts (e.g., Kaput, 1986). In particular, the Principles and Standards for School Mathematics (NCTM, 2000) includes a representation standard and attests to the importance of multiple representations in mathematics teaching and learning in pre-K-12 classrooms.

The affordability and convenience of graphing calculators, in particular, opened new possibilities for using and creating links among numeric, graphic, and symbolic representations. Most middle and secondary school curricula encourage students to use graphing calculators on a regular basis in order to explore mathematical concepts in their various representations. Indeed, the use of multiple representations appears to have a positive impact on student learning. A number of studies (e.g. Demana, Shoen, & Waits, 1993; Ellington, 2003; Kieran, 2001; Moreno, Rojano, Bonilla, & Rerrusquia, 1999) demonstrate that calculator environments induce students to conceive of algebra as a language for representing general phenomena and relations. Computer environments with graphing capabilities are also being used successfully as tools for representing the relationships of problem situations and as visual support for understanding symbolic expressions (e.g., Stylianou & Shapiro, 2002).

However, at the same time some studies have brought new insights related to students' understanding of multiple representations to our attention by revealing that multiple linked-environments do not always help students acquire a complete understanding of functions. For example, Schoenfeld, Smith, and Arcavi (1994) investigated students' behavior when engaged in the study of multiple representations of functions. In this study, it was shown that despite the presence of multiple representations and the student's ability to move flexibly between these representations, the student found functions difficult to understand as objects. It became clear that students need experiences on which to ground their understanding of new functions. Hence, the physical action, or the actual phenomenon that is being represented by symbols, graphs, and numbers is currently attracting the interest of researchers in the field as a fourth type of representation.

New studies are beginning to uncover the important roles of physical motion in understanding mathematical representations (Kaput & Rochelle, 2000; Nemirovsky, 1993; Nemirovsky, Tierny & Write, 1998; Stylianou & Kaput, 2003). In studying their own movement, students confront subtle relations among their kinesthetic sense of motion, interpretations of other objects' motions, and graphical, tabular and even algebraic notations. It is becoming clear that the fourth perspective, the phenomenon that is being represented in parallel as well as in an interaction with the three (now widely accepted as "traditional") types of representations is a necessary addition to the study of new, mathematical concepts.

In the past decade or so, the introduction of microcomputer-based laboratory (MBL) equipment (1) that allows one to import physical data to a computer has opened the possibility of actually representing phenomena. More recently, new equipment available to educators allows for the importing of temperature and motion data into inexpensive and broadly used calculators--the calculator-based-laboratory (CBL) and the calculator-based-ranger (CBR). This new technology supports the notion that representations are about something, especially something with which the student has first hand experience. The student may now learn about both the phenomena represented and the mathematics that is used to represent their quantitative aspects. This serves to reflect the connection of mathematics with experience and its power as a sense-making tool.

Once again, educators expressed an interest in the use of the new technology in mathematics classrooms (e.g., Beckman & Rozanski, 1999; Doerr, Rieff & Tabor, 1999; Graham & Sharp, 1999) and, naturally so, even more enthusiastically in science classrooms (e.g., Linn, Layman, & Nachmias, 1987; Mokros & Tinker, 1987; Nachmias & Linn, 1987). However, despite their growing popularity, little is known about how these tools are to be used in the mathematics classroom or the potential benefits of CBL and CBR instruction to students' competence in graphing, in particular, and understanding of the mathematics involved, in general.

CBRs as Tools in Teacher Education--The Purpose of This Study

As is the case with all new technologies, CBRs and the idea of modeling phenomena are new to many pre-service and in-service teachers. Further, teachers are asked to familiarize themselves with these devices in a short amount of time. This "familiarization" means the ability to use the technology, demonstrate its use, as well as foresee and troubleshoot all the possible ways it can potentially go astray in the hands of students. Then, these novice teachers are expected to appropriately incorporate the use of this technology in their classroom instruction. Hence, an environment that incorporates the use of CBRs for the study of motion graphs provides us with the opportunity to study pre-service teachers in their dual roles of learners (new technology and the mathematics involved in it) and teachers (instructional issues associated with the mathematics of motion and the pedagogical implications that the new technology may have). Specifically, this study aims to investigate two issues:

1. In what ways does pre-service teachers' own understanding of the graphical representations of motion functions develop when using the CBR technological tools?

