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The Status of Proving Among US Secondary Mathematics Teachers

Abstract

This report examines teachers’ self-espoused attitudes and beliefs on proving in the secondary mathematics classroom. Conclusions were based on a questionnaire of 78 US mathematics teachers who had completed at least 2 years of teaching mathematics at the secondary level. While these teachers placed importance on proving as a general mathematical skill, when they discuss their own classrooms, procedural skill consistently surpasses proof-related activities in importance for a majority of high school teachers. Furthermore, teachers labeling their own past experiences in proving as causing anxiety are predictably more likely to put less value on proving. Interestingly, the quantity of past college mathematics courses is a reverse predictor indicating that further study should consider how students perceptions of proving change as they pass through a mathematics major.

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References

  • Almeida, D. (1996) Justifying and proving in the mathematics classroom. Philosophy of mathematics education newsletter, 9. Retrieved from http://people.exeter.ac.uk/PErnest/pome/pompart8.htm.

  • Almeida, D. (2000). A survey of mathematics undergraduates' interaction with proof: some implications for mathematics education. International Journal of Mathematical Education in Science and Technology, 31(6), 869–890.

    Article  Google Scholar 

  • Ashton, P. (1985). Motivation and the teacher’s sense of efficacy. Research on Motivation in Education, 2, 141–174.

    Google Scholar 

  • Ball, D. L. (1988). Unlearning to teach mathematics. For the Learning of Mathematics, 8(1), 40–48.

    Google Scholar 

  • Ball, D. L. (1991). Teaching mathematics for understanding: what do teachers need to know about subject matter? In M. Kennedy (Ed.), Teaching Academic Subjects to Diverse Learners. New York: Teachers College Press.

    Google Scholar 

  • Barkatsas, A. T. & Malone, J. (2005). A typology of mathematics teachers’ beliefs about teaching and learning mathematics and instructional practices. Mathematics Education Research Journal, 17(2), 69–90.

    Article  Google Scholar 

  • Battista, M. T. (1994). Teacher beliefs and the reform movement in mathematics education. Phi Delta Kappan, 75(6), 462–470.

    Google Scholar 

  • Bell, A. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7(1–2), 23–40.

    Article  Google Scholar 

  • Beswick, K. (2012). Teachers’ beliefs about school mathematics and mathematicians’ mathematics and their relationship to practice. Educational Studies in Mathematics, 79(1), 127–147.

    Article  Google Scholar 

  • Bieda, K. N. (2010). Enacting proof-related tasks in middle school mathematics: Challenges and opportunities. Journal for Research in Mathematics Education, 41, 351–382.

    Google Scholar 

  • Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D. & Agard, P. C. (1992). Learning to teach hard mathematics: Do novice teachers and their instructors give up too easily? Journal for Research in Mathematics Education, 23(3), 194–222.

    Article  Google Scholar 

  • Borko, H., Peressini, D. Romagnano, L., Knuth, E., Willis-Yorker, C., Wooley, C., Hovermill, J. & Masarik, K. (2000). Teacher education does matter: A situative view of learning to teach secondary mathematics. Educational Psychology, 35(3), 193–206.

  • Bruner, J. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press.

    Google Scholar 

  • Buchmann, M. & Schwille, J. (1983). Education: The overcoming of experience. American Journal of Education, 92(1), 30–51.

    Article  Google Scholar 

  • Clements, D. H. & Battista, M. T. (1992). Geometry and spatial reasoning. In D. A. Grouws (Ed.),  Handbook of research on mathematics teaching and learning (pp. 420-464). New York, NY: Macmillan.

  • Common Core State Standards for Mathematics (2010). Washington, D.C.: Council of Chief State School Officers and National Governors Association.

  • Conner, A. M., Edenfield, K. W.,Gleason, B. & Ersoz, F. A. (2011). Impact of a content and methods course sequence on prospective secondary mathematics teachers’ beliefs. Journal of Mathematics Teacher Education, 14(6), 483–504.

  • Cross, D. I. (2009). Alignment, cohesion, and change: Examining mathematics teachers’ belief structures and their influence on instructional practices. Journal of Mathematics Teacher Education, 12(5), 325–346.

    Article  Google Scholar 

  • De Villiers, M. (1999). Rethinking proof with geometer’s sketchpad. Emeryville, CA: Key Curriculum Press.

    Google Scholar 

  • De Villiers, M. (2004). Using dynamic geometry to expand mathematics teachers’ understanding of proof. International Journal of Mathematical Education in Science and Technology, 35(5), 703–724.

    Article  Google Scholar 

  • Dreyfus, T. (1990). Advanced mathematical thinking. Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education. P. Nesher and J. Kilpatrick. Great Britain, Cambridge University Press: 113–134.

  • Ernest, P. (1989). The impact of beliefs on the teaching of mathematics. In P. Ernest (Ed.), Mathematics teaching: The state of the art (pp. 247–254). New York, NY: Falmer Press.

