pp. 77-95 | Article Number: mathedu.2015.006
Published Online: August 02, 2015
Article Views: 369 | Article Download: 620
What is mathematics? The difficulty of having a precise, universal definition of mathematics has led prospective teachers to define the term in ways that make sense to them. This paper is part of a larger research project conducted in 2000 in an Ontarian university, Canada. The objectives were to identify and discuss conceptualizations of mathematics that prospective teachers brought to their preparation program and to explore the implications of such conceptualizations in terms of teaching and learning. It was believed that both the identification tools and understandings of prospective teachers’ conceptualizations of mathematics were significant for designing an effective pedagogy in accordance with mathematics reform-based perspectives. The research sample consisted of ten prospective teachers enrolled in a one-year bachelor of education program at an Ontarian university. The research used mathematics autobiographies of the respondents and semi-structured interviews of them as sources of data. Guided by the theory of personal construct for analysis of the data, the results showed that the respondents conceptualized mathematics in terms of metaphor, metonymy and combination of the two. The conclusion explores implications of such conceptualizations for mathematics teaching, learning and assessment.
Keywords: conceptions of mathematics, language, metaphoric, metonymic, pre-service teachers, teachers’ math autobiographies
Alagic, M. & Emery, S. (2003). Differentiating instruction with marbles: Is this algebra or what? International Journal for Mathematics Teaching and Learning,(July,8th). Accessed Dec.20, 2007 from http://www.cimt.plymouth.ac.uk/journal/algicemery
Andrews, P. And Hatch, G. (1999). A new look at secondary teachers’ conceptions of mathematics and its teaching, British Educational Research Journal, 25(2), 203-223.
Baig, S. & Halai, A. (2006). Learning mathematical rules with reasoning. Eurasia Journal of Mathematics, Science, and Technology Education, 2(2), 15-39.
Bako', M. (2002) Why we need to teach logic and how we can teach it. International Journal of Mathematics Teaching and Learning. Accessed April 20, 2005 from.http://www.cimt plymouth.ac.uk/journal/bakom
Ball, D. L. (1988). Unlearning to teach mathematics. For The Learning of Mathematics, 8(2), 40-46.
Ball, D. L. (1990). Breaking with experience in learning to teach mathematics: The role of a preservice methods course. For The Learning of Mathematics, 10(2), 10-16.
Ball, D. (1991). Research on teaching mathematics: Making subject-matter knowledge part of the equation. In J. Brophy (Ed.), Advances in research on teaching (Vol. 2, pp. 1–48) Greenwich, CN: JAI Press
Ball, D.L., Goffney, I. M. & Bass, H. (2005). The role of mathematics instruction in building a socially just and diverse democracy. The Mathematics Educator, 15(1), 2-6.
Barcelona, A. (2011). Reviewing the properties and prototype structure of metonymy. In RékaBenczes,
Antonio Barcelona and Francisco José Ruiz de Mendoza Ibáňez, (eds). Defining Metonymy in Cognitive Linguistics: Towards a Consensus View (pp. 7-57). Amsterdam/Philadelphia: John Benjamins.
Brown, A. (1999). A mathematics teaching even that changed my belief. Philosophy of Mathematics Education Journal # 12. Accessed Nov. 12, 2007 from http://www.people.ex.ac
Borba, M.C. (1992). Teaching mathematics: Challenging the sacred cow of mathematical certainty. The Learning House,65(6), 332-333.
Burns, M. (1994). Arithmetic: The last holdout. Phi Delta Kappan, 75(6) 471-476.
Catalano, T. and Waugh, R.L (2013). The language of money: How verbal and visual metonymy shapes public opinion about financial events. International Journal of Language Studies, 7(2), 11-60.
Countryman, C. (1992). Writing to learn mathematics. Portsmouth, NH: Heinemann Educational Books Inc.
Davis, A. (1994). ‘Constructivism’. In A. Davis and D. Pettit (Eds.).Developing Understanding in primary mathematics (pp.11-13). London: The Falmer Press.
Devlin, K. (1994). Mathematics: The science of patterns: The search for order in life, mind, and the universe. New York: Scientific American Library.
Dossey, J. (1992). The nature of mathematics: Its role and its influence. In D. A. Grouws (Ed.),
Handbook of research on mathematics teaching and learning (pp. 39-48). New York: Macmillan.
