How Students Think When Math Gets Hard: Three Problem-Solving Patterns Revealed

When a mathematics problem refuses to yield to a memorized formula, something interesting happens inside a student’s head. Doubt creeps in, strategies get tested and discarded, and prior knowledge is dragged out of storage and pressed into service. A new study published in Discover Education has now mapped this hidden cognitive drama in remarkable detail, following high school students as they wrestled with non-routine arithmetic series problems and revealing three distinct patterns in how they think, struggle, and ultimately succeed.

The research, led by Muhammad Noor Kholid of Universitas Muhammadiyah Surakarta together with colleagues from Indonesia, Vietnam, and Singapore, focused on reflective thinking, a concept with a long intellectual pedigree stretching back to John Dewey. Reflective thinking, in the framework used by the team, is a mental activity that begins with a state of perplexity, whether doubt, difficulty, or error, and is resolved by drawing on prior knowledge and experience through repeated evaluation until the confusion is overcome. In mathematics, this means students do not simply apply procedures; they monitor their reasoning, adjust strategies mid-course, and integrate conceptual understanding as they work toward a solution.

To capture this process as it unfolded, the researchers adopted a descriptive qualitative design involving 35 high school students. Participants were selected precisely because they were reflective thinkers: students who experienced doubt or difficulty when solving mathematical problems but persisted until they reached a correct solution. Each student tackled non-routine problems on arithmetic series, a topic chosen because it sits at a critical juncture in secondary mathematics, underpinning later work on linear equations, statistics, and mathematical modeling. Crucially, the problems could not be solved by plugging numbers into a formula; they demanded interpretation of the problem’s structure and the invention of an appropriate strategy.

The methodological centerpiece was the think-aloud protocol. Students verbalized their reasoning while solving the problems, and the sessions were recorded using audiovisual methods. Answer sheets, observation sheets, and recordings were then analyzed through the constant comparison method, an inductive procedure in which data incidents are compared with one another and with emerging categories until stable classifications emerge. The instruments themselves were rigorously vetted: three experts in mathematics education, assessment, and evaluation rated the tasks, yielding an Aiken’s V index of 1, above the minimum criterion of 0.95, and 100 percent inter-rater agreement on reliability.

The analysis, anchored in the reflective thinking framework of Zehavi and Mann, which identifies four components, techniques, monitoring, insight, and conceptualisation, revealed three clearly distinguishable problem-solving patterns. The first, labeled Employ Variable, describes students who translate the problem’s key quantities into algebraic symbols. The second, Visual Representation, captures students who sketch the situation to build understanding. The third, Direct, characterizes students who plunge straight into manual calculation without constructing any symbolic or visual model of the problem.

In the Employ Variable pattern, exemplified by a student coded PS1, the reflective journey began with a systematic inventory of what was known and what was asked. The student assigned variables, n for the number of rows, b for the difference in seats between rows, and a for the seats in the first row, precisely because the initial reading of the problem produced confusion. In an interview, the student explained that equating critical information with variables made the problem easier to understand and easier to solve. This symbolic scaffolding then allowed the student to apply the arithmetic series sum formula, re-examine each step, and, when confusion resurfaced over how many additional rows of seats were needed, reread the problem repeatedly until the correct first step re-emerged. Every component of reflective thinking was activated: techniques in selecting the strategy, monitoring in re-checking the work, insight in breaking through the impasse, and conceptualisation in linking the variables to the underlying structure.

The Visual Representation pattern told a different but equally complete story. A student coded PS2 responded to the same problem by sketching the seating arrangement of a performance hall, row by row, writing in the seat counts from 10 in the first row to 22 in the last. The sketch was not decoration; it was a core cognitive tool. As the student put it during the interview, drawing the rows allowed him to imagine what the situation actually looked like. From the sketch he computed total seats, multiplied by the ticket price, discovered the revenue fell short of the target, and then extended his tally row by row until the target was met, concluding that six additional rows were required. Here, too, all four reflective components were engaged, with visualization serving to stabilize understanding, reduce confusion, and guide monitoring of each calculation.

The Direct pattern was strikingly different. A student coded PS3 simply wrote down the seat counts and added them up manually, with no formula, no variables, and no sketch. Perplexity did appear, but only later, at the stage of interpreting intermediate results, when the student paused over whether the first revenue figure represented the full answer. Reflection in this pattern was contextual and partial: the student re-checked calculations, adjusted steps when results mismatched the target, and verified the final answer, but never restructured the problem symbolically or visually. The analysis found that Direct-pattern students missed several key reflective indicators, including articulating what was being asked, restructuring information, revisiting the problem when confused, and integrating conceptual understanding. Notably, the researchers caution that this does not signal low ability; rather, it reflects a situational, minimal level of reflection confined largely to monitoring.

The broader significance of the findings lies in what they mean for classrooms. Prior research cited by the team shows that teachers often struggle to recognize the specific strategies students use, tending to identify only general difficulties with comprehension or procedure. By classifying problem-solving patterns through the lens of reflective thinking, the study offers teachers a diagnostic vocabulary: a student stuck in the Direct pattern may need experiences that encourage representation and conceptual reconstruction, while students already employing variables or sketches may benefit from challenges that deepen their monitoring and evaluation. The authors acknowledge limitations, including the modest sample size and the single topic of arithmetic series, and call for future work with open-ended and ill-structured problems, as well as instructional models designed to infuse reflective learning. For now, the study delivers a vivid message: the path a student chooses through a hard problem, whether symbolic, visual, or direct, shapes not just the answer they reach, but the quality of the thinking they do along the way.

Subject of Research: Reflective thinking patterns in students' problem-solving of non-routine arithmetic series tasks

Article Title: Students’ problem-solving patterns in arithmetic series tasks viewed through reflective thinking

Article References: Kholid, M. N., Febriyanti, R., Santosa, Y. T., Faiziyah, N., Hidayati, Y. M., Budi, A. B. S., Sukmawati, T., Suharna, H., Tong, D. H., & Lam, T. T. (2026). Students’ problem-solving patterns in arithmetic series tasks viewed through reflective thinking. Discover Education, 5(1), Article 991. https://doi.org/10.1007/s44217-026-02176-4

Image Credits: AI Generated

DOI: 10.1007/s44217-026-02176-4

Keywords: reflective thinking, arithmetic series, problem-solving, mathematics education, non-routine problems, think-aloud protocol, qualitative research, metacognition, visualization, algebraic variables, high school students, pedagogy