# The development of a culture of problem solving with secondary students through heuristic strategies

- 737 Downloads
- 4 Citations

## Abstract

The article reports the results of a longitudinal research study conducted in three mathematics classes in Czech schools with 62 pupils aged 12–18 years. The pupils were exposed to the use of selected heuristic strategies in mathematical problem solving for a period of 16 months. This was done through solving problems where the solution was the most efficient if heuristic strategies were used. The authors conducted a two-dimensional classification of the use of heuristic strategies based on the work of Pólya (2004) and Schoenfeld (1985). We developed a tool that allows for the description of a pupil’s ability to solve problems. Named, the Culture of Problem Solving (CPS), this tool consists of four components: intelligence, text comprehension, creativity and the ability to use existing knowledge. The pupils’ success rate in problem solving and the changes in some of the CPS factors pre- and post-experiment were monitored. The pupils appeared to considerably improve in the creativity component. In addition, the results indicate a positive change in the students’ attitude to problem solving. As far as the teachers participating in the experiment are concerned, a significant change was in their teaching style to a more constructivist, inquiry-based approach, as well as their willingness to accept a student’s non-standard approach to solving a problem. Another important outcome of the research was the identification of the heuristic strategies that can be taught via long-term guided solutions of suitable problems and those that cannot. Those that can be taught include systematic experimentation, guess–check–revise and introduction of an auxiliary element. Those that cannot be taught (or can only be taught with difficulty) include the strategies of specification and generalization and analogy.

## Keywords

Problem solving Heuristic strategies Culture of problem solving Intelligence Creativity## Notes

### Acknowledgments

The research was supported by Czech Science Foundation project P407/12/1939.

