Principles and Applications of APS

I am currently in Salamanca (Spain), attending the conference Tools for Teaching Logic III. My talk was on teaching logic through algorithmic problem solving and it went quite well, I think. In particular, it seems that the audience enjoyed the examples that I have used and the teaching scenarios that I have shown. As a result, I have promised that I would upload my PhD thesis into this website. Since the thesis can also be useful for other people, I have decided to write a new blog post. I hope you enjoy!

Abstract

Algorithmic problem solving provides a radically new way of approaching and solving problems in general by using the advances that have been made in the basic principles of correct-by-construction algorithm design. The aim of this thesis is to provide educational material that shows how these advances can be used to support the teaching of mathematics and computing.

We rewrite material on elementary number theory and we show how the focus on the algorithmic content of the theory allows the systematisation of existing proofs and, more importantly, the construction of new knowledge in a practical and elegant way. For example, based on Euclid’s algorithm, we derive a new and efficient algorithm to enumerate the positive rational numbers in two different ways, and we develop a new and constructive proof of the two-squares theorem.

Because the teaching of any subject can only be effective if the teacher has access to abundant and sufficiently varied educational material, we also include a catalogue of teaching scenarios. Teaching scenarios are fully worked out solutions to algorithmic problems together with detailed guidelines on the principles captured by the problem, how the problem is tackled, and how it is solved. Most of the scenarios have a recreational flavour and are designed to promote self-discovery by the students.

Based on the material developed, we are convinced that goal-oriented, calculational algorithmic skills can be used to enrich and reinvigorate the teaching of mathematics and computing.

Download the PDF

Principles and Applications of Algorithmic Problem Solving (PhD Thesis, João F. Ferreira, 345 pages)