Why Math Teachers Blur the Line Between Arguing and Explaining

In the daily hum of a secondary-school mathematics classroom, two kinds of talk happen constantly: students and teachers argue, and they explain. To an outside observer the two can look nearly identical—both involve sentences, both involve mathematics, and both are supposed to build understanding. Yet in the philosophy of language they are fundamentally different speech acts, with different purposes, different starting points, and different conditions for success. A new study published in Discover Education by Juan Gabriel Rave-Agudelo and José Antonio Fernández-Plaza has taken a close look at how ten Colombian secondary-school mathematics teachers distinguish between these two discourse practices, and the results reveal a striking gap between what teachers say about arguing and explaining and what formal theory says those practices should be.

The stakes are higher than they might first appear. Argumentation and explanation are not decorative features of mathematics teaching; researchers in mathematics education have long linked them to the social construction of mathematical knowledge through communicative interaction between teachers and students. When a teacher understands that argumentation exists to justify a contested claim while explanation exists to clarify how or why a phenomenon arises, that understanding can shape instructional decisions in the moment—which questions to pose, which student contributions to press further, and when to supply supporting reasons rather than background information. Conversely, when the boundary between the two practices blurs, opportunities to foster either mathematical justification or genuine student understanding can quietly disappear from classroom discourse.

To probe this boundary, the researchers adopted an unusual theoretical lens: Speech Act Theory, one of the earliest pragmatic theories in the philosophy of language, combined with the Linguistic-Normative Model of Argumentation developed by Lilian Bermejo-Luque. In this framework, every utterance is analysed on three levels—the locutionary act of what is said, the illocutionary act of what the speaker intends by saying it, and the perlocutionary act of the effect produced on the listener. Speech acts succeed only when certain felicity conditions hold, including preparatory conditions, a propositional content condition, a sincerity condition, and, crucially, an essential condition: what conventionally counts as the act being performed. It is this essential condition that lets theorists say when a stretch of classroom talk constitutes an argument rather than an explanation.

Within this model, argumentation is defined as a complex speech act linking two simpler acts: adducing and concluding. A claim—called the target-claim—is in doubt, and the speaker adduces reasons to show that it is correct. The implicit question argumentation answers is: why is this claim correct? Explanation, by contrast, is modelled as a relationship between pointing out and corroborating. Here the correctness of the phenomenon is not in question; what is in doubt is its origin or nature, and the speaker points out relevant information that makes that origin intelligible. The implicit question explanation answers is different: why is this fact or phenomenon the way it is and not otherwise? The distinction, in short, lies in the communicative function—argumentation responds to challenges about the validity of claims, while explanation serves to clarify aspects of reasoning that may not be immediately evident.

The study itself was a basic interpretative qualitative investigation, exploratory and descriptive in design. Ten in-service secondary-school mathematics teachers, each with professional qualifications under Colombian regulations and more than two years of classroom experience, were recruited from different educational institutions to capture a range of professional contexts. Data were collected through a semi-structured, open-ended questionnaire hosted on a Moodle platform, carefully piloted with an experienced teacher-researcher so that the wording avoided the specialised register of Speech Act Theory. Instead of asking directly for definitions, the instrument asked teachers what they understood by each practice and what they thought its purpose was—questions designed to elicit personal conceptions without intimidating participants who had never studied argumentation theory.

Analysis used a hybrid of deductive and inductive thematic methods, segmenting responses into phrases and sentences that could carry multiple codes. The findings showed that teachers describe both practices using three broad character types. Argumentation and explanation were both framed at times as epistemic-mathematical practices, involving procedures, algorithms, and justification of mathematical knowledge; as enunciative-informative practices, centred on showing, communicating, and making mathematical content known, often moving between concrete, pictorial, and symbolic representations; and as instructional practices, aimed at making content understandable and promoting participation and communication in the classroom. When it came to essential conditions or constitutive goals, teachers attributed cognitive goals such as comprehension, appropriation of knowledge, and development of reasoning; epistemic-applicative goals such as validating results and applying theory to practice; and informative goals such as making rules and content explicit.