## References

- Arslan, Ç., & Altun, M. (2007). Learning to solve non-routine mathematical problems.
*Elementary Education Online, 6*(1), 50–61.Google Scholar - Australian Curriculum and Assessment Reporting Authority [ACARA]. (2014). Foundation to year 10 curriculum: mathematics. Retrieved from http://www.australiancurriculum.edu.au/mathematics/curriculum/f-10?layout=1.
- Babakhani, N. (2011). The effect of teaching the cognitive and meta-cognitive strategies (self-instruction procedure) on verbal math problem-solving performance of primary school students with verbal problem-solving difficulties.
*Procedia - Social and Behavioral Sciences, 15*, 563–570.CrossRefGoogle Scholar - Bahar, A. L., & Maker, C. J. (2011). Exploring the relationship between mathematical creativity and mathematical achievement.
*Asia-Pacific Journal of Gifted and Talented Education, 3*(1), 33–48.Google Scholar - Brousseau, G. (1997).
*Theory of didactical situations in mathematics*. Dordrecht: Kluwer.Google Scholar - Calder, N., Brown, T., Hanley, U., & Darby, S. (2006). Forming conjectures within a spreadsheet environment.
*Mathematics Education Research Journal, 18*(3), 100–116.CrossRefGoogle Scholar - Chamberlin, S. A., & Moon, S. M. (2005). Model-eliciting activities as a tool to develop and identify creatively gifted mathematicians.
*Journal of Secondary Gifted Education, 17*(1), 37–47.Google Scholar - Clarke, D., Goos, M., & Morony, W. (2007). Problem solving and working mathematically: an Australian perspective.
*ZDM Mathematics Education, 39*, 475–490.CrossRefGoogle Scholar - Eisenmann, P., & Přibyl, J. (2014). Properties of problem solving strategies. In M. Houška, I. Krejčí, & M. Flégl (Eds.),
*Proceedings of efficiency and responsibility in education 2014*(pp. 623–630). Prague: Czech University of Life Sciences.Google Scholar - Eisenmann, P., Novotná, J., & Přibyl, J. (2014). Culture of solving problems – one approach to assessing pupils’ culture of mathematics problem solving. In D. Velichová (Ed.),
*Aplimat 2014 – 13th conference on applied mathematics*(pp. 115–122). Bratislava: Publishing House of STU.Google Scholar - Eisenmann, P., Novotná, J., & Přibyl, J. (2015). The heuristic strategy introduction of an auxiliary element. In D. Szarková, D. Richtáriková, & Ľ. Balko (Eds.),
*Proceedings of 14th conference on applied mathematics (APLIMAT 2015)*(pp. 232–245). Bratislava: Slovak University of Technology in Bratislava.Google Scholar - Elia, I., Van den Heuvel-Panhuizen, M., & Kolovou, A. (2009). Exploring strategy use and strategy flexibility in non-routine problem solving by primary school high achievers in mathematics.
*ZDM Mathematics Education, 41*(5), 605–618.CrossRefGoogle Scholar - Fan, L., & Zhu, Y. (2007). Representation of problem-solving procedures: a comparative look at China, Singapore, and US mathematics textbooks.
*Educational Studies in Mathematics, 66*(1), 61–75.CrossRefGoogle Scholar - Getzels, J. W., & Jackson, P. W. (1962).
*Creativity and intelligence: explorations with gifted students*. New York: John Wiley.Google Scholar - Haspekian, M. (2014). Teachers’ instrumental geneses when integrating spreadsheet software. In A. Clark-Wilson, O. Robutti, & N. Sinclair (Eds.),
*The mathematics teacher in the digital era: an international perspective on technology focused professional development*(pp. 241–275). Dordrecht: Springer.CrossRefGoogle Scholar - Hensberry, K. K. R., & Jacobbe, T. (2012). The effects of Polya’s heuristic and diary writing on children’s problem solving.
*Mathematics Education Research Journal, 24*(1), 59–85.CrossRefGoogle Scholar - Herl, H. E., O’Neil, H. F., Jr., Chung, K. W. K., Bianchi, C., Wang, S., Mayer, R., & Tu, A. (1999).
*Final report for validation of problem-solving measures: CSE technical report 501*. Los Angeles: National Center for Research on Evaluation, Standards, and Student Testing.Google Scholar - Higgins, K. M. (1997). The effect of year-long instruction in mathematical problem solving on middle-school students’ attitudes, beliefs, and abilities.
*The Journal of Experimental Education, 66*(1), 5–28.CrossRefGoogle Scholar - Hite, S. (2009).
*Improving problem solving by improving reading skills*. Grant: University of Nebraska-Lincoln. Summative Projects for MA Degree.Google Scholar - Hrabal, V., & Hrabal, V. m. (2002).
*Pedagogickopsychologická diagnostika žáka s úvodem do diagnostické aplikace statistiky*. Praha: Karolinum.Google Scholar - Jacobse, A. (2012).
*Can we improve children’s thinking? A metacognitive approach to problem solving in mathematics*. Groningen: GION.Google Scholar - Jeřábek, J., Lisnerová, R., Smejkalová, A., & Tupý, J. (Eds.). (2013).
*Rámcový vzdělávací program pro základní vzdělávání: (verze platná od 1. 9. 2013) úplné znění upraveného RVP ZV*. Praha: MŠMT.Google Scholar - Jonassen, D. H. (2011).
*Learning to solve problems: a handbook for designing problem-solving learning environments*. New York: Routledge.Google Scholar - Kline, P. (2000).
*The handbook of psychological testing*(2nd ed.). London and New York: Routledge.Google Scholar - Larson, L. C. (1983).
*Problem-solving through problems*. New York: Springer.CrossRefGoogle Scholar - Lester, F. K., Garofalo, J., & Kroll, D. L. (1989).
*The role of metacognition in mathematical problem solving: a study of two grade seven classes. Final report*. Bloomington: School of Education.Google Scholar - Lubart, T. I. (1994). Creativity. In R. J. Sternberg (Ed.),
*Thinking and problem solving*(pp. 290–323). San Diego: Academic.Google Scholar - Martinsen, Ø., & Kaufmann, G. (1991). Effect of imagery, strategy and individual differences in solving insight problems.
*Scandinavian Journal of Educational Research, 35*(1), 69–76.CrossRefGoogle Scholar - McLeod, D. B. (1989). The role of affect in mathematical problem solving. In D. B. McLeod & V. M. Adams (Eds.),
*Affect and mathematical problem solving: a new perspective*(pp. 20–36). Springer-Verlag: New York.CrossRefGoogle Scholar - Meador, K. S. (1997).
*Creative thinking and problem solving for young learners*. Englewood: Teacher Ideas Press.Google Scholar - Meier, S. L. (1992). Evaluating problem-solving processes.
*Mathematics Teacher, 85*(8), 664–666.Google Scholar - Michalewitz, Z., & Fogel, D. B. (2000).
