# Interesting and difficult mathematical problems: changing teachers’ views by employing multiple-solution tasks

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## Abstract

The study considers mathematical problem solving to be at the heart of mathematics teaching and learning, while mathematical challenge is a core element of any educational process. The study design addresses the complexity of teachers’ knowledge. It is aimed at exploring the development of teachers’ mathematical and pedagogical conceptions associated with systematic employment of multiple-solution tasks (MSTs) in a “problem-solving” course for prospective mathematics teachers (PMTs). Our attention to teachers’ mathematical conceptions focused on the development of PMTs’ problem-solving competences. Our attention to teachers’ meta-mathematical and pedagogical conceptions focused on changes in teachers’ views concerning the level of interest and level of difficulty of the mathematical tasks. We differentiated between the systematic and craft modes of professional development integrated in the course. Systematic mode involved problem-solving sessions and reflective discussions on collective solution spaces. Craft mode involved interviewing school students. The study demonstrates the effectiveness of MSTs for PMTs’ professional development.

## Keywords

Multiple-solution tasks (MSTs) Problem solving Problem difficulty Problem interest Teacher’s professional development Prospective mathematics teachers (PMTs)## References

- Abrahamson, D., & Cigan, C. (2003). A design for ratio and proportion.
*Mathematics Teaching in the Middle School,**8*(9), 493–501.Google Scholar - Arcavi, A., & Friedlander, A. (2007). Curriculum developers and problem solving: the case of Israeli elementary school projects.
*Zentralblat für Didaktik de Mathematik,**39*(5–6), 355–364.CrossRefGoogle Scholar - Ball, D. (1992). Teaching mathematics for understanding: What do teachers need to know about the subject matter? In M. Kennedy (Ed.),
*Teaching academic subjects to diverse learners*(pp. 63–83). New York: Teaching College Press.Google Scholar - Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?
*Journal of Teacher Education,**59*(5), 389–407.CrossRefGoogle Scholar - Behr, M., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio and proportion. In D. Grouws (Ed.),
*Handbook on research of teaching and learning*(pp. 296–333). New York: McMillan.Google Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics*. Dordrecht, The Netherlands: Kluwer.Google Scholar - Callejo, M. L., & Vila, A. (2009). Approach to mathematical problem solving and students’ belief systems: two case studies.
*Educational Studies in Mathematics,**72*(1), 111–126.CrossRefGoogle Scholar - Charles, R., & Lester, F. (1982).
*Teaching problem solving: What, why and how*. Palo Alto, CA: Dale Seymour Publications.Google Scholar - Clandinin, D. J., & Connelly, F. M. (1996). Teachers’ professional knowledge landscapes: Teacher stories/stories of teachers/school stories/stories of school.
*Educational Researcher,**25*(3), 24–30.Google Scholar - Cooney, T. J., & Krainer, K. (1996). Inservice mathematics teacher education: The importance of listening. In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & C. Laborde (Eds.),
*International handbook of mathematics education*(pp. 1155–1185). Dordrecht. The Netherlands: Kluwer.Google Scholar - Davidson, J. E., & Sternberg, R. J. (2003).
*The psychology of problem solving*. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar - Davydov, V. V. (1996).
*Theory of developing education*. Intor (In Russian): Moscow.Google Scholar - De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Improper use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students’ errors.
*Educational Studies in Mathematics,**50*(3), 311–334.CrossRefGoogle Scholar - English, L., & Sriraman, B. (2010) Problem solving for the 21st Century. In Sriraman, B., English, L. (Eds.) Theories of mathematics education: Seeking new frontiers. monograph 1 of advances in mathematics education (pp 263–290). Springer: Berlin.Google Scholar
- Goldenberg, E. P. (1996). ‘Habits of mind’ as an organizer for the curriculum.
