Rate of Change in the Calculus I Pathway
A critical foundation for reasoning about rates of change is conceiving of changes in quantities as quantities in their own right and distinguishing such changes from the original quantities. From this foundation, students may begin to understand, distinguish, and use the meanings of constant rate of change and average rate of change in various contexts and representations. These concepts, in turn, form a foundation for students’ understanding and reasoning about instantaneous rate of change in calculus. In particular, constant rate of change entails a proportional relationship between corresponding changes in the measures of the two quantities (e.g., see the conceptual analysis in Thompson, 1994). Reasoning about these changes and their proportional relationship across multiple representations can build an important foundation for further development of average and instantaneous rates. Non-quantitative interpretations of constant and average rate of change restrict students to iconic images, such as “steeper is faster.” Although such pseudo-structural reasoning may be sufficient for many procedural applications of derivatives, they prevent students from productively unpacking these ideas when necessary in problem-solving situations.
Faculty at the MIP Calculus I Initiation Workshop in August, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Rate of Change for success in the Oklahoma Calculus I Pathway should:
1. Help students conceive of changes in quantities as meaningful quantities in their own right (e.g., see early tasks involving describing and reasoning about changes in quantities in Carlson, Oehrtman, & Moore, 2016).
2. Engage students in interpreting average rates of change as a constant rate for an auxiliary scenario with the same total changes in both quantities. These materials could reinforce and motivate the use of function notation in algebraic representations of average rates, developing the difference quotient.
3. Informally introduce instantaneous rates through a context that necessitates finding average rates over progressively smaller intervals.
4. Unpack rate of change statements in terms of coordinating amounts of change. Such tasks may ask students to analyze amounts of change in the function for constant increments of the independent variable (e.g., see MA3 reasoning in Carlson et al., 2002).
5. Draw diagrams that represent changes in the output variable corresponding to successive increments in the input variable to help students conceptualize varying rates more robustly. Students should subsequently represent these relationships graphically and algebraically and interpret them in terms of rate of change in the problem context.
Addressing Components of Inquiry
Participants of the MIP Workshop on Calculus I suggested the following ways resources about Rate of Change could address the three MIP components of mathematical inquiry:
Active Learning: Students in a Calculus I course will have significant experience applying procedures to solve routine problems about constant or average rate of change. Thus, it is particularly important that modules engage students in tasks that challenge these rote applications and require them to explore the underlying meanings, especially in terms of invariant multiplicative relationships between corresponding amounts of change of two quantities that vary simultaneously.
Meaningful Applications: Students should engage in rates of change as a natural entry point to understand, represent, and explain, how quantities change in actual situations. Correspondingly, identifying and applying key rate of change characteristics of various function types can help reinforce broader understanding of these functions and their value in appropriate modeling scenarios. Varying the contexts promotes students’ development of a generalized concept of rate of change that is not bound to any single situation or representation.
Academic Success Skills: Exploring rate of change in-depth and in meaningful applications can help students reinforce their academic identity. Supporting students in constructing meaning for foundational mathematical ideas like rate of change allows them to develop the expectation that their understandings enable them to reason about novel tasks and contexts. This expectation has the potential to reduce or even eliminate the reflexive interpretations of mathematical stimuli as potential threats to one’s identity, and which tend to initiate unproductive behavioral reactions (e.g., task avoidance; memorization; the unreasoned employment of coping mechanisms). In addition to supporting productive meanings for rate of change grounded in quantitative and covariational reasoning, CoRD modules could attend to reinforcing a growth mindset and encouraging perseverance by providing scaffolding that keeps students engaged without preempting their ability to develop significant solutions on their own.
References
Byerley, C. & Thompson, P. W. (2017). Secondary teachers’ meanings for measure, slope, and rate of change. Journal of Mathematical Behavior, 48, 168-193.
Carlson, M., Jacobs, S., Coe, E., Larsen, S., & Hsu, E. (2002). Applying covariational reasoning while modeling dynamic events: A framework and a study. Journal for Research in Mathematics Education, 33(5), 352–378.
Carlson, M. P., Smith, N., & Persson, J. (2003). Developing and connecting calculus students’ notions of rate-of-change and accumulation: The fundamental theorem of calculus. In N. Patemen (Ed.), Proceedings of the 2003 Meeting of the International Group for the Psychology of Mathematics Education–North America (Vol. 2, pp. 165–172). Honolulu, HI: University of Hawaii.
Thompson, P. W. (1994). Images of rate and operational understanding of the Fundamental Theorem of Calculus. Educational Studies in Mathematics, 26(2-3), 229-274.
Thompson, P. W. (1994b). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 181-234). Albany, NY: SUNY Press.
Thompson, P. W., & Thompson, A. G. (1992). Images of rate. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco.
Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.), Research in Collegiate Mathematics Education IV. (Vol 8, pp. 103-127). Providence, RI: American Mathematical Society.