LFCS Seminar: 16 November 2021 - Ian Benson

Title: Early Algebra, Domain-Specific Languages and “Cryptomorphism”.

Abstract:

Recent educational research has highlighted the contribution that early algebra and reasoning about equivalence can make to school mathematics. As a result there is an opportunity for conceptual mathematics and functional programming to play a part in in-service teacher training. However, these subjects have a reputation for being difficult to learn and teach. In this seminar we will demonstrate classroom evidence that this need not be the case. We will build on Gattegno Mathematics early algebra text-books and the Cuisenaire staircase to explore key computational concepts such as function, algebraic data type, pattern matching and “cryptomorphism". 

The Cuisenaire-Gattegno (Cui) approach to early algebra and equivalence uses colour coded rods of unit increment lengths embedded in a systematic curriculum designed to guide learners as young as age five from exploration of ratio through to formal algebraic writing. The term "cryptomorphism" describes an equivalence between objects in a category that is not obvious. I wish to suggest that there is a functorial relationship between, on the one hand, cryptomorphisms in Haskell as a category, and on the other hand, the experience of number systems gained by Cuisenaire rod construction and the structured drawing and writings that Gattegno suggested could serve as metaphors for algebraic operation. This relationship can provide the basis for a reformed school mathematics curriculum as envisaged by Davydov, Gattegno and their colleagues.

A pre-print of paper submitted for publication that describes our longitudinal evaluation of Cui can be found at:  https://psyarxiv.com/zasb9/

Affiliation: Dr Ian Benson School of Education, University of Roehampton. tizard.stanford.edu