It is well-known in mathematics education that the transition from school to university is very challenging for many mathematics students and preservice mathematics teachers. One of the main reasons identified in the literature is that students face difficulties with proof-based mathematics to which they are commonly introduced when they start university (e.g., Gueudet, 2008; A. Selden, 2012). Over the past 30 years, research on first-year students’ proof skills has increased significantly. However, several proof-related activities are still under-researched and the specific relations between these activities are not fully understood yet. In particular, no prior research on students’ understanding of the generality of mathematical statements—which can be seen as an essential part of the comprehension of statements and students’ understanding of proof—and the specific relations to proof reading and construction had been conducted. Therefore, the main goal of the present study was to close this gap. Since no definition of understanding the generality of statements could be identified in the literature, I provided a clear definition for understanding the generality of statements myself by relating students’ estimation of truth to that of the existence of counterexamples. A correct understanding was then defined as consistent responses.

Further, to highlight the relevance of the statement itself for proof-related activities, I suggested an adapted version of the framework on proof-related activities by Mejía Ramos and Inglis (2009b), which distinguishes activities related to the reading of the statement, for which a proof needs to be constructed or a proof has to be read, and activities related to the respective reading and construction of arguments. The research questions were then mainly guided by this framework.

Since previous studies have shown differences in students’ understanding and evaluation of different types of arguments (e.g., Healy & Hoyles, 2000; Kempen, 2018, 2021; Tabach, Barkai, et al., 2010), I analyzed the influence of reading different types of arguments (no argument, empirical argument, generic proof, and ordinary proof) on students’ understanding of generality and other proof-related activities. Additionally, I considered the familiarity with the statement and its truth value as important characteristics that might also influence students’ performance in proof-related activities, as suggested in the literature (e.g., Barkai et al., 2002; Dubinsky & Yiparaki, 2000; Hanna, 1989; Stylianides, 2007; Weber & Czocher, 2019).

Through the experimental design of my study, I provided detailed results on the influence of the type of argument and statement on students’ understanding of the generality of statements, proof reading and construction, and on relations between these activities. The data was thereby analyzed using mainly generalized linear mixed models. My results extend prior research in that

  • in a comparatively large percentage of observations (about one third), students lacked understanding of the generality of mathematical statements,

  • students with a correct knowledge of mathematical generality are more likely to have a correct understanding of the generality of statements,

  • students’ usage and conviction of empirical arguments is negatively related to their understanding of the generality of statements,

  • students’ level of conviction of the truth of statements is significantly related to the reading and construction of different types of arguments; in particular, empirical arguments (and to a lesser degree generic proofs) support students in successfully estimating the truth value of (true) universal statements—but ordinary proofs do not,

  • the familiarity with the statement and the truth value influence students’ understanding of the generality of statements and performance in proof-related activities.

Further, my results (experimentally) confirm prior research findings that

  • first-year university students lack basic mathematical knowledge and therefore have difficulties with the comprehension of statements (e.g., Dubinsky & Yiparaki, 2000; Ferrari, 2002) and proofs (e.g., Conradie & Frith, 2000; Dubinsky & Yiparaki, 2000; Moore, 1994; Reiss & Heinze, 2000) as well as with proof evaluation (e.g., Harel & Sowder, 1998; Healy & Hoyles, 2000; Kempen, 2019; Recio & Godino, 2001; Weber, 2010),

  • most students find generic and, in particular, ordinary proofs convincing (e.g., Kempen, 2018, 2021; Ko & Knuth, 2013; Weber, 2010),

  • about half of the students used empirical arguments (Barkai et al., 2002; Bell, 1976; Healy & Hoyles, 2000; Lee, 2016; Recio & Godino, 2001) to justify unfamiliar universal statements, while most of them used external arguments (i.e., based on authorities or a rule) (Harel & Sowder, 1998; Sen & Guler, 2015; Sevimli, 2018; Stylianou et al., 2006) to justify familiar universal statements.

My findings can be used to develop future university courses in a manner that eases and promotes the transition to proof-based mathematics. Particular attention should be put on students’ comprehension of statements, including understanding their generality, before they are confronted with proof reading or construction. Empirical arguments and potentially generic proofs can support students’ understanding of statements and their success in estimating the truth value, but their limitations and the necessity of proof should be made clear.

Lastly, based on my findings, I provided several suggestions for future research on students’ proof skills and students’ understanding of the generality of statements. In particular, further research on students’ self-reports regarding proof-related activities and their actual understanding and proof skills would be very valuable. My study further highlights the benefits of and need for more experimental studies in mathematics education and in particular in research on proof and argumentation.