  • Fennema, E. & Sherman, J. (1976). Fennema-Sherman mathematics attitudes scales: instruments designed to measure attitudes toward the learning of mathematics by females and males. Journal for Research in Mathematics Education, 7(5), 324–326.

    Article  Google Scholar 

  • Fischbein, E. & Kedem I. (1982). Proof and certitude in the development of mathematical thinking. Proceedings of the Sixth International Conference for the Psychology of Mathematical Education, Antwerp, Belgium, Universitaire Instelling Antwerpen.

  • Frost, J. H. (2010). Looking through the lens of a teacher's life: The power of prototypical stories in understanding teachers' instructional decisions in mathematics. Teaching and Teacher Education, 26(2), 225–233.

    Article  Google Scholar 

  • Furinghetti, F. & Morselli, F. (2011). Beliefs and beyond: Hows and whys in the teaching of proof. ZDM, 43(4), 587–599.

    Article  Google Scholar 

  • Fuys, D., Geddes, D. & Tischler, R. (1988). The van Hiele model of thinking in geometry among adolescents. Journal of Research in Mathematics Education Monograph, 2.

  • Galbraith, P. L. (1981). Aspects of proving: A clinical investigation of process. Educational Studies in Mathematics, 12(1), 1–28.

    Article  Google Scholar 

  • Hanna, G. (1989). Proofs that prove and proofs that explain. Proceedings of the International Group for the Psychology of Mathematics Education, Paris.

  • Hanna, G. & Barbeau, E. (2008). Proofs as bearers of mathematical knowledge. ZDM, 40(3), 345–353.

    Article  Google Scholar 

  • Harel, G. & Sowder, L (2007). Toward a comprehensive perspective on proof, In F. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning, National Council of Teachers of Mathematics.

  • Hart, L. C. (2002). Preservice teachers’ beliefs and practice after participating in an integrated content/methods course. School Science and Mathematics, 102(1), 4–14.

    Article  Google Scholar 

  • Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.

    Article  Google Scholar 

  • Herbst, P. G. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49(3), 283–312.

    Article  Google Scholar 

  • Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399.

    Article  Google Scholar 

  • Hiebert, J. & Carpenter T. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 65-97). New York NY, Macmillan.

  • Jones, K. (2000). The student experience of mathematical proof at university level. International Journal of Mathematical Education in Science and Technology, 31(1), 53–60.

    Article  Google Scholar 

  • Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405.

  • Kuhs, T. M. & Ball, D. L. (1986). Approaches to teaching mathematics: Mapping the domains of knowledge, skills, and dispositions. Lansing, MI: Michigan State University.

    Google Scholar 

  • Lampert, M. (1990). When the problem is not the question and the solution is not the answer. American Educational Research Journal, 27(1), 29–63.

    Article  Google Scholar 

  • Levenson, E. (2013). Exploring one student’s explanations at different ages: the case of Sharon. Educational Studies in Mathematics, 83(2), 181–203.

    Article  Google Scholar 

  • Liljedahl, P. (2008). Teachers’ insights into the relationship between beliefs and practice. Beliefs and attitudes in mathematics education: New research results, 33–44.

  • Lortie, D. C. (1975). School teacher: A sociological inquiry. Chicago: University of Chicago Press.

    Google Scholar 

  • Maher, C. & Martino, A. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27(2), 194–214.

    Article  Google Scholar 

  • Martin, T. S. & McCrone S. S. (2001). Investigating the teaching and learning of proof: First year results. Proceedings of the Annual Meeting of the North American Chapter of the International group for the Psychology of Mathematics Education, Snobird, Utah.

  • Martino, A. M. & Maher, C. A. (1999). Teacher questioning to promote justification and generalization in mathematics: What research practice has taught us. The Journal of Mathematical Behavior, 18(1), 53–78.

    Article  Google Scholar 

  • McGinnis, J. R., Kramer, S., Roth-McDuffie, A. & Watanabe, T. (1998). Charting the attitude and belief: Journeys of teacher candidates in a reform-based mathematics and science teacher preparation program. San Diego, CA: American Educational Research Association.

  • Mingus, T. T. Y. & Grassl, R. M. (1999). Preservice teacher beliefs about proofs. School Science and Mathematics, 99(8), 438–444.

    Article  Google Scholar 

  • Movshovitz-Hadar, N. (2001). Proof. In L. S. Grinstein and S. I. Lipsey (Eds.), Encyclopedia of mathematics education (pp. 585-591). New York, NY, Routledge.

  • National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

  • National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.

  • Peterson, P. (1989). Teachers’ Belief Questionnaire. (Available from P. Peterson, 2004, Fall).

  • Peterson, P. & Fennema E. (1989). Teachers’ pedagogical content beliefs in mathematics. Cognition and Instruction, 6(1):1-40.

