Mathematics Education Research Journal

, Volume 28, Issue 3, pp 441–455 | Cite as

Developing mathematical practices through reflection cycles

  • Daniel L. ReinholzEmail author
Original Article


This paper focuses on reflection in learning mathematical practices. While there is a long history of research on reflection in mathematics, it has focused primarily on the development of conceptual understanding. Building on notion of learning as participation in social practices, this paper broadens the theory of reflection in mathematics learning. To do so, it introduces the concept of reflection cycles. Each cycle begins with prospective reflection, which guides one’s actions during an experience, and ends with retrospective reflection, which consolidates the experience and informs the next reflection cycle. Using reflection cycles as an organizing framework, this paper synthesizes the literature on reflective practices at a variety of levels: (1) metacognition, (2) self-assessment, (3) noticing, and (4) lifelong learning. These practices represent a spectrum of reflection, ranging from the micro level (1) to macro level (4).


Reflection Metacognition Self-assessment Teacher noticing Self-regulation 


  1. Alrø, H., & Skovsmose, O. (2003). Dialogue and learning in mathematics education: intention, reflection, critique (Vol. 29). Dordrecht: Kluwer Academic Publishers.Google Scholar
  2. Atkins, S., & Murphy, K. (1993). Reflection: a review of the literature. Journal of Advanced Nursing, 18(8), 1188–1192.CrossRefGoogle Scholar
  3. Australian Curriculum Assessment and Reporting Authority. (2009). Shape of the Australian curriculum: mathematics. Sydney: National Curriculum Board.Google Scholar
  4. Australian Education Council. (1990). A national statement on mathematics for Australian schools. Carlton: Australian Education Council.Google Scholar
  5. Averill, R., Drake, M., Anderson, D., & Anthony, G. (2016). The use of questions within in-the-moment coaching in initial mathematics teacher education: enhancing participation, reflection, and co-construction in rehearsals of practice. Asia-Pacific Journal of Teacher Education, 1–18. doi: 10.1080/1359866X.2016.1169503.
  6. Black, P., Harrison, C., & Lee, C. (2003). Assessment for learning: putting it into practice. Berkshire: Open University Press.Google Scholar
  7. Boud, D., & Walker, D. (1991). Experience and learning: reflection at work. EAE600 adults learning in the workplace: part A. Victoria: Deakin University.Google Scholar
  8. Boud, D., Keogh, R., & Walker, D. (1996). Promoting reflection in learning: a model. In Boundaries of adult learning (Vol. 1, pp. 32–56). New York, NY: Routledge.Google Scholar
  9. Boyd, E. M., & Fales, A. W. (1983). Reflective learning: key to learning from experience. Journal of Humanistic Psychology, 23(2), 99–117.CrossRefGoogle Scholar
  10. Brown, A. L. (1987). Metacognition, executive control, self-regulation, and other more mysterious mechanisms. In F. E. Weinart & R. H. Kluwe (Eds.), Metacognition, motivation, and understanding (pp. 65–116). Hillsdale: Lawrence Erlbaum Associates.Google Scholar
  11. Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective discourse and collective reflection. Journal for Research in Mathematics Education, 28(3), 258–277.CrossRefGoogle Scholar
  12. Dewey, J. (1933). How we think: a restatement of the relation of reflective thinking to the educative process. New York: D.C. Heath and Company.Google Scholar
  13. Dubinsky, E., & McDonald, M. A. (2002). APOS: a constructivist theory of learning in undergraduate mathematics education research. In The teaching and learning of mathematics at university level: an ICMI study (Vol. 7, pp. 273–280).CrossRefGoogle Scholar
  14. Dubinsky, E., & Wilson, R. T. (2013). High school students’ understanding of the function concept. The Journal of Mathematical Behavior, 32(1), 83–101.CrossRefGoogle Scholar
  15. Dunning, D., Johnson, K., Ehrlinger, J., & Kruger, J. (2003). Why people fail to recognize their own incompetence. Current Directions in Psychological Science, 12(3), 83.CrossRefGoogle Scholar
  16. Dunning, D., Heath, C., & Suls, J. M. (2004). Flawed self-assessment: implications for health, education, and the workplace. Psychological Science in the Public Interest, 5(3), 69–106. doi: 10.1111/j.1529-1006.2004.00018.x.
  17. Fennema, E., Franke, M. L., Carpenter, T. P., & Carey, D. A. (1993). Using children’s mathematical knowledge in instruction. American Educational Research Journal, 30(3), 555–583.CrossRefGoogle Scholar
  18. Flavell, J. H. (1979). Metacognition and cognitive monitoring: a new area of cognitive–developmental inquiry. American Psychologist, 34(10), 906.CrossRefGoogle Scholar
  19. Gandhi, P. R., Livezey, J., Zaniewski, A. M., Reinholz, D. L., & Dounas-Frazer, D. R. (in press). Attending to experimental physics practices and lifelong learning skills in an introductory laboratory course. American Journal of Physics.Google Scholar
  20. Garofalo, J., & Lester, F. K., Jr. (1985). Metacognition, cognitive monitoring, and mathematical performance. Journal for Research in Mathematics Education, 16(3), 163–176. doi: 10.2307/748391.
  21. Goos, M. (1994). Metacognitive decision making and social interactions during paired problem solving. Mathematics Education Research Journal, 6(2), 144–165.CrossRefGoogle Scholar
  22. Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition: creating collaborative zones of proximal development in small group problem solving. Educational Studies in Mathematics, 49(2), 193–223. doi: 10.1023/A:1016209010120.