*Journal of Education,**178*, 13–34.Google Scholar - Goldin, G. A. (2009). The affective domain and students’ mathematical inventiveness. In R. Leikin, A. Berman, & B. Koichu (Eds.),
*Creativity in mathematics and the education of gifted students*(pp. 181–194). Rotterdam: Sense Publishers.Google Scholar - Jaworski, B. (1992). Mathematical teaching: What is it?
*For the Learning of Mathematics,**12*(1), 8–14.Google Scholar - Jaworski, B. (1994).
*Investigating mathematics teaching: A constructivist inquiry*. London: Falmer.Google Scholar - Kennedy, M. M. (2002). Knowledge and teaching.
*Teacher and Teaching: Theory and practice,**8*(3), 355–370.CrossRefGoogle Scholar - Kilpatrick, J. (1985). A retrospective account of the past twenty-five years of research on teaching mathematical problem solving. In E. A. Silver (Ed.),
*Teaching and learning mathematical problem solving: Multiple research perspectives*(pp. 1–15). Hillsdale, NJ: Erlbaum.Google Scholar - Krainer, K. (1993). Powerful tasks: A contribution to a high level of acting and reflecting in mathematics instruction.
*Educational Studies in Mathematics,**24*(1), 65–93.CrossRefGoogle Scholar - Lampert, M. (2001).
*Teaching problems and the problems in teaching*. New Haven, CT: Yale University Press.Google Scholar - Leder, G. C., Pehkonen, E., & Torner, G. (2002).
*Beliefs: A Hidden Variable in Mathematics Education?*Dordrecht, the Netherlands: Kluwer.Google Scholar - Leikin, R. (2004). Towards high quality geometrical tasks: Reformulation of a proof problem. In Hoines M. J., & Fuglestad, A. B. (Eds.).
*Proceedings of the 28th international conference for the psychology of mathematics education*(Vol. 3, pp. 209–216).Google Scholar - Leikin, R. (2006a). Learning by teaching: The case of 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. (2006b). Getting Aware: Secondary school teachers learn about ‘Mathematics Challenges in Education’. The paper presented at the 16th
*ICMI Study:**Mathematical challenges in and out of the classroom*. http://www.amt.canberra.edu.au/icmis16pisrleikin.pdf. - Leikin, R. (2007). Habits of mind associated with advanced mathematical thinking and solution spaces of mathematical tasks. In
*The**Fifth Conference of the European Society for Research in Mathematics Education*—*CERME*-*5*(pp. 2330–2339). (CD-ROM and On-line). Available: http://ermeweb.free.fr/Cerme5.pdf. - Leikin, R. (2008). Teams of prospective mathematics teachers: Multiple problems and multiple solutions. In Wood, T. (Series Editor), & K. Krainer (Volume Editor),
*International handbook of mathematics teacher education: Vol. 3. Participants in mathematics teacher education: individuals, teams, communities, and networks*(pp. 63–88). Rotterdam, The Netherlands: Sense Publishers.Google Scholar - Leikin, R. (2009). Bridging research and theory in mathematics education with research and theory in creativity and giftedness. In Leikin, R., Berman, A., & Koichu, B. (Eds.).
*Creativity in mathematics and the education of gifted students.*(Part IV—synthesis, Ch. 23, pp. 385–411). Rotterdam, The Netherlands: Sense Publisher.Google Scholar - Leikin, R., & Dinur, S. (2007). Teacher flexibility in mathematical discussion.
*Journal of Mathematical Behavior,**26*, 328–347.CrossRefGoogle Scholar - Leikin, R., & Levav-Waynberg, A. (2007). Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks.
*Educational Studies in Mathematics,**66*, 349–371.CrossRefGoogle Scholar - Leikin, R., & Levav-Waynberg, A. (2008). Solution spaces of multiple-solution connecting tasks as a mirror of the development of mathematics teachers’ knowledge.
*Canadian Journal of Science, Mathematics and Technology Education,**8*(3), 233–251.CrossRefGoogle Scholar - Leikin, R., & Levav-Waynberg, A. (2009). Development of teachers’ conceptions through learning and teaching: Meaning and potential of multiple-solution tasks.
*Canadian Journal of Science, Mathematics and Technology Education,**9*(4), 203–223.CrossRefGoogle Scholar - Lesh, R., & Zawojewski, J. S. (2007). Problem solving and modeling. In F. Lester (Ed.),
*Second handbook of research on mathematics teaching and learning*(pp. 763–804). Charlotte, NC: Information Age Publishing.Google Scholar - Levav-Waynberg A, Leikin R (2009). Multiple solutions to a problem: A tool for assessment of mathematical thinking in geometry. In Durand-Guerrier, V., Soury-Lavergne, S., & F. Arzarello (Eds.).
