## Abstract

In the study presented, as we report in this paper, we describe our theoretical and practical consideration to engage first-year pre-service teachers in proving activities in the context of a transition-to-proof course. We investigated how students argued to verify a claim of elementary number theory on entering university and compared the results to their performance in the final examination of the course. Subsequently, we elaborate on the following results: On entering university, students do not seem to be capable of using algebraic variables as a heuristic to engage in reasoning. However, after learning about different kinds of proofs and the symbolic language of mathematics, students give evidence of starting to value mathematical language and of enhancing their proof competencies.

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In this paper, “reasoning” is used as a superordinate concept that comprises all kinds of possible approaches to verify a claim.

The testing of the claim helps the students to understand the claim and to get an idea about its truth [cf. the structure of the proving process given in Boero (1999, p. 2)].

Definition 1.1: \(n\in \mathbb{N}\) is divisible by \(a\in \mathbb{N}\), iff there is a number \(q\in \mathbb{N}\) with \(n = a \cdot q\).

\({D}_{n}\) is the nth triangular number: \(1+2+3+\dots +n=\frac{n(n+1)}{2}\).

In this paper, the word “argumentation” is meant to describe an approach, where a person applies some warrant on data to support a conclusion (in the sense of Toulmin 1958).

An explanation for this result may be that we did not use many tasks where students had to make use of two variables to prove a given claim. This problem should be considered in the next implementation of the course.

A complete solution would have been for example\(\dots =6n+15=2\cdot \left(3n+7\right)+1.\) Since \(\left(3n+7\right)\in \mathbb{N}\), the result is an odd number (Theorem 2.1).

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Kempen, L., Biehler, R. Fostering first-year pre-service teachers’ proof competencies.
*ZDM Mathematics Education* **51**, 731–746 (2019). https://doi.org/10.1007/s11858-019-01035-x

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DOI: https://doi.org/10.1007/s11858-019-01035-x