Much research over the last twenty years has documented the difficulties that students encounter when creating and interpreting models of changing phenomena. Several studies have examined the role that students' concepts of function and rate of change play in students' abilities to represent and to reason about dynamic events (Carlson et al., 2002; Monk, 1992; Thompson, 1994). As Oerhtman, Carlson and Thompson (2008) argue, in order to reason dynamically, students must be able to simultaneously attend to both the changing values of the output of a function and the rate of that change as the input values vary over intervals in the domain. The complexity of such reasoning has proven difficult even for high achieving undergraduate students (Carlson, 1998).

The broad goal of our research is to understand the inter-related development of students' abilities to create and interpret models of phenomena where change is occurring and the development of students' concepts of rate of change. In this paper, we describe tasks that engaged students in creating and exploring models of motion. We are interested in two research questions: (1) how did the modeling tasks support the development of students' concepts of rate of change and (2) to what extent were students able to interpret graphical representations of the rate of change?

Modeling approaches to the teaching and learning of mathematics encompass a wide range of theoretical perspectives (Kaiser & Sriraman, 2006). In this paper, we draw on the modeling perspective developed by Lesh and colleagues (Lesh et al., 2003).  In this perspective, model eliciting activities confront the student with the need to develop a model that can be used to describe the behavior of a familiar situation. Such tasks are followed by model exploration activities that focus on the underlying structure of the model and its representations.

DESIGN AND METHODOLOGY.  We began instruction with a model-eliciting activity, using bodily motion along a straight line. Students created position graphs using a motion detector and their own motion. The graphs included comparative situations of faster and slower constant speed, changing speed and changing direction, and graphs where the motion was not physically possible. Following the model-eliciting activity, the students engaged in several model exploration activities. These activities were designed to develop students' understandings of the representational systems for describing motion, using SimCalc Mathworlds (Kaput & Roschelle, 1996). This simulation environment reversed the representational space of the model-eliciting activity where bodily motion created a position graph and extended that space by using velocity graphs to create the motion of a simulated character. From this motion, the students created position graphs, thus providing an opportunity to develop students' abilities to interpret position information from a velocity graph and velocity information from a position graph.

DATA COLLECTION, SETTING, PARTICIPANTS AND ANALYSIS. To measure students' understanding of average rate of change, we designed a “Rate of Change Concept Inventory” consisting of items in four categories: algebraic expressions, graphical interpretation, symbolic interpretation, and contextual. Fourteen of the items were drawn from the research literature on students' conceptions of rate of change. Three items were developed to test the students' mastery of algebraic expressions of average rates of change. The development of this concept inventory is part of our on-going research.

The modeling tasks took place during a six-week course, taught by the second author, to students who were about to enter the university. There were 33 subjects from this course who volunteered to participate in the study. All but one participant had completed four years of study of high school mathematics; 18 students had studied calculus in high school. All participants completed the 17 item pre- and post-test “Rate of Change Concept Inventory.” The overall pre- and post-test scores and the four sub-scores were analyzed using t-tests.

RESULTS. The post-test results show that there was a significant (p<0.001) improvement in the students' understanding of the concept of average rate of change from an overall score of 52% correct to 75% correct. There were four graphical items on the inventory for which the improvement was greater than 30%. Three of these items addressed interpreting information about velocity when given a position graph. The fourth item involved reasoning about position when given a velocity (or speed) graph, a well-known source of difficulty for calculus students (Monk, 1992). We found a 43% improvement in the number of students who were able to correctly interpret the relative position of two cars, starting from the same position and travelling in the same direction, when given the speed graph shown in Figure 1.

Figure 1.  Interpreting the relative position of two cars given their speed.

DISCUSSION AND CONCLUSIONS.  The results of this study provide evidence that the modeling tasks had a positive impact on the students' understanding of average rate of change, as measured by the statistically significant overall gain on the “Rate of Change Concept Inventory” and by the gain on the graphical item subscore. This latter gain may be due to the model exploration tasks that focused on interpreting graphical representations of motion. By shifting between position and velocity graphs, students distinguished between a function's graph and the graph of its rate of change. Thus, the students' attention focused simultaneously on the quantity that was measured and on how that quantity was changing with respect to some other quantity. A coordinated understanding of these two measurements is at the crux of representing and reasoning about dynamic events (Oerhtman et al., 2008).  Overall, the majority of students were able to give meaningful interpretations and descriptions of change in the context of motion. At the end of the study, a small number of students still had difficulty in distinguishing between changes in the function values and changes in the values of the average rate of change. The difficulty in expressing this distinction suggests the need for further research on the development of students' understandings of the representational systems for describing changing phenomena.

REFERENCES

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Kaput, J. & Roschelle, J. (1996). SimCalc:  MathWorlds. [Computer software].

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