Abstract:

Dice and card games formed a significant part of the social life of nobles in the 16th and 17th centuries. This led to a great development of combinatorics and to writing of the first major works in this area (Blaise Pascal, Gottfried Wilhelm Leibniz, Jacob Bernoulli). The arithmetic triangle (Pascal's triangle) played a crucial role in all these works.

Both combinatorics and arithmetic are very important components of mathematics education, so it seems to be useful to develope the combinatorial thinking by means of non-standard arithmetic problems.

In our paper, we will present several non-standard arithmetic problems requiring combinatorial considerations. Among the others, we recall the problem named Sum and Product Puzzle which was published by Hans Freudethal in 1969. We show the modifications of this problem that can be accessible to pupils of different age groups.

We will also show how to use MS Excel during solution of some non-standard arithmetic problems (requiring combinatorial considerations), thus helping to develop digital literacy of pupils.

Using two tests, one focused on combinatorics and the other on arithmetic, we conducted research among prospective lower secondary school teachers to determine the dependence between their ability to solve combinatorial and arithmetic problems. We will use Spearmann's rank correlation coefficient for the collected data.

Keywords:

Non-standard, problem, arithmetic, combinatorial thinking, development, teacher.