1 Introduction

Interest is a unique cognitive and affective motivational variable that develops (Renninger & Hidi, 2022). It describes individuals’ participation in mathematics: their psychological state during engagement, as well as their motivation to reengage mathematics voluntarily over time (Hidi & Renninger, 2006). The presence of interest has repeatedly been found to benefit learning as it increases attention, sustains engagement, and improves performance (e.g., Bernacki & Walkington, 2018; Jansen et al., 2016). When learners are interested in mathematics, they search for relevant information, seek deeper understanding, persevere when faced with difficulty, and engage in meaningful learning (Renninger & Hidi, 2016).

During the 1980s and 1990s, numerous investigations were conducted which underscored the importance of interest in comprehension of text (see reviews in Hidi, 2001; Schraw & Lehman, 2001), and research has shown that supporting learners to work with personalized features of mathematics text has a positive influence on performance (e.g., Clinton & Walkington, 2019). Studies have also demonstrated that educators are uniquely positioned to support their students to develop their interest in and seriously engage with mathematics through the selections and design choices that they make about instruction, as well as in their follow-through to support learners to make connections to the content with which they are working (e.g., Bernacki & Walkington, 2018; Crouch et al., 2018; Xu et al., 2012). However, investigations that consider interest, other learner characteristics, and students’ comprehension of mathematical argumentation in text are lacking. This type of information is critical for the design of mathematics resources that are accessible for learners who may be in different stages in the development of their abilities to identify metamathematical concepts and synthesize mathematical arguments.

Learners also have been shown to successfully build complex and formalized ways of comprehending mathematical argumentation with practice, instructional support, and mathematical cognitive development (e.g., Blanton & Stylianou, 2014; Harel & Sowder, 1998; Segal, 1999; Tall & Mejía-Ramos, 2010). However, studies have suggested that secondary and undergraduate learners can encounter difficulty, or tripping points, in (a) understanding what constitutes mathematical justification or proof (Harel & Sowder, 1998; Segal, 1999), (b) identifying the types of declarations that need to be justified (Edwards & Ward, 2004; Zaslavsky & Shir, 2005), and (c) synthesizing the argument of a justification (Inglis & Alcock, 2012; Weber & Alcock, 2005). These challenges are particularly salient for learners as they engage mathematical argumentation in text or written proofs (e.g., Inglis & Alcock, 2012).

The present research addresses whether and how existing mathematics interest, and other learner characteristics, relate to comprehension of mathematical argumentation and triggered interest in mathematical text. It is use-inspired (Stokes, 1997) in that methods and findings from basic research are used to address questions of practice. This is an approach that can also contribute to the field’s empirical and theoretical understanding (e.g., Cai et al., 2019). We collaborated with mathematicians, mathematics educators, and mathematics students (henceforth, practitioners) who were developing online informational resources for a full range of potential users. They anticipated these users would differ in their learner characteristics—specifically, mathematics interest, level of prior course work (which in the United States is not regularized), and proof scheme (or interpretive frames for mathematical arguments; Harel & Sowder, 1998). The practitioners conjectured that understanding these characteristics’ joint impact on comprehension of mathematical argumentation in different types of text would enable them to use this information to promote interest and comprehension. In our research, following Dowling’s (1998, 2001) distinction of two types of mathematical text, we focused on how participants comprehended argumentation and developed interest in two types of text domains: (1) text featuring concrete, real-world applications (public domain) and (2) text with abstract and generalized modes of expression and content (abstract domainFootnote 1).

2 Theoretical background

To provide context, we review research on mathematical argumentation, then turn to studies of text domain, mathematics interest, prior coursework, and proof scheme.

2.1 Comprehension of mathematical argumentation

Following Conner et al. (2014), we define mathematical argumentation as goal-oriented activity that substantiates a claim, not unlike mathematical reasoning: “When an individual is creating an argument, he or she is reasoning, and when an individual is reasoning, he or she is creating an argument; thus we contend that, when considered as individual activities, argumentation and reasoning refer to the same process in mathematics” (Conner et al., 2014, p. 183). As such, mathematical argumentation is understood to broadly encompass both informal argument and formal proof, as both share foundational logical structures (Aberdein, 2019; Knipping, 2008). Here, we conceptualize the comprehension of mathematical argumentation in text as consisting of two specific competencies: (a) the ability to distinguish between metamathematical structures (such as definition, theorem, and proof), thereby establishing goals for argumentation, and (b) the ability to synthesize an argument through interpretation of its data-warrant-claim structures.

