• Po-Hung LiuEmail author


The present study observed how Taiwanese college students’ epistemological beliefs about mathematics evolved during a year-long historical approach calculus course. On the basis of the characteristics of initial accounts, seven students were invited to participate in this study and were divided into two groups. An open-ended questionnaire, mathematics biographies, in-class reports, and follow-up semi-structured interviews served as instruments for identifying their epistemological beliefs. Furthermore, four randomly selected students from another calculus class constituted the control group. Results indicated that most of the students receiving this course exhibited relatively significant changes in their epistemological beliefs of mathematics, but trends and extents in their epistemological development varied across groups as well as individuals. This study identifies the potential relationships among the course features, initial beliefs, and the tendency of belief development, followed by a discussion of the mechanism of belief change and an afterthought on HPM approach.

Key words

calculus epistemological belief history of mathematics HPM nature of mathematics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ball, D.L. (2000). Working on the inside: using one’s own practice as a site for studying teaching and learning. In A.E. Kelly & R.A. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 365–402.) Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  2. Baxter-Magolda, M.B. (2004). Evolution of a constructivist conceptualisation of epistemological reflection. Educational Psychologist, 39(1), 31–42.CrossRefGoogle Scholar
  3. Bendixen, L.D. & Rule, D.C. (2004). An integrative approach to personal epistemology: a guiding model. Educational Psychologist, 39(1), 69–80.CrossRefGoogle Scholar
  4. Beth, E.W. & Piaget, J. (1966). Mathematical epistemology and psychology. Dordrecht: D. Reidel Publishing Company.Google Scholar
  5. Butterfield, H. (1931). The Whig interpretation of history. London: G. Bell and Sons.Google Scholar
  6. Carlson, M.P. (1999). The mathematical behavior of six successful mathematics graduate students: influences leading to mathematical success. Educational Studies in Mathematics, 40(3), 237–258.CrossRefGoogle Scholar
  7. Dennis, D. (2000). The role of historical studies in mathematics and science educational research. In D. Lesh & A. Kelly (Eds.), Research design in mathematics and science education. Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  8. Fried, M. (2001). Can mathematics education and history of mathematics coexist? Science and Education, 10, 391–408.CrossRefGoogle Scholar
  9. Furinghetti, F. & Pehkonen, E. (2002). Rethinking characterizations of beliefs. In G.C. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: a hidden variable in mathematics education? (pp. 39–58. ) Dordrecht: Kluwer.Google Scholar
  10. Garofalo, J. & Lester, F.K. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16(3), 163–176.CrossRefGoogle Scholar
  11. Glas, E. (1998). Fallibilism and the use of history in mathematics education. Science and Education, 7, 361–379.CrossRefGoogle Scholar
  12. Grugnetti, L. & Rogers, L. (2000). Philosophical, multicultural and interdisciplinary issues. In J. Fauvel & J. van Maanen (Eds.), History in mathematics education (pp. 39–62.) Dordrecht: Kluwer.Google Scholar
  13. Hofer, B. (1999). Instructional context in the college mathematics classroom: epistemological beliefs and student motivation. Journal of Staff, Program, & Organizational Development, 16(2), 73–82.Google Scholar
  14. Hofer, B. (2004). Epistemological understanding as a metacognitive process: thinking aloud during online searching. Educational Psychologist, 39(1), 43–55.CrossRefGoogle Scholar
  15. Hofer, B. & Pintrich, P. (1997). The development of epistemological theories: beliefs about knowledge and knowing and their relation to learning. Review of Educational Research, 67, 88–140.Google Scholar
  16. Horng, W.-S. (2000). Euclid versus Liu Hui: a pedagogical reflection. In V. Katz (Ed.), Using history of mathematics in teaching mathematics (pp. 37–47.) Washington, DC: Mathematical Association of America.Google Scholar
  17. Kitcher, P. (1984). The nature of mathematical knowledge. New York: Oxford University Press.Google Scholar
  18. Kline, M. (1980). Mathematics: the loss of certainty. New York: Oxford University.Google Scholar
  19. Kloosterman, P. & Stage, F.K. (1991). Relationships between ability, belief and achievement in remedial college mathematics classrooms. Research and Teaching in Developmental Education, 8(1), 27–36.Google Scholar
  20. Lakatos, I. (1976). Proofs and refutations: the logic of mathematical discovery. N.J.: Cambridge University Press.Google Scholar
