Session Information
99 ERC SES 07 H, Research on Arts Education
Paper Session
Contribution
Problem-solving is an important part of the primary school mathematics curriculum. The purpose of problem-solving activities in the classroom is to apply abstract mathematical concepts to real-world situations. (Verschaffel et. al, 2000; Mellone, et. al, 2014) Riley et. al (1983) described the conceptual knowledge required to solve simple addition or subtraction word problems in terms of semantic relations residing between quantitative information existing in problem text: compare, combine and change.
For primary school pupils, comprehending word problems at the early stages of learning to read, is a difficult task. Despite their lack of understanding, some pupils still engage in the solving process, employing arbitrary strategies, such as randomly combining numbers existing in the problem into mathematical operations suggested by specific keywords in the problem i.e. “more” for addition and “less” for subtraction (Schoenfeld, 1991).
Research on mathematical reasoning evidenced that mental representations of abstract mathematical concepts appear to be visual, originating in one's visually sensed experiences. (Bishop, 1989) Arcavi (2003) described visualization as the ability, process and product of creation, interpretation, use and reflection upon pictures, images, and diagrams in our minds or paper or with other technological means to describe and communicate information, develop thinking, and advance understanding of new ideas. Dreyfus (1991) described visual reasoning in mathematics as a process of expressing verbal information in concrete visual representations that illustrate the relationships between mathematical expressions and concepts. By advancing the use of visual reasoning in mathematics learning, comprehension is translated into one's ability to use the given information to solve problems. (Mayer, 1989)
To improve pupils' problem-solving abilities, recent studies explored different methods of facilitating the understanding of mathematical relations in word problems. In his research, Glenberg et al. (2012) improved elementary school pupils' problem-solving performance by having them physically manipulate objects that recreated the problem situation, which supported forming accurate mental representations of the relations between quantitative information in the problem. Dewolf et. al (2017) investigated the effect of representational illustrations that accompanied problematic word problems in solving process, expecting to help pupils mentally imagine the situation and solve the problems more realistically by employing everyday life knowledge. The findings evidenced no positive effect on realistic problem-solving.
Research question/ hypothesis
The current study investigates the effect of visual reasoning on the solving process of a mathematical word problem which involves part-whole relations between sets of elements. To test our hypothesis, we worked on word problems commonly encountered in the first-grade mathematics curriculum, which required addition and subtraction operations to determine the problem solution. We also aimed to investigate and describe the correlation between students' reading comprehension abilities and their visual and mathematical reasoning performances.
We expect that visual reasoning will help pupils form accurate mental representations of the mathematical relations in the problem, improving their comprehension of the problem situation and increasing the number of correct problem solutions afterwards.
- Is it possible to improve first-graders' word problem comprehension by asking them to create visual representations of the problem situation by drawing?
- Will the number of correct problem solutions increase if pupils create correct visual representations of the problem situation, by drawing?
- Will pupils with average reading comprehension abilities create accurate visual representations of the problem situation, by drawing and determining the correct problem solution afterwards?
We hypothesized that asking first-grade pupils to create visual representations of the problem situation by drawing will improve comprehension, determining an increased number of correct problem solutions.
We also predicted that pupils with average reading comprehension abilities would create correct visual representations of the problem situation leading them to perform the appropriate operations to determine the correct problem solution.
Method
Design: quasi-experiment, one group pretest-post-test. Participants: 45 first-grade pupils (22 boys and 23 girls) with ages of 7 and 8 years old (mean age 7.13). The pupils belonged to two first-grade classes from the same urban primary school in Cluj-Napoca, Romania. The pupils were assigned to each class randomly, following the Romanian class formation legislation in 2021. At the beginning of the experiment, pupil's mathematics performances and reading comprehension abilities (RCA) were globally assessed by their teacher, by completing an individual form. The individual mathematical abilities (IMP) ranged from very good (= 27 participants, 15 boys and 12 girls), good (= 10 participants, 2 boys and 8 girls); sufficient (= 6 participants, 4 boys and 2 girls), to insufficient (= 2 participants, 1 male and 1 female). The reading comprehension abilities ranged from 1 (poor) to 5 (high) as follows: 1 (= 5 participants, 3 boys and 2 girls), 2 (= 3 participants, 1 boy and 2 girls), 3 (= 8 participants, 3 boys and 2 girls), 4 (= 16 participants, 8 boys and 8 girls), to 5 (= 13 participants, 7 boys and 6 girls). The participants were tested in two different contexts: In normal context, pupils received the following word problem, containing compare and combine semantic relations between sets of objects, in an individual paper-and-pencil task during a usual mathematics class: Radu has 3 pencils, and Tudor has 4 more pencils than Radu. How many pencils do children have altogether? The problem was read aloud once by the teacher. Pupils were instructed to read the problem again and solve it independently, writing down the solution procedure and the answer on paper. In visual context, a similar word problem was given during another regular mathematics class: 5 frogs are sitting on a water lily leaf and 3 less frogs are sitting on the leaf nearby. How many frogs are sitting on the lily leaves altogether? The problem was written on the board and read aloud once, by the teacher. The pupils were instructed to individually read and illustrate the problem situation by drawing, following the information in the problem statement. Afterwards, they were required to perform the mathematical operations and determine the numerical solution of the problem, on the back of the page.