We adopt Zaslavsky and Shir’s (2005) use of the term metamathematical structures to describe how learners distinguish between three types of mathematical communication: definitions, theorems, and argumentation. Definitions, theorems, and argumentation are distinct from one another in how and whether they are mathematically argued. Definitions structure important concepts and support argumentation but do not in and of themselves require support with mathematical reasoning (Levenson, 2012). Theorems are statements that highlight a conclusion that has been drawn from a mathematical proof (Hanna & Barbeau, 2010; Rav, 1999). Finally, argumentation can be conceptualized as existing along a spectrum with explanation at the opposite end, separated by characteristics such as strength of the reasoning involved and integration of more metamathematical details (Dreyfus, 1999; Duval, 1992; Sierpinska, 1994). Thus, although each metamathematical structure has features that make it distinct, they operate together in complex ways to support ideas in a text.

In our analyses, we also draw on Toulmin’s (1958) and Krummheuer’s (1995) work to conceptualize synthesis of argument structure as the comprehension of the elements of data (or evidence), claim (or conclusion), and warrant (or the logical relationship between these components). Researchers in mathematics education (e.g., Weber & Alcock, 2005) have used this scheme to study proof comprehension. For example, Mejía-Ramos et al. (2012) developed one such assessment model that included an element involving three specific skills related to components of Toulmin’s structure. Importantly, Toulmin's (1958) structure has been applied to a range of argumentation patterns beyond deductive reasoning, including inductive and abductive styles (e.g., Conner et al., 2014) and less formal meaning-making justification processes within classroom conversations (e.g., Knipping, 2008). This suggests that it also could be used to analyze argument structures across different types of content.

2.2 Text domain

Comprehension of text-based mathematical argumentation has also been linked to the features of the text, as mathematics texts differ in terms of their audience and purposes (Dowling, 1998, 2001; Österholm & Bergqvist, 2013). Here, we focus on the linguistic features of text as originally discussed by Dowling (1998), who distinguished between public domain texts, which include frequent use of real-world applications and are oriented toward less-experienced mathematics audiences, and abstract domain texts, which involve abstract and generalized modes of expression and content.

Text in the abstract domain is characterized by contextless, generalized symbolic notation, and work within this domain often represents the goal of mathematical activity (Dowling, 1998). However, Dörfler (2000) and Sfard (2000) have both suggested that texts that fail to provide scaffolding for learners to develop their comprehension of symbols can confuse and limit comprehension, resulting in only superficial understandings of abstract mathematical ideas. Comprehension of symbol-heavy, abstract text has also been found to be more developed among those with more mathematics experience, and this has been particularly salient in comparisons of undergraduates and mathematicians (Lew & Mejía-Ramos, 2019; Shepherd & van de Sande, 2014).

As Dowling (2001) observed, however, public domain texts often require abstract strategies and symbolic fluency for problem solving that can only be obtained through interaction with the abstract domain. Similarly, Schleppegrell (2007) pointed out that many word problems intended to be more generally accessible to a wide range of learners contain grammatically unique mathematical structures and vocabulary. Therefore, students need to be supported to develop their capacity to work with both text domains, as comprehension in one may reinforce comprehension in another.

2.3 Learner characteristics

According to Weinberg and Wiesner (2011), analysis of text comprehension also involves considering the characteristics of the learner—that is, how the reader relates to the text with which they are working. Here we overview previous research specific to three learner characteristics (interest, prior coursework in mathematics, and proof scheme) and their relation to comprehension of mathematical argumentation in text.

