The researchers framed their analysis using a well-known process-oriented model developed by Annie and John Selden, which divides any proof into two distinct components. The formal-rhetorical part consists of the largely technical work of unpacking a statement’s logical structure, identifying definitions, and setting up a proof framework — for example, recognizing that a conditional statement might be handled by proving its contrapositive, or that a universal claim can be refuted by a single counterexample. The problem-centered part, by contrast, is the content-specific mathematical work needed to fill that framework: the algebra, the reasoning about functions, the manipulation of odd and even integers. The distinction matters because a student can fail at either stage for entirely different reasons — one may build a correct framework but lack the mathematical substance to complete it, or possess the substance but never manage to start.

Participants were selected through maximum variation sampling, with six students from each of five levels: newly admitted, first, second, third, and fourth year. Each took part in a clinical interview lasting roughly 90 minutes, during which they were asked to prove or disprove six statements involving functions and numbers — deliberately familiar territory, chosen to minimize conceptual barriers and expose the structure of proving itself. The six tasks were designed to elicit six different proof methods: proof by exhaustion, direct proof, proof by contraposition, proof by counterexample, proof by contradiction, and existence proof. Quantifiers were written out in words rather than symbols, so that any difficulty with logical structure reflected the participants’ reading of mathematical language rather than their decoding of formal notation.

The results revealed a clear pattern in one half of the framework and a stubborn persistence of problems in the other. The clearest cross-year differences concerned the formal-rhetorical part. Newly admitted students often did not know how to begin a proof at all, did not accept that checking every case in a finite domain constitutes a valid proof, and misread existential statements as universal ones. By the third and fourth years, most participants could initiate an appropriate framework for the exhaustion, counterexample, and existence tasks. But the problem-centered part told a different story: even final-year students failed to use the uniqueness condition in the definition of a function, or fell into the trap of representing a single integer as both odd and even using the same variable, which makes a contradiction argument impossible.

The six methods were far from equal in difficulty. Proof by counterexample — familiar to Turkish students since primary school — was the clear winner, with 23 of the 30 participants both able to use the method and able to complete a valid proof, at every year level. Proof by contraposition sat at the opposite extreme: only five participants could use the method, and only seven produced valid proofs of the targeted statement, some of them by switching to proof by contradiction instead. Direct proof of a function statement was nearly as problematic; not a single newly admitted, first-year, or second-year participant completed it, and only two students each in the third and fourth years succeeded. The stumbling block was conceptual: participants checked that the image of an input lies in the codomain but neglected to show that the image is unique, which is half of what it means to be a function.

The interview transcripts bring these failures vividly to life. Yonca, a newly admitted student, looked at the images of a finite set, confirmed they all landed in the codomain, and declared the proof finished — missing the uniqueness requirement altogether. Gizem, asked whether her exhaustive checking counted as a proof, replied that it did not: “what I did here is actually just substituting and checking.” Naz, a first-year student, similarly dismissed her complete verification as “going through an example.” Even in the counterexample task, where success was highest, hesitation surfaced: Melis, having found a number that refuted the claim that every real number is less than its square, insisted her work was not a proof but merely “trying.” This reluctance, the authors note, is less a cognitive deficit than an epistemological one — a shaky understanding of what a proof actually is.

The contraposition task exposed a second family of errors rooted in logic. The statement to be proved was: for a natural number n, if n² is odd, then n is odd. The most common mistake was the converse error — assuming the conclusion and trying to derive the hypothesis. Leyla, a fourth-year student, fell into exactly this trap, writing n as 2k + 1, squaring it, and announcing she had proved the statement, before realizing mid-interview that she had “gone the exact opposite way.” Others chose direct proof, which superficially matches the conditional structure but is not workable here. Nihal, one of the few third-year students to succeed, articulated the contrapositive cleanly: “if n is an even number, then n² is an even number. If I prove this, I will have proved that as well.” Meanwhile, in the contradiction task, the same-variable misconception proved remarkably durable: only two of the thirty participants — one third-year and one fourth-year student — used distinct variables when representing a number as simultaneously odd and even.

Quantifiers caused their own quiet carnage. In the existence task — prove that there is a real number greater than 5 whose square is less than 26 — all six newly admitted participants recognized the existential structure, yet only two actually produced a number. Several read “there exists” as “for all” and confidently declared the statement false, offering this misreading as a finished proof. Others, like Ece, argued that a suitable value could surely be found — “even if I cannot give it, someone certainly can” — without ever producing one or justifying its existence. And in perhaps the most striking display of fragile epistemology, Seda, a newly admitted participant, counted her failure to find a counterexample as proof of truth: “since I could not find one, this is an indication of its truth.”

The authors are careful about what these patterns do and do not show. Because the design was cross-sectional — different students at each level, interviewed at the same time — the year-level differences reflect group comparisons, not tracked individual development, and cannot be read as developmental trajectories. Each level contained only six participants, so the numerical patterns are descriptive characterizations of this small sample rather than population estimates. The findings are also limited to six statements built on functions and numbers. Within those bounds, however, the contrast between the two halves of a proof is instructive: formal-rhetorical skill, which the Seldens characterize as a learnable technique, improved markedly with exposure to university mathematics, while problem-centered difficulties tied to conceptual understanding persisted to the very end of the program.

The implications reach beyond the participants themselves. Pre-service teachers need proving competence twice over — as learners who must construct valid arguments, and as future teachers who must judge the mathematical validity of their students’ reasoning and respond productively. A teacher who cannot unpack a conditional statement may fail to see the structure of a child’s argument; one with weak conceptual knowledge may be unable to diagnose why an argument fails. Barak and Kabael therefore recommend that proof not be confined to a single course but woven explicitly and systematically through all four years: unpacking quantified statements and comparing proof methods in early mathematics courses, analyzing student-generated arguments in methods courses, and embedding reasoning and proving in ordinary topics during the school practicum. More exposure to formal proof alone, the study suggests, is not the answer — coordinated attention to frameworks, logic, concepts, and language is.

Subject of Research: Proving processes and difficulties of pre-service middle school mathematics teachers across all year levels of a teacher education program

Article Title: Examining the proving processes of pre-service middle school mathematics teachers

Article References: Barak, B., & Kabael, T. (2026). Examining the proving processes of pre-service middle school mathematics teachers. Frontiers in Education, 11, 1937865. https://doi.org/10.3389/feduc.2026.1937865

Image Credits: AI Generated

DOI: 10.3389/feduc.2026.1937865

Keywords: mathematics education, mathematical proof, pre-service teachers, teacher education, proof methods, formal-rhetorical part, problem-centered part, clinical interviews, qualitative research, counterexample, contraposition, Türkiye