Paper: L. Radford, On theories in mathematics education and their conceptual differences. In: B. Sirakov, P. de Souza, & M. Viana (Eds.), Proceedings of the International Congress of Mathematicians (Vol. 4, pp. 4055–4074). Singapore: World Scientific Publishing Co. 2018.

Abstract: In this article I discuss some theories in mathematics education research. My goal is to highlight some of their differences. How will I proceed? I could proceed by giving a definition, T, of the termtheoryand by choosing some differentiating criteria such as c1, c2, etc. Theories, then, could be distinguished in terms of whether or not they include the criteria c1, c2, etc. However, in this article I will take a different path. In the first part I will focus on a few well-known theories in Mathematics Education and discuss their differences in terms of their theoretical stances. In the last part of the article, I will comment on a sociocultural emergent trend.

**Review:** The paper contains 87 occurrences of words *know / knowing / knowledge* and only 13 of *think / thinking*. This makes me question the author’s own “theoretical stance”: in my humble opinion, he misses a significant aspect of mathematics: it is all about **thinking**. This is what mathematicians do: they **think**. Learning mathematics is learning to **think**. Teaching mathematics is teaching to **think**. Of course, both teaching and learning frequently fail — and reasons for that deserve some discussion.