2.3.1 Interest

Although interest has not been studied in relation to comprehension of mathematical argumentation previously, interest has been shown to develop as readers engage with text and to relate to comprehension and performance. An individual’s interest in reading and their interest in the topic of the text being read both influence learners' attention, comprehension, sustained engagement, and performance (e.g., Ainley et al., 2002; McDaniel et al., 2000). McDaniel et al. (2000), for example, demonstrated that undergraduate readers required fewer cognitive resources to comprehend high-interest texts, and more for low-interest texts. Their findings, together with those showing that the level of learners’ prior knowledge (Lehman et al., 2007) and the nature of the task (Park et al., 2011) affect learning from the text, provided a basis for Magner et al.’s (2014) study of situational interest in geometry tasks. Magner et al. studied the impact of illustrations in near- and far-transfer tasks and concluded that existing interest and the nature of the task both need to be considered as factors benefitting learning.

The process of interest development begins with an initial triggering of interest in the situation, or situational interest, which may be supported to develop into a well-developed, or individual interest (Hidi & Renninger, 2006). As Renninger and Hidi (2016) explained, the connections to the mathematics content that learners are positioned to make are essential to the development of their interest, and support from discussion and activity that helps them to begin making their own connections to mathematics are critical. Examples include the use of personalization of the mathematics context to enable learners to understand the problem posed (Bernacki & Walkington, 2018). Moreover, when the connections that they make are their own, this can lead them to sustain engagement, gain practice and facility in working to understand, and begin asking and seeking answers to their own questions. Once learners begin posing and seeking answers to their own questions, they also voluntarily re-engage with the content in order to continue to think about it. This juncture in the process of interest development has been associated with the activation of the reward circuitry in the brain (e.g., Gottlieb et al., 2013). Seeking to understand mathematics becomes rewarding, and compared to the other things that they do, individuals increasingly become more attentive to, willing to expend greater effort on, and better able to set and pursue goals for themselves in mathematics.

Jansen et al. (2016) showed that having a developed interest in mathematics predicted student achievement; this was an effect that held both between (students with higher interest also had higher achievement) and within (students had higher achievement in the subjects in which they had more interest) persons. Although topics of interest (e.g., calculus, playing the piano) may vary, the phases of interest development (whether learners have less-developed or more-developed interest, and that less-developed precedes more-developed interest) have consistently been found to progress in similar ways, regardless of content area (Renninger & Hidi, 2022).

2.3.2 Prior coursework in mathematics

Carlson et al. (2015) reported that the mathematics courses learners have taken are associated with their level of understanding and reasoning. Moreover, the ways in which learners interact with mathematics over time accumulates as part of their interpretive frame for comprehension of mathematical argumentation; that is, learners’ “mathematical met-befores” influence their subsequent readiness to work with the same structures at higher levels of mathematics (Tall, 2004; see also McGowen & Tall, 2010). Their “met-befores” in secondary contexts may come to either undermine or support their work with mathematical argumentation as an undergraduate (see related discussion in Stylianides et al., 2017).

Prior research shows that students’ experience working with metamathematical structures is limited when entering undergraduate mathematics. For instance, students learn about definitions primarily through examples and experience (Edwards & Ward, 2004; Zaslavsky & Shir, 2005), which has led many secondary learners to believe that definitions can be proved (Edwards & Ward, 2004; Levenson, 2012). Exposure to the metamathematical structure of proof is similarly sparse prior to college—often consisting of little beyond a very brief introduction to two-column proofs in high school geometry—despite attempts at reform (Stylianides et al., 2017). Without an articulated understanding of the nature and purpose of these structures, students transitioning from secondary education into higher education might not distinguish between the structures fundamental to mathematical argumentation.

Individuals’ synthesis of data, warrant, and claim argumentative structures also can vary depending on their level of mathematical coursework. Selden and Selden (2003) reported that the undergraduate mathematics students they studied focused on “surface features,” such as mathematical symbols and notation, in their descriptions of proof. By contrast, the mathematicians that Inglis and Alcock (2012) studied were focused on identifying implicit warrants of claims and comprehending the overall structures of arguments when reading proofs. Comprehension of the data-warrant-claim system can be complicated because warrants are often implicit within mathematical texts, and data and claims are many times not explicitly linked or identified (Weber, 2010; Weber & Alcock, 2005). Prior coursework seems to differentiate learners’ synthesis of argument structures and the skills they apply to do so. Thus, we posit prior coursework contributes to differences in comprehension of mathematical argumentation.