2. How do pre-service teachers use the CBR technological tools to develop insights into pedagogical issues related to the graphical representation of motion functions?

Theoretical Perspective

The theoretical framework that guides our work is based on a socio-cultural perspective where learning is situated in everyday practice. Knowledge is a construct of the social and cultural environment surrounding the individual, implying that the development of knowledge cannot be understood apart form the social context in which it occurs (Vygotsky, 1962/1934). From this perspective, the learning of mathematics is viewed as situated in day-to-day practice and in communities of practice; it is a tool in the pursuit of everyday goals in life. Consequently, as Greeno (1988) argues, "learning mathematics involves acquiring aspects of an intellectual practice, rather than just acquiring some information and skills" (p. 481). The critical element in this perspective is that learners acquire understanding of mathematics through their participation in mathematical practices--including using mathematical tools and taking part in mathematical discussions.

Graphing--as a mathematical practice--takes meaning from context. In our classroom, the use of graphing was introduced in the context of an activity with the calculator-based-ranger (CBR) where pre-service teachers were asked to use graphs as a means to describe motion. The graphs were used as tools to understand motion and as means of communicating about motion. These uses of graphs fit relatively well with those found among mathematicians as well as the applied mathematics community (e.g., scientists and engineers). Roth and McGinn (1997) suggest that graphs serve three major purposes:

(a) Semiotic (representational) objects that constitute and represent other aspects of reality. This is the most common educational usage.

(b) Rhetorical objects in scientific communication; a graph can be manipulated (e.g., change of scale) in a way that allows certain not-so-obvious patterns to emerge. In this case, the graph-user makes choices during the graph presentation in order to show an alternative perspective.

(c) Conscription devices that mediate collective scientific activities (talking, or constructing facts). Here graphs are central to interaction among scientists. Graphs constitute a shared interactive space that facilitates communication, as a graph may be used as a common language tool.

The study described in this article was based on the premise that learning about graphing can emerge as an aspect of mathematical practice. As such, it allows learners to use a graph as a tool to complete a task and as an object of learning itself. Hence, the learners in our environment were given a (relatively) contextualized situation and asked to control the graph-generating tools so as to arrive at a group solution. The goal was to shift the learners' attention from learning skills to participating in mathematical practices.

METHODS

Participants: There were 28 participants--pre-service teachers who attended a mathematics course for elementary school teachers. All involved consented to the instructor's request to participate in a 2-week study (four classroom sessions). During the 2 weeks, the participants worked in four groups of 5 pre-service teachers and in two groups of 4 pre-service teachers. The number of groups (six) was determined based on the number of available CBRs and graphing calculators. All participants were comfortable with the idea of working in groups, and allowed to choose working partners.

Technological tools and environment: At the center of our investigation lies the use of Calculator-Based Rangers (CBRs) and graphing calculators. CBRs are one form of CBLs (discussed earlier) designed to collect specifically motion data. Specifically, CBRs, similar to sonic motion detectors, send out ultrasonic pulses and then measure how long it takes for each pulse to return after bouncing off an object. As the measurements are collected, the CBR calculates its distance from the object and imports these motion data into the graphing calculator (2). The graphing calculator then displays these data as a graph. Graphing calculators used in this study were equipped with the MathWorlds (3) software environment. MathWorlds allows the user not only to see the graph of the motion (as captured by the CBR), but also to replay the motion as an animated simulation.

Instruments and Data Collection Procedures: The pre-service teachers were first asked to respond to two questions--these questions formed the study's pre-test. The two questions (shown in Figure 1) asked for a story that would describe each of the two given graphs. Our goal was to examine some aspects of the participants' pre-existing understanding of graphs of motion--hence the questions used in the instrument were variations of test items frequently used in studies on graph comprehension (e.g., Berg & Smith, 1995; Mokros & Tinker, 1987; Parke, Lane, Silver, & Magone, 2003). Participants were allowed to work in groups while responding to the two pre-test questions and videotaped while they did so.

