# Secondary mathematics teachers’ knowledge of slope

- 256 Downloads
- 13 Citations

## Abstract

This study, conducted in the United States, investigated secondary mathematics teachers’ concept definitions, mathematical understanding, and pedagogical content knowledge of slope. Surveys were collected from 18 preservice and 21 inservice teachers; 8 teachers from each group were also interviewed. Geometric ratios dominated teachers’ concept definitions of slope. Problems involving the recognition of parameters, the interpretation of graphs, and rate of change challenged teachers’ thinking. Teachers’ descriptions of classroom instruction included physical situations more often than functional situations. Results suggest that mathematics teacher education programs need to specifically address slope as a fundamental concept, emphasising its connection to the concept of function.

## Keywords

Preservice Teacher Pedagogical Content Knowledge Procedural Knowledge Inservice Teacher Mathematical Understanding## Preview

Unable to display preview. Download preview PDF.

## References

- Azcarate, C. (1992). Estudio de los esquemas conceptuales y de los perfiles de unos alumnos de segundo de bup en relacion con el concepto de pendiente de una recta [A study of the conceptual schemes and the profiles of some BUP students in relation to the concept of the slope gradient].
*Epsilon, 24*, 9–22.Google Scholar - Barr, G. (1980). Graphs, gradients and intercepts.
*Mathematics in School, 9*(1), 5–6.Google Scholar - Barr, G. (1981). Some student ideas on the concept of gradient.
*Mathematics in School, 10*(1), 14–17.Google Scholar - Bell, A., Janvier, C. (1981). The interpretation of graphs representing situations.
*For the Learning of Mathematics, 2*(1), 34–42.Google Scholar - Cooney, T. J. (1994). Teacher education as an exercise in adaptation. In D. B. Aichele (Ed.),
*Professional development for teachers of mathematics*(pp. 9–22). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Day, R. P. (1995). Using functions to make mathematical connections. In P. A. House & A. F. Coxford (Eds.),
*Connecting mathematics across the curriculum*(pp.54–64). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Erickson, F. (1986). Qualitative methods in research on teaching. In M. C. Wittrock (Ed.),
*Handbook of research on teaching*(3rd ed., pp. 119–161). New York, NY: Macmillan.Google Scholar - Even, R. (1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept.
*Journal for Research in Mathematics Education, 24*, 94–116.CrossRefGoogle Scholar - Fey, J. T. (1990). Quantity. In L. A. Steen (Ed.),
*On the shoulders of giants*(pp. 61–94). Washington, DC: National Academy Press.Google Scholar - Fey, J. T. & Good, R. A. (1985). Rethinking the sequence and priorities of high school mathematics curricula. In C. R. Hirsch (Ed.),
*The secondary school mathematics curriculum*(pp. 43–52). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Hart, K. M. (Ed.) (1981).
*Children’s understanding of mathematics: 11–16*. London: John Murray.Google Scholar - Hiebert, J., & Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.),
*Conceptual and procedural knowledge: The case of mathematics*(pp. 1–28). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Heller, P. M., Post, T. R., Behr, M., Lesh, R. (1990). Qualitative and numerical reasoning about fractions and rates by seventh- and eighth-grade students.
*Journal for Research in Mathematics Education, 21*, 388–402.CrossRefGoogle Scholar - Hughes-Hallett, D., Osgood, B. G., Gleason, A. M., Pasquale, A., Gordon, S. P., Tecosky-Feldman, J., Lomen, D. O., Thrash, J. B., Lovelock, D., Thrash, K. R., McCallum, W. G., Tucker, T. W. (1992).
*Calculus*. New York, NY: John Wiley.Google Scholar - Janvier, C. (1981). Use of situations in mathematics education.
*Educational Studies in Mathematics, 12*, 113–122.CrossRefGoogle Scholar - Janvier, C. (1987). Translation processes in mathematics education. In C. Janvier (Ed.),
*Problems of representation in the teaching and learning of mathematics*(pp. 27–32). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Kaput, J. J., & West, M. M. (1994). Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns. In G. Harel & J. Confrey (Eds.),
*The development of multiplicative reasoning in the learning of mathematics*(pp. 235–287). Albany, NY: SUNY Press.Google Scholar - Karplus, R., Pulos, S., & Stage, E. K. (1983). Early adolescents’ proportional reasoning in “rate” problems.
*Educational Studies in Mathematics, 14*, 219–233.CrossRefGoogle Scholar - Kerslake, D. (1981). Graphs. In K. Hart (Ed.),
*Children’s understanding of mathematics: 11–16*(pp. 120–136). London: John Murray.Google Scholar - Lamon, S. J. (1995). Ratio and proportion: Elementary didactical phenomenology. In J. T. Sowder & B. P. Schappelle (Eds.),
*Providing a foundation for teaching mathematics in the middle grades*(pp. 167–198). Albany, NY: SUNY Press.Google Scholar - Livingston, C., Borko, H. (1990). High school mathematics review lessons: Expert-novice distinctions.
