The Status of Proving Among US Secondary Mathematics Teachers

Article

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.

Keywords

Justification Proofs Secondary mathematics teachers Teacher beliefs Teacher preparation 

Notes

Acknowledgments

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

References

  1. 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.
  2. 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.CrossRefGoogle Scholar
  3. Ashton, P. (1985). Motivation and the teacher’s sense of efficacy. Research on Motivation in Education, 2, 141–174.Google Scholar
  4. Ball, D. L. (1988). Unlearning to teach mathematics. For the Learning of Mathematics, 8(1), 40–48.Google Scholar
  5. 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
  6. 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.CrossRefGoogle Scholar
  7. Battista, M. T. (1994). Teacher beliefs and the reform movement in mathematics education. Phi Delta Kappan, 75(6), 462–470.Google Scholar
  8. Bell, A. (1976). A study of pupils’ proof-explanations in mathematical situations. Educational Studies in Mathematics, 7(1–2), 23–40.CrossRefGoogle Scholar
  9. Beswick, K. (2012). Teachers’ beliefs about school mathematics and mathematicians’ mathematics and their relationship to practice. Educational Studies in Mathematics, 79(1), 127–147.CrossRefGoogle Scholar
  10. 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
  11. 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.CrossRefGoogle Scholar
  12. 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.Google Scholar
  13. Bruner, J. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press.Google Scholar
  14. Buchmann, M. & Schwille, J. (1983). Education: The overcoming of experience. American Journal of Education, 92(1), 30–51.CrossRefGoogle Scholar
  15. 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.Google Scholar
  16. Common Core State Standards for Mathematics (2010). Washington, D.C.: Council of Chief State School Officers and National Governors Association.Google Scholar
  17. 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.Google Scholar
  18. 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.CrossRefGoogle Scholar
  19. De Villiers, M. (1999). Rethinking proof with geometer’s sketchpad. Emeryville, CA: Key Curriculum Press.Google Scholar
  20. 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.CrossRefGoogle Scholar
  21. 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.Google Scholar
  22. 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.Google Scholar
  23. 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.CrossRefGoogle Scholar
  24. 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.Google Scholar
  25. 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.CrossRefGoogle Scholar
  26. Furinghetti, F. & Morselli, F. (2011). Beliefs and beyond: Hows and whys in the teaching of proof. ZDM, 43(4), 587–599.CrossRefGoogle Scholar
  27. 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.Google Scholar
  28. Galbraith, P. L. (1981). Aspects of proving: A clinical investigation of process. Educational Studies in Mathematics, 12(1), 1–28.CrossRefGoogle Scholar
  29. Hanna, G. (1989). Proofs that prove and proofs that explain. Proceedings of the International Group for the Psychology of Mathematics Education, Paris.Google Scholar
  30. Hanna, G. & Barbeau, E. (2008). Proofs as bearers of mathematical knowledge. ZDM, 40(3), 345–353.CrossRefGoogle Scholar
  31. 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.Google Scholar
  32. 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.CrossRefGoogle Scholar
  33. Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396–428.CrossRefGoogle Scholar
  34. 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.CrossRefGoogle Scholar
  35. Hersh, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24(4), 389–399.CrossRefGoogle Scholar
  36. 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.Google Scholar
  37. 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.CrossRefGoogle Scholar
  38. Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379–405.Google Scholar
  39. 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
  40. 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.CrossRefGoogle Scholar
  41. Levenson, E. (2013). Exploring one student’s explanations at different ages: the case of Sharon. Educational Studies in Mathematics, 83(2), 181–203.CrossRefGoogle Scholar
  42. Liljedahl, P. (2008). Teachers’ insights into the relationship between beliefs and practice. Beliefs and attitudes in mathematics education: New research results, 33–44.Google Scholar
  43. Lortie, D. C. (1975). School teacher: A sociological inquiry. Chicago: University of Chicago Press.Google Scholar
  44. 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.CrossRefGoogle Scholar
  45. 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.Google Scholar
  46. 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.CrossRefGoogle Scholar
  47. 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.Google Scholar
  48. Mingus, T. T. Y. & Grassl, R. M. (1999). Preservice teacher beliefs about proofs. School Science and Mathematics, 99(8), 438–444.CrossRefGoogle Scholar
  49. 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.Google Scholar
  50. National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.Google Scholar
  51. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.Google Scholar
  52. Peterson, P. (1989). Teachers’ Belief Questionnaire. (Available from P. Peterson, 2004, Fall).Google Scholar
  53. Peterson, P. & Fennema E. (1989). Teachers’ pedagogical content beliefs in mathematics. Cognition and Instruction, 6(1):1-40.Google Scholar
  54. Piaget, J. & Inhelder, B. (1969). The early growth of logic in the child. New York, NY: Norton.Google Scholar
  55. Polya, G. (1957). How to solve it: A new aspect of mathematical method. Garden City, NY: Doubleday.Google Scholar
  56. 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.CrossRefGoogle Scholar
  57. Raimi, R. A. (2001). Standards in school mathematics. Notices of the American Mathematical Society, 48(2), 166.Google Scholar
  58. 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.CrossRefGoogle Scholar
  59. Ross, K. A. (2000). The MAA and the New NCTM Standards. from http://www.maa.org/features/pastfeatures.html. Accessed 30 June 2012.
  60. Schoenfeld, A. H. (1998). Toward a theory of teaching-in-context. Issues in Education, 4(1), 1–94.CrossRefGoogle Scholar
  61. Senk, S. L. (1985). How well do students write geometry proofs? Mathematics Teacher, 78(6), 448–456.Google Scholar
  62. Silver, E. A. & Kenney, P. A. (2000). Results from the seventh mathematics assessment. Reston, VA: NCTM.Google Scholar
  63. 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.CrossRefGoogle Scholar
  64. Skemp, R. R. (1978). Relational understanding and instrumental understanding. Arithmetic Teacher, 26, 9–15.Google Scholar
  65. Smith, J. P. (1996). Efficacy and teaching mathematics by telling: a challenge for reform. Journal for Research in Mathematics Education, 27, 387–402.CrossRefGoogle Scholar
  66. 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.Google Scholar
  67. 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.CrossRefGoogle Scholar
  68. Stein, M. K., Baxter, J. A. & Leinhardt, G. (1990). Subject-matter knowledge and elementary instruction. American Educational Research Journal, 27(4), 639–663.CrossRefGoogle Scholar
  69. Stevens, J. P. (2002). Applied multivariate statistics for the social sciences. Mahwah, NJ: Erlbaum.Google Scholar
  70. 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
  71. 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.Google Scholar
  72. Stylianides, A. J. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38, 289–321.Google Scholar
  73. 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
  74. 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.CrossRefGoogle Scholar
  75. Thompson, A. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15(2), 105–127.CrossRefGoogle Scholar
  76. 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
  77. 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
  78. Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101–119.CrossRefGoogle Scholar
  79. 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.Google Scholar

Copyright information

© Ministry of Science and Technology, Taiwan 2015

Authors and Affiliations

  1. 1.Fordham UniversityNew YorkUSA

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