The Impact of Visual Representation on Solving Mathematical Word Problems: A Comparative Study Across Number, Proportion, and Geometry Domains

INTRODUCTION
Mathematical word problems represent one of the most challenging aspects of mathematics education. Over the past 50 years, they have been the focus of extensive research as educators increasingly emphasize the application of mathematical knowledge in real-life situations (Verschaffel et al., 2020; Kilpatrick, 1996; OECD, 1999). The successful solution of mathematical problems requires understanding relationships, recognizing data, and comprehending concepts, where mental-visual representations play a crucial role (Haghverdi & Goya, 2010). Visual representation is one method for presenting mathematical problems. While word problems are often accompanied by images that can enhance understanding, questions remain about whether visual representations are beneficial for the learning process itself (Noori Afshar et al., 2017; Seufert, Jänen & Brünken, 2007). This research examines how changing mathematical word problems from descriptive to visual representation affects student performance across different educational levels (elementary sixth grade and lower secondary) and content domains (numbers, proportions, and geometry).
THEORETICAL FRAMEWORK
Visual representations are typically an indirect representation of the real world for students (Beaumont et al., 2017). According to Diezmann and English (2003), visual representation in mathematics education is important for two reasons: it supports reflection of problem data and provides meaning for connecting mathematical ideas. Hegarty and Kozhevnikov (1999) distinguished schematic representation (concrete visualization) from diagrammatic representation (abstract, emphasizing relationships). Elia and Philippou (2004) classified images into four functional types: (1) Decorative images with no connection to the problem; (2) Representational images describing the problem scene; (3) Organizational images providing instructions for problem-solving; and (4) Informational images providing essential information for understanding or solving the problem.
METHODOLOGY
This quasi-experimental study used paired equivalent problems presented in both visual and verbal forms. The sample consisted of 60 female students from Borujerd city (20 elementary sixth grade, 40 lower secondary), selected through convenience sampling. The research instrument comprised 21 paired problems equally distributed across numbers (7), proportions (7), and geometry and measurement (7). Each problem was designed in both descriptive and visual formats. For each student, 12 of the 21 problems were randomly selected as descriptive problems (Version A), while the remaining 9 appeared as visual problems (Version B), presented in random order. Students had 60 minutes to complete the test, with calculator use permitted. Data were analyzed using paired and independent t-tests, with responses scored as incorrect/correct (0 or 1).
FINDINGS
Results showed that changing word problems from descriptive to visual representation significantly affected student performance. Students' average scores in all three content domains were significantly higher in the visual version. The paired t-test confirmed this difference was statistically significant (t = -5.90, p<0.05), with a negative Pearson correlation indicating an inverse relationship between descriptive and visual problem difficulty. In elementary school, descriptive problem scores averaged 10.65 (SD=2.39) versus 16.50 (SD=3.14) for visual problems—a difference of 5.85 points. In lower secondary, descriptive scores averaged 12.40 (SD=2.12) versus 14.58 (SD=2.27) for visual problems—a difference of 2.18 points. Independent t-tests confirmed significant performance differences between educational levels (p<0.05). Notably, lower secondary students performed better on descriptive problems due to developed language skills, while elementary students excelled with visual problems, suggesting stronger visual-spatial thinking at younger ages. Content domain analysis revealed varying effects: In numbers, descriptive scores averaged 10.17 (SD=5.44) versus 14.67 (SD=5.20) for visual—a 4.5-point increase. In proportions, scores were 12.42 (SD=5.48) versus 15.67 (SD=6.92)—a 3.25-point increase. In geometry, scores were 13.00 (SD=5.38) versus 16.00 (SD=4.94). Although the geometry domain showed a 3-point numerical difference, the lower standard deviation in visual problems indicated more stable and consistent effects, aligning with the spatial-visual nature of geometric problems.
DISCUSSION AND CONCLUSION
This research demonstrated that changing problem representation from descriptive to visual significantly improves students' problem-solving ability, consistent with previous research (Azizi Mahmoudabad et al., 2019; Beaumont et al., 2017). Excessive verbal content often negatively affects performance (Boonen et al., 2014; Fuchs et al., 2015), explaining the positive effect of visual representation. A key finding is the differential effect across content domains. Visual representation was most effective in geometry and measurement, consistent with Hoogland et al. (2016) and Schultz et al. (2010), likely due to the spatial-visual nature where visual representation directly supports mental model construction. In proportions, the effect was relatively less, possibly due to inherent complexity requiring quantity-to-whole comparisons. Elementary students benefited more from visual representation, suggesting greater reliance on visual-spatial thinking, while lower secondary students performed better on descriptive problems due to developed language skills. These findings have important implications for designing educational materials: balanced use of descriptive and visual representations according to content domain and educational level can enhance learning and problem-solving. Study limitations include the restriction to female students and convenience sampling. Future research should employ larger, more diverse samples and qualitative methods such as interviews and think-aloud protocols to investigate factors affecting visual representation effectiveness more deeply.