Article: The notion of historical "parallelism" revisited: historical evolution and students' conception of the order relation on the number line

Yannis Thomaidis and Constantinos Tzanakis: The notion of historical "parallelism" revisited: historical evolution and students' conception of the order relation on the number line, Educational studies in mathematics (2007) 66: 165-183.

This article combines historical study with classroom study, to examine the "relation between historical evolution of mathematical concepts and the process of their teaching and learning". They wanted to look at both whether there is a "parallelism" here and what such a parallel "between a creative mathematician of the past and a student learning mathematics in a modern classromm" might consist of.

The mathematical topic considered is the order relation on the number line - that is (for instance); is -4 greater than or smaller than -2? In section 2 of the article, the authors trace the development of mathematicians' thought on this. For instance, Newton wrote that "the greatest negative [root]" was the one "most remote [from zero]", that is -4 > -2. Also, Bolzano used the notation e<±1 for which we would write -1< e< 1.

The study of students posed three questions, of which the first one was:
"What are the solutions of the inequality x² > 9 when x ∈ ℜ?"

Lots of interesting answers were given to the questions. For instance, it did turn out that some students had the same basic ideas as Newton and Bolzano mentioned above.

The authors argue that there are parallelisms that could be exploited to

"forsee possible persistent difficulties of the students; and to make teachers more tolerant towards their students' errors, by increasing their awareness that these errors and difficulties do not simply mean that "the student has not studied enough" but may have deeper epistemological roots which should be explored and understood thoroughly."

but also

"become more tolerant towards non-conventional, but essentially correct, views of their students, even though they may be wrong or insufficient by modern standards of logical rigor and clarity; [...] encourage these and other students to express their possibly idiosyncratic views on specific mathematical problems and in this way to implicitly guide them towards conceiving mathematics as an exciting human activity".

Reason enough to include history of mathematics in teacher education, in my opinion...