Teaching the skill of writing a mathematical argument in a first-year linear algebra course

The transition to university-level mathematics is challenging for first-year students,
particularly on the topic of mathematical reasoning and proving. This note describes
the design of the course ‘Linear Structures 1’ at the University of Twente, developed
with the goal to teach first-year students the skill of writing mathematical proofs,
next to the topics of linear algebra. Unlike traditional courses, where this skill is
often only an implicit learning objective, our course aligns all activities to make
it tangible, explicit, and achievable for the students. The proposed course design
incorporates blended learning strategies, such as micro lectures, digital quizzes and
interactive in-class activities. We introduced the Proof of the week: an activity where
students write a short proof every week in a low-stakes and safe environment: the
only requirement is students show effort. We provide students with feedback on
their proof-writing through in-class discussions with peers and comments from the
teacher. Two notable outcomes are the improved quality of students’ proofs and the
75% reduction in grading time, which makes the course highly scalable.