Article: Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking

Issue number 2 of Educational studies in mathematics 2007 was a special issue on history of mathematics. For some reason, I haven't read it in full before, but now is as good a time as any. In the following days I will blog about the articles in this special issue.

Luis Radford and Luis Puig: Syntax and Meaning as Sensuous, Visual, Historical forms of Algebraic Thinking, Educational studies in mathematics (2007) 66: 145-164.

There are a few main concepts in this article, and by pointing to these, I will give an idea of what the article is about:
The Embedment Principle: "our cognitive mechanisms (e.g. perceiving, abstracting, symbolizing) are related, in a crucial manner, to a historical conceptual dimension ineluctably embedded in our social practices and in the signs and artifacts that mediate them."

Zones of proximal development of the culture: "new mathematical ideas are answers worked out in the historically situated zones of proximal development of their cultures".

The authors argue that "phylogenesis cannot recapitulate ontogenesis". "To learn algebra is not to construct the objects of knowledge (for they have already been constructed) but to make sense of them."

The article goes on to look at how this can be explored in an 8th grade class' work on fractional equations. The authors look at a word problem, in which students successfully writes the algebraic equation. In this form, the students recognize every part of the equation as a representation of a corresponding part of the word problem. However, when trying to solve the equation, the correspondence with the word problem is lost, and the operations take over as the "main personages".

This is an interesting point which I will try to keep in mind next time I teach equations.

The point about learners not "constructing" the objects of knowledge ("for they have already been constructed") is not entirely convincing to me, however. Why not "construct anew"? I agree that when students learn algebra, they do not on their own construct the objects of knowledge from nothing, but the word "construct" still makes sense to me. Or maybe I would prefer to think of it as a combination - partly (re)constructing and partly making sense?

(Obviously, I need to think more about this...)