Educational Studies in Mathematics

, Volume 66, Issue 2, pp 185–201 | Cite as

Stages in the History of Algebra with Implications for Teaching

  • Victor J. KatzEmail author
  • Bill Barton


In this article, we take a rapid journey through the history of algebra, noting the important developments and reflecting on the importance of this history in the teaching of algebra in secondary school or university. Frequently, algebra is considered to have three stages in its historical development: the rhetorical stage, the syncopated stage, and the symbolic stage. But besides these three stages of expressing algebraic ideas, there are four more conceptual stages which have happened along side of these changes in expressions. These stages are the geometric stage, where most of the concepts of algebra are geometric ones; the static equation-solving stage, where the goal is to find numbers satisfying certain relationships; the dynamic function stage, where motion seems to be an underlying idea, and finally, the abstract stage, where mathematical structure plays the central role. The stages of algebra are, of course not entirely disjoint from one another; there is always some overlap. We discuss here high points of the development of these stages and reflect on the use of these historical stages in the teaching of algebra.


algebra abstract stage equation-solving stage function stage geometric stage rhetorical stage symbolic stage syncopated stage 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.University of the District of Columbia Washington DCColumbiaUSA
  2. 2.The University of AucklandAucklandNew Zealand

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