2.3.3 Proof scheme

Harel and Sowder (1998) used the concept of proof scheme to describe the development of a learner’s frame for interpreting what constitutes a warrant (or justification). Among other levels the authors describe, individuals with an empirical proof scheme express justification as founded on the generalization of specific cases or the perception of certain qualities of a figure, whereas those with an analytical proof scheme create and understand arguments using a variety of fundamental axioms. For instance, to support a claim that the sum of two odd numbers is even, a learner with an empirical proof scheme may determine that calculating an even sum for several additions of odd numbers may justify the claim; a learner with an analytical proof scheme may instead require reasoning with the properties of odd and even numbers to feel convinced.

Importantly, Harel and Sowder (1998, 2005) suggested that learners could be supported to develop their proof scheme, and many studies (e.g., Blanton & Stylianou, 2014; Segal, 1999) have suggested that this often occurs in the transition to higher education. For example, Segal (1999) documented changes in first-year undergraduate mathematics students’ perceptions of what constituted valid and convincing proofs. She found that as students progressed through coursework, they were increasingly likely to identify empirical arguments as non-proofs, suggesting that they were developing concrete links between their prior coursework and proof schemes. Blanton and Stylianou (2014) also reported on changes in higher education classroom discourse over a semester that reflected shifts from empirical to analytical frames. They showed how interactions between teachers and students enabled criticism, clarification, or elaboration, and eventually encouraged students’ use of more analytical frameworks in their reasoning. Although no studies have previously explicitly connected proof scheme with both interest and prior coursework in mathematics, the connections described in the literature suggest that they may also develop in tandem to support work with mathematical argumentation.

3 The present study

Here we leverage constructs and findings from prior research to address practitioners’ questions about potential differences among readers of mathematical text, which is posted online for a general readership. The practitioners were concerned that differences among individuals’ interest in mathematics, level of prior coursework, and interpretation of justification could collectively impact text readability, interest, and comprehension. In order to address their questions, we undertook a person-centered analysis. Whereas a more traditional variable-centered analysis would have led to examine relationships among the variables studied, the person-centered analysis enabled us to identify potential similarities among the participants studied. We created profiles we could then relate to learners’ comprehension of, as well as their interest in, mathematical text (see Bergman & Magnusson, 1997; Woo et al., 2018).

We developed the Assessment of Mathematical Comprehension (AMC), an interactive online assessment. We used the AMC to explore undergraduate students’ comprehension of mathematical argumentation in relation to specific text and learner characteristics. To evaluate comprehension of mathematical argumentation, we assessed (1) participants’ distinctions between metamathematical structures in terms of their ability to identify definitions, theorems, and proofs in text and (2) their synthesis of argument structures, defined as the ability to chain together a declaration’s function (as data, warrant, or claim) within a presented argument. We also examined the role of text domain (public, abstract) in the comprehension process and in relation to learner characteristics.

Given the breadth of learner characteristics involved in this approach, our first and second research questions sought to make sense of the connections between various learner characteristics and then relate those connections to the comprehension of mathematical argumentation by text domain. In these cases, learners’ interest was treated as one of many independent variables. Specifically, we explored:

  • (RQ1) Are there different profiles of learners based on interest, prior coursework in mathematics, and proof scheme?

  • (RQ2) Does comprehension of mathematical argumentation differ across learner profiles and by text domain?

In conjunction with the analyses of RQ1 and RQ2, we also assessed participant interest in particular sections of text and the rationale for their interest across text domains: (RQ3) Is interest differentially triggered across the profiles of learners and by text domain?

Given the ways that triggered interest can support, transform, and sustain lasting interest (Renninger & Hidi, 2022), these questions taken together allow consideration of the role of learner interest in ongoing engagement with public and abstract text.

4 Methods

4.1 Participants

A purposeful random sample of 64 undergraduate students was selected for study participation based on the amount of their prior coursework in mathematics (either “less” prior coursework if the prospective participant had not taken single-variable calculus and linear algebra, or “more” if the prospective participant had completed both of these courses) and gender (male or female). Four assessment groups of participants were selected: 16 males with more prior coursework, 16 females with more prior coursework, 16 males with less prior coursework, and 16 females with less prior coursework. Consistent with the Institutional Review Board (IRB) guidelines protecting the rights and welfare of research participant, identities were anonymized following assignment to an assessment group. Participants were recruited using email, notices on bulletin boards, and by word of mouth.

