# The effects of using drawings in developing young children’s mathematical word problem solving: A design experiment with third-grade Hungarian students

## Abstract

The present study aims to investigate the effects of a design experiment developed for third-grade students in the field of mathematics word problems. The main focus of the program was developing students’ knowledge about word problem solving strategies with an emphasis on the role of visual representations in mathematical modeling. The experiment involved five experimental and six control classes (*N* = 106 and 138, respectively) of third-grade students. The experiment comprised 20 lessons with 73 word problems, providing a systematic overview of the basic word problem types. Teachers of the experimental classes received a booklet containing lesson plans and overhead transparencies with different types of visual representations attached to the word problems. Students themselves were invited to make drawings for each task, and group work and teacher-led discussion shaped their beliefs about the role of visual representations in word problem solving. The effect sizes of the experiment were calculated from the results of two tests: an arithmetic skill and a word problem test, and the unbiased estimates for Cohen’s *d* proved to be 0.20 and 0.62. There were significant changes also in experimental group students’ beliefs about mathematics. The experiment pointed to the possibility, feasibility, and importance of learning about visual representations in mathematical word problem solving as early as in grade 3 (around age 9–10).

## Keywords

Word problem Visual representation Design experiment Elementary school mathematics## Notes

### Acknowledgments

This research was supported by a grant from the Hungarian Scientific Research Fund (OTKA 63360) awarded to Csaba Csíkos and by the MTA-SZTE Research Group on the Development of Competencies. Thanks are due to Gabriella Pataky for her help in the use of the Clark Drawing Test. We would also like to thank Paul Andrews, Andrea Kárpáti, Julