Abstract
This paper associates the findings of a historical study with those of an empirical one with 16 years-old students (1st year of the Greek Lyceum). It aims at examining critically the much-discussed and controversial relation between the historical evolution of mathematical concepts and the process of their teaching and learning. The paper deals with the order relation on the number line and the algebra of inequalities, trying to elucidate the development and functioning of this knowledge both in the world of scholarly mathematical activity and the world of teaching and learning mathematics in secondary education. This twofold analysis reveals that the old idea of a “parallelism” between history and pedagogy of mathematics has a subtle nature with at least two different aspects (metaphorically named “positive” and “negative”), which are worth further exploration.
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Notes
“[The] mathematical development in the individual retraces the history of mathematics itself” (Fauvel 1991, p. 3).
Glaeser claims that “... [Euler’s “Algebra”] reveals... another obstacle, concerning the incomprehensibility of unifying the number line, that Euler ... did not surpass”, thus detecting here an epistemological obstacle (Glaeser 1981, p. 320; our translation). However, this interpretation is risky. Probably, Euler considered the unified number line too “advanced” for his introductory algebra, because the calculus of inequalities was not still an established part of it; the order relation in his book was mainly a didactic choice, also supported by Euler’s clear presentation of ordering on the (unified) number line, in his earlier “Analysis” (Euler 1748/1990, p. 4).
E.g., the ε–δ techniques in analysis resulted from systematically using inequalities in error estimations of solving equations (Grabiner 1981; cf. previous paragraph above).
In Greece the same textbook (written according to the official syllabus for the respective school subject) is used in all schools and is issued free of charge. This textbook is the primary (if not the only) resource for the teachers’ planning of the lesson.
This point, however, should be further explored to be clarified completely.
This indirectly supports the conclusion in (a) above.
E.g.: answer F3 to Q1 in G2 in connection with Bolzano’s conception and notation of ordering; answer II3 to Q3 in G1 (and in general answers II, III in both groups) and Descartes’ greatest “false” root; answer I3 to Q3 in G2 and Newton’s root most remote from zero.
We do not blame modern teaching for not following the complicated historical evolution, but we want to stress its essential differences from genuine mathematical activity, as registered in history, and challenge the uncritical acceptance of their strict parallelism.
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This paper is dedicated to the memory of J.G. Fauvel.
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Thomaidis, Y., Tzanakis, C. The notion of historical “parallelism” revisited: historical evolution and students’ conception of the order relation on the number line. Educ Stud Math 66, 165–183 (2007). https://doi.org/10.1007/s10649-006-9077-6
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DOI: https://doi.org/10.1007/s10649-006-9077-6