Supervisors info:
Ζαχαριάδης Θεοδόσιος Καθηγητής ΕΚΠΑ (Επιβλέπων), Πόταρη Δέσποινα Αναπλ. Καθηγήτρια ΕΚΠΑ, Θωμαΐδης Γιάννης Δρ. Στη Διδακτική των Μαθηματικών-Σχολικός Σύμβουλος
Summary:
This thesis argues in favor of using the Ηistory of Μathematics in teaching and
in particular, in teaching the rule: «( – )·( – ) = +». The basic principles of
Realistic Mathematics Education, both sides of "Parallelism" and the Conceptual
Change, which is required for a student to understand the operations with
negative numbers, were studied in the theoretical framework. While teaching the
rule: «The product of two negative numbers equals a positive number» in seventh
grade students, justifications of great mathematicians of the past, such as
Euler’s and Viete’s, were used, reaching more recent mathematicians and
researchers, like Freudenthal and Rapke. Although the literature suggests that
teaching has invented countless intuitive – real life models to justify the
rule of signs, which are being criticized in the paper, historically a
significant conceptual change and the development of a theoretical way of
thinking was required to provide a consistent mathematical interpretation of
the rule. The purpose of the research can be summarized in the following
questions: Is it possible for seventh grade students to achieve this kind of
thinking? Is it possible to combine the practice in carrying out operations
with negative numbers correctly and comprehend the meaning of these operations?
Do students consider as essential the knowledge to justify the rule and how the
use of original historical sources contributes to this purpose? Is it necessary
to add more teaching hours? For the systematic review of these, an empirical
study was conducted with seventh grade students. The survey’s results argue
that the development of a theoretical way of thinking and achieving a
conceptual change, which is required for the interpretation of the rule of
signs, is possible at the first classes of high school level, with the strong
assistance of historical texts, in a context of “guided discovery”. In this
context, several students understood the rule of signs in multiplication of
rational numbers, not as a result of some intuitive models, which usually
contradict with common sense, but as a consequence of the extension of
fundamental properties of arithmetic operations, such as the distributive
property.
Keywords:
History in Mathematics Education, Realistic Mathematics Education, Rule of signs, Μultiplication of rational numbers, Distributive Property
