Mathematical research generally results in statements that are–once they have been validly proven–true without any exceptions. In this respect, mathematics differs from other sciences: The unrestricted generality and absolute conviction of truth as well as the method, namely proof, with which respective results are achieved, are unique (e.g. Poincaré, 1952; Toulmin, 2003). Mathematics is therefore sometimes called a proving science (in German beweisende Wissenschaft), particularly in German literature (e.g., Dreher & Heinze, 2018; 2000; A. Heinze, Anderson, & Reiss, 2004; Hilbert, Renkl, Kessler, & Reiss, 2008; Kirsten, 2021; Reiss & Ufer, 2009). Because of its fundamental role in mathematics, proof has been a research focus in philosophy of mathematics and mathematics education for many decades, but the research area saw a particular rise of interest in the last 10–15 years (Sommerhoff & Brunner, 2021). Even though proof and argumentation are internationally seen as important learning goals in mathematics and are incorporated in many national curricula (e.g., Department of Basic Education, 2011; Kultusministerkonferenz, 2012; National Council of Teachers of Mathematics, 2000), students seemingly do not gain sufficient experience with proof during high school (e.g., Hemmi, 2008; Kempen & Biehler, 2019). Consequently, students of different school levels and forms seem to lack fundamental proof skills and understanding of proof (e.g., Dubinsky & Yiparaki, 2000; Harel & Sowder, 1998; Healy & Hoyles, 2000; Kempen, 2019; Recio & Godino, 2001; Weber, 2001). Difficulties with proof are diverse and include insufficient proof comprehension and difficulties with proof construction and validation (a good overview can be found in Reid & Knipping, 2010, pp. 59–72, for instance). The lack of proof skills is particularly relevant for students entering university, because in many countries it coincides with the transition to proof-based mathematics. Students’ insufficient proof skills and understanding are in fact often identified as main reasons for students’ difficulties with mathematics at the transition from school to university (e.g., Gueudet, 2008; A. Selden, 2012). High drop out rates in mathematics, in particular compared to other fields, seem to be one of the consequences (e.g., Dieter, 2012; Heublein et al., 2022). Research on university students’ proof skills has therefore increased significantly over the past 30 years, especially at the transition (e.g., Alcock, Hodds, Roy, & Inglis, 2015; Gueudet, 2008; Kempen & Biehler, 2019; Moore, 1994; Rach & Ufer, 2020; Recio & Godino, 2001; Sommerhoff, 2017; A. Selden & Selden, 2003; Stylianides & Stylianides, 2009; Stylianou, Chae, & Blanton, 2006).

While a large body of research has focused on students’ proof construction, the reading of proof, which includes proof evaluation and comprehension, is still under-researched (e.g., Mejía Ramos & Inglis, 2009a; Sommerhoff, Ufer, & Kollar 2015). For instance, studies have repeatedly provided evidence that many students find empirical arguments convincing (e.g., Gholamazad, Liljedahl, & Zazkis, 2004; Healy & Hoyles, 2000; Knuth, 2002; Martin & Harel, 1989; Segal, 1999), but most do not consider them to be proofs (e.g., Healy & Hoyles, 2000; Lesseig, Hine, Na, & Boardman, 2015; Stylianou, Blanton, & Rotou, 2019; Tabach, Levenson, et al., 2010). However, it is unclear and therefore an open research question what level of conviction students gain by reading or constructing these types of arguments (Weber & Mejia-Ramos , 2015). Moreover, research on aspects that influence students’ conviction, for instance, understanding the argument, the perceived generality of the argument, or being familiar with the type of argument, is still scarce (Ko & Knuth, 2013; Sommerhoff & Ufer, 2019). Further, only few studies have investigated students’ proof comprehension and further research is needed to find ways to better assess it and to identify specific difficulties students encounter when trying to comprehend a proof (e.g., Mejía Ramos, Fuller,Weber, Rhoads, & Samkoff, 2012; Neuhaus-Eckhardt, 2022).

To make proofs more accessible to students, different types of arguments have been considered in research on proof comprehension as well as proof evaluation. In particular, generic proofs are considered to be potentially useful in the learning of proof and argumentation (Dreyfus, Nardi, & Leikin, 2012; Mason & Pimm, 1984; Rowland, 2001). However, little is known yet regarding the influence of the type of argument on students’ understanding of proof (see, e.g., Lew, Weber, & Mejía-Ramos, 2020; Malek & Movshovitz-Hadar, 2011; Mejía Ramos et al., 2012).

