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Review: Ghosts of Diophantus

6 September 2011

Harper, E. (1987). Ghosts of Diophantus. Educational Studies in Mathematics, 18, 75-90

 The main issue of this article is the relationship between the historical evolution of algebraic ideas and their conceptual development that indicates how an historical analysis can guide and inform teaching. It is based on a conjecture that the ontogenetic development of mathematical ideas, development of ideas during the life time of the individual mathematician, might parallel with its phylogenetic evolution, evolution throughout the historical time. Related to this idea, some mathematics education experts such as Poincare, Branford, and Polya suggested that curriculum content in mathematics, in order to make it useful for students, should be presented in precisely the same order in which that content evolved during the history of mathematics. Furthermore, Polya suggested that students should be facilitated to ‘rediscover’ all the ‘great steps’ taken by mathematicians throughout history. This idea is similar with Freudenthal’s suggestion that students should be encouraged to ‘re-invent’ mathematics ideas.

 The study conducted by Harper has an aim to identify the major steps taken in the historical development of algebra, with particular reference to the use made of letters, and then to decide if evidence might be obtained to indicate that these same steps are taken by students in the classroom. He proposes three major stages that algebra has passed through the history of its development: rhetorical (long written argument with no symbol for unknown), syncopated (symbol for unknown quantities), and symbolic (letters also for given quantities). He gives a problem that was worked in the history of algebra to 144 secondary grammar school students and he finds that how the students do the problem is similar with the historical development of algebra in that lower students work in rhetorical stage, and as the levels of students increase they start work with syncopated and symbolic.

 This article’s suggestion that an historical analysis can guide and inform teaching is really need to be applied in designing a research project. Knowing how a mathematical idea develops through the times can lead into a better understanding of the materials and also provides opportunity to develop meaningful context that could facilitate students to reinvent the intended mathematics.

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