# Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks

- 815 Downloads
- 28 Citations

## Abstract

This study is grounded in the theoretical position that solving problems in different ways creates mathematical connections when learning and teaching mathematics. It acknowledges the central role teachers play in providing students with learning opportunities, and it is based on the empirical finding that mathematics teachers are reluctant to solve problems in different ways in the classroom. In this paper we address the contradiction between theory-based recommendations and school mathematics practice. Based on analysis of individual interviews and two group meetings with 12 Israeli secondary school mathematics teachers, we demonstrate that in the context of multiple-solution connecting tasks this discrepancy is caused by the situated nature of the teachers’ knowledge. We also reveal the complex relationship between different types of teacher knowledge and argue the significance of developing a common language between members of the mathematics education community, including teacher educators and researchers.

### Keywords

Mathematics teachers Multiple solution connecting tasks Teacher knowledge Teaching practice## Notes

### Acknowledgment

This research was made possible by grant #891/03 from the Israel Science Foundation. We wish to thank Irena Gurevich for her assistance in data collection. We are indebted to the teachers who participated in the study for their collaboration and goodwill. We would like to thank Anna Sfard and the anonymous reviewers for their insightful and stimulating comments on the earlier version of the paper.

### References

- Askew, M. (2001). Policy, practices and principles in teaching numeracy: What makes a difference? In P. Gates (Ed.),
*Issues in mathematics teaching*(pp. 105–119). London: Routledge.Google Scholar - Ball, D. L., Hill, H. C., & Bass, H. (2005). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide?
*American Educator*, fall 2005, 14–17, 20–22, 43–46.Google Scholar - Brown, J. S., Collins, A., & Diguid, P. (1989). Situated cognition and the culture of learning.
*Educational Researcher, 1*, 32–41.Google Scholar - Cobb, P. (2000). Conducting teaching experience in collaboration with teachers. In A. E. Kelly & R. A. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 307–333). Mahwah, NJ: Erlbaum.Google Scholar - Cooney, T. J. (1994). Teacher education as an exercise in adaptation. In D. B. Aichele & A. F. Coxford (Eds.),
*Professional development for teachers of mathematics. 1994 yearbook*(pp. 9–22). Reston, VA: NCTM.Google Scholar - Cooney, T. J., & Krainer, K. (1996). Inservice mathematics teacher education: The importance of listening. In A. J. Bishop, et al. (Eds.),
*International handbook of mathematics education*(pp. 1155–1185). Dordrecht, The Netherlands: Kluwer.Google Scholar - Dewey, J. (1933).
*How we think: A statement of the relation of reflective thinking to the educative process*. Boston, MA: D.C. Heath and Co.Google Scholar - Ervynck, G. (1991). Mathematical creativity. In D. Tall (Ed.),
*Advanced mathematical thinking*(pp. 42–53). Dordrecht, The Netherlands: Kluwer.Google Scholar - Fennema, E., & Romberg, T. A. (Eds.) (1999).
*Classrooms that promote mathematical understanding*. Mahwah, NJ: Erlbaum.Google Scholar - Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.),
*Handbook of research on mathematics teaching and learning*(pp. 65–97). New York: Macmillan.Google Scholar - House, P.A., & Coxford, A. F. (1995).
*Connecting mathematics across the curriculum: 1995 Yearbook*. Reston, VA: NCTM.Google Scholar - Kennedy, M. M. (2002). Knowledge and teaching.
*Teacher and Teaching: Theory and Practice, 8*, 355–370.CrossRefGoogle Scholar - Krainer, K. (1993). Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction.
*Educational Studies in Mathematics, 24*, 65−93.CrossRefGoogle Scholar - Lampert, M., & Ball, D. L. (1999). Aligning teacher education with contemporary K-12 reform visions. In L. Darling-Hammond & G. Sykes (Eds.),
*Teaching as the learning profession: Handbook of policy and practice*(pp. 33–53). San Francisco: Jossey-Bass.Google Scholar - Lave, J. (1996). Teaching, as learning, in practice.
*Mind, Culture and Activity, 3*, 149–164.CrossRefGoogle Scholar - Lave, J., & Wenger, E. (1991).
*Situated learning: Legitimate peripheral participation.*Cambridge: Cambridge University Press.Google Scholar - Leikin, R. (2003). Problem-solving preferences of mathematics teachers.
