Using Ancient Ideas and Digital Tools in Mathematics Teacher Education

The 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 from elementary theory of numbers. Explorations deal with subsequences of integer sequences through step-by-step elimination of every other term obtained on the previous step. This process, resembling the sieve of Eratosthenes, is applied to tetrahedral numbers appearing in the social context of the family therapy triangulation method. It is demonstrated that symbolic computations of Wolfram Alpha enable generalization in the construction of the sieves that is confirmed by Maple and a spreadsheet. The 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 those approaches. The activities advocate for the value of integrating ancient ideas, digital tools, and elementary number theory in the education of mathematics teachers. Reflective comments by teacher candidates are included as appropriate.