Key Mathematical Concepts in the Transition from Secondary School to University

Abstract

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.

Mathematical Prerequisites for STEM Programs: What do University Instructors Expect from New STEM Undergraduates?

“…In school, typically, the solving of real-life problems is emphasized (see the PISA studies, e.g., OECD 2016), whereas in university mathematics the focus is on inner-mathematical problems (e.g., Engelbrecht 2010;Leviatan 2008;De Guzmán et al 1998). Similarly, the focus shifts from describing concepts to precisely defining concepts or from convincing via argumentation to mathematical proving (Tall 1991;Thomas et al 2015).…”

“…The before mentioned findings concerning the high dropout rates in STEM degree programs suggest a missing congruence or fit between students and universities with respect to mathematics in STEM programs. Approaches for a better alignment between students and the academic environment at the universities include: (a) the facilitation of choices of study degree programs congruent to students' interests and abilities (see Lubinski and Benbow 2000), (b) mathematical transition courses that support mathematical abilities of new STEM students (e.g., Leviatan 2008;Griese 2017), and (c) better communication and coordination between schools and universities (recommended by Thomas et al 2015). All such approaches for better alignment require transparent information on both the person and environment side.…”

“…Additionally, the results of this study may support students to choose study degree programs congruent with their interests and abilities by increasing the transparency of mathematical requirements for STEM degree programs. Finally, the identified mathematical requirements could be employed to initiate a negotiation between schools and universities about commonly accepted requirements that ensure transparency for all stakeholders and future undergraduates (see Thomas et al 2015). Such negotiations would also help to create a common meaning of the expected prerequisites among the university instructors and the school representatives.…”

“…The transition from secondary to tertiary education is of crucial importance for student retention and success in higher education (Gueudet 2008;Rach and Heinze 2017). Students' transitions come with difficulties, in particular for students taking mathematics courses in their first semester at university (Gueudet 2008;Thomas et al 2015;Engelbrecht 2010). As a result, many studentsespecially in STEM degree programsfail the transition and drop out (e.g., Australia: Rylands and Coady 2009;Germany: Heublein et al 2014;UK: National Audit Office 2007 andCox 2001;US: McCormick and Lucas 2011).…”

The Effectiveness of CCMPedia in Teaching and Learning Calculus

“…Mathematics also called hierarchical build-up of concepts that must be woven together, skills and facts by Sarwadi and Shahril for the successful learners of mathematics [1]. An unrevealed error, which is rooted in the mind of students, is therefore a major threat to the construction of students' mathematical knowledge [2]. Meanwhile, student errors may reveal the erroneous problem-solving process and thus provide information on the understanding of and the attitudes towards mathematical problems [1].…”