*Journal of Mathematics Teacher Education, 6*, 297–329.CrossRefGoogle Scholar - Leikin, R. (2006). Learning by teaching: The case of the Sieve of Eratosthenes and one elementary school teacher. In R. Zazkis & S. Campbell (Eds.),
*Number theory in mathematics education: Perspectives and prospects*(pp. 115–140). Mahwah, NJ: Erlbaum.Google Scholar - Leikin, R., Levav-Vineberg, A., Gurevich, I., & Mednikov, L. (2006). Implementation of multiple solution connecting tasks: Do students’ attitudes support teachers’ reluctance?
*FOCUS on Learning Problems in Mathematics, 28*, 1–22.Google Scholar - Leinhardt, G. (1993). On teaching. In R. Glaser (Ed.),
*Advances in instructional psychology*(Vol. 4, pp. 1–54). Hillsdale, NJ: Erlbaum.Google Scholar - Ma, L. (1999).
*Knowing and teaching elementary mathematics: Teacher’s understanding of fundamental mathematics in China and the United States*. Hillsdale, NJ: Erlbaum.Google Scholar - National Council of Teachers of Mathematics (NCTM) (2000).
*Principles and standards for school mathematics*. Reston, VA: NCTM.Google Scholar - Neubrand, M. (2005). “COACTIV”: Cognitive activation in the classroom and the professional knowledge of teachers of Mathematics. Personal communication at
*ICMI-Study 15: The Professional Education and Development of Teachers of Mathematics*, Brazil, 15–21 May 2005.Google Scholar - Polya, G. (1973).
*How to solve it. A new aspect of mathematical method*. Princeton, NJ: Princeton University Press.Google Scholar - Rogoff, B. (1990).
*Apprenticeship in thinking: Cognitive development in social context.*New York: Oxford University Press.Google Scholar - Roth, W.-M. (1998).
*Designing communities.*Boston: Kluwer Academic Publication.Google Scholar - Scheffler, I. (1965).
*Conditions of knowledge. An introduction to epistemology and education*. Glenview, IL: Scott, Foresman.Google Scholar - Schoenfeld, A. H. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses.
*Educational Psychologist, 23*, 145–166.CrossRefGoogle Scholar - Schoenfeld, A. H. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J. Voss, D. N. Perkins, & J. Segal (Eds.),
*Informal reasoning and education*(pp. 311–343). Hillsdale, NJ: Erlbaum.Google Scholar - Shulman, L. S. (1986). Those who understand: Knowing growth in teaching.
*Educational Researcher, 5*(2), 4–14.Google Scholar - Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & Font Strawhun, B. T. (2005). Moving from rhetoric to praxis: Issues faced by teachers in having students consider multiple solutions for problems in the mathematics classroom.
*Journal of Mathematical Behavior, 24*, 287–301.CrossRefGoogle Scholar - Simon, A. M. (1997). Developing new models of mathematics teaching: An imperative for research on mathematics teacher development. In E. Fennema & B. Scott-Nelson (Eds.),
*Mathematics teachers in transition*(pp. 55–86). Mahwah, NJ: Erlbaum.Google Scholar - Skemp, R. R. (1987).
*The psychology of learning mathematics*. Hillsdale, NJ: Erlbaum.Google Scholar - Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers.
*Journal of Mathematics Teacher Education, 1*(2), 157–189.CrossRefGoogle Scholar - Stigler, J. W., & 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 - Tirosh, D., & Graeber, A. (2003). Challenging and changing mathematics teaching classroom practices. In A. J. Bishop, M. A. Clements, D. Brunei, C. Keitel, J. Kilpatrick, & F. K. S. Leung (Eds.),
*The second international handbook of mathematics education*(pp. 643–687). Dordrecht, The Netherlands: Kluwer.Google Scholar - Watson, A., & Mason, J. (2005).
*Mathematics as a constructive activity: Learners generating examples*. Mahwah, NJ: Erlbaum.Google Scholar - Wilson, S., Shulman, L., & Richert, A. E. (1987). “150” Different ways of knowledge in teaching. Representations of knowledge in teaching. In J. Calderhead (Ed.),
*Exploring teachers’ thinking*(pp. 1–37). London, UK: Cassell.Google Scholar - Yerushalmy, M., & Chazan, D. (2002). Flux in school algebra: Curricular change, graphing technology, and research on student learning and teacher knowledge. In L. English (Ed.),
*Handbook of international research in mathematics education*(pp. 725–755). New Jersey, NJ: Erlbaum.Google Scholar - Zazkis, R., & Leikin, R. (in press). Learner generated examples: From pedagogical to a research tool.
*For the Learning of Mathematics*.Google Scholar