It was assumed that the study participants were familiar with neither CBRs nor graphing calculators. They were given an introduction to the use of the two devices as well as a handout with detailed instructions. For example, they were shown how to connect the CBRs to graphing calculators, collect motion data, and display their data on the calculators. Subsequently, working in groups they were asked to complete a mathematical task using the CBRs to collect data and represent mathematics in motion. This task was divided into six subtasks and included questions for reflection among group members during the indicated activities (Figure 2). The main idea of the position of subtasks and questions was to engage the pre-service teachers in an investigation of a simple position graph (positive constant slope). The series of subtasks focused the students' attention on matching simple graphs. Additionally it asked for the creation of graphs that interact with the given graphs in pre-determined ways. It was hoped that the issues of velocity, starting point, distance, positive and negative distance time would all come out during the investigations. The second series of questions was equivalent to the first series except that it dealt with the topic of velocity. Note, however, that this study did not aim to teach algebraic notation and representation of motion functions. Rather, the two-week intervention focused on graphical representations of motion functions and aimed at the development of graph-associated skills. This was a decision based on the well-documented finding that individuals with a weak understanding of functions when given the choice to work with either algebraic notation or graphical representations prefer the former--resorting to the use of procedures without connection to meaning or other representations.

[FIGURE 1 OMITTED]

Pre-service teachers engaged in this study were encouraged to familiarize themselves with the CBRs in order to understand their function and to explore any questions that may arise during group explorations and discussions. The researchers kept a stance of non-interference, allowing students to pursue the questions they found interesting. The CBR activities and the following class discussion were completed during the four class- sessions (approximately 75 minutes each session) and videotaped by two roaming-video-camera operators. The first two sessions focused on position graphs, and the remaining two sessions focused on velocity graphs. The end of the second week, each group was asked to respond to the two questions shown in Figure 3.

Finally, the pre-service teachers were asked to write about the mathematics they had been doing and to provide an instructional activity they would use with 5th or 6th grade students using the CBRs.

[FIGURE 3 OMITTED]

Data Analysis: The pre-test and post-test were first coded for correctness. Errors were then categorized and described based on the findings of earlier studies that had used similar tasks. Similarly, the reflections provided by the participants were coded and analyzed based on correctness with respect to the mathematics.

The videos were transcribed and subsequently coded and analyzed using a qualitative methodology. The method of analysis was to derive inductively the descriptions and explanations of how the study participants interacted with the CBR devices both as learners of mathematics and as pre-service teachers and approached the three topics that form the core of this study: (a) using graphs as representations, (b) using graphs as rhetoric devices, and (c) using graphs as conscription devices. Including related behaviors and comments within each topic, we wrote descriptions of the students' operations and actions. These descriptions formed the findings of the study.

RESULTS

The main goal of our work was to examine how pre-service teachers use technological tools to develop their understanding of graphical representations of motion functions in terms of both mathematical and pedagogical content. The results of the pre-test suggested that pre-service teachers' mathematical understanding varied substantially. While four of the six groups were able to respond to the pre-test tasks, correctly providing a story that described the given motion graphs sufficiently, the remaining two groups responded incorrectly. Two major type of errors, both of which have been widely observed within secondary school and college populations, were documented in the analysis of the pre-test data: (1) a graph-as-picture misconception; and (2) a slope/height misconception. Both errors are, in fact, variations of the well-documented error of the "iconic" interpretation of a graph. In the first case, pre-service teachers interpreted the graphs as if they were actual pictures of the path traveled by "Tony"--including turns and names of streets. In the second case, positive slope was interpreted as running up the hill, and negative slope was interpreted as a down-hill path. Additionally, video data suggested that (at least) three other pre-service teachers in the groups that provided correct responses held similar misconceptions (iconic interpretations of graphs). Finally, no one in the six groups indicated a connection between the velocity and distance graphs.

The pre-service teachers' responses to the pre-test questions provided us with a better appreciation of their general understandings of graphs and graphs of motion functions. Our participants varied considerably with respect to their ability to read and use graphs--a characteristic of most elementary pre-service teachers' classes. This variability in previous experience and understanding of graphs provided a context for their group explorations and discussions during our experiment.

