The complex system of problem solving - providing the conditions to develop proficiency

Problem solving in mathematics has been a central topic of research in mathematics education, stemming mainly from Pólya’s (1945) work on problem solving heuristics (Lesh and Zawojewski, 2007). Despite the attention this topic has received, the research conducted in problem solving has not easily translated into improving school practice (Lester and Kehle, 2003). Foster et al. (2014) found that the teachers’ knowledge and ability to employ “effective strategies for teaching problem solving processes was particularly underdeveloped” (p.7). Felmer and Diaz (2016, p.289) also noted that there is little research on “teachers as problem solvers”, particularly at second level. Evidence on pre-service secondary teachers’ performance on non-routine problem solving tasks and experience with solving such problems led Felmer and Diaz (2016) to recommend that action be taken on pre-service mathematics teacher education courses. The aim of this research was to develop a framework for teaching and assessing problem solving in mathematics and to implement this framework in the form of a teaching intervention with a sample of pre-service secondary level mathematics teachers. The author developed a Framework for Teaching and Assessing Problem Solving (F-TAPS) in mathematics, consistent with findings on significant issues identified from the review of literature. The author’s F-TAPS in mathematics integrates significant knowledge and affective factors with mathematical thinking in order to develop proficiency in problem solving in mathematics through the coherent construction of meaningful knowledge. The author integrated her F-TAPS in mathematics with the Modified Moore Method using the Modified 4C-ID model to design and implement an educational intervention in problem solving. This intervention was implemented with sixteen secondary pre-service mathematics teachers. Two assessments in problem solving were developed to assess problem solving ability before and after the intervention. The mindset (growth/fixed) of the pre-service teachers who participated in this study was also evaluated before and after the intervention. Educational Design Research methodology which incorporates a proof of concept approach underpinned this research. Pre and post-test findings showed statistically significant increases in the growth mindset and problem solving ability of the pre-service teachers who participated in this study. The intervention (and the integrated F-TAPS) had a moderate effect on the mindset of the pre-service teachers and a strong effect on the problem solving ability of the pre-service teachers. The proof of concept approach provides evidence that the author’s F-TAPS could contribute to improved teaching of problem solving in mathematics at secondary level.