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

\bibitem{asee_peer_19823}
  Doerr, H. M., \& Arleback, J. B., \& O'Neil, A. H. (2013, June), \emph{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

Helen M Doerr, Jonas Bergman Arleback, and AnnMarie H O'Neil.  "Interpreting and Communicating about Phenomena with Negative Rates of Change".  2013 ASEE Annual Conference & Exposition, Atlanta, Georgia, 2013, June.  ASEE Conferences, 2013.  https://peer.asee.org/19823 Internet.  14 Dec, 2024

\bibitem{asee_peer_19823}
  Helen M Doerr, Jonas Bergman Arleback, and AnnMarie H O'Neil.
  "Interpreting and Communicating about Phenomena with Negative Rates of Change".
  \emph{2013 ASEE Annual Conference \& Exposition, Atlanta, Georgia, 2013, June}.
  ASEE Conferences, 2013.
  https://peer.asee.org/19823 Internet.  14 Dec, 2024

@INPROCEEDINGS{asee_peer_19823
author = "Helen M Doerr, Jonas Bergman Arleback, and AnnMarie H O'Neil"
title = "Interpreting and Communicating about Phenomena with Negative Rates of Change"
booktitle = "2013 ASEE Annual Conference \& Exposition"
year = "2013"
month = "June"
address = "Atlanta, Georgia"
publisher = "ASEE Conferences"
note = {https://peer.asee.org/19823}
number = {10.18260/1-2--19823}
}

TY  - CPAPER
AB  -                 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.
AU  - Helen M Doerr
AU  - Jonas Bergman Arleback
AU  - AnnMarie H O'Neil
CY  - Atlanta, Georgia
DA  - 2013/06/23
PB  - ASEE Conferences
TI  - Interpreting and Communicating about Phenomena with Negative Rates of Change
UR  - https://peer.asee.org/19823
DO  - 10.18260/1-2--19823
ER  -