Here, we describe the ways in which pre-service teachers first came to develop both mathematical and pedagogical insights into certain core ideas of graphs of motion functions. in each of the three main contexts in which graphs are usually used: (a) graphs as representations, (b) graphs as rhetoric devices, and (c) graphs as conscription devices (as shown in Table 1). In the discussion that follows, we describe our findings in each of these six categories in more detail.

Pre-Service Teachers as Learners of Mathematics

We describe here the insights pre-service teachers developed with respect to graphs in their roles as learners of mathematics.

(a) Graphs as representations: The most common use of graphs in school mathematics is as semiotic objects (representations) that constitute other aspects of reality. The CBR sessions revolved around reading and creating graphs of motion functions. Hence, pre-service teachers had to face their misconception of iconically interpreting graphs. During the first session, they were asked to "match" a position-time graph consisting of one segment of positive constant slope by walking a similar motion. We focused our attention primarily on the actions (and reactions) of the groups whose pre-test responses suggested misconceptions of graph-reading.

As the groups were first asked to discuss a given graph and then come up with a plan for their own motion, the initial reactions of group members were mixed varying from attempts to "read" the graph to approaching the task using a trial-and-error methodology. One suggestion that occurred frequently was for the "walker" to walk "faster and faster" to match the positive constant slope. After overcoming some technical difficulties, the groups started implementing their plans. As explained earlier, the MathWorlds software environment allows the user not only to see the graph of their own motion (as captured by the CBR), but also to re-play the motion. This feature facilitated the discussion among participants in that their arguments did not have to rely solely on the memory of their "walk" or on the difficult-to-understand static graph. Instead the walk could be re-played and matched piece-by-piece to the corresponding graph, as illustrated below:

Jen: Which one is our graph? [graph of the group's motion]

Mike: This one [pointing at their graph].

Sheryl: Isn't it the one above? [pointing at the given graph]

Kim: Let's re-play it.

Jen: What's that lump?

[They replay the motion watching the two dots moving according to the two graphs. Dot A, corresponding to the given motion, moves at a constant velocity, while Dot B, corresponding to Sheryl's motion moves slower, stops moving briefly, and then resumes the motion again.]

Sheryl: Oh, I stopped--that's why we get that lump. That's me!

Mike: And you walked slow. So, a lower graph is walking slower?

The group initially could not identify the graph that represented their motion. The relationship between indicators such as velocity, distance traveled, and a brief stop in the motion, and the graph was not immediately seen by the groups. It was only when the motion of the group's walker, Sheryl, was re-enacted that the relationship started becoming obvious. Now, the group was ready to read this bi-directional relationship and construct a graph.

Pre-service teachers who had responded correctly to the pre-test tasks were more confident in their enactments of the given motion. They knew they had to walk a motion of constant velocity. However, they were soon faced with other issues not initially considered. For example, to find the actual velocity, they had to determine the exact distance traveled:

Nick: The lines are parallel.

Tammy: I was going faster.

Nick: No, it starts a little higher, a little farther away, like ... two feet

Tammy: Maybe I stared walking before she said "go"?

[...]

Katelyn: Here you have distance at zero time and here you have time at zero distance.

What does it mean for two position graphs to be parallel? What does it mean for two position graphs to have different slopes? And how does one determine the velocity of a motion by looking at a position graph? These are the sort of questions our participants did not ask themselves while responding to the pre-test tasks. Yet, these questions had to be answered in order to enact the given motion. Not only were these pre-service teachers faced with their misconceptions on graphs, but they were also faced with gaps in their understanding.

The following week, the pre-service teachers were asked to work with velocity graphs. By this time, they had had a better understanding of graphs as representations of distance. However, velocity graphs brought forth new issues for discussion. In particular, the pre-service teachers found it difficult to infer distance from a velocity graph. Indeed, realizing that a velocity graph determines a family of position-graphs has been reported as difficult to grasp in a number of studies. As Bowers and Doerr (2001) found in their study, pre-service teachers (even those who are being prepared to teach secondary mathematics) are not always fully aware of the fact that one single velocity graph may represent a whole family set of position graphs with varying starting points. In our sessions, one group brought to the classroom discussion their finding that when attempting to match the velocity graph, it did not make a difference where the "walker" started her/his walk; different starting points resulted in the same velocity graph. This conjecture motivated other groups to experiment further with velocity graphs, and spurred rich classroom discussions.

