Ghanaian pupils see squares but miss rectangle properties

Ghanaian Grade 6 pupils readily recognized squares in familiar visual forms, but most failed to identify rectangles in non-standard orientations and showed even weaker understanding of the properties that define both shapes. In a survey of 112 pupils from three public schools in Kwadaso Municipality, 93.7% correctly identified a square in one appearance-based item and 79.5% did so in another. By contrast, only 7.1% selected both rectangles in a pair of non-standard figures, while 13.4% answered another appearance-based rectangle item correctly. The results indicate that visual familiarity, rather than a stable concept of the figures, drove much of the pupils’ performance.

The study examined whether learners retained geometry content expected from the end of Grade 5 in Ghana’s National Council for Curriculum and Assessment, or NaCCA, curriculum. Grade 5 learners are expected to sort quadrilaterals by their attributes and understand properties of squares and rectangles involving sides, angles and diagonals. The research focused on Grade 6 pupils because understanding these shapes is expected to provide a foundation for later work in area, perimeter and more advanced geometry. It also asked how accurately pupils could name the figures from their appearance and what foundational knowledge they had about their defining attributes.

Researchers used an adapted version of the Van Hiele geometric thinking test. The first 10 of its 25 multiple-choice items were adapted to the Ghanaian curriculum: five assessed visual recognition and five addressed geometric properties. Two senior mathematics teachers and a university lecturer specializing in mathematics education reviewed the instrument for curriculum alignment, cognitive demand and content validity. It was pilot tested with 20 pupils at a school outside the study, after which ambiguous questions were revised. A test-retest reliability coefficient of 0.81 was reported for the instrument.

The participants were in their first term of the 2025–2026 academic year and had spent one month in Grade 6 when tested. The sample included 51 boys, or 45.5%, and 61 girls, or 54.5%. Ages were concentrated in two groups: 55 pupils, or 49.1%, were aged 10–11, while 56, or 50.0%, were aged 12–13; one participant, or 0.9%, was 14–15. The pupils came from three conveniently selected public schools within a learner population of 312. Their responses were scored descriptively, with each of the 10 items receiving either one point for a correct answer or zero for an incorrect one.

The strongest visual recognition did not extend to rectangles. In the item containing Shapes S and T, only 8 pupils, or 7.1%, correctly identified both as rectangles, compared with 104, or 92.9%, who answered incorrectly. The most common error, selected by 79.6% of pupils, was to recognize only Shape T as a rectangle, suggesting that an elongated, familiar outline carried more weight than the defining properties. On the second rectangle item, 15 pupils, or 13.4%, gave the correct answer, while 97, or 86.6%, did not. These results show a marked contrast with the high rates of success on the two square-identification items.

Property-based questions produced uniformly low performance. When identifying a rectangle from combinations of attributes such as equal opposite sides, equal diagonals, parallel opposite sides and equal angles, 14 pupils, or 12.5%, selected the correct combination on Item 6. Fifteen pupils, or 13.4%, answered Item 8 correctly, and 30, or 26.8%, answered Item 10 correctly. Even on the best-performing rectangle item, fewer than three in ten pupils demonstrated the expected knowledge. The pattern indicates that a pupil could sometimes recognize a conventional rectangle without being able to describe or apply the properties that make a figure rectangular.

Squares presented a similar problem when pupils had to work from properties rather than appearance. Correct responses reached 19.6% on Item 5, with 22 of 112 pupils answering correctly, and 24.1% on Item 7, with 27 answering correctly. Item 9 was the most difficult: only 8 pupils, or 7.1%, selected the correct answer. The authors reported that most pupils did not recognize that the diagonals of a square bisect one another at right angles, a feature closely connected with the low success on that item. Thus, familiarity with the square’s outward appearance was not matched by knowledge of its internal structure.

Overall scores confirmed the separation between shape recognition and conceptual understanding. Pupils received between zero and 6 correct answers out of 10; none scored 7 or more. The mean score was 2.34, the median was 2 and the mode was also 2. Using the study’s proficiency categories, 59.8% of learners were classified as having low proficiency, 38.4% had moderate proficiency, and 1.8% reached the highest category. Because the instrument included both appearance and property items, the totals combined different kinds of reasoning. Still, the distribution shows that high recognition of familiar squares did not produce near-mastery of the tested geometry content.

The findings were interpreted through the Van Hiele model of geometric thinking. At Level 1, known as visualization, learners recognize figures as visual wholes and depend heavily on prototypes and appearance. Level 2, or analysis, requires them to identify and characterize figures through their properties and spatial relationships. The pupils’ concentration on elongated forms for rectangles, difficulty with non-standard orientations and low success on property items placed their demonstrated reasoning predominantly at Level 1, below the property-based analysis described in the curriculum expectations. The survey did not test the higher levels of the model involving hierarchical classification, formal deduction or axiomatic reasoning, so it cannot establish whether any pupils had progressed beyond this developmental stage.

The study has important limitations. It examined only squares and rectangles, restricting what can be inferred about learners’ understanding of the wider family of quadrilaterals. Although 112 pupils formed a sizable sample, all attended three schools within one municipality, and the schools were conveniently selected, limiting representativeness and generalizability. The researchers also relied on a single adapted test rather than combining it with interviews, classroom observation or other evidence of reasoning. The results are therefore best understood as a snapshot of one cohort in one part of Ghana, not a national estimate or proof that a particular teaching method caused the performance pattern.

Despite those constraints, the data identify a clear instructional gap: pupils could often name familiar shapes without reliably applying their defining properties. The authors recommend concept-based teaching that asks learners to explore, compare and classify quadrilaterals by attributes, together with activities that make hierarchical relationships explicit. Such activities could help pupils examine why a square belongs to the broader class of rectangles, distinguish elongated prototypes from the general category and connect sides, angles and diagonals. The study did not test those interventions, so their effects remain unknown, but the low property-based scores show why moving beyond visual naming is a central challenge for primary geometry in the surveyed Ghanaian classrooms.

Subject of Research: Grade 6 learners’ visual and property-based understanding of squares and rectangles in Ghana

Article Title: An analysis of Grade 6 learners’ geometric understanding of rectangles and squares

Article References: Enu, J., & Larbi, E. (2026). An analysis of Grade 6 learners’ geometric understanding of rectangles and squares. Frontiers in Education, 11, Article 1964887. https://doi.org/10.3389/feduc.2026.1964887

Image Credits: AI Generated

DOI: 10.3389/feduc.2026.1964887

Keywords: geometry education, primary mathematics, Grade 6 learners, Ghana, squares, rectangles, quadrilaterals, geometric reasoning, Van Hiele model, prototype effect, analysis, Grade