The Teaching of Irrational Numbers in Basic Education and in Mathematics Teacher Education: a vicious circle is underway?

Abstract This article presents a reflection about the lack of articulation between the teaching of irrational numbers in Basic Education and in the initial mathematics teacher education. In basic education, the subject is treated with superficiality and little conceptual deepening, basically by way of examples, while in the mathematics teacher initial formation, there is the prevalence of the formalist approach to irrational numbers, which does not enable the future teacher to teach the subject in basic education. Felix Klein had already pointed out the result of this break between undergraduate and basic education in mathematics over a century ago - there is a double discontinuity in teaching. In the case of irrational numbers, more specifically, this mismatch can provoke a vicious cycle: the teacher leaves the university without adequate formation to address the subject in Basic Education, causing the students to reach the university without an adequate image of the irrational number, and the cycle repeats itself.