Scheming and Re-scheming: Secondary Mathematics Teachers’ Use and Re-use of Resources

In this article, we examine secondary mathematics teachers’ work with resources using the Documentational Approach to Didactics lens. Specifically, we look at the resources and a teacher’s scheme of use (aims, rules of actions, operational invariants, and inferences) of these resources across a set of lessons (macro-level analysis) that aim towards students’ preparation for the examinations and how this use emerges in a set of three lessons on the same topic (micro-level analysis) as a response to contingent moments. We propose the terms scheming—a teacher’s emerging scheme of use related to the same set of resources used for the same aim—and re-scheming, namely, shifts in such scheming. Our analysis of lesson observations and the teacher’s reflections on his actions from a post-observation interview demonstrate the interplay between the stable characteristics of the scheme of use and the scheming and re-scheming in individual lessons. We conclude this article with a discussion on the methodological potential of using both macro- and micro-level analyses in the investigation of teachers’ use of resources.

Preliminary note on the operational invariants of a binary quantic

My object is to present a theory of operational invariants for the binary quantic of order n . Many of the results that are here given have been explicitly or implicitly given in the works of Cayley, Clebsch, Gordan, and other writers, but not from the present point of view or in the convenient notation that will be adopted. Some quite new results are reached, which will be found to throw much light upon the work of others. 1. In the algebraic theory we take the quantic to be a n x = ( a 1 x 1 + a 2 x 2 ) n , and b, c, d , . . . as alternative symbolic letters to a . We then have symbolic factors of two types, viz., the suffix type, a x , and the bracketed or determinant type ( ab ).