4.2 Measure and procedure

4.2.1 AMC text

Text from the Math Images wiki page collection (https://mathimages.swarthmore.edu/index.php/Main_Page) was used to provide the context for AMC items. The Math Images wiki pages describe the mathematics associated with images and/or explain mathematical concepts using a combination of images and text. The Math Images wiki page about Fibonacci numbers (https://mathimages.swarthmore.edu/index.php/MILS_04B_hlv5) was selected for study purposes because its content included mathematical argumentation in both public and abstract domains (see Fig. 1). The page selected was shortened and adapted for study purposes. Public domain text sections included “Origin” and “Fibonacci Numbers in Nature.” Abstract domain text sections included “Symbolic Definition of Fibonacci Sequence” and “Greatest Common Divisor Property.”

Fig. 1
figure 1

Math Images Wiki Page for Fibonacci Numbers

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4.2.2 Measure

The AMC consisted of 41 items that provided descriptive information about participants’ interest in mathematics and text domain, as well as their comprehension of mathematical argumentation (see description of AMC sections in Table 1; to view AMC items, see https://works.swarthmore.edu/fac-education/169/). It was developed through multiple rounds of pilot testing.

Table 1 Sections of the Assessment of Mathematical Comprehension (AMC)
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Items assessing comprehension of mathematical argumentation referenced the web page/text on Fibonacci numbers. Section 0 consisted of three open-ended items used to assess which sections of the AMC were “the most interesting” subsections for the participant, two of which were relevant to our line of inquiry about interest. In AMC Sections A–D, participants were presented with two panels. The right panel presented them with AMC items, and the left panel provided them with continuous access to the text on Fibonacci numbers. Text at the top of each page encouraged participants explicitly to avoid random guessing on the assessment, and there was an option on all items for them to indicate if their response was a random guess.

Section A was included as a check for algorithmic fluency; however, for this analysis, we focused on the results of Section B and Section C/D to assess comprehension of mathematical argumentation. Section B items asked participants to identify whether a given statement, extracted from the text, was a definition, an unjustified theorem, or a completely justified theorem. Statements were presented as variations in the necessity and sufficiency of “mathematical justification.” A sample item from Section B is displayed in Fig. 2.

Fig. 2
figure 2

Sample question from AMC Section B, which assessed participants’ distinction between metamathematical structures

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Section C/D required participants to analyze the structure of two complete mathematical arguments, specifically examining the connections between data, warrants, and claims (see sample item, Fig. 3). These items drew upon Toulmin's (1958) argumentation theory and are consistent with Mejía-Ramos et al.’s (2012) dimension of “justification of claims.” In order to control for order effect and/or possible intimidation effects of the abstract text, two forms of the AMC Section C/D were developed: Form X presented an abstract domain argument prior to a public domain argument, and Form Y presented the public domain argument prior to the abstract argument. Participants within their assessment groups were randomly assigned to either Form X or Form Y for the study. A question assessing participant proof scheme (Section E) and demographic questions concluded the assessment.

Fig. 3
figure 3

Sample question from AMC Section C/D, which assessed participants’ synthesis of argument structure

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4.3 Analysis

Before undertaking our analyses, we developed a scoring guide for assessing AMC Sections A, B, and C/D. A continuous score for interest in mathematics was computed by summing participants’ responses to the four Likert items in the introductory section of the AMC. Information from the application for study participation was used to identify participants’ prior coursework as more or less. Finally, participants’ proof schemes were identified as analytical if a participant identified only “an algebraic derivation” as constituting “mathematical justification”; proof scheme was classified as empirical if another option was also or exclusively selected.

The question addressing participants’ interest in a section of the text (Section 0) was coded as “public domain text” or “abstract domain text.” Two researchers coded the follow-up request to explain their response using Dowling’s (2001) definitions of text domain: a “public domain explanation” referred to their stated interest in real-world applications, connections, and utility; an “abstract domain explanation” described their interest in thinking about mathematical properties, abstraction, or generalized symbolic representation. Following training to agreement, consensual qualitative analysis (Hill, 2012) was used to categorize participants’ explanations.