Ellis, M. W. and Berry III, R. Q. (2005). The paradigm shift in mathematics education: Explanation and implications of reforming conceptions of mathematics teaching and learning. The Mathematics Educator, 15(1), 1-17.
Ernest, P. (1989). The knowledge, beliefs, and attitudes of the mathematics teacher: A model. Journal of Education for Teaching, 15, 13-33.
Ernest, P. (1996).The nature of mathematics and teaching. Philosophy of Mathematics Education Newsletter, 9, November
Ernest, P. (2002). Empowerment in mathematics education. Philosophy of Mathematics Education Journal, # 15. Accessed Nov. 2007 from http://www.people.ex.ac.uk/PErnest/pome15
Ernest, P. (2010). Mathematics and metaphor: A response to Elizabeth Mowat & Brent Davis . Complicity: An International Journal of Complexity and Education, 7(1), 98-104
Fass, D. (1991). Met: A method for discriminating metonymy and metaphor by computer. Computational Linguistics, 17(1), 49-90.
Feiman-Nemser, S. (2001). From preparation to practice: Designing a continuum to strengthen and sustain teaching.Teacher College Record, 103, 1013-1055.
Gates, P. (2006). Going Beyond Belief Systems: Exploring a Model for the Social Influence on Mathematics Teacher Beliefs.Educational Studies in Mathematics 63(3). 347–369
Goulding, M., Rowland, T. & Barber, P. (2002). Does it matter? Primary trainees’ subject knowledge in mathematics. British Educational Research Journal, 28(5), 689-704.
Hersh, R. (1986). Some proposals for revising the philosophy of mathematics. In T. Tymoczko(Ed.), New Directions in the philosophy of mathematics. Boston: Birkhauser.
Hewitt, D. (1987) Mixed ability mathematics: Losing the building block metaphor. Forum 1, 39(2), 46-49.
Jamnik, M. (2001). Mathematical reasoning with diagrams. Sanford, U.S.A: CSLI Press.
Jeffery, B. (1997). Metaphors and representation: Problems and heuristic possibilities in ethnography and social science writing. International Education, 27(1), 26-28.
Kelly,G. A. (1995). The psychology of personal constructs. New York: W.W. Norton& Comp . Inc.
Kovecses, Z, (2006). Language, mind, and culture. Oxford: Oxford University Press
Krulik, S. & Rudnick, J. A. (1982). Teaching problem solving too preservice teachers. Arithmetic Teacher, 23(2), 42-47.
Lakoff, G. & Johnson, M. (1980). Metaphors we live by. Chicago: The University of Chicago Press.
Leatham, K. R. (2006). Viewing teachers’ beliefs as sensible systems. Journal of Mathematics Teacher Education, 9, 91-102.
Lehrer, R. & Franke, M. L(1992). Applying personal construct psychology to the study of teachers’ knowledge of fractions.Journal for Research in Mathematics Education, 23(3), 223-241
McQualter, J.W. (1986). Becoming a mathematics teacher: The use of personal construct theory Educational Studies in Mathematics, 17, 1-14
Merseth, K. (1993). How old is the shepherd? An essay about mathematics education. Phi Delta Kappan, 74(7), 548-554.
Murphy, C. (2007). The constructive role of conceptual metaphor in children arithmetic: A comparison and contrast of Piagetian and embodied learning perspectives. Philosophy of Mathematics Education Journal, Nov., 22.
National Council of Teachers of Mathematics.(2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
Nebesniak, A. L (2012). Learning to teach mathematics with reasoning and sense-making. Unpublished doctoral dissertation. Lincoln, Nebraska: University of Nebraska.
Nerlich, B. (2006). Metonymy. Encyclopedia of Language and Linguistics, 109-113
Nesher, P. (1986) Are mathematical understanding and algorithmic performance related? For The Learning of Mathematics, 3, 7-11.
New Zealand Ministry of Education (1992). Mathematics in the New Zealand curriculum. Wellington: N.Z: Author.
Noyes, A. (2006). Using metaphor in mathematics teacher preparation. Teaching and Teacher Education, 22, 898-909
Ontario Ministry of Education (2005a) Mathematics: The Ontario curriculum grades 1-8. Toronto, Ontario: Author.
