University Geometry Errors Stem Primarily from Process Skill and Transformation Breakdowns

When university mathematics education students tackle geometry word problems, their breakdowns occur far more frequently in executing mathematical procedures and translating problem statements into algebraic formulations than in basic reading comprehension, according to an explanatory sequential mixed-methods investigation conducted at Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development in Kumasi, Ghana. Published in Frontiers in Education, the study evaluated where cognitive obstacles emerge as post-secondary students attempt to bridge spatial conditions and symbolic manipulation. Working with a cohort of first-year undergraduate students, researchers Patience Sumalia Asie, Ernest Larbi, Kwaku Appiah, and Francis Amenorfe applied the Newman Error Analysis framework to classify student mistakes across five sequential stages: reading, comprehension, transformation, process skills, and encoding. Rather than viewing geometry performance purely through aggregate scores or final numerical correctness, the research team isolated the structural junctions where mathematical problem-solving breaks down.

The study was designed around an explanatory sequential mixed-methods architecture, a methodology in which quantitative test evaluations precede and inform qualitative interview investigations. To establish their cohort, the researchers drew from a target population of 240 Level 100 Mathematics Education students enrolled in the Department of Mathematics Education. Using stratified random sampling to guarantee balanced representation across genders, the quantitative phase engaged 80 undergraduate students, split evenly between 40 male students and 40 female students. Within this sample of 80 participants, demographic records indicated that 50 students, or 62.5 percent, were between the ages of 18 and 20 years, while 30 participants, representing 37.5 percent, were aged 20 years or older. This initial quantitative phase provided a structured foundation to assess error distributions across standardized test conditions before exploring individual learner perspectives in qualitative interviews.

To evaluate problem-solving stages, the authors developed the Geometry Word Problem Test, a diagnostic instrument comprising three open-ended word problems requiring the integration of geometric concepts, symbolic formulation, and algebraic reasoning. Unlike closed-ended or multiple-choice formats, this instrument mandated that participants show every intermediate step and computational calculation within a 90-minute testing window administered in regular classroom environments. To verify content and construct validity, two university professors of mathematics education reviewed the test items, providing critical feedback to refine their clarity, relevance, and alignment with research objectives. The diagnostic rubric scored items using an open-ended five-point scale spanning zero to five marks per question. Prior to the main administration, a pilot study involving comparable undergraduate students yielded continuous partial-credit scores that, when analyzed using SPSS version 27, demonstrated high internal consistency, producing a Cronbach’s alpha coefficient of 0.82.

Across the quantitative phase, the assessment generated 240 question-level student responses across the 80 participants, with every response systematically coded according to the five Newman Error Analysis tiers. The diagnostic tally revealed that process skill errors constituted the single largest category of student difficulty, accounting for 95 occurrences, or 39.6 percent of all coded responses. Transformation errors formed the second most prevalent category, appearing 65 times and making up 27.1 percent of the total. When combined, these two procedural and representational categories comprised 66.7 percent of all identified errors, representing fully two-thirds of the recorded mistakes. In contrast, difficulties at earlier and later stages occurred with markedly lower frequency: comprehension errors accounted for 32 occurrences, or 13.3 percent, encoding errors appeared 30 times, or 12.5 percent, and reading errors were the least frequent, registering just 18 occurrences, or 7.5 percent.

The researchers also investigated whether mathematical problem-solving performance differed significantly across student gender. Before undertaking comparative inferential testing, the team verified the parametric assumption of normality across score distributions using SPSS version 27. The Kolmogorov-Smirnov test showed non-significant results for male students, with D equal to 0.050 and p equal to 0.200, as well as for female students, with D equal to 0.065 and p equal to 0.200. Similarly, the Shapiro-Wilk test yielded non-significant departures from normality among male participants, with W equal to 0.988 and p equal to 0.698, and among female participants, with W equal to 0.987 and p equal to 0.572. Because all p-values exceeded the conventional 0.05 threshold, the performance score distribution satisfied criteria for parametric evaluation.