Three analyses were used to investigate our research questions: (1) an exploratory K-prototypes cluster analysis on the learner characteristics of interest, prior coursework in math, and proof scheme to identify learner profiles, (2) t-tests on AMC Sections B and C/D results to consider within and between cluster differences in performance, and (3) chi-square analyses to investigate whether interest was differentially triggered for clusters by text domain. To address RQ1, we chose K-prototypes cluster analysis as a person-centered analytical approach that enabled analyses of the similarities and differences across and within groups. We selected K-prototypes analysis for clustering because of the nominal and ordinal nature of our independent variables (Huang, 1998), and our sample of 64 participants fell within the range of sample sizes described for cluster analyses in Niemivirta et al.’s (2019) review of person-centered studies.

In order to determine the power of our sample to respond to RQ2, an a priori power analysis was conducted for a one-way ANOVA using two groups and two-tailed p-values. Power was set to 0.80 for the ANOVA, meaning there would be an 80% probability of reaching statistical significance if the obtained sample differences were truly present in the population. Furthermore, we ran the analysis to determine a sample size that would reasonably detect only large effect sizes [i.e., Cohen’s (1988) d = 0.80] given the pre-existing literature positing the impacts of interest, prior coursework, and proof scheme on comprehension of mathematical argumentation (e.g., Crouch et al., 2018; Segal, 1999). Results revealed that a sample size of 52 for ANOVA would be sensitive to differences in ranks in this case. This finding confirms that our sample of 64 met our criteria for RQ2.

5 Results

5.1 Identification of clusters

In order to address RQ1, we applied a K-prototypes cluster method on the learner characteristics of interest, prior coursework, and proof scheme. Validation testing using the silhouette method suggested that two groups would be optimal for this analysis, and this decision was also confirmed with a scree plot (see Fig. 4). We note that the optimal index was 0.467 for two groups and then decreased when adding more groups.

Fig. 4
figure 4

Scree plot for cluster analysis

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The cluster analysis identified two groups of participants: (a) a more-mathematically immersed (MMI) cluster—those, generally, with more interest in mathematics, more prior mathematics coursework, and who had an analytical proof scheme and (b) a less-mathematically immersed (LMI) cluster—those, generally, with less interest in mathematics, less prior coursework in mathematics, and who had an empirical proof scheme. A chi-square test revealed no statistically significant difference in the distribution of gender across clusters. A breakdown of cluster membership with respect to each independent variable as well as additional demographic and assessment information can be found in Table 2.

Table 2 Demographic information for MMI and LMI clusters
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5.2 Within- and across-cluster performance in comprehension of mathematical argumentation

To address RQ2, we conducted a series of t-tests to examine potential differences in performance on AMC sections assessing comprehension of mathematical argumentation across clusters by text domain. First, we ran three groups of independent samples t-tests to assess differences between clusters in performance on questions: (1) distinguishing between metamathematical structures versus synthesizing argument structures, (2) referring to abstract domain text versus those referring to public domain text, and (3) distinguishing/synthesizing in abstract/public domain text types. Due to running multiple t-tests with the same dependent variables, alphas are inflated leading to an increased probability of committing a Type I error. We performed a Bonferroni correction to protect against Type I error (α = 0.05/8 = 0.00625).

As depicted in Table 3, those in the MMI cluster outperformed those in the LMI cluster in their synthesis of argument structures (Section C/D, t[62] = – 3.02, p = 0.004, d = 0.75). There was no statistically significant difference ibetween the clusters in the students' abilities to distinguish between metamathematical structures. When text domain was examined, statistical differences were found for questions addressing the abstract domain; the MMI cluster consistently had higher performance (t[62] = – 3.56, p < 0.001, d = 0.89). We did not discover statistically significant differences between the public domain scores of each cluster.

Table 3 Means and standard deviations of participant scores on the AMC by cluster assignment
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Given the overlap between categories in the above analyses (for example, a question can belong both to the abstract domain and it also may assess distinction between metamathematical structures), we also conducted analyses on each combination of section and text domain (see Table 3). Statistically significant differences in cluster performance were found in the synthesis of argument structures in the abstract domain (t[62] = – 3.36, p < 0.001, d = 0.84), with higher performance in the MMI cluster. No differences were found for either the distinction between metamathematical structures in the public or abstract domain or the synthesis of argument structures in the public domain.