Ontario Ministry of Education (2005b). Mathematics: The Ontario curriculum grades 9-10. Toronto, Ontario: Author
Ontario Ministry of Education (2007) Mathematics: The Ontario curriculum grades 11-12. Toronto, Ontario: Author
Padmanabahan, R. (2000). Logical reasoning in mathematics vs. brute force calculations. Manitobamath Links, 1(1). Accessed December 12, 2005 from http://www.umanitoba.ca/Faculties/science/mathematics/new/issue1.pdf
Paradis, C. (2004). Where does metonymy stop? Senses, facets and active zones. Metaphors and Symbol, 19(4), 245-264
Presmeg, N. (2002). Beliefs about the nature of mathematics in the bridging of everyday and school mathematical practices. In G. Leder, E. Pehkonen, & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education? (pp. 293-312). Dordrecht: Kluwer
Quilter, D. & Harper, E. (1988). Why we didn’t like mathematics and why we can’t do it. Educational Research, 30(2), 121-128.
Radden, G. & Kovecses, Z. (1999). Towards a theory of metonymy. In Panther-U & Radden, G. (eds). Metonymy in language and thought (pp.17-60). Amsterdam/Philadelphia: John Benjamins.
Ruthven, K. (1987).Ability stereotyping in mathematics. Educational Studies in Mathematics, 18, 243-253.
Sam, C.L. (1999). Using metaphor analysis to explore adults’ images of mathematics. Philosophy of Mathematics Education Journal, 12. Accessed December 2,2007 from http://www.people.ex.ac.uk/PErnest/pome12/article9.htm
Silver, E.A, Kilpatrick, J, & Schlesinger, B. (1990). Thinking through mathematics. New York: College Entrance Examination Board.
Searle, J. (1979). Metaphor. In A Ortony (Ed.) metaphor and thought (pp.92-123). Cambridge: Cambridge University Press.
Song, S, (2011). Metaphors and metonymy- A tentative research into modern cognitive linguistics. Theory and Practice in Language Studies, 1(1), 68-73
Steel, D. and Widman, T. F. (1997). Practitioner’s research: A study in changing perspective teachers’ conceptions about mathematics and mathematics teaching and learning. School Science and Mathematics, 97(3), 192-199.
Steen, L.A.(1989). Teaching mathematics for tomorrow world. Educational Leadership, 47(1), 18-22
Steen, L. A. (1999). Twenty questions about mathematical reasoning. .L Stiff (ed.) in Developing mathematical reasoning in grades k-12 (pp.270-285), Reston, VA: National Council of Teachers of Mathematics.
Steinbring, H. (1989) Routine and meaning in mathematics classroom. For the Learning of Mathematics, 9(1), 24-33.
Stinson, D.W. (2004). Mathematics as “gate-keeper”(?): Three theoretical perspectives that ai towardempowering all children with the key to the gate. The Mathematics Educator, 14(1), 8-18.
Strauss, A.L, & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage.
Tapson, F. (2002). The language of mathematics. International Journal of Mathematics Teaching and Learning. Accessed November 25, 2009 frohttp://www.plymouth.ac.uk/journal/ftlag
The Alliance education Organization (2006). Closing the achievement gap: But practices in teaching mathematics.Charleston, West Virgin: Author. Accessed Dec.20, 2007 from http:///www.educationalliance.org
Thompson, A. G. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15(2), 105-127.
Timmerman, M. A. (2004). The influence of three interventions on perspective elementary teachers’ beliefs about the knowledge base needed for teaching mathematics. School Science and Mathematics, 104(8), 369-382.
Wood, T. (2001). Teaching differently: Creating opportunities for learning mathematics. Theory into Practice(Spring), 40(2), 110-117.
Yetkin, E. (2003). Student difficulties in learning elementary mathematics. ERIC Clearinghouse for Science, Mathematics and Environmental Education, Reproduction No ED482727.
Zazkis, R. & Campbell, S. (1996). Prime composition: Understanding uniqueness. The Journal of Mathematics Behavior,15(2), 207-218.
Zazkis, R. & Gunn, C. (1997). Sets, subsets and the empty set: Student constructions and mathematics conventions. Journal of Computers in Mathematics and Science, 16(1), 133-169.
Zazkis, R. (1999). Challenging basic assumptions: Mathematical experience for preservice teachers. Journal of Mathematical Education in Science and Technology, 30(5), 631-650.