Vol.:(0123456789) 1 3 ZDM https://doi.org/10.1007/s11858-019-01087-z ORIGINAL ARTICLE Student perspectives on proof in linear algebra Sepideh Stewart 1  · Michael O. J. Thomas 2 Accepted: 6 September 2019 © FIZ Karlsruhe 2019 Abstract Proof has a prominent place in the linear algebra curriculum, teaching and learning but in frst-year courses it continues to be challenging for both instructors and students. While an introduction to new concepts through defnitions and theorems adds to the complexity of the course, proof remains the number one hurdle for many students. How do students view proof in linear algebra? Do they distinguish argumentation and proof, and if so how? are among many questions that are still unanswered. Although research on proof in mathematics education is increasing, systematic studies on proof in linear algebra are still scarce. In this study, we examined responses to a set of interview questions on proof by a group of 16 frst-year undergradu- ate students shortly after their fnal examination. This paper opens the case for a pedagogy of proof in linear algebra and examines students’ reactions to, and voices on, proof in a frst-year course in linear algebra. In particular, it addresses areas such as student views on understanding of proof, the purpose of a proof, and when and how proofs communicate to them. We employed Tall’s Three Worlds as well as Harel’s intellectual need to analyse the data. Although, these models are often applied to what students construct, we argue they can also be applied to how students perceive proofs. The results revealed that understanding a proof in order to gain personal conviction was a major concern of students. Keywords Proof · Linear algebra · Convincing · Comprehension 1 Background Proof is considered by mathematicians to be central to doing mathematics (Thurston 1995). Hence, there has been much research into student ability with respect to proof, comprising three broad strands: constructing proofs; vali- dating proofs; and proof comprehension. In this paper we examine student perspectives on the purposes of proof and their preferences for the kind of proofs they think meet these purposes. In order to set the scene, we frst review what the literature tells us about undergraduate students’ reading, comprehension and construction of proofs in mathematics in general, and then consider their role in the teaching of linear algebra. Since proof is held in such high regard, Weber and Mejía- Ramos (2015, p. 15) note that, often, “a primary goal of mathematics instruction is for students to adopt the stand- ards for proving and conviction that mathematicians hold.” This may explain why considerable efort has been put into research on how proof construction (Mejía-Ramos and Inglis 2009), such as students’ ability to reproduce or construct certain proofs (Lockwood et al. 2016). Three requirements for successful engagement with proof given by Stylianides and Stylianides (2007) are: to recognize the need for a proof; to understand the role of defnitions in the development of a proof; and the ability to use deductive reasoning. However, among the conclusions from research is that students do not have the experiences to support building rigorous, deduc- tive arguments (Stylianou et al. 2015). While mathemati- cians may employ diferent strategies for proof construction (Lockwood et al. 2016) research has suggested some possi- ble ways to assist students to construct proofs. These include the use of conjectures (Pedemonte 2008), strategic examples (Lockwood et al. 2016), and counterexamples (Zazkis and Chernof 2008). In addition to learning how to construct proofs research- ers such as Harel (1997) have proposed that students should be encouraged to learn how to read proofs. Recently studies on student reading of proofs have focused on the manner in which they read proofs (Inglis and Alcock 2012; Panse et al. 2018) and how this compares with the ways mathematicians * Sepideh Stewart sepidehstewart@ou.edu 1 University of Oklahoma, Norman, USA 2 The University of Auckland, Auckland, New Zealand