(b) Graphs as rhetoric devices: In scientific communication, graphs are tools used in the problem-solving process. Mathematicians often manipulate graphs for alternative perspectives or to allow patterns that may not otherwise be obvious to emerge (Stylianou, 2002). Students' limited understanding of graphs, on the other hand, often lead them to view graphs as static snapshots--there's little one can do to change a snapshot and allow hidden features to emerge.

As the pre-service teachers in our study spent more time working with position and velocity graphs their increased familiarity with graphs as objects that can be constructed, talked about, and described, allowed them to begin to think of ways to use graphs as tools that might aid them in exploring different perspectives and, presenting these new views to their peers. The following episodes illustrate the ease with which graphs were beginning to be used to resolve questions and confirm tentative hypotheses:

Episode I: [Laura, David and Carolyn are working with velocity graphs and they wonder what will the graph look like if they walked backwards]

Laura: He should go backwards ...

Carolyn: Yes, but the time keeps moving and he keeps moving and he keeps moving in a constant velocity, and we agreed that position doesn't matter.

Laura: Yeah, but it's negative velocity. Let's walk it. Shall we go slow, fast, stop and backwards?

David: No, we only want to try going forward and then backward. We only need to clarify this part of the graph here. OK, ready? Go!

Episode II: [Nick and Tammy are trying to determine if a certain part of the graph should be smooth ("rounded") or sharp]

Nick: Is it rounded here? We should get a peak.

Tammy: We stopped, paused and then went back.

Nick: Let's do it slowly so we can see this point here.

In both episodes, pre-service teachers suggested ways in which graphs can be manipulated to allow for different views and arguments. In the first episode, David suggested that his group did not need to enact the entire scenario; they only needed to enact the part of the motion that was in dispute, and focus on that to understand velocity graph behavior in general. In the second episode, Nick suggested that his group repeat the motion around the disputed point slowly--an attempt to enact a "zoom in" or to magnify a portion of the graph that seemed unclear. It is important to note that the participants were not exactly manipulating the graph; they were manipulating the motion to allow for a different view of the graph. In other words, our participants did not choose to change the scale, nor did they use the "zoom in" feature of the graphing calculator to act immediately on the graph. They approached their exploration by linking the graph back to the phenomenon that the graph represented. The manipulation of the motion (and the direct feedback from the graphing calculator) was not only easier, but also more meaningful to them.

(c) Graphs as conscription devices: Graphs are often at the center of communication among mathematicians and scientists. All pre-service teachers in our sessions acted in a similar manner by participating in discussions around graphs. They first developed their understanding of the graphs--a common communication code--and then used the graphs to communicate with their peers and explain their reasoning by pointing to data points, lines, and axes, and gesturing to graph trends:

Episode I.

Erin: Did he get the right distance?

Stacey: Yeah, look it stops right there, on the 12th mark [pointing at x-axis].

Kelly: Yeah, he stopped right where the TV is [which is about 12 feet away]

Erin: Did you walk fast enough?

Bruce: I walked faster at the end. The slope is going up [gestures with his hand]

Episode II.

Kara: The previous time we did one step a second, now I'm going to try to do one step in two seconds for four seconds, and then two steps a second the last two seconds.

Jody: No, I think the first two seconds we should do one step a second and then the next two seconds we should do two steps a second.

Amy: What? Wait!

Emily: Why don't you graph it for us?

In both episodes, the graphs were at the center of the group's discussion. Once again, the motion provided context for the graph, but the group members pointed at the graph while linking the motion to characteristics (e.g., in the first episode Bruce mentioned his increasing velocity while pointing at the increasing slope). As the graphs were produced in a group effort, it became natural to use the graphs to facilitate the discussion. In this case, they were not talking about a graph as a foreign object, but about the outcome of a joint labor based on a shared understanding of what the graph actually represents.

The second episode illustrates yet a different instance of the use of graphs as conscription devices in the hands of pre-service teachers. The members of the group were not simply using a pre-established graph as a context for their discussions, they were asking Jody to graph her plan for the group as a way to communicate her reasoning. For this group, it was graphs in general and not a particular distance or velocity graph that was considered common understanding. That is, any graph at all could be the basis for shared understanding because, as a group, the participants shared an understanding of the process of constructing and reading a graph of this type.

