A Mathematical problem

 Here we shall try to answer the following academic questions about a mathematical problem.

  1) What is a mathematical problem? 

 2) What type of mental activities are involved in the problem-solving process? 

3) Factors influencing success in problem solving.

 4) What makes a problem difficult?

 5) Polya's problem-solving techniques 

 6) How should learners be taught to be better problem solvers

1.          A mathematics problem
A mathematics problem is a mathematical task whose solution is not immediately available from a memorized procedure. It usually demands interpretation, selection of a method, and some degree of reasoning beyond routine computation.

a thinking process in which a solver tries to make sense of a problem situation using mathematical knowledge she/he has and attempts to obtain new information about that situation till she/he can ‘resolve the tension or ambiguity” Kazuhiko Nunokawa, 2005. A mathematical problem exists when the solver’s mathematical knowledge cannot be directly applied to the situation at hand, requiring transformation of the situation or finding new perspectives.

 

2.       Mental activities involved
The main mental activities are understanding the problem, planning, carrying out the plan, and checking the result. More finely, the literature describes these as cognitive, metacognitive, and affective activities: monitoring progress, judging whether a strategy is working, managing working-memory load, and controlling anxiety or frustration. Good problem solvers also notice structure and separate relevant from irrelevant information.

3.       Factors influencing success
Success in problem solving is influenced by prior knowledge, metacognition, working memory, mathematics anxiety, beliefs, and confidence. The RAMPS review treats these as interacting cognitive, metacognitive, and affective factors, while classroom studies found that beliefs about mathematics and problem solving are related to performance.

4.       What makes a problem difficult?
A problem becomes difficult when it is unfamiliar, non-routine, or poorly translated into mathematical form; when it hides the relevant structure; or when it requires linking several concepts before the answer can be checked. In the sources I found, the biggest errors cluster around understanding the problem and "looking back," and elementary students especially struggled with choosing formulas, connecting concepts, and checking answers.

5.       Pólya’s problem-solving techniques
Pólya’s core heuristic is: understand the problem, devise a plan, carry it out, and look back. In practice, that means encouraging learners to identify what is known and unknown, look for related problems, simplify, work backward when useful, and verify whether the answer makes sense.

6.       How learners should be taught to become better problem solvers
Learners should be taught problem solving explicitly, not left to pick it up from routine exercises. The studies and reviews here point to three classroom moves: model the thinking aloud, give non-routine tasks that require strategy choice, and require reflection, checking, and discussion of multiple strategies; intervention studies reported better performance when Pólya’s steps were taught deliberately and supported with activities.

References
Fajemidagba, M., & Olawoye, F. A. (2009). Effects of Polya and Schoenfeld problem-solving instructional strategies on students’ beliefs about mathematics and mathematical problem solving. Journal of Curriculum and Instruction.

Jacobbe, T. (2007). Connecting research to teaching: Using Pólya to overcome translation difficulties. Mathematics Teacher: Learning and Teaching PK-12. https://doi.org/10.5951/mt.101.5.0390

Kilpatrick, J. (1987). George Pólya's influence on mathematics education. Mathematics Magazine. https://doi.org/10.1080/0025570X.1987.11977328

Liljedahl, P., Santos-Trigo, M., Malaspina, U., & Bruder, R. (2016). Problem solving in mathematics education. Springer. https://doi.org/10.1007/978-3-319-40730-2

Logoglu, P. K., & Ücredi, L. (2017). The effect of mathematics teaching through Polya’s problem solving steps upon 4th grade students’ success of solving mathematic problems. European Journal of Education Studies. https://doi.org/10.46827/EJES.V0I0.992

Nurhalimah, A., Mandailina, V., Mahsup, & Syaharuddin. (2022). Measuring the difficulty level of mathematical problems based on Polya criteria. Journal of Education Research and Evaluation. https://doi.org/10.23887/jere.v6i4.46316

Nurkaeti, N. (2018). Polya’s strategy: An analysis of mathematical problem solving difficulty in 5th grade elementary school. EduHumaniora: Jurnal Pendidikan Dasar Kampus Cibiru. https://doi.org/10.17509/eh.v10i2.10868

Scheibe, D. A., Was, C. A., Dunlosky, J., & Thompson, C. A. (2023). Metacognitive cues, working memory, and math anxiety: The regulated attention in mathematical problem solving (RAMPS) framework. Journal of Intelligence, 11(6), 117. https://doi.org/10.3390/jintelligence11060117

Snyder, R. (1998). A clinical study of three high school problem solvers. The High School Journal.