Abstract
In this article, we question the prevalent assumption that teaching and learning mathematics should always entail movement from the concrete to the abstract. Such a view leads to reported difficulties in students moving from manipulatives and models to more symbolic work, moves that many students never make, with all the implications this has for life chances. We propose working in ‘symbolically structured environments’ as an alternative way of characterising students’ direct engagement with the abstract and exemplify two such environments, both of which involve early number learning. We additionally propose some roles for the teacher working in a symbolically structured environment.
Résumé
Dans cet article, nous remettons en question l’hypothèse, la plus répandue, selon laquelle l’enseignement et l’apprentissage des mathématiques doit toujours impliquer un passage du concret à l’abstrait. Une telle optique conduit à certaines difficultés signalées chez de nombreux étudiants qui doivent passer des manipulations et des modèles à un travail plus axé sur les symboles, passage que beaucoup d’entre eux n’arrivent jamais à faire, avec tout ce que cela implique pour leurs perspectives de réussite. Nous proposons le travail dans des « environnements structurés symboliquement » comme une façon différente de caractériser l’engagement direct des étudiants dans le raisonnement abstrait, et nous présentons deux environnements de ce type qui impliquent tous deux l’apprentissage précoce des nombres. Nous proposons en outre certains rôles pour les enseignants qui œuvrent dans un environnement structuré symboliquement.
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Notes
This resonates with how the mathematician Jacques Hadamard is said to have opined, the concrete is the abstract made familiar by time.
Thus using what Fowler (1979) calls “ratio numbers”.
While the terms generalisation and abstraction are often used as synonyms of sorts, our characterization of the abstract as attending to relations clearly distinguishes these two types of activity.
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Coles, A., Sinclair, N. Re-thinking ‘Concrete to Abstract’ in Mathematics Education: Towards the Use of Symbolically Structured Environments. Can. J. Sci. Math. Techn. Educ. 19, 465–480 (2019). https://doi.org/10.1007/s42330-019-00068-4
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DOI: https://doi.org/10.1007/s42330-019-00068-4