Subsequent independent-samples t-test evaluations indicated that gender did not serve as a statistically significant determinant of geometry problem-solving performance. Descriptive scoring revealed that male students achieved a mean score of 7.5625, with a standard deviation of 3.20139 and a standard error mean of 0.58221, while female students achieved a mean score of 6.7333, with a standard deviation of 3.18329 and a standard error mean of 0.58319. Although descriptive performance varied slightly across discrete score brackets—with 37.5 percent of male students and 32.5 percent of female students falling into the 10 to 12 point range—the mean difference was not statistically significant, yielding a t-value of 1.54 across 78 degrees of freedom and a p-value of 0.127. Furthermore, the calculated Cohen’s d effect size was 0.27, representing a small magnitude that underscores comparable performance across male and female undergraduates.

Following the quantitative analysis, the researchers initiated the qualitative phase to explore the underlying cognitive challenges and personal perceptions shaping these error patterns. Using purposive sampling based on the diversity of errors observed in the written test scripts, the researchers selected five students representing distinct performance levels and error profiles spanning comprehension, transformation, process skills, and encoding. Individual semi-structured interviews lasting approximately ten minutes were conducted within the participants’ familiar classroom setting. These discussions were audio-recorded, supplemented by observational field notes, and transcribed verbatim for inductive thematic analysis following the Braun and Clarke framework. Qualitative rigor and trustworthiness were maintained through an audit trail documenting analytical decisions, peer debriefing conducted with two school mathematics facilitators, and formal member checking wherein synthesized interpretations were returned to participants for verification.

The thematic analysis identified four primary qualitative themes, beginning with pronounced difficulties in understanding problem statements. Many participants reported that complex sentence structures, unfamiliar technical phrasing, and dense information loads hindered their ability to extract the core question, even when they possessed foundational mathematical knowledge. As one participant noted, reading the text allowed them to identify the general topic, but they remained uncertain about what specific quantity the problem asked them to determine. Another participant remarked that textual wording generated confusion because they struggled to differentiate essential mathematical information from irrelevant contextual details. These comprehension breakdowns caused students to focus on extraneous elements, creating misconceptions that persisted downstream into procedural calculations.

A second major theme centered on the acute challenges students encountered when translating descriptive verbal statements into formal mathematical representations. Even when participants grasped the general narrative context, they struggled to establish variables, select appropriate symbolic notations, and construct correct algebraic equations. As one interviewee explained, reading and understanding the situation was manageable, but converting verbal relationships into formal equations presented an imposing hurdle, while another cited determining which formula or equation to apply as the most difficult task. These difficulties extended directly into spatial visualization, where learners frequently failed to draft accurate geometric diagrams, thereby creating substantial obstacles at the transformation stage before algebraic computation could proceed.

Finally, the qualitative findings documented that while students widely perceived geometry word problems as complex, cognitively demanding, and anxiety-inducing, they simultaneously recognized their educational value. Several participants noted that word problems compelled deeper mathematical reasoning beyond the passive memorization of formulas, demonstrating how classroom mathematics applies to authentic contexts outside educational institutions. Integrating their findings with constructivist learning theory, the authors concluded that errors in geometry problem-solving do not merely stem from isolated computational slips, but instead reflect fragmented conceptual integration between geometric relationships and algebraic symbolism. To address these systemic hurdles, the study recommends that mathematics educators implement guided instructional approaches that emphasize algebraic reasoning, problem comprehension, and multi-representational translation.

Subject of Research: Undergraduate algebraic error patterns in geometry word problem solving

Article Title: Analysis of algebraic errors university students make in solving geometry problems using Newman error analysis

Article References: Asie, P. S., Larbi, E., Appiah, K., & Amenorfe, F. (2026). Analysis of algebraic errors university students make in solving geometry problems using Newman error analysis. Frontiers in Education, 11, Article 1870459. https://doi.org/10.3389/feduc.2026.1870459

Image Credits: AI Generated

DOI: 10.3389/feduc.2026.1870459

Keywords: geometry education, word problems, Newman error analysis, mathematics education, algebraic reasoning, process skill errors, transformation errors, explanatory sequential design, undergraduate mathematics, Ghanaian education, constructivist theory, error analysis, mathematical problem solving, spatial visualization, gender differences in mathematics, student perceptions