This report1 from the ICME12 Survey Team 4 examines issues in the transition from secondary school to university mathematics with a particular focus on mathematical concepts and aspects of mathematical thinking. It comprises a survey of the recent research related to: calculus and analysis; the algebra of generalised arithmetic and abstract algebra; linear algebra; reasoning, argumentation and proof; and modelling, applications and applied mathematics. This revealed a multi-faceted web of cognitive, curricular and pedagogical issues both within and across the mathematical topics above. In addition we conducted an international survey of those engaged in teaching in university mathematics departments. Specifically, we aimed to elicit perspectives on: what topics are taught, and how, in the early parts of university-level mathematical studies; whether the transition should be smooth; student preparedness for university mathematics studies; and, what university departments do to assist those with limited preparedness. We present a summary of the survey results from 79 respondents from 21 countries.

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This chapter aims to show the impact of culture on the learning of mathematics and consequently that studies of mathematics for teaching require strong theoretical frameworks that foreground the relationship between culture and pedagogy. For this purpose, we describe two different research projects in Southern Africa, each focused on the notion of mathematics for teaching. The first study analyses teacher learning of the mathematical concept of limits of functions through participation in a research community in Mozambique, and is framed by Chevallard's anthropological theory of didactics. The second, the QUANTUM project, studies what and how mathematics is produced in and across selected mathematics and mathematics education courses in in-service mathematics teacher education programmes in South Africa, and is shaped by Bernstein's theory of pedagogic discourse. We argue that separately and together these two studies demonstrate that mathematics for teaching can only be grasped through a language that positions it as structured by, and structuring of, the pedagogic discourse (in Bernstein's terms) or the institution (in Chevallard's terms) in which it 'lives'.