Finally, the extensive talk about graphs resulted in rich individual reflections about the mathematics of graphs. In their journal reflections, pre-service teachers spent a substantial amount of time describing their experiences with the CBRs and graph reading and construction. While our study did not measure the changes in our participants' ability to talk about graphs or their ease in doing so, our informal observation corroborates an earlier finding of Mokros and Tinker (1987) who had reported that younger students working with MBLs showed significant changes in their competence to talk about and use graphs.

Pre-Service Teachers as Teachers in Transition

In this section, we continue to view the participants as students learning mathematics. However, we now consider the ways in which they developed pedagogical insights as they worked with the CBR activities. Note, that our study participants were pre-service teachers, that is, individuals with little (if any at all) experience in teaching mathematics and working with students. Thus, the data that were used and analyzed for this study were not classroom data; we did not have any evidence on how our study participants used or would use their knowledge on graphs to actually teach a lesson. Our data was based on the videotaped group discussions as well as on the journal reflections. While we recognize the limitations of self-reports and responses to potential instructional scenarios, we also take these reports as indications of pre-service teachers' beliefs and attitudes. Indeed, our role as instructors of pre-service mathematics education classes is often limited to exposing our pre-service teachers to a broad spectrum of topics on pedagogy and, whenever possible, attempting to change long-held beliefs on the nature of mathematics and its teaching.

(a) Graphs as representations: One of the most widely reported insights by participants was the difference that the body motion made in their own understanding of graphs. Pre-service teachers recognized the value of building on students' kinesthetic experience and linking this experience to other representations of motion functions:

      "I usually get confused with graphs, as was the case at first when
      the class did this activity. Using the motion detectors and
      actually walking to match the position and velocity graphs really
      made me understand through trial and error what was happening"

All of the study participants mentioned that they personally benefited from the use of CBRs and consider it important to incorporate what they had learned about physical motion into the study of graphs in their own classrooms. This insight is supported by previous research on the student use of MBL in science classes (Mokros & Tinker, 1987) which concluded that "learning through MBL provides a real-time link between a concrete experience and the symbolic representation of the experience" (p. 381).

(b) Graphs as rhetoric devices: Pre-service teachers discussed the difference between local and global interpretation of graphs as tools for different types of arguments. While all pre-service teachers had been asked to "read" graphs in their tenure as students of mathematics, they recognized that their own previous graph interpretations were limited to general or global comments (at best). However, their experiences with the CBR devices helped them recognize the value of local interpretations. These experiences involved paying attention to specific point-wise features such as points of discontinuity (e.g., Can a person physically walk a motion that involves a sharp change in position or velocity?), intercepts of graphs (e.g., What does it mean for position graphs of two characters to cross? Does it mean the two characters run into each other? How about intersections of velocity graphs?). Our elementary school pre-service teachers may not fully appreciate the value of the habit of mind of looking at local behaviors of graphs. However, we claim that those students trained to develop such a habit of mind will be more inclined to look for discontinuities and relative extrema in graphs when the begin studying calculus later on in their program.

I. "Usually, learning about graphs is limited to overall graph interpretation, which causes some aspects and factors to be overlooked as in the case with my group. Therefore incorporating an activity like this would be beneficial in developing specific concepts."

II. "Not only are we seeing how movement is detected, we are also able to really work with graphs. By doing this we start to see and understand how graphs work and what different lines or patterns mean, and we can use them to explain different parts of the problem."

(c) Graphs as conscription devices: Finally, the journal reflections indicated a recognition of the importance for students to talk about mathematics (in general) with their peers. Further, practically all of our study participants appeared to recognize the need for teachers to provide their students with appropriate environments that allow for discussion and communication and, particularly, with appropriate contexts that would captivate their students' attention and make the study of graphs appealing:

I. "I will give [my future students] realistic and interesting scenarios like [for example] a race and ask them to talk about their assumptions and share their thoughts."

II. "When the graph was on paper it looked self-explanatory, but after going through the motion it allowed us to see our mistakes, to talk about what it all meant, and it led us to a better understanding of the graph overall."

