Summary

The aim of this work was to show by means of a case study how algorithmic thinking and mathematical thinking can be integrated in high school or college curriculum. The idea is to introduce a subject into the mathematics or the computer science curricula, and to then discuss it in a way that leads to small research questions, and encourages the algorithmic approach.

From the computer science and mathematics curricula we discussed topics like algorithmic problems, and algorithms, decision problems and recursion, and touched upon issues like the correctness of algorithms, and data structures. However, we did not expect to cover or complete in any way all the aspects of the given example, or the issues we mentioned or pointed out for discussion. There are many more questions to be discussed, which may lead to interesting ``mathematical discoveries" made by the students, and may serve as an opening to more profound theoretical issues.

We can ask for example: What happens if we change the three axioms to:
A1. $0 \in \cal{S}$,
A2. $x \in \cal{S} \rightarrow$ $f(x) \in \cal{S}$ and $g(x) \in \cal{S}$,
A3. $\cal{S}$ is the minimal set satisfying A1 and A2.

This will lead to the more complicated topics of computer science and mathematics, asking what if $f(x)$ and $g(x)$ are computable, and what if they are not.

There are more profound issues that can be discussed. We mentioned the complexity and correctness of algorithms, but we did not go into detail, not even analyzing the complexity of the algorithms we presented. However as we mentioned above, we definitely prepared the ground for further study of these subjects, in a later stage of the students learning.

It should be emphasized, however, that even presenting in detail only what we have in this paper, might constitute a crucial contribution to the students' mathematical education as a well-established basis for further studies. It might not be suitable for the entire student population of high schools and colleges; it was not meant to. It was designed for students with a greater and deeper interest in mathematics and computer science, or in logical thinking. We are sure that the students who have a strong interest in mathematics will find this example most attractive.

Acknowledgment
We wish to thank our colleague A. Arcavi with whom we had very stimulating and fruitful discussions.