What do we mean by mean?

In order to be successful with a certain concept at school, it is often sufficient to master some types of tasks, without having to be aware of the different perspectives that a concept can be viewed from. It is, however, definitely not enough when the goal is to understand mathematics and become flexible and creative in solving mathematical tasks. Not only school students, but also prospective teachers should be encouraged to reflect on what particular concepts actually mean and how they can be thought of. In this article we focus our attention on the concept of mean. In the introduction we outline the gaps that our paper partially fills. Then we provide a brief review of how the Polish core curriculum and two very popular mathematics textbook series (one each for primary and secondary school) treat the topic of means. Next, we present results from a study conducted on a group of first-year students of mathematics and a group of teachers undertaking postgraduate studies qualifying for teaching mathematics as a second subject. We show the results obtained in two tasks concerning the arithmetic and geometric mean. The paper ends with several teaching suggestions and recommendations for mathematics teachers’ educators.