III. "I will make my students write a paragraph about what is being represented in each graph [...] they need to be able to explain clearly."

The pre-service teachers' own experiences during group work with the CBR technology led them to recognize the value of communication (oral and written) in learning mathematics. As the second excerpt suggests, pre-service teachers often do not initially see all that there is to be explored and discussed in mathematics. Graphs appear to be self-explanatory, and students do not realize their own limitations until they are faced with them. Hence, as the first excerpt suggests, pre-service teachers need to recognize the need to provide contexts for mathematical discussions and opportunities for students to face their own misconceptions.

DISCUSSION

This study aimed to investigate the use of CBRs as tools in the development of mathematical and pedagogical knowledge by pre-service teachers with respect to the graphical representation of motion functions. Approaching the two research questions from a sociocultural perspective, we were interested in identifying the ways in which CBRs facilitate the development of pre-service teachers' mathematical and pedagogical knowledge of graphs. Our results suggest that pre-service teachers who are provided with the opportunity to work with these tools make gains in their ability to use graphs in their three main functions: as representations, rhetoric devices, and conscription devices. The study participants, who possessed diverse backgrounds in their knowledge of graphs of motion functions, ended the study by using graphs in ways that suggest that they were able to overcome some of their initial misconceptions about graphs (including iconic interpretations of graphs) and use graphs as a means for communication and argumentation with their peers. This is a critical issue, since these pre-service teachers will be faced with the demand to teach in a "reformed" way that demands a stronger and deeper mathematical foundation wherein "the conception of content is more uncertain than a traditional view of mathematics as skills and rules, the view of children as thinkers more unpredictable" (Ball, 1993, p. 394).

Furthermore, the pre-service teachers developed pedagogical insights while learning the mathematics of motion functions. Pre-service teachers referred to their own learning experiences with the CBRs as valuable in understanding the potential difficulties of their future students, suggesting that the new experiences and insights they had developed as learners contributed to their pedagogical knowledge--a finding that corroborates earlier work by Bowers and Doerr (2001).

While it was tempting to attribute the learning that the study participants experienced to the actual CBR devices, we stepped back and looked at the context in which these gains in mathematical knowledge and changes in attitude towards teaching practices took place. Each time we recorded an individual pre-service teacher experiencing a new mathematical insight, the experience was the outcome of a discussion with her/his peers about graphs. Hence, we join Roth and McGinn (1997) and Mokros and Tinker (1987) in their proposals that the benefits of the use of CBRs (and potentially other similar technology tools) may be due to the fact that the CBRs make "graphs the central means of communication" (Mokros & Tinker, 1987, p. 369). This argument fits well in the sociocultural framework within which this study is situated; Vygotsky maintained that "higher voluntary forms of human behavior have their roots in social interaction" (Minick, 1996, p. 33). The development of higher order thinking (and it is safe to claim that reasoning about functions falls within this category) finds its origins in community discussions and is, then, internalized by the individual.

The findings of this study can also be discussed in the context of the constructivist perspective--where learning is viewed as a process of working to resolve dissonance by developing one's explanations for a particular phenomenon (von Glaserfeld, 1995). Viewed from this perspective, the cognitive changes that the pre-service teachers experienced in their understanding of graphs of motion functions can be seen as a result of their work with the CBRs--the immediate feedback they received when viewing the graphs produced by the CBRs and graphing calculators elicited perturbations in the pre-service teachers' previous conceptions of graphs of motion functions. However, this would be a very general statement. For, each pre-service teacher participated in a group activity, and the perturbations in the pre-service teachers' conceptions were often resolved (causing the cognitive change) within these group discussions. That is, even though the technology tools may have assisted the pre-service teachers in recognizing a gap in their own mathematical knowledge, it was within the group discussions that new insights were formed.

Overall our results suggest that pre-service teachers embraced the use of the technology tools and used them to build new and deeper understandings on both a mathematical and a pedagogical level. Indeed, we see in these tools an opportunity to approach the education of future teachers in graphical representation--an important topic that is far too often overlooked in pre-service teacher programs.