  21. Lampert, M. (1990). When the problem is not the question and the solution is not the answer: mathematical knowing and teaching. American Educational Research Journal, 27, 29–63.Google Scholar
  22. Leder, G.C., Pehkonen, E. & Törner, G. (Eds.) (2002). Beliefs: a hidden variable in mathematics education? Dordrecht: Kluwer.Google Scholar
  23. Legrand, M. (2001). Scientific debate in mathematics courses. In D. Holton (Ed.), The teaching and learning of mathematics at university level: an ICMI study (pp. 127–135.) Netherlands: Kluwer.Google Scholar
  24. Leng, N.-W. (2006). Effects of an ancient Chinese mathematics enrichment programme on secondary students’ achievement in mathematics. International Journal of Science and Mathematics Education, 4(2), 485–511.CrossRefGoogle Scholar
  25. Lester, F.K., Garofalo, J. & Kroll, D.L. (1989). Self-confidence, interest, beliefs, and metacognition. In D.B. McLeod & V.M. Adams (Eds.), Affect and mathematical problem solving (pp. 75–88.) Berlin Heidelberg New York: Springer.Google Scholar
  26. Liu, P.-H. (2007). Exploring the relationship between a history oriented calculus course and the development of students’ views on mathematics. Chinese Journal of Science Education, 15(6), 703–723.Google Scholar
  27. Liu, P.-H. & Niess, M. (2006). An exploratory study of college students’ views of mathematical thinking in a historical approach calculus course. Mathematical Thinking and Learning, 8(4), 373–406.CrossRefGoogle Scholar
  28. MacLeod, D.B. (1992). Research on affect in mathematics education: a reconceptualization. In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 575–596.) New York: Macmillan.Google Scholar
  29. National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  30. National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  31. Op ‘t Eynde, P., De Corte, E. & Verschaffel, L. (2002). Framing students’ mathematics-related beliefs: a quest for conceptual clarity and a comprehensive categorization. In G.C. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: a hidden variable in mathematics education? (pp. 13–37.)Dordrecht: Kluwer.Google Scholar
  32. Pajares, F. (1992). Teachers’ beliefs and educational research: cleaning up a messy construct. Review of Educational Research, 62(3), 307–332.Google Scholar
  33. Piaget, J. & Garcia, R. (1989). Psychogenesis and the history of science. New York: Columbia University Press.Google Scholar
  34. Pintrich, P.R. (2002). Future challenges and directions for theory and research on personal epistemology. In B.K. Hofer & P.R. Pintrich (Eds.), Personal epistemology: the psychology of beliefs about knowledge and knowing (pp. 389–414.) Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  35. Presmeg, M. (2002). Beliefs about the nature of mathematics in the bridging of everyday and school mathematical practice. In G.C. Leder, E. Pehkonen & G. Törner (Eds.), Beliefs: a hidden variable in mathematics education? (pp. 293–311.) Dordrecht: Kluwer.Google Scholar
  36. Radford, L. (1997). On psychology, historical epistemology, and the teaching of mathematics: towards a socio-cultural history of mathematics. For the Learning of Mathematics, 17(1), 26–33.Google Scholar
  37. Schoenfeld, A.H. (1985). A framework for the analysis of mathematical behavior. In A.H. Schoenfeld (Ed.), Mathematical problem solving (pp. 11–45.) New York: Academy Press.Google Scholar
  38. Schoenfeld, A.H. (1987). What’s all the fuss about metacognition? In A.H. Schoenfeld (Ed.), Cognitive science and mathematics education (pp. 189–215.) Hillsdale, NJ: Lawrence Erlbaum.Google Scholar
  39. Schoenfeld, A.H. (1992). Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 340–370.). New York: Macmillan.Google Scholar
  40. Sierpinska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18, 371–397.CrossRefGoogle Scholar
  41. Sierpinska, A. & Lerman, S. (1996). Epistemologies of mathematics and of mathematics education. In A.J. Bishop, K. Clements, C. Keitel, J. Kilpatrick & C. Laborde (Eds.), International handbook of mathematics education (pp. 827–876). Dordrecht: Kluwer.Google Scholar
  42. Thompson, A. (1984). The relationship of teachers’ conceptions of mathematics and mathematics teaching to instructional practice. Educational Studies in Mathematics, 15, 105–127.CrossRefGoogle Scholar
  43. Waldegg, G. (1997) Histoire, épistémologie et méthodologie dans la recherche en didactique. For the Learning of Mathematics, 17(1), 43–46.Google Scholar
  44. Whitemire, E. (2004). The relationship between undergraduates’ epistemological beliefs, reflective judgment, and their information-seeking behavior. Information Processing Management, 40, 97–111.CrossRefGoogle Scholar

Copyright information

© National Science Council, Taiwan 2008

Authors and Affiliations

  1. 1.General Education CenterNational Chin-Yi University of TechnologyTaichung CountyTaiwan

Personalised recommendations