*Journal for Research in Mathematics Education, 21*, 372–387.CrossRefGoogle Scholar - Lloyd, G. M., Wilson, M. (1998). Supporting innovation: The impact of a teacher’s conceptions of functions on his implementation of a reform curriculum.
*Journal for Research in Mathematics Education, 29*, 248–274.CrossRefGoogle Scholar - Mathematical Association of America. (1991).
*A call for change: Recommendations for the mathematical preparation of teachers of mathematics*. Washington, DC: Author.Google Scholar - McConnell, J. W., Brown, S., Eddins, S., Hackworth, M., & Usiskin, Z. (1990).
*UCSMP: Algebra*. Glenview, IL: Scott, Foresman.Google Scholar - 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*, 503–512.CrossRefGoogle Scholar - McDiarmid, G. W., Ball, D. L., & Anderson, C. W. (1989). Why staying one chapter ahead doesn’t really work: Subject-specific pedagogy. In M. C. Reynolds (Ed.),
*Knowledge base for the beginning teacher*(pp. 193–205). Oxford, UK: Pergamon.Google Scholar *Merriam-Webster’s collegiate dictionary*(10th ed.). (1993). Springfield, MA: Merriam-Webster.Google Scholar- Moschkovich, J. (1990). Students’ interpretations of linear equations and their graphs.
*Proceedings of the 14th annual conference of the International Group for the Psychology of Mathematics Education*(Vol. 2, pp. 109–116). Oaxtepex, Mexico: Program Committee.Google Scholar - Moschkovich, J., Schoenfeld, A. H., & 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 representation of functions*(pp. 69–100). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - National Council of Teachers of Mathematics. (1989).
*Curriculum and evaluation standards for school mathematics*. Reston, VA: Author.Google Scholar - Norman, A. (1992). Teachers’ mathematical knowledge of the concept of function. In G. Harel & E. Dubinsky (Eds.),
*The concept of function: Aspects of epistemology and pedagogy*(MAA Notes Vol. 25, pp. 215–232). Washington, DC: Mathematical Association of America.Google Scholar - Orton, A. (1984). Understanding rate of change.
*Mathematics in School, 23*(5), 23–26.Google Scholar - Rizzuti, J. M. (1991). Students’ conceptualizations of mathematical functions: The effects of a pedagogical approach involving multiple representations (Doctoral dissertation, Cornell University, 1991).
*Dissertation Abstracts International*, 52-10A, 3549.Google Scholar - Schoenfeld, A. H., Smith, J. P., & Arcavi, A. (1993). Learning: The microgenetic analysis of one student’s evolving understanding of a complex subject matter domain. In R. Glaser (Ed.),
*Advances in instructional psychology*(Vol. 4, pp. 55–175). Hillsdale, NJ: Lawrence Erlbaum.Google Scholar - Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching.
*Educational Researcher, 15*(2), 4–14.Google Scholar - 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*,183–197.CrossRefGoogle Scholar - Singer, J. A., Resnick, L. B. (1992). Representations of proportional relationships: Are children part-part or part-whole reasoners?
*Educational Studies in Mathematics, 23*, 231–246.CrossRefGoogle Scholar - Stein, M. K., Baxter, J. A., Leinhardt, G. (1990). Subject-matter knowledge and elementary instruction: A case from functions and graphing.
*American Educational Research Journal, 27*, 639–663.Google Scholar - Stump, S. L. (1996).
*Secondary mathematics teachers’ knowledge of the concept of slope*. Unpublished doctoral dissertation/Illinois State University, Normal, IL.Google Scholar - Tall, D. & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity.
*Educational Studies in Mathematics, 12*, 151–169.CrossRefGoogle Scholar - Thompson, A. G., Thompson, P. W. (1996). Talking about rates conceptually, Part II: Mathematical knowledge for teaching.
*Journal for Research in Mathematics Education, 27*, 2–24.CrossRefGoogle Scholar - Thompson, P. W. (1994). The development of the concept of speed and its relationship to the concepts of rate. In G. Harel & J. Confrey (Eds.),
*The development of multiplicative reasoning in the learning of mathematics*(pp. 179–234). Albany, NY: SUNY Press.Google Scholar - Thompson, P. W., Thompson, A. G. (1994). Talking about rates conceptually, Part I: A teacher’s struggle.
*Journal for Research in Mathematics Education, 25*, 279–303.CrossRefGoogle Scholar - Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.),
*Acquisition of mathematics concepts and processes*(pp. 127–173). New York: Academic Press.Google Scholar - Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.),
*Number concepts and operation in the middle grades*(pp. 141–161). Reston, VA: National Council of Teachers of Mathematics.Google Scholar - Wilson, M. R. (1994). One preservice secondary teacher’s understanding of function: The impact of a course integrating mathematical content and pedagogy.
*Journal for Research in Mathematics Education, 25*, 346–370.CrossRefGoogle Scholar