In research on students’ proof construction and reading, the comprehension of the statements which are to be proven–or for which a proof has to be read–has largely been neglected, even though it can be considered as a prerequisite for proof comprehension. The comprehension of a statement involves understanding the statement’s generality, which is defined in this study as consistent evaluation of both the truth value of the statement and the existence of so-called counterexamples. Understanding generality is essential for the comprehension of mathematical statements and students’ proof skills, because it is the mathematical generality that is the defining element of mathematical proof and what distinguishes mathematics from other disciplines, as already mentioned above. Further, without understanding the generality of statements it might be difficult to develop an intellectual need for proof (see, e.g., Harel, 2013). As such, the importance of understanding generality has repeatedly been emphasized in the literature (e.g., Conner, 2022; Ellis, Bieda, & Knuth 2012; Fischbein, 1982; Kunimune, Kumakura, Jones, & Fujita, 2009; Lesseig et al., 2019). However, virtually no studies have explicitly investigated students’ or teachers’ understanding of the generality of statements. The few studies that have been conducted so far in this direction, most of them qualitative, relate students’ understanding of the generality of mathematical statements to the understanding of the generality of proof. For instance, some researchers have reported that students and (preservice) teachers, who seemed to be absolutely convinced of the truth of a statement and the correctness of its proof, were at the same time not fully convinced that no counterexample to the statement can exist (Chazan, 1993; Knuth, 2002). Others have investigated students’ awareness that one counterexample disproves a universal statement (Buchbinder & Zaslavsky, 2019; Galbraith, 1981) or that a proof holds for any given subset of cases (Healy & Hoyles, 2000). With respect to the understanding of the generality of proof, some researchers argue that students’ usage or conviction of empirical arguments may indicate an insufficient understanding of generality (e.g., Conner, 2022). However, this relation has not been explicitly investigated so far, in particular, with respect to the understanding of generality of statements. Overall, neither the extent to which students lack understanding of the generality of mathematical statements nor the specific relations to the construction, evaluation, and comprehension of proofs have explicitly been researched yet.

The main goal of the present study is therefore to investigate the understanding of the generality of mathematical statements and the relation to the reading and construction of proofs in first-year university students. 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). Thus, I focused on 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).

To investigate students’ understanding of generality, proof comprehension, evaluation and construction, their relations, and, in particular, the influence of the type of argument and statement on students’ performance in these activities, I designed an experimental study which was conducted online during two first-semester lectures in November 2020. 430 preservice teachers and mathematics students from a German university completed the questionnaire. They were randomly divided into four groups. All participants were asked questions about the same five universal statements (two of them familiar, two of them unfamiliar, and one of them false). The first group received no arguments to justify the statements but was instead asked to produce justifications themselves. The second group was provided with empirical arguments, the third group with so-called generic proofs, and the fourth group with ordinary proofs (those that are typically constructed by mathematicians). I analyzed the data using mainly generalized linear mixed models. Results confirm prior research findings that students lack basic mathematical knowledge and therefore have difficulties with proof comprehension and evaluation, and that many students use empirical arguments to justify universal statements. They further extent findings in that a comparatively large percentage of students lacks understanding of the generality of mathematical statements, that students level of conviction of the truth of statements is related to the reading and construction of different types of arguments, and how the familiarity with the statement and the truth value influence students’ understanding of generality and performance in proof-related activities. Furthermore, the results of this study suggest relations between students’ proof evaluation, proof comprehension, the estimation of truth, and their comprehension of statements, particularly their understanding of the generality of statements. Based on these findings, implications for future university courses, especially at the transition from school to university, and directions for future research can be derived.

The theoretical basis for the conceptualizations of understanding generality and proof is build in chapter 2 by highlighting characteristics of mathematical statements and generality, in particular how mathematical generality differs from generality in other sciences, how proof is shaped historically by so-called socio-mathematical norms, and different views and characteristics of proof. Further, this chapter also discusses usages and relations of the terms reasoning, argumentation, and proving, before different types of arguments widely used in mathematics education are introduced. The second theoretical chapter (chapter 3) provides an overview of the current state of research on students’ understanding of generality of statements and proofs as well as proof-related activities relevant for this thesis. Thereby, the framework on proof-related activities proposed by Mejía Ramos and Inglis (2009b) is adapted and used for structuring the discussion of prior research findings. Resulting research desiderata are summarized in chapter 4, from which the research questions of the present thesis are then derived and specified. In chapter 5, the design of the study, the construction of instruments, and the collection and analysis of data are thoroughly described and justified. Subsequently, the results are presented comprehensively in chapter 6, guided by the research questions. Research findings are interpreted and discussed in detail with respect to prior research in chapter 7, where I also reflect on methodological decisions and the adapted framework on proof-related activities, identify limitations, and outline implications for the teaching of proof and further research. Lastly, in chapter 8, I conclude the thesis with a short summary and outlook regarding theoretical and practical implications.