15. september 2016

Teaching infinitesimal calculus in high school

Specialerapport. IND's studenterserie nr. 53. Matematik.

Mikkel Mathias Lindahl and Jonas Kyhnæb, august 2016.

"Teaching infinitesimal calculus in high school" [4,7Mb]


In high school students have their first encounter with the notion of infinitesimal calculus. A common problem encountered when introducing this subject is for the students to understand the theory behind the techniques they develop when being taught this. In this thesis the use of infinitesimals will be reinstated to teach this particular theme. They were used to teach infinitesimal calculus even after their abolishment, when the real numbers were constructed. With the dawn of the nonstandard analysis, through the construction of the hyperreal numbers, the infinitesimals were reinstalled as a mathematical sound quantity. This thesis will give reasons for or against the use of infinitesimals in the form of nonstandard analysis by employing the anthropological theory of didactics. The thesis will give a short introduction of how the hyperreal numbers are constructed and how the definition of the differential quotient and integral is equivalent statements to the definitions usually encountered in high school.
A teaching course was done in order to survey if the didactical reasons suggested by the theories are noticeable when introducing infinitesimal calculus, with infinitesimals, in high school. As such a textbook material had to be developed in order for the students to prepare for the exam(s). This textbook material is void of the construction of the hyperreal numbers, as introduced in the thesis, but hinges on an intuitive construction of infinitesimal quantities to generate them.
An analysis of the teaching employing the anthropological theory of didactics was done in order to determine if the merits of the infinitesimals are on par with the theoretical reasons. The thesis concludes that teaching based on infinitesimals are realistic and helps engage the students by letting their intuition guide them to a great degree.