*Proceedings of the Sixth Conference of the European Society for Research in Mathematics Education*—*CERME*-*6*(pp. 776–785). http://ife.ens-lyon.fr/editions/editions-electroniques/cerme6/working-group-5. - Mason, J. (2010). Attention and intention in learning about teaching through teaching. In R. Leikin & R. Zazkis (Eds.),
*Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice*(pp. 23–47). Berlin: Springer.Google Scholar - National Council of Teachers of Mathematics - NCTM (2000).
*Principles and standards for school mathematics*. Reston, VA: NCTM.Google Scholar - Pajares, F. (1992). Teachers’ beliefs and educational research: Cleaning up a messy construct.
*Review of Educational Research,**62*(3), 307–332.Google Scholar - Polya, G. (1973).
*How to solve it: A new aspect of mathematics method*. Princeton, NJ: Princeton University Press.Google Scholar - Polya, G. (1981).
*Mathematical discovery: On understanding, learning, and teaching problem solving*. New York: Wiley.Google Scholar - Ponte, J. P., & Chapman, O. (2006). Mathematics teachers’ knowledge and practices. In A. Gutierrez & P. Boero (Eds.),
*Handbook of research on the psychology of mathematics education: Past, present and future*(pp. 461–494). Rotterdam, The Netherlands: Sense Publishers.Google Scholar - Scheffler, I. (1965).
*Conditions of knowledge. An introduction to epistemology and education*. Glenview, IL: Scott, Foresman and Company.Google Scholar - Schoenfeld, A. H. (1983).
*Problem solving in the mathematics curriculum: A report, recommendations, and an annotated bibliography*. Washington, DC: The Mathematical Association of America.Google Scholar - Schoenfeld, A. H. (1985).
*Mathematical problem solving*. Orlando, Fla.: Academic Press.Google Scholar - Shulman, L. S. (1986). Those who understand: Knowing growth in teaching.
*Educational Researcher,**15*(2), 4–14.Google Scholar - Silver, E. A. (1987). Foundations of cognitive theory and research for mathematics problem solving instruction. In A. H. Schoenfeld (Ed.),
*Cognitive science and mathematics education*(pp. 33–60). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Silver, E. A. (1997). Fostering creativity through instruction rich in mathematical problem solving and problem posing.
*ZDM,**3*, 75–80.CrossRefGoogle Scholar - Silver, E. A., Ghousseini, H., Gosen, D., Charalambous, C., & 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*(3–4), 287–301.CrossRefGoogle Scholar - Silver, E. A., & Marshall, S. P. (1990). Mathematical and scientific problem solving: Findings, issues, and instructional implications. In B. F. Jones & L. Idol (Eds.),
*Dimensions of thinking and cognitive instruction*(pp. 265–290). Hillsdale, NJ: Lawrence Erlbaum Associates.Google 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: Lawrence Erlbaum Associates.Google Scholar - Star, J. R., & Newton, K. J. (2009). The nature and development of experts’ strategy flexibility for solving equations.
*ZDM,**41*, 557–567.CrossRefGoogle Scholar - Steinbring, H. (1998). Elements of epistemological knowledge for mathematics teachers.
*Journal of Mathematics Teacher Education,**1*(2), 157–189.CrossRefGoogle Scholar - Taylor, P. (2006) Challenging mathematics and its role in the learning process,
*ICMI Study 16: Challenging mathematics in and beyond the classroom, Trondheim, Norway.*Retrieved from: http://www.amt.edu.au/icmis16ptaylor.pdf. - Thompson, A. G. (1985). Teacher’s conceptions of mathematics and the teaching of problem solving. In E. A. Silver (Ed.),
*Teaching and learning mathematical problem solving: Multiple research perspectives*(pp. 281–294). Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar - Thompson, A. G. (1992). Teachers’ beliefs and conceptions: A synthesis of the research. In D. A. Grouws (Ed.),
*Handbook for research on mathematics teaching and learning*(pp. 127–146). New York: McMillan.Google Scholar - Tzur, R. (2010). How and what might teachers learn through teaching mathematics: Contributions to closing an unspoken gap. In R. Leikin & R. Zazkis (Eds.),
*Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice*(pp. 49–67). Berlin: Springer.Google Scholar