Should the historical evolution of math concepts inform teaching?

By | December 10, 2011

Should the history of a math concept inform the way we should teach it? Some camps, especially those that strongly object to the usual axiomatic-deductive style of teaching, advocates the use of a “genetic” teaching model that takes seriously into account the historical roots of mathematical knowledge. Here are some studies that support this approach.

Harper (2007) compared the historical analysis with students’ empirical data and found a parallelism between the evolution of algebraic symbolism and the way students understand the use of letters in school algebra, concluding that “… the sequencing of conceptual acquisition appears to parallel that which is to be detected through the study of the history of mathematics.”

Moreno and Waldegg (1991) found that “… in situations involving the concept of infinity, the student response schemes are similar to the different response schemes given by mathematicians throughout the history of mathematics,…, when faced with the same kind of questions”

However, there are also those who contradicts this conclusions: For example, on solving linear equations, Arcavi argues that,

….solution methods generated throughout history are quite different from the usual methods generated by students. Consequently, we cannot assert that a reason for the study of linear equations is based on or inspired by parallels between history and psychology – these parallels do not seem to exist (Arcavi 2004, p. 26).

Herscovics acknowledges that while obstacles in the nature and evolution of knowledge are in parallel with some of those met by the learner and are associated with his/her cognitive evolution, she also warns that this parallelism should not be taken too literally, since learning environments in the past are significantly different from those of our learners now (Herscovics 1989, p. 82).

In their investigation of the parallelism between historical evolution and students’ conceptions of order in the number line, Thomaidis & Tzanakis (2007) has this to say:

If room is left for genuine problems to help the emergence of the new concepts, motivate students to appreciate their necessity, or formulate their own alternative ideas (as it happened historically), teaching will not be restricted to the presentation of formal constructs in their polished final form, as it is often the case under the additional pressure of factors peculiar to the modern educational system itself, but will help students conceive mathematics as a creative, adventurous human activity.

Like in most issues related to teaching and learning, there is no clear cut answer here,  but it will always pay to know for teachers to have a sense of how specific math concepts evolved in history. It could provide valuable information both in the design of instruction, in anticipating cognitive obstacles and, for making sense of students difficulties in learning the concept. Teachers must also always remember that the evolution of a math concept is always tending towards abstraction. And because definitions of math concepts are already abstractions of those concepts, starting with definition in teaching is a no-no. Read why I think it is bad practice to teach a math concept via its definition.

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