  • Piaget, J. & Inhelder, B. (1969). The early growth of logic in the child. New York, NY: Norton.

    Google Scholar 

  • Polya, G. (1957). How to solve it: A new aspect of mathematical method. Garden City, NY: Doubleday.

    Google Scholar 

  • Putnam, R. (1992). Teaching the "hows" of mathematics for everyday life: A case study of a fifth-grade teacher. The Elementary School Journal, 93(2), 163–177.

    Article  Google Scholar 

  • Raimi, R. A. (2001). Standards in school mathematics. Notices of the American Mathematical Society, 48(2), 166.

    Google Scholar 

  • Raymond, A. (1997). Inconsistency between beginning elementary school teacher's mathematics beliefs and teaching practice. Journal for Research in Mathematics Education, 28(5), 550–576.

    Article  Google Scholar 

  • Ross, K. A. (2000). The MAA and the New NCTM Standards. from http://www.maa.org/features/pastfeatures.html. Accessed 30 June 2012.

  • Schoenfeld, A. H. (1998). Toward a theory of teaching-in-context. Issues in Education, 4(1), 1–94.

    Article  Google Scholar 

  • Senk, S. L. (1985). How well do students write geometry proofs? Mathematics Teacher, 78(6), 448–456.

    Google Scholar 

  • Silver, E. A. & Kenney, P. A. (2000). Results from the seventh mathematics assessment. Reston, VA: NCTM.

    Google Scholar 

  • Simon, M. A. & Blume, G. W. (1996). Justification in the mathematics classroom: A study of prospective elementary teachers. Journal of Mathematical Behavior, 15, 3–31.

    Article  Google Scholar 

  • Skemp, R. R. (1978). Relational understanding and instrumental understanding. Arithmetic Teacher, 26, 9–15.

    Google Scholar 

  • Smith, J. P. (1996). Efficacy and teaching mathematics by telling: a challenge for reform. Journal for Research in Mathematics Education, 27, 387–402.

    Article  Google Scholar 

  • Smith, E. P. & Henderson K. B. (1959). The growth of mathematical ideas, grades K-12. In P. S. Jones (Ed.), Yearbook of the NCTM (pp.111-181). Washington, DC: NCTM.

  • Speer, N. (2005). Issues of methods and theory in the study of mathematics teachers’ professed and attributed beliefs. Educational Studies in Mathematics, 58, 361–391.

    Article  Google Scholar 

  • Stein, M. K., Baxter, J. A. & Leinhardt, G. (1990). Subject-matter knowledge and elementary instruction. American Educational Research Journal, 27(4), 639–663.

    Article  Google Scholar 

  • Stevens, J. P. (2002). Applied multivariate statistics for the social sciences. Mahwah, NJ: Erlbaum.

  • Stigler, J. & Hiebert, J. (1999). The teaching gap: Best ideas from the world’s teachers for improving education in the classroom. New York, NY: The Free Press.

    Google Scholar 

  • Stigler, J. W., Gonzales, P., Kwanaka, T., Knoll, S. & Serrano, A. (1999). The TIMSS Videotape Classroom Study: Methods and Findings from an Exploratory Research Project on Eighth-Grade Mathematics Instruction in Germany, Japan, and the United States. A Research and Development Report.

  • Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38, 289–321.

    Google Scholar 

  • Swan, P. (December, 2004). I hate mathematics. Paper presented in MAV Annual Conference, Monash University, Melbourne, Australia. Retrieved from http://www.mav.vic.edu.au/files/conferences/2004/Swan.pdf

  • Swars, S. L., Smith, S. Z., Smith, M. E. & Hart, L. C. (2009). A longitudinal study of effects of a developmental teacher preparation program on elementary prospective teachers’ mathematics beliefs. Journal of Mathematics Teacher Education, 12(1), 47–66.

    Article  Google Scholar 

  • Thompson, A. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15(2), 105–127.

    Article  Google Scholar 

  • Thompson, A. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 127–146). New York, NY: Macmillan.

    Google Scholar 

  • Von Glasersfeld, E. (1987). Learning as a constructive activity. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 3–18). Hillsdale, NJ: Lawrence Erlbaum Associates.

    Google Scholar 

  • Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.

    Article  Google Scholar 

  • Yackel, E. & G. Hanna (2003). Reasoning and proof. A research companion for NCTM standards. J. Kilpatrick, G. Martin and D. Schifter. Reston, VA, NCTM: 227–236.

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Acknowledgments

I would like to thank Dr. Henry Pollak for his advice, support, and guidance through this study.

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Correspondence to Usha Kotelawala.

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Kotelawala, U. The Status of Proving Among US Secondary Mathematics Teachers. Int J of Sci and Math Educ 14, 1113–1131 (2016). https://doi.org/10.1007/s10763-015-9638-1

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Keywords

  • Justification
  • Proofs
  • Secondary mathematics teachers
  • Teacher beliefs
  • Teacher preparation