  23. Gore, J. M., & Zeichner, K. M. (1991). Action research and reflective teaching in preservice teacher education: a case study from the United States. Teaching and Teacher Education, 7(2), 119–136.CrossRefGoogle Scholar
  24. Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity, and flexibility: a “proceptual” view of simple arithmetic. Journal for Research in Mathematics Education, 25(2), 116–140.CrossRefGoogle Scholar
  25. Hatton, N., & Smith, D. (1995). Reflection in teacher education: towards definition and implementation. Teaching and Teacher Education, 11(1), 33–49.CrossRefGoogle Scholar
  26. Jacobs, V. R., Lamb, L. L. C., & Philipp, R. A. (2010). Professional noticing of children’s mathematical thinking. Journal for Research in Mathematics Education, 41(2), 169–202.Google Scholar
  27. Kennison, M. M., & Misselwitz, S. (2002). Evaluating reflective writing for appropriateness, fairness, and consistency. Nursing Education Perspectives, 23(5), 238–242.Google Scholar
  28. Kolb, D. A. (1984). Experiential learning: experience as the source of learning and development (Vol. 1). Upper Saddle River: Prentice-Hall.Google Scholar
  29. Lave, J. (1996). Teaching as learning, in practice. Mind, Culture, and Activity, 3(3), 149–164.CrossRefGoogle Scholar
  30. Lyons, N. (Ed.). (2010). Handbook of reflection and reflective inquiry: mapping a way of knowing for professional reflective inquiry. New York: Springer.Google Scholar
  31. Mackintosh, C. (1998). Reflection: a flawed strategy for the nursing profession. Nurse Education Today, 18(7), 553–557.CrossRefGoogle Scholar
  32. Mann, K., Gordon, J., & MacLeod, A. (2009). Reflection and reflective practice in health professions education: a systematic review. Advances in Health Sciences Education, 14(4), 595–621.CrossRefGoogle Scholar
  33. Moon, J. A. (1999). Learning journals: a handbook for academics, students and professional development. New York: Routledge.Google Scholar
  34. Niss, M. (2010). Modeling a crucial aspect of students’ mathematical modeling. In R. Lesh, P. L. Galbraith, C. R. Haines, & A. Hurford (Eds.), Modeling students’ mathematical modeling competencies (pp. 43–59). New York: Springer.CrossRefGoogle Scholar
  35. Niss, M. (2011). The Danish KOM project and possible consequences for teacher education. Cuadernos de Investigación Y Formación En Educación Matemática, 6(9), 13–24.Google Scholar
  36. Pavlovich, K. (2007). The development of reflective practice through student journals. Higher Education Research and Development, 26(3), 281–295.CrossRefGoogle Scholar
  37. Piaget, J. (1972). The principles of genetic epistemology (Vol. 7). London: Routledge & Kegan Paul.Google Scholar
  38. Piaget, J. (2001). Studies in reflecting abstraction. Sussex, England: Psychology Press.Google Scholar
  39. Polya, G. (1945). How to solve it. Princeton: Princeton University Press.Google Scholar
  40. Reinholz, D. L. (2015a). Peer-assisted reflection: a design-based intervention for improving success in calculus. International Journal of Research in Undergraduate Mathematics Education. doi: 10.1007/s40753-015-0005-y.
  41. Reinholz, D. L. (2015b). The assessment cycle: a model for learning through peer assessment. Assessment & Evaluation in Higher Education, 1–15. doi: 10.1080/02602938.2015.1008982.
  42. Reinholz, D. L., Cox, M., & Croke, R. (2015). Supporting graduate student instructors in calculus. International Journal for the Scholarship of Teaching and Learning, 9(2), 1–8.CrossRefGoogle Scholar
  43. Schoenfeld, A. H. (1985). Mathematical problem solving. New York: Academy Press.Google Scholar
  44. 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: Lawrence Erlbaum Associates.Google Scholar
  45. Schön, D. A. (1983). The reflective practitioner: how professionals think in action (Vol. 5126). New York: Basic books.Google Scholar
  46. Schön, D. A. (1992). Designing as reflective conversation with the materials of a design situation. Knowledge-Based Systems, 5(1), 3–14.CrossRefGoogle Scholar
  47. Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1–36.CrossRefGoogle Scholar
  48. Sfard, A. (1998). On two metaphors for learning and the dangers of choosing just one. Educational Researcher, 27(2), 4–13.CrossRefGoogle Scholar
  49. Sherin, M., Jacobs, V., & Philipp, R. (2011). Mathematics teacher noticing: seeing through teachers’ eyes. Routledge.Google Scholar
  50. Simon, M. A., Tzur, R., Heinz, K., & Kinzel, M. (2004). Explicating a mechanism for conceptual learning: elaborating the construct of reflective abstraction. Journal for Research in Mathematics Education, 35(5), 305–329.CrossRefGoogle Scholar
  51. Tanner, H., & Jones, S. (2000). Scaffolding for success: reflective discourse and the effective teaching of mathematical thinking skills. Research in Mathematics Education, 2(1), 19–32.CrossRefGoogle Scholar
  52. Thorpe, K. (2004). Reflective learning journals: from concept to practice. Reflective Practice, 5(3), 327–343.CrossRefGoogle Scholar
  53. Vygotsky, L. S. (1978). Mind in society: the development of higher mental process. Cambridge: Harvard University Press.Google Scholar
  54. Wilson, J., & Clarke, D. (2004). Towards the modelling of mathematical metacognition. Mathematics Education Research Journal, 16(2), 25–48.CrossRefGoogle Scholar
  55. Zimmerman, B. J. (2002). Becoming a self-regulated learner: an overview. Theory Into Practice, 41(2), 64–70. doi: 10.1207/s15430421tip4102_2.

Copyright information

© Mathematics Education Research Group of Australasia, Inc. 2016

Authors and Affiliations

  1. 1.Department of MathematicsSan Diego State UniversitySan DiegoUSA

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