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.摘要 本文就基础教育阶段无理数教学与数学教师职前教育之间的衔接缺失问题展开反思。在基础教育阶段，该内容的教学流于表面，概念深化不足，基本仅通过举例展开；而在数学教师职前培养中，无理数教学普遍采用形式主义进路，这使得未来教师无法胜任基础教育阶段的该内容教学。一个多世纪前，菲利克斯·克莱因（Felix Klein）就已指出数学本科教育与基础教育之间的这种割裂现象——数学教学存在双重断裂问题。具体到无理数教学这一场景，这种脱节会引发恶性循环：教师毕业离校时未接受足够的培养以胜任基础教育阶段的该内容教学，进而导致学生进入大学时仍未建立对无理数的正确认知，如此循环往复。