Finally, we also conducted four paired-sample t-tests to examine performance between sections and text domain by cluster. We again performed a Bonferroni correction to interpret the significance of the results (α = 0.05/4 = 0.0125). There were no differences between the LMI and MMI clusters in either the distinction of metamathematical structures or the synthesis of argument structures. However, while participants in the MMI cluster performed similarly in their responses to items concerning both public and abstract domain text, participants in the LMI cluster were more accurate in their responses to items associated with public domain than abstract domain text (t[32] = – 3.11, p = 0.004, d = 0.56).

5.3 “Most interesting” text and explanation

To address RQ3, we analyzed participant responses to questions about the “most interesting” section of text to determine if the text may have triggered participant interest differently with respect to cluster membership. As depicted in Table 4, the MMI cluster was more likely to report interest in text sections identified as abstract domain text, whereas the LMI cluster was more likely to choose as interesting text sections identified as public domain text (χ2[1, N = 64] = 7.28, p = 0.007). In both clusters, a majority of participants reported interest in sections identified as public domain text.

Table 4 “Most interesting” subsection analyses
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We also found that participants in the two clusters differed in their responses to the question, “Why did you find this subsection most interesting?” Some respondents’ explanations (n = 8) did not provide a clear reference to text domain (e.g., “[it was] something that I didn't know before”). However, with the results that could be categorized, a chi-square test revealed that participants in the MMI cluster were more likely to report that they preferred sections of the assessment for abstract reasons than those in the LMI cluster (χ2[1, N = 64] = 13.19, p = 0.001). Participants’ responses that were coded as abstract domain explanations included:

  • “The fact that the [greatest common divisor] of any two Fibonacci numbers is also a Fibonacci number and that it can be found by simply manipulating the indices of the two Fibonacci numbers.” (MMI participant)

  • “I liked the proof and the mathematical explanation. As I read the first two sections I was really hoping to know why the rules were true.” (MMI participant)

  • “Because I would think that sequences that appear in nature normally involve multiplication rather than addition.” (MMI participant)

    Examples of responses that were coded as public domain explanations included:

  • “I rarely think of mathematical patterns or expressions describing anything in nature—especially in such a universal way.” (LMI participant)

  • “It's a concrete, rather than abstract way of looking at math.” (LMI participant)

  • “It applies to the physical world. Theory without application doesn't interest me.” (LMI participant)

6 Discussion

To the best of our knowledge, this study is the first to jointly consider interest, prior coursework, and proof scheme in individuals’ comprehension of mathematical argumentation. Our findings revealed that two groups of learners could be identified: an LMI group (less interest for mathematics, less prior coursework, and an empirical proof scheme) and an MMI group (more interest for mathematics, more prior coursework, and an analytic proof scheme). We also learned that learners identified as LMI and MMI differed in their reading of public and abstract domain text. Whereas participants in the MMI cluster comprehended argumentation in both public and abstract domain text similarly, those in the LMI cluster were able to better comprehend public domain text than abstract domain text, so much so that their performance in the public domain did not differ significantly from that of the MMI cluster. In addition, we found that the sections of the AMC identified as public domain text were of interest to participants in both clusters, suggesting the utility of public domain text for accessible resource design.

Our findings contribute to the literature on interest development as well as that on comprehension of mathematical argumentation. The identification of two clusters and the coordination of learners’ interest, prior coursework, and proof scheme in these data suggest that, in general, development of knowledge (proof scheme) and taking additional coursework co-occur with increases in interest. Although prior studies have suggested that it is the content of the connections that learners make to the discipline that provide a basis for the development of interest (Renninger & Hidi, 2022), specificity about differences in knowledge as interest develops, such as that provided here about proof scheme and prior coursework, has not been available.

