Article: Stages in the History of Algebra with Implications for Teaching

Victor J. Katz (with Bill Barton): Stages in the History of Algebra with Implications for Teaching, Educational studies in mathematics (2007) 66: 185-201.

In this article, Victor Katz gives "highlights" from the history of algebra. He does not spend time on the well-known three stages in the expression of algebra: the rhetorical stage, the syncopated stage, and the symbolic stage. Instead, he looks at four conceptual stages: the geometric stage, the static equation-solving stage, the dynamic function stage and the abstract stage.

The by far most long-lasting of these stages (so far) was the geometric stage. Euclid's algebra was geometrical, but so was Babylonian algebra from about 4000 years ago. Al-Khwarizmi (about 825) still justified the methods by geometrical means, but the reader was supposed to learn the algorithm without needing recourse to the geometry. For Katz, Al-Khwarizmi marks the move to the sttatic equation-solving stage.

Although Sharaf al-Din (died 1213) were using methods that could have been the start of the dynamic function stage, it instead had to wait until the early 1600s to take hold. With Fermat and Descartes, and later Newton, algebra moved from being mostly concerned with solving equations to be a method for determining curves, for instance.

Then, during the 1800s mathematicians worked on more general concepts, such as the group (Galois and Cayley are important names in that development).

Katz asks whether these stages should have pedagogical implications. Should geometric figures play a bigger part in the beginning of work on algebra? Should we work more on equations before introducing functions? And so on.

As usual, there are no easy answers when it comes to pedagogical issues, but at least it seems a good idea to know the history before designing the teaching of the future.

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...)