Using Ancient Ideas and Digital Tools in Mathematics Teacher Education

Abstract

This paper shows how the ideas of Archimedes about integrating “mechanical methods” and formal reasoning can be connected with the modern-day use of three computer programs—Wolfram Alpha, Maple, and Excel—in exploring topics related to the elementary theory of numbers. Explorations deal with the subsequences of integer sequences through the step-by-step elimination of every other term obtained in the previous step. This process, resembling the sieve of Eratosthenes and some modern-day sieving algorithms, is applied to tetrahedral numbers stemming from an ancient tradition to associate integers with geometric shapes and social contexts. It is demonstrated that symbolic computations in Wolfram Alpha enable generalization in the construction of sieves that is confirmed by Maple and a spreadsheet. This conceptual paper addresses one of the aims of the Special Issue by demonstrating the duality of mathematics and technology in the sense that whereas the latter facilitates new approaches to knowledge acquisition, the former can be used to improve the efficiency of computations by reflecting on the results made possible by these approaches. The activities conducted advocate for the value of a pedagogical framework that integrates ancient ideas, digital tools, and elementary number theory in the education of mathematics teachers. Reflective comments by teacher candidates are included as appropriate and linked to established educational frameworks to illustrate conformity.