Prospective Teachers’ Understanding of the Constant π andtheir Knowledge of How to Prove its Constant Nature through the Concept of Linearity
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When taught the precise definition of π, students may be simply asked to memorize its approximate value without developing a rigorous understanding of the underlying reason of why it is a constant. Measuring the circumferences and diameters of various circles and calculating their ratios might just represent an attempt to verify that π has an approximate value of 3.14, and will not necessarily result in an adequate understanding about the constant or formally proves that it is a constant. In this study, we aim to investigate prospective teachers' conceptual understanding of π, and as a constant and whether they can provide a proof of its constant property. The findings show that prospective teachers lack a holistic understanding of the constant nature of π, and reveal how they teach students about this property in an inappropriate approach through a proving activity. We conclude our findings with a suggestion on how to improve the situation.
Abstract
When taught the precise definition of π, students may be simply asked to memorize its approximate value without developing a rigorous understanding of the underlying reason of why it is a constant. Measuring the circumferences and diameters of various circles and calculating their ratios might just represent an attempt to verify that π has an approximate value of 3.14, and will not necessarily result in an adequate understanding about the constant or formally proves that it is a constant. In this study, we aim to investigate prospective teachers' conceptual understanding of π, and as a constant and whether they can provide a proof of its constant property. The findings show that prospective teachers lack a holistic understanding of the constant nature of π, and reveal how they teach students about this property in an inappropriate approach through a proving activity. We conclude our findings with a suggestion on how to improve the situation.
참고문헌 (31)
- Balacheff, N., "Radical constructivism in mathematics education", pp.89~110 (1991), Kluwer
- Balacheff, N., "Explanation and proof in mathematics: Philosophical and educational perspectives", pp.115~135 (2010), Springer
- Ball, D., "Effects of Teachers Mathematical Knowledge for Teaching on Student Achievement", American Educational Research Journal, 42(2), pp.371~406 (2005)
- Ball, D., "Handbook of research on teaching", pp.433~456 (2001), American Educational Research Association
- Belfort, E., "The influence of subject matter on teachers' practices: Two case studies", Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, 1, pp.73~80 (2002)
- Blum, W., "Preformal proving: Examples and reflections", Educ. Stud.Math, 22(2), pp.183~203 (1991)
- "Nordic Studies in Mathematics Education NOMAD", 2007, 1~4
- Curriculum Development Committee, Hong Kong (CDC), "Syllabus for secondary schools: mathematics (secondary 1-5)" (1999), Government Printer
- Curriculum Development Council, Hong Kong & Hong Kong Examinations and Assessment Authority (CDC & HKEAA), "Mathematics education key learning area: mathematics curriculum and assessment guide (Secondary 4-6)" (2007), Government Printer
- Dreyfus, T., "Why Johnny Can't Prove", Educ. Stud. Math, 38(1-3), pp.85~109 (1999)
- Griffiths, P.A., "Mathematics at the Turn of the Millennium", Am. Math. Mon, 107(1), pp.1~14 (2000)
- Gruenwald, N., "Using Counter-Examples in Teaching Calculus: Students' Attitudes", N. Z. Math. Mag, 40(2), pp.22~41 (2003)
- Hill, H. C., "Unpacking Pedagogical Content Knowledge: Conceptualizing and Measuring Teachers' Topic-Specific Knowledge of Students", J. Res. Math. Educ, 39(4), pp.372~400 (2008)
- Leung, K. C. I., "Prospective Teachers' Understanding of the Constant ππ andtheir Knowledge of How to Prove its Constant Nature through the Concept of Linearity", Proceedings of the 2013 Joint International Conference on Mathematics Education held at Seoul National University, pp.151~742 (2013), Korean Society of Mathematical Education
- Li, J., "A framework on mathematical knowledge for teaching", Proceedings of the 5th East Asia Regional Conference on Mathematics Education (EARCOME 5), 2, pp.1~8 (2010), Japan Society of Mathematics Education
- Li, Y., "Chinese elementary mathematics teachers' knowledge in mathematics and pedagogy for teaching: the case of fraction division", ZDM, Int. J. Math. Educ, 40(5), pp.845~859 (2008)
- Li, Y., "Knowledge and confidence of pre-service mathematics teachers: the case of fraction division", ZDM, Int. J. Math. Educ, 40(5), pp.833~843 (2008)
- Meel, D. E., "Prospective teachers' understandings: Function and composite function. Issues Undergrad", Math. Prep. Sch. Teach. J, 1(1), pp.1~12 (1999)
- Moore, R. C., "Making the Transition to Formal Proof", Educ. Stud. Math, 27(3), pp.249~226 (1994)
- Park, K., "Mathematics Teacher Education in East Asian Countries-From the Perspective of PedagogicalContent Knowledge", Paper presented at the 3rd East Asia Regional Conference on Mathematics Education (EARCOME 3) (2005)
- The preparation gap: teacher education for middle school mathematics in six countries (MT21 report), "Schmidt, W.", Michigan State University, ~, pp.MT21 report (2007)
- Sanchez, V., "Four Student Teachers' Pedagogical Reasoning on Function", J. Math. Teach. Educ, 6(1), pp.5~25 (2003)
- Schwarz, B., "Future teachers' professional knowledge on argumentation and proof: a case study from universities in threecountries", ZDM, Int. J. Math. Educ, 40(5), pp.791~811 (2008)
- Shulman, L. S., "Those who understand: Knowledge growth in teaching", Educational Researcher, 57(1), pp.1~22 (1986)
- Styliandes, G. J., "Preservice Teachers' Knowledge of Proof by Mathematical Induction", J. Math. Teach. Educ, 10(3), pp.145~166 (2007)
- Stylianides, A. J., "Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving", J. Math. Teach. Educ, 11(4), pp.307~332 (2008)
- Tall, D., "Developments in School Mathematics Education around the World Volume 4", pp.117~136 (1999), NCTM
- Weber, K., "Student difficulty in constructing proof: The need for strategic knowledge", Educ. Stud. Math, 48(1), pp.101~119 (2001)
- Wong, N. Y., "How Chinese learn mathematics: perspectives from insiders", pp.27~70 (2004), World Scientific
- Wu, H., "The Mis-Education of Mathematics Teachers", Notices of the American Mathematical Society, 58(3), pp.372~384 (2011)
- Zandieh, M., "Defining as a Mathematical Activity: A Framework for Characterizing Progress from Informal to More Formal Ways of Reasoning", J. Math. Behav, 29(2), pp.57~75 (2010)