Students’ Dichotomous Experiences of the Illuminating and Illusionary Nature of Pattern Recognition in Mathematics

Abstract

The concept of pattern recognition lies at the heart of numerous deliberations concerned with new mathematics curricula, because it is strongly linked to improved generalised thinking. However none of these discussions has made the deceptive nature of patterns an object of exploration and understanding. Yet there is evidence showing that pattern recognition has both positive and negative effects on learners’ development of concepts. This study investigated how pattern recognition was both illuminating and illusionary for Grade 11 learners as they factorised quadratic trinomials. Psillos's four-conditions model was used to judge the reasonableness of learners’ generalisations in six selected examples. The results show that pattern recognition was illuminating in the first three examples where learners made use of localised pattern recognition. In one example, pattern recognition was coincidental but not beneficial in terms of conceptual understanding. In the last two examples localised pattern recognition was at the centre of learner confusion as they failed to extend its application beyond the domain of the examples that generated the pattern. Learners’ confusion with pattern recognition could be attributed to teachers’ failure to meet four important conditions for good generalisations. Results from this study confirm earlier studies showing that abduced generalisations developed out of a few localised instances might be illuminating at first but might not provide the best explanation when extended beyond the localised domain. Further studies are needed that assist in developing pattern-aware teachers.

Keywords:

• Mathematical patterns
• trinomials
• abduction
• generalisation
• factorisation