*How to solve it: modern heuristics*. Berlin: Springer.CrossRefGoogle Scholar - Nadjafikhah, M., Yaftian, N., & Bakhshalizadeh, S. (2012). Mathematical creativity: some definitions and characteristics.
*Procedia - Social and Behavioral Sciences, 31*, 285–291.CrossRefGoogle Scholar - National Council of Teachers of Mathematics. (2000).
*Principles and standards for school mathematics*. Reston: The National Council of Teachers of Mathematics, Inc.Google Scholar - Nelsen, R. (1993).
*Proofs without words: exercises in visual thinking*. Washington: Mathematical Association of America.Google Scholar - Nelsen, R. (2000).
*Proofs without words II: more exercises in visual thinking*. Washington: Mathematical Association of America.Google Scholar - Nováková, H., & Novotná, J. (2015). A priori analysis and its role (2015). In G. Aldon, B. Di Paola, & C. Fazio (Eds.),
*Proceedings CIEAEM 66: mathematics and realities*(pp. 165–172). Palermo: University of Palermo.Google Scholar - Novotná, J., Eisenmann, P., & Přibyl, J. (2014). Impact of heuristic strategies on pupils’ attitudes to problem solving. In M. Houška, I. Krejčí, & M. Flégl (Eds.),
*Proceedings of efficiency and responsibility in education 2014*(pp. 514–520). Prague: Czech University of Life Sciences.Google Scholar - Novotná, J., Eisenmann, P., & Přibyl, J. (2015a). Analogy – a friend or fiend when solving math problems? In D. Tan (Ed.),
*Engineering technology, engineering education and engineering management: Proceedings of the International Conference on Engineering Technologies, Engineering Education and Engineering Management (ETEEEM 2014), Hong Kong, 15–16 November 2014*(pp. 103–108). Boca Raton: CRC Press.Google Scholar - Novotná, J., Eisenmann, P., & Přibyl, J. (2015b). Impact of heuristic strategies on pupils’ attitude to problem solving.
*Journal on Efficiency and Responsibility in Education and Science, 8*(1), 15–23.CrossRefGoogle Scholar - Pajares, F., & Kranzler, J. (1995). Self-efficacy beliefs and general mental ability in mathematical problem-solving.
*Contemporary Educational Psychology, 20*(4), 426–443.CrossRefGoogle Scholar - Pape, S. J. (2004). Middle school children’s problem-solving behavior: a cognitive analysis from a reading comprehension perspective.
*Journal for Research in Mathematics Education, 35*(3), 187–219.CrossRefGoogle Scholar - Pólya, G. (2004).
*How to solve it: a new aspect of mathematical method (Expanded Princeton Science Library ed.)*. Princeton: Princeton University Press.Google Scholar - Reiss, K., & Törner, G. (2007). Problem solving in the mathematics classroom: the German perspective.
*ZDM Mathematics Education, 39*, 431–441.CrossRefGoogle Scholar - Sak, U., & Maker, C. J. (2005). Divergence and convergence of mental forces of children in open and closed mathematical problems.
*International Education Journal, 6*(2), 252–260.Google Scholar - Schoenfeld, A. H. (1982). Measures of problem-solving performance and of problem-solving instruction.
*Journal for Research in Mathematics Education, 13*(1), 31–49.CrossRefGoogle Scholar - Schoenfeld, A. H. (1985).
*Mathematical problem solving*. London: Academic Press Inc. (London) Ltd.Google Scholar - 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. 334–370). New York: Macmillan.Google Scholar - Silver, E. A. (1985).
*Teaching and learning mathematical problem solving: multiple research perspectives*. Hillsdale: Lawrence Erlbaum Associates, Inc., Publishers.Google Scholar - Simon, H. A., & Newell, A. (1971). Human problem solving: the state of the theory in 1970.
*American Psychologist, 22*(6), 145–159.CrossRefGoogle Scholar - Sriraman, B. (2005). Are giftedness and creativity synonyms in mathematics?
*Journal of Secondary Gifted Education, 17*(1), 20–36.Google Scholar - Sternberg, R. J. (2005). Creativity or creativities?
*International Journal of Human-Computer Studies, 63*(4–5), 370–382.CrossRefGoogle Scholar - Sullivan, P., & McDonough, A. (2007). Eliciting positive student motivation for learning mathematics. In J. Watson & K. Beswick (Eds.),
*Mathematics: essential research, essential practice*(Vol. 1, pp. 698–707). Adelaide: MERGA, Inc.Google Scholar - Sullivan, P., Mousley, J., & Zevenbergen, R. (2003). The contexts of mathematics tasks and the context of the classroom: are we including all students?
*Mathematics Education Research Journal, 15*(2), 107–121.CrossRefGoogle Scholar - Szetela, W. (1987). The problem of evaluation in problem solving: can we find solutions?
*Arithmetic Teacher, 35*(November), 36–41.Google Scholar - Szetela, W., & Nicol, C. (1992). Evaluating problem solving in mathematics.
*Educational Leadership, 49*(8), 42–45.Google Scholar - Tiong, J. Y. S., Hedberg, J. G., & Lioe, L. T. (2005). A metacognitive approach to support heuristic solution of mathematical problems.
*Proceedings of the Redesigning Pedagogy: Research, Policy, Practice Conference*. Singapore: NIE. Retrieved from https://repository.nie.edu.sg/bitstream/10497/141/1/2005a15.pdf. - Wenke, D., Frensch, P. A., & Funke, J. (2005). Complex problem solving and intelligence. In R. J. Sternberg & J. E. Pretz (Eds.),
*Cognition and intelligence: identifying the mechanisms of the mind*(pp. 160–187). Cambridge: Cambridge University Press.Google Scholar - Wittmann, E. C. (1995). Mathematics education as a “design science”.
*Educational Studies in Mathematics, 29*(4), 355–374.CrossRefGoogle Scholar - Wu, M., & Adams, R. (2006). Modelling mathematics problem solving item responses using a multidimensional IRT model.
*Mathematics Education Research Journal, 18*(2), 93–113.CrossRefGoogle Scholar - Yimer, A., & Ellerton, N. F. (2006). Cognitive and metacognitive aspects of mathematical problem solving: An emerging model. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.),
*Identities, cultures and learning spaces: conference proceedings 2006*(pp. 575–582). Canbera: Merga.Google Scholar - Zanzali, N. A. A., & Nam, L. L. (2000). Evaluating the levels of problem solving abilities in mathematics. In A. Rogerson (Ed.),
*International conference on mathematics education into the 21st century: mathematics for living*. Third World Forum: Amman. Retrieved from http://math.unipa.it/~grim/Jzanzalinam.PDF.Google Scholar