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.

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