Turn on your calculator and choose the application "CBR & Position" in
the "CBR Animator" menu. On your screen your will see the position graph
of a person's motion (we'll call this person "B"). In the activities
described below, you will be asked to act out physically this motion,
that is, to walk in such a way that your walk will re-create the same
graph as the one shown on your screen. The CBR will collect the data
representing your motion and will display it next to this graph. Follow
the instructions below:
1. Before you begin the activity, discuss in your group and describe
   below what this graph represents. Talk about the distance covered by
   person B whose motion is represented here and the velocity with
   which they moved.
2. Now, press the blue button under the indication "GO", to see this
   motion being acted out by a dot. Does this animation help you
   understand the motion better? Do you want to change the description
   you wrote above?
3. Now you will walk the physical motion so that you match the graph
   displayed on the computer screen. To do that have one person be
   ready to do the walking. Put the motion detector in place as
   explained in class. Press 2nd LINK (yellow button and the button on
   the right of the green button). Go down the menu to the "Begin
   Sampling" and as soon as you press it, the CBR starts the data
   collection. When the data collection is completed press ENTER The
   new graph represents your motion. Copy it here and discuss whether
   you are satisfied with your performance! If not, try again, and
   again, and again.... Choosing "GO" you will have a chance to see
   your motion being "acted" by the top dot.
4. Do you want to adjust your initial description of what the graph
   represents?
5. Now walk a physical motion so that you start off at the same place
   as person B, but move slower than B, and catch up to B at the end of
   the motion.
   - Record both your graph and B's graph below.
   - Describe how the differences between the two graphs match specific
     differences in their motions. For example "From 0 to 2 seconds, B's
     graph was below/above mine and this meant that B was moving
     faster/slower...."
6. Now, here's a different scenario: You are walking with your friend
   towards the school, but halfway through the distance from home to
   school, you realize you forgot your homework, so you immediately
   turn back and go home. Your friend continues walking towards the
   school. You arrive at home the same moment your friend arrives at
   school.
   - Below draw the graph of your path (before you actually do it)
   - Now walk this motion and record it with the CBR
   - Below record both your graph and your friend's graph as shown on
     the screen.
   - Describe how the differences between the two graphs match specific
     differences in the motions.

Figure 2. CBR activity instructions.

Table 1 Pre-service Teachers' Mathematical and Pedagogical Insights on
Graphs of Functions When Working With the CBR Devices.

                  Pre-service teachers        Pre-service teachers
                  as learners of mathematics  as teachers in transition

Graphs as         * Facing their              * Recognizing the value of
representations     misconceptions in           building on students'
                    interpreting graphs         kinesthetic experience
                    "iconically"
                  * Realizing that a          * Recognizing the need to
                    velocity graph              link the concrete
                    determines a family of      experience to symbolic
                    position graphs             representations of that
                                                experience
Graphs as         * Realizing that graphs     * Differentiating local
rhetoric devices    can be manipulated to       and global
                    allow for different         interpretation of graphs
                    views and arguments         as tools for different
                                                types of arguments
Graphs as         * Using graphs as a means   * Recognizing the need to
conscription        for communication.          provide environments
devices                                         that allow for
                                                discussion and
                                                communication around the
                                                topic of graphs

Acknowledgments

The research reported here was supported in part by the National Science Foundation under Grant #REC-0087573. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Notes

(1) The Microcomputer-Based Lab (MBL) was invented in the late 1970's by Tinker and Thornton at Technical Education Research Center (TERC) in Cambridge Massachusetts, as part of an effort to improve science education by developing curriculum materials that use the computer in the laboratory for real-time data gathering and analysis (Mokros & Tinker, 1987).

(2) For more information about CBRs and a guide on their use please visit http://education.ti.com/downloads/guidebooks/eng/cbr-eng.pdf

(3) The Math Worlds software environment is a simulation world developed at the University of Massachusetts at Dartmouth, under the direction of James Kaput. It is a dynamic microworld for exploring one-dimensional motion (Kaput & Rochelle, 1997). A Java version of Math Worlds for the computer is available at http://www.simcalc.umassd.edu

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DESPINA A. STYLIANOU AND BEVERLY SMITH

City College, The City University of New York 

USA

[e-mail]

JAMES J. KAPUT

University of Massachusetts, Dartmouth

USA