Our findings are consistent with those who have described learners as having interpretive frames that influence their work with mathematical text (e.g., Weinberg & Wiesner, 2011). We add to this literature by showing differences in the frames of learners based on their mathematical immersion—their interest, prior coursework, and proof scheme. Uniquely, we investigated this proposition with undergraduate learners who were and were not pursuing majors in mathematics. Our finding that textual features (namely, public or abstract text domain) were comprehended differently by learners based on how immersed they were in mathematics highlights the relation of comprehension of mathematical argumentation to the domain of text.

6.1 Implications

We undertook this use-inspired study (Stokes, 1997) to address practitioners’ questions about the role of learner interest in resource design and its potential to enhance their work with mathematical argumentation. Interest together with other learner characteristics (prior coursework and proof scheme) was related to learners’ comprehension of mathematical argumentation in text. In previous studies, learners have been able to develop all three of these characteristics when provided with appropriate instructional support (e.g., Blanton & Stylianou, 2014; Segal, 1999; Xu et al., 2012). If support were provided to trigger learners’ interest, we conjecture this would enable them to make connections to the texts with which they work. Text could also be designed to facilitate learners’ pursuit of additional coursework that challenges their “met-befores” (McGowen & Tall, 2010) and provide support for the development of their proof schemes (Harel & Sowder, 1998). In turn, we theorize this would positively benefit learners’ comprehension of mathematical argumentation in text.

This research also underscores the importance of text domain in learning. Not only is public domain text likely to be accessible to both LMI and MMI learners, but we found that public domain text triggered interest for many students (and particularly those in the LMI cluster). It appears that educators could consider leveraging comprehension and interest with public domain text as a scaffold for working with abstract domain text (e.g., Dörfler, 2000; Sfard, 2000). Prior research suggests that such support could improve abstract domain comprehension and would reciprocally improve public domain comprehension, as well (e.g., Dowling, 2001; Schleppegrell, 2007). We conjecture that text that promotes the triggering and sustaining of interest will encourage learners’ continued engagement with mathematics, support the development of more analytical proof scheme, and influence how learners are positioned to read and comprehend mathematical argumentation.

For the practitioners with whom we worked, these findings underscored the importance of providing online mathematics resources as public domain text, as these could be expected to be accessible for those with less and those with more immersion in mathematics. The practitioners also developed the practice of providing hot links, or highlighted words, in the text that provided further in-depth explanation and representations for those seeking additional detail.

6.2 Study limitations and directions for future research

Although we undertook the present study with a purposeful sample of participants, we suggest that replication of study findings be undertaken with a larger sample that would also have greater power to address RQ3. In so doing, optimal combinations of public and abstract domain text could be explored for both LMI and MMI learners. We also encourage research to investigate the variables reported in Table 1 that were used to describe participants (such as prior knowledge of Fibonacci numbers and fluency in applying algorithms), as their means suggest that this could provide further detail about LMI and MMI learners’ strengths and needs in developing their comprehension of mathematical argumentation. Validation of the AMC measure we developed could be undertaken, and/or study questions could be examined using already validated assessments of mathematics comprehension.

Moreover, future assessment could consider additional text variables (e.g., representations or imagery) that, like text domain, provide a context for learning. Other specific skills of comprehension of mathematical argumentation (beyond distinguishing between metamathematical structures and synthesizing argument structures) are warranted, as well. Such study could clarify how classroom instruction and text might promote interest and assist students’ developing comprehension of mathematical text. Longitudinal consideration similar to that provided by Segal (1999) also would offer useful insight about how mathematical thinking changes in relation to developing interest and mathematical experience. Finally, we suggest further examination of the relationship of the variables of the present study in a study design that would permit consideration of variable-centered comparison using regression analysis.

6.3 Concluding thoughts

Our findings suggest that interest is integrally related to the comprehension of mathematical argumentation. By studying existing interest, as well as the triggering of interest, we demonstrate that the development of interest is coordinated with taking additional coursework and making connections to analytical proof schemes. We underscore the need to recognize that not all learners in higher education are similarly immersed in mathematics. More specifically, our findings (a) suggest that interest informs learners’ interpretive frames as they read and work with mathematical text, (b) point to the domain (public or abstract) of the texts that learners are provided as having the potential to trigger and sustain interest in mathematics, and (c) indicate that text domain and interest potentially scaffold comprehension of mathematical argumentation.