Gaining Mathematical Understanding: The Effects of Creative Mathematical Reasoning and Cognitive Proficiency

Jonsson, Bert; Granberg, Carina; Lithner, Johan · 2020 · Crossref

DOI: 10.3389/fpsyg.2020.574366

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Summary

This study investigates whether students develop deeper mathematical understanding through Creative Mathematical Reasoning (CMR), where they must construct their own solution methods, or through Algorithmic Reasoning (AR), where they follow predetermined instructions. The research addresses a longstanding debate in mathematics education regarding whether "productive struggle" facilitates conceptual learning more effectively than rote application of algorithms. The authors hypothesized that CMR would yield superior performance on both practiced tasks and transfer tasks, and that cognitive proficiency would influence learning outcomes. The researchers conducted three experiments involving 270 upper secondary school students in Sweden. Participants were assigned to either AR or CMR practice conditions using computer-based tasks involving pattern recognition and formula construction. AR tasks provided explicit formulas and examples, whereas CMR tasks required students to derive the methods independently. To assess conceptual understanding, the study utilized both practiced test tasks (similar to practice items) and transfer test tasks (requiring new reasoning sequences). The study also measured individual differences in cognitive proficiency, combining scores from a working memory task (operation span) and a fluid intelligence test (Raven’s Advanced Progressive Matrices). Students were further categorized by their mathematics track (basic vs. advanced) to account for prior knowledge differences. The results demonstrated that practicing with CMR tasks was significantly superior to practicing with AR tasks for both practiced and transfer test performance. This indicates that the effortful struggle inherent in CMR facilitates better retention and the ability to apply reasoning to novel problems. Cognitive proficiency was found to have a significant effect on learning outcomes for both CMR and AR conditions, with more cognitively proficient students outperforming their less proficient peers. However, the students' mathematics track (basic versus advanced) showed no significant effect on performance, suggesting that the intervention's benefits were not dependent on prior curriculum level. The findings support the theoretical framework that mechanical repetition and algorithmic instruction can hinder the development of conceptual understanding. Instead, the study concludes that engaging students in activities that require them to construct their own mathematical methods is essential for fostering deep comprehension. The results imply that educational practices should prioritize tasks that induce productive struggle over those that provide step-by-step solutions, as the former better supports the transfer of mathematical reasoning to new contexts.

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• mental demand