Atlanta, Georgia
June 23, 2013
June 23, 2013
June 26, 2013
2153-5965
Mathematics
16
23.809.1 - 23.809.16
10.18260/1-2--19823
https://peer.asee.org/19823
333
Interpreting and Communicating about Phenomena with Negative Rates of ChangeMuch research over the last twenty years has documented the difficulties that students encounterwhen reasoning about and interpreting rates of change1-3. The complexity of such reasoning hasproven difficult for high achieving undergraduate mathematics students4 and students studyingphysics5, 6. To reason about rates of change, students must be able to simultaneously attend toboth the changing values of the outputs of a function and changing values of the inputs to thefunction7, 8. Such covariational reasoning is a foundational understanding for average rates ofchange in pre-calculus and instantaneous rates of change in calculus. An equally importanteducational objective for engineering students is the ability to interpret and communicate thismathematical reasoning in terms of the context of physical phenomena9. However, relatively littleresearch has attended to the particular challenges encountered when reasoning about negativerates of change. In this study, we examined students’ abilities to interpret and communicate theirreasoning about negative rates of change in the context of physical phenomena.Using a modeling perspective on teaching and learning mathematics10-14 and a design-basedresearch methodology15, we implemented a sequence of modeling tasks16 to support thedevelopment of students’ understandings of average rates of change. This sequence formed thebasis for a six-week course for students who were preparing to enter their university engineeringstudies. A total of 101 subjects (29 female and 72 male) participated in the study; 58 students hadstudied calculus in high school and 43 had not studied any calculus. The students worked in smallgroups to complete three sets of model development tasks: a model eliciting activity using motiondetectors to initially elicit the notion of negative velocity with motion along a straight path;several model exploration activities using a computer simulation to explore changing negativerates17 and structured exercises18 to explore representations of negative rates; and two modelapplication activities, one to create a model of the intensity of light with respect to the distancefrom the light source, and the other to model the rate at which a fully charged capacitor in asimple resistor-capacitor circuit discharged with respect to time. Class discussion following thetasks focused on the relationships among different representations of negative rates of change andstudents’ interpretations of change in different contexts. All the lessons were video taped andstudents’ written work was collected and analyzed. Following the principles of grounded theory19,codes were developed to categorize the students’ reasoning about of negative rates of change andtheir interpretations in context.The results show that interpreting and communicating about negative rates of change wasparticularly difficult for the students. We found that many students were able to clearlydistinguish between the function values and the values of the rates of change when the rates werepositive, but then confounded these quantities for negative rates. Additional obstacles for studentsarose when describing negative rates of change that were increasing, such as with the dischargingcapacitor. In this case, the magnitude of the change decreased, while the signed change increasedas it became less negative. For many students, language for describing this change in terms of thecapacitance appeared in conflict with formal mathematical language for describing that change.The model development sequence supported many students in interpreting and describingphenomena with positive rates of change, but many students experienced difficulties incommunicating these concepts when the rates of change were negative. This suggests the need forcloser attention to developing and expressing the concept of a negative rate of change in differentcontexts.REFERENCES[1] 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.[2] Monk, S. (1992). Students’ understanding of a function given by a physical model. In E. Dubinsky & G. Harel (Eds.), The concept of function: Aspects of epistemology and pedagogy (pp. 175-194). Washington, DC: Mathematical Association of America.[3] 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.[4] Carlson, M. P. (1998). A cross-sectional investigation of the development of the function concept. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), CBMS Issues in Mathematics Education: Research in Collegiate Mathematics Education III, 7, 114–162.[5] Beichner, R. J. (1994). Testing student interpretation of kinematics graphs. American Journal of Physics, 62, 750-762[6] McDermott, L. C., Rosenquist, M. L., & van Zee, E. H. (1987). Student difficulties in connecting graphs and physics: Examples from kinematics. American Journal of Physics, 55(6), 503-513.[7] Oehrtman, M., Carlson, M., & Thompson, P. (2008). Foundational reasoning abilities that promote coherence in students’ function understanding. In M. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics education (pp. 27-41). Washington, DC: Mathematical Association of America.[8] Johnson, H. (2012). Reasoning about variation in the intensity of change in covarying quantities involved in rate of change. The Journal of Mathematical Behavior, 31(3), 313- 330.[9] ABET. Criteria for Accrediting Engineering Programs, 2011-2012. General Criteria 3. Students Outcome, http://www.abet.org/eac-current-criteria/, accessed 8/23/12.[10] Lesh, R. A., & Doerr, H. M. (2003). Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching. Mahwah, NJ: Lawrence Erlbaum Associates.[11] Hamilton, E., Besterfield-Sacre, M., Olds, B., & Siewiorek, N. (2010). Model-eliciting activities in engineering: A focus on model building. In American Society for Engineering Education Annual Conference and Exposition, Conference Proceedings.[12] Bursic, K., Shuman, L., & Besterfield-Sacre, M. (2011). Improving student attainment of ABET outcomes using model-eliciting activities (MEAS). In American Society for Engineering Education Annual Conference and Exposition, Conference Proceedings.[13] Ridgely, J., & Self, B. (2011). Model-eliciting activities in a mechanical engineering experimental methods course. In American Society for Engineering Education Annual Conference and Exposition, Conference Proceedings.[14] Kean, A., Miller, R., Self, B., Moore, T., Olds, B., & Hamilton, E. (2008). Identifying robust student misconceptions in thermal science using model-eliciting activities. In American Society for Engineering Education Annual Conference and Exposition, Conference Proceedings.[15] Kelly, A., Lesh, R. A., & Baek, J. Y. (2008). Handbook of design research methods in education: Innovations in science, technology, engineering and mathematics learning and teaching. NY: Routledge.[16] Lesh, R. A., Cramer, K., Doerr, H. M., Post, T., & Zawojewski, J. S. (2003). Model development sequences. In R. A. Lesh & H. M. Doerr (Eds.) Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 35–57). Mahwah, NJ: Lawrence Erlbaum Associates.[17] Kaput, J. & Roschelle, J. (1996). SimCalc: MathWorlds. [Computer software].[18] Watson, A., & Mason, J. (2006). Seeing an exercise as a single mathematical object: Using variation to structure sense-making. Mathematical Thinking and Learning, 8(2), 91–111.[19] Strauss, A., & Corbin, J. (1998). Basics of qualitative research: Techniques and procedures for developing grounded theory. Thousand Oaks, CA: Sage Publications.
Doerr, H. M., & Arleback, J. B., & O'Neil, A. H. (2013, June), Interpreting and Communicating about Phenomena with Negative Rates of Change Paper presented at 2013 ASEE Annual Conference & Exposition, Atlanta, Georgia. 10.18260/1-2--19823
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