The asymmetries in these attributions are telling. Argumentation was predominantly understood as an epistemic-mathematical practice with epistemic-applicative goals—one teacher defined it as justifying, with mathematical and logical rules, algorithmic procedures and interpretations of mathematical statements, while another described the purpose as demonstrating the validity of an obtained result. Explanation, meanwhile, was chiefly conceived as an enunciative-informative and instructional practice oriented toward cognitive goals—making concepts and properties known so that students understand. Notably, no evidence emerged that any teacher regarded explanation as an epistemic practice concerned with legitimising mathematical knowledge, meaning that notions from the philosophy of science such as the explanans-explanandum relationship simply played no role in their professional discourse. The teachers’ distinctions tended to track the question each practice answers: several noted that explanation responds to how questions while argumentation responds to why questions.

Yet the study also found that these distinctions were frequently blurred. Some teachers assigned the same practice type and the same constitutive goals to both—defining argumentation as making oneself understood through examples and explanation as making a topic understandable, for instance—and unsurprisingly then reported little difference between the two, with comments such as one participant’s admission that honestly, explanation already involves argumentation. The analysis also uncovered divergence between the type of practice teachers described and the goals they attributed to it: a teacher might frame argumentation as a matter of carrying out procedures to solve problems, yet assign it goals of supporting learning and encouraging observation of results. In most cases, however, recognisable pairings emerged—epistemic-mathematical practices with epistemic-applicative goals, enunciative-informative practices with informative goals, and instructional practices with cognitive goals.

Measured against the Linguistic-Normative Model, the alignment was limited. The teachers did not appear to recognise the felicity conditions or the distinct communicative functions that, in theory, cleanly demarcate argumentation from explanation, even though they characterised the practices in rich and varied ways. The authors argue that this partial alignment reflects genuine complexity in teachers’ professional knowledge of discourse practices, and that the way teachers describe a practice is closely tied to how they ultimately distinguish it from its neighbour. The researchers are careful to note the limits of their design: with ten participants recruited by availability, the findings offer an in-depth initial account rather than a statistically representative picture of Colombian mathematics teachers, and future work with larger samples, other educational levels, and complementary methods—particularly studies of the lexical differences between everyday and specialised meanings of arguing and explaining—is needed.

Even so, the implications for teacher education are concrete. If recognising and distinguishing discourse practices fosters their deliberate enactment in classrooms, then initial teacher training and professional development programmes may benefit from explicitly introducing the pragmatic-linguistic distinction between justifying a contested claim and revealing the origin of a phenomenon. The study also demonstrates the value of importing frameworks from the philosophy of language into mathematics education, where communication is increasingly recognised as the medium through which mathematical meaning is built. Behind the apparently simple question of when a classroom utterance is an argument and when it is an explanation lies a whole theory of how speakers commit themselves, how listeners interpret intentions, and how mathematical certainty and clarity are negotiated one sentence at a time.

Subject of Research: How secondary-school mathematics teachers distinguish between argumentation and explanation as classroom discourse practices

Article Title: Distinctions between argumentation and explanation by secondary-school mathematics teachers

Article References: Rave-Agudelo, J. G., & Fernández-Plaza, J. A. (2026). Distinctions between argumentation and explanation by secondary-school mathematics teachers. Discover Education, 5(1), Article 1166. https://doi.org/10.1007/s44217-026-02137-x

Image Credits: AI Generated

DOI: 10.1007/s44217-026-02137-x

Keywords: argumentation, explanation, mathematics education, Speech Act Theory, Linguistic-Normative Model of Argumentation, classroom discourse, secondary school teachers, teacher conceptions, qualitative research, Colombia, pragmatics, pedagogy