## Abstract

This paper extends work about quantitative reasoning related to covarying quantities involved in rate of change. It reports a multiple case study of three students’ reasoning about quantities involved in rate of change when working on tasks incorporating multiple representations of covarying quantities. When interpreting relationships between associated amounts, students identified sections (e.g., an interval on a graph) in which they could make comparisons between amounts of change in quantities. Although such reasoning is useful for interpreting a Cartesian graph as a representation of covarying quantities, it does not foster attention to variation in the intensity of change in covarying quantities (e.g., a decreasing increase). Focusing on the kinds of relationships students make between amounts of change in covarying quantities might provide further insight into how students could develop a robust understanding of rate of change.

This is a preview of subscription content, access via your institution.

## Notes

In using variationally, I intend to communicate an envisioning of variables as varying, not to distinguish between variation and covariation.

This choice, however, does not preclude the possibility of a learning effect.

By task, I mean a problem that has been purposefully designed for a particular audience (Sierpinska, 2004).

Due to scheduling constraints, Jacob was interviewed twice during 1 week.

For more detail regarding students’ work on other tasks, see Johnson (2010).

See Johnson (2010) for a more comprehensive discussion.

## References

Bezuidenhout, J. (1998). First-year university students' understanding of rate of change.

*International Journal of Mathematical Education in Science and Technology, 29*(3), 389–399.Borman, K. M., Clarke, C., Cotner, B., & Lee, R. (2006). Cross-case analysis. In J. L. Green, G. Camilli, & P. B. Elmore (Eds.),

*Handbook of complementary methods in education research*(pp. 123–139). Mahwah: Erlbaum.Cantoral, R., & Farfán, M. (1998). Pensamiento y lenguaje variacional en la introducción al análisis [Thought and variational language in the introduction to analysis].

*Épsilon, 42*(3), 353–369.Cantoral, R., & Farfán, M. (2003). Mathematics education: A vision of its evolution.

*Educational Studies in Mathematics, 53*(3), 255–270.Carlson, M. P., 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., Larsen, S., & Lesh, R. A. (2003). Integrating a models and modeling perspective with existing research and practice. In R. A. Lesh & H. Doerr (Eds.),

*Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching*(pp. 465–478). Mahwah: Erlbaum.Castillo-Garsow, C. (2010).

*Teaching the Verhulst model: A teaching experiment in covariational reasoning and exponential growth*(Unpublished doctoral dissertation). Arizona State University, Phoenix, AZ.Castillo-Garsow, C. (2012). Continuous quantitative reasoning. In R. Mayes & L. L. Hatfield (Eds.),

*Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context*(Vol. 2, pp. 55–73). Laramie: University of Wyoming College of Education.Castillo-Garsow, C., Johnson, H. L., & Moore, K. C. (2013). Chunky and smooth images of change.

*For the Learning of Mathematics, 33*(3), 31–37.Clagett, M. (1968).

*Nicole Oresme and the medieval geometry of qualities and motions*. Madison: University of Wisconsin Press.Clement, J. (1989). The concept of variation and misconceptions in cartesian graphing.

*Focus on Learning Problems in Mathematics, 11*(1–2), 77–87.Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. A. Lesh (Eds.),

*Handbook of research design in mathematics and science education*(pp. 547–589). Mahwah: Erlbaum.Confrey, J., & Smith, E. (1995). Splitting, covariation, and their role in the development of exponential functions.

*Journal for Research in Mathematics Education, 26*(1), 66–86.Corbin, J., & Strauss, A. (2008).

*Basics of qualitative research: Techniques and procedures for developing grounded theory*(3rd ed.). London: Sage.Goldin, G. A. (2000). A scientific perspective on structured, task-based interviews in mathematics education research. In A. E. Kelly & R. A. Lesh (Eds.),

*Handbook of research design in mathematics and science education*(pp. 517–545). Mahwah: Erlbaum.Heid, M. K., Lunt, J., Portnoy, N., & Zembat, I. O. (2006). Ways in which prospective secondary mathematics teachers deal with mathematical complexity. In S. Alatorre, J. L. Cortina, M. Sáiz, & A. Méndez (Eds.),

*28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 2-9). Mérida, Mexico.Heinz, K. (2000).

*Conceptions of ratio in a class of preservice and practicing teachers*(Unpublished doctoral dissertation). The Pennsylvania State University, University Park, PA.Herbert, S., & Pierce, R. (2012). Revealing educationally critical aspects of rate.

*Educational Studies in Mathematics, 81*(1), 85–101.Janvier, C. (1998). The notion of chronicle as an epistemological obstacle to the concept of function.

*Journal of Mathematical Behavior, 17*(1), 79–103.Johnson, H. L. (2010).

*Making sense of rate of change: Secondary students' reasoning about changing quantities*(Unpublished doctoral dissertation). The Pennsylvania State University, University Park, PA.Johnson, H. L. (2012). Reasoning about variation in the intensity of change in covarying quantities involved in rate of change.

*Journal of Mathematical Behavior, 31*(3), 313–330.Johnson, H. L. (2015). Secondary students’ quantification of ratio and rate: A framework for reasoning about change in covarying quantities.

*Mathematical Thinking and Learning, 17*(1), 64–90.Lamon, S. J. (2007). Rational numbers and proportional reasoning. In F. K. Lester (Ed.),

*Second handbook of research on mathematics teaching and learning*(Vol. 1, pp. 629–667). Charlotte: Information Age.Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning and teaching.

*Review of Educational Research, 60*(1), 1–64.Lesh, R. A., Post, T. R., & Behr, M. (1988). Proportional reasoning. In J. Hiebert & M. Behr (Eds.),

*Number concepts and operations in the middle grades*(Vol. 2, pp. 93–118). Reston: National Council of Teachers of Mathematics.Lobato, J., Ellis, A. B., & Muñoz, R. (2003). How "focusing phenomena" in the instructional environment support individual students' generalizations.

*Mathematical Thinking and Learning, 5*(1), 1–36.Lobato, J., & Siebert, D. (2002). Quantitative reasoning in a reconceived view of transfer.

*Journal of Mathematical Behavior, 21*, 87–116.Monk, S., & Nemirovsky, R. (1994). The case of Dan: Student construction of a functional situation through visual attributes. In E. Dubinsky, A. H. Schoenfeld, & J. Kaput (Eds.),

*Research in Collegiate Mathematics Education, I*(pp. 139-168).Moore, K. C., Paoletti, T., & Musgrave, S. (2013). Covariational reasoning and invariance among coordinate systems.

*The Journal of Mathematical Behavior, 32*(3), 461–473.Piaget, J. (1970).

*Genetic epistemology*. New York: Columbia University Press.Roschelle, J., Kaput, J. J., & Stroup, W. (2000). SIMCALC: Accelerating students' engagement with the mathematics of change. In M. J. Jacobsen & R. B. Kozma (Eds.),

*Innovations in science and mathematics education: Advanced designs for technologies of learning*(pp. 47–76). Mahwah: Erlbaum.Saldanha, L., & Thompson, P. W. (1998). Re-thinking covariation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berenson, K. R. Dawkins, M. Blanton, W. N. Coloumbe, J. Kolb, K. Norwood, & L. Stiff (Eds.),

*Proceedings of the 20th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education*(Vol. 1, pp. 298–303). Raleigh: North Carolina State University.Shell Centre for Mathematical Education (University of Nottingham). (1985).

*The language of functions and graphs: An examination module for secondary schools*: Nottingham, UK: JMB/Shell Centre for Mathematical Education.Sierpinska, A. (1992). On understanding the notion of function. In G. Harel & E. Dubinsky (Eds.),

*The concept of function: Aspects of epistemology and pedagogy*(pp. 25–58). Washington, DC: Mathematical Association of America.Sierpinska, A. (2004). Research in mathematics education through a keyhole: Task problematization.

*For the Learning of Mathematics, 24*(2), 7–15.Simon, M. A. (2006). Key developmental understandings in mathematics: A direction for investigating and establishing learning goals.

*Mathematical Thinking and Learning, 8*(4), 359–371.Simon, M. A., & Blume, G. W. (1994). Mathematical modeling as a component of understanding ratio-as-measure: A study of prospective elementary teachers.

*Journal of Mathematical Behavior, 13*(2), 183–197.Stake, R. E. (2005). Qualitative case studies. In N. K. Denzin & Y. S. Lincoln (Eds.),

*The SAGE handbook of qualitative research*(pp. 443–466). Thousand Oaks: Sage.Steen, L. A. (Ed.). (1990).

*On the shoulders of giants: New approaches to numeracy*. Washington, DC: National Academy Press.Stroup, W. (2002). Understanding qualitative calculus: A structural synthesis of learning research.

*International Journal of Computers for Mathematical Learning, 7*, 167–215.Stroup, W. (2005). Learning the basics with calculus.

*Journal of Computers in Mathematics and Science Teaching, 24*(2), 179–196.Stump, S. L. (2001). High school precalculus students' understanding of slope as measure.

*School Science and Mathematics, 101*(2), 81–89.Thompson, P. W. (1994). 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: State University of New York Press.Thompson, P. W. (2012). Invited commentary—Advances in research on quantitative reasoning. In R. Mayes & L. L. Hatfield (Eds.),

*Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context*(Vol. 2, pp. 143–148). Laramie: University of Wyoming College of Education.Ubuz, B. (2007). Interpreting a graph and constructing its derivative graph: stability and change in students' conceptions.

*International Journal of Mathematical Education in Science and Technology, 38*(5), 609–637.Wilhelm, J. A., & Confrey, J. (2003). Projecting rate of change in the context of motion onto the context of money.

*International Journal of Mathematical Education in Science and Technology, 34*(6), 887–904.Yin, R. K. (2006). Case study methods. In J. L. Green, G. Camilli, & P. B. Elmore (Eds.),

*Handbook of complementary methods in education research*(pp. 111–122). Mahwah: Erlbaum.Zandieh, M., & Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example.

*Journal of Mathematical Behavior, 25*, 1–17.

## Acknowledgments

This research was completed to fulfill the dissertation requirement for a doctoral degree at The Pennsylvania State University under the advisement of Rose Mary Zbiek. Results reported in this paper are based on subsequent analysis of dissertation data. I am grateful to Evan McClintock for his thoughtful comments on prior versions of this paper and for the insights that resulted from our conversations related to this paper. This paper is supported in part by the National Science Foundation under Grant ESI-0426253 for the Mid-Atlantic Center for Mathematics Teaching and Learning (MAC-MTL). Any opinions, findings, or conclusions expressed in this document are my own and do not necessarily reflect the views of the National Science Foundation.

A previous version of this article appeared in:

Johnson, H. L. (2012). Reasoning about quantities involved in rate of change as varying simultaneously and independently. In R. Mayes & L. L. Hatfield (Eds.), *Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context* (Vol. 2, pp. 39-53). Laramie, WY: University of Wyoming College of Education.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

## About this article

### Cite this article

Johnson, H.L. Together yet separate: Students’ associating amounts of change in quantities involved in rate of change.
*Educ Stud Math* **89**, 89–110 (2015). https://doi.org/10.1007/s10649-014-9590-y

Published:

Issue Date:

DOI: https://doi.org/10.1007/s10649-014-9590-y

### Keywords

- Quantitative reasoning
- Rate of change
- Quantity
- Covariational reasoning
- Cartesian graphs