Expected Outcomes
Problem solutions and visual representations were categorised as follows:: Correct Problem Solution (= CPS): participants determined the number of elements required by the problem statement, performing one or two operations; Solution Error (= SE/ pSE): participants only performed subtraction 5–3=2 (=partial solution error, pSE) or provided other numeric solution than CPS; No Answer (= N/A) Correct Visual Representation (= CVR): accurate illustration of numeric information and of the relations between the two sets of elements; Representation Error (= RE): incorrectly illustrates the sets of elements that must be combined to determine the whole value. In normal context, we assumed that understanding the problem situation was associated with the amount of CPS. Solving problems in visual context revealed increased comprehension of the problem situation, reflected by the amount of CVR. Data analysis in SPSS revealed a significant correlation (p=0.044<0.05) between the problem solutions determined in normal context and the problem solutions determined in visual context. Findings evidenced significantly improved problem solutions when pupils solved the problem in visual context compared to problem solutions determined in normal context. Pupils with higher RCA and IMP levels who determined CPS in normal context maintained their performance in visual context. About a third of pupils that provided pSE in normal context, most of them with very good IMP and medium RCA, determined CPS in visual context. Despite the positive effect of using visual reasoning in solving problems, about half of the participants with CVR couldn’t determine CPS. Participants with CVR who provided SE couldn’t associate mathematical operations required to determine the numeric solution and combined numbers in the problem into a subtraction suggested by the keyword “less”. Therefore, illustrating the problem situation by drawing can be a helpful tool in current teaching practice because of its positive effect on problem comprehension and solving process.
References
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52, 215–241. https://doi.org/10.1023/A:1024312321077 Bishop, A. J. (1988). A review of research on visualization in mathematics education. In A. Borbás (Ed.), Proceedings of the 12th PME International Conference (vol. 1, pp. 170–176). OOK Printing House. Dewolf, T., Dooren, W., & Verschaffel, L. (2017). Can visual aids in representational illustrations help pupils to solve mathematical word problems more realistically? European Journal of Psychology of Education, 32(3), 335–351. https://doi.org/10.1007/s10212-016-0308-7 Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the 15th PME International Conference, 1, 33-48. Glenberg, A., Willford, J., Gibson, B., Goldberg, A., & Zhu, X. (2012). Improving Reading to Improve Math. Scientific Studies of Reading, 16(4), 316–340. 10.1080/10888438.2011.564245 Riley, M. S., Greeno, J. G., & Heller, I. J. (1983). Development of Children’s Problem-Solving Ability in Arithmetic. In H. P. Ginsburg (Ed.), The Development of Mathematical Thinking (pp. 153–196). Academic Press. Mayer, R. E. (1989). Models for Understanding. Review of Educational Research, 59(1), 43–64. https://doi.org/10.3102/00346543059001043 Mellone, M., Verschaffel, L., & Van Dooren, W. (2014). Making sense of word problems: The effect of rewording and dyadic interaction. In P. Liljedahl, S. Oesterle, C. Nicol & D. Allan (Eds.), Proceedings of the Joint Meeting of PME 38 and PME-NA 36, Vol. 4, (pp. 201‒208). https://www.pmena.org/pmenaproceedings/PMENA%2036%20PME%2038%2020 14%20Proceedings%20Vol%204.pdf Schoenfeld, A. H. (1991). On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics. In J. F. Voss, D. N. Perkins, & J. W. Segal (Eds.), Informal reasoning and education (pp. 311–343). Lawrence Erlbaum Associates. Verschaffel, L., Greer, B., & De Corte, E. (2000). Making sense of word problems. Swets and Zeitlinger.
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