Abstract
This study focuses on connections between linear functions and their graphs that were made by tertiary remedial algebra students. In particular, we describe students’ work on a Task designed to examine the connection between points on a graph and the equation of a line. The data consist of 63 responses to a written questionnaire and individual interviews with three participants. The results indicate that visual approaches impede students’ solutions and point to incomplete connections between algebraic and graphical representation. While algebraic approaches point to various connections used in approaching the Task, students’ ability to work with algebraic representation did not necessarily result in capitalizing on these connections. Furthermore, interpretations of the graph based on visual inspection appeared most useful when used in support of the algebraic approach.
Similar content being viewed by others
References
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241.
Davis, J. (2007). Real-world contexts, multiple representations, student-invented terminology, and y-intercept. Mathematical Thinking and Learning, 9, 387–418.
Bardini, C., Pierce, R. U., & Stacey, K. (2004). Teaching linear functions in context with graphics calculators: Students’ responses and the impact of the approach on their use of algebraic symbols. International Journal of Science and Mathematics Education, 2(3), 353–376.
Ginsburg, H. P. (1981). The clinical interview in psychological research on mathematical thinking: Aims, rationales, techniques. For the Learning of Mathematics, 1(3), 4–11.
Ginsburg, H. P. (1997). Entering the child's mind: The clinical interview in psychological research and practice. Cambridge: Cambridge University Press.
Hattikudur, S., Prather, R., Asquith, P., Alibali, M., Knuth, E., & Nathan, M. (2012). Constructing graphical representations: Middle schoolers’ intuitions and developing knowledge about slope and y-intercept. School Science and Mathematics, 112, 230–240.
Knuth, E. J. (2000a). Student understanding of the Cartesian Connection: An exploratory study. Journal for Research in Mathematics Education, 31(4), 500–507.
Knuth, E. J. (2000b). Understanding connections between equations and graphs. The Mathematics Teacher, 93(1), 48–53.
Knuth, E., Alibali, M., McNeil, N., Weinberg, A., & Stephens, A. (2005). Middle school students' understanding of core algebraic concepts: Equivalence & variable. Zentralblatt für Didaktik der Mathematik, 37, 68–76.
Leinhardt, G., Zaslavsky, O., & Stein, M. K. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching. Review of Educational Research, 60(1), 1–64.
Lima, R. N., & Tall, D. (2008). Procedural embodiment and magic in linear equations. Educational Studies in Mathematics, 67(1), 3–18.
Mesa, V. (2017). Mathematics education at U.S. public two-year colleges. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 949-967). Reston: NCTM.
Moschkovich, J. (1996). Moving up and getting steeper: Negotiating shared descriptions of linear graphs. The Journal of the Learning Sciences, 5, 239–277.
Moschkovich, J., Schoenfeld, A., & Arcavi, A. (1993). Aspects of understanding: On multiple perspectives and representations of linear relations and connections among them. In T. A. Romberg, E. Fennema, & T. P. Carpenter (Eds.), Integrating research on the graphical representations of functions (pp. 69–100). Hillside: Erlbaum.
Moon, K., Brenner, M. E., Jacob, B., & Okamoto, Y. (2013). Prospective secondary mathematics teachers’ understanding and cognitive difficulties in making connections among representations. Mathematical Thinking and Learning, 15(3), 201–227.
Patton, M. Q. (2002). Qualitative research and evaluation methods. Thousand Oaks: Sage.
Pierce, R., Stacey, K., & Bardini, C. (2010). Linear functions: Teaching strategies and students' conceptions associated with y= mx+ c. Pedagogies: An International Journal, 5(3), 202–215.
Weber, R. P. (1990). Basic content analysis. Thousand Oaks: Sage.
Zazkis, R., Dubinsky, E., & Dautermann, J. (1996). Coordinating visual and analytic strategies: a study of student's understanding of the group D4. Journal for Research in Mathematics Education, 27(4), 435–457.
Author information
Authors and Affiliations
Corresponding author
Additional information
Notes
All student names are pseudonyms.
Rights and permissions
About this article
Cite this article
Glen, L., Zazkis, R. On Linear Functions and Their Graphs: Refining the Cartesian Connection. Int J of Sci and Math Educ 19, 1485–1504 (2021). https://doi.org/10.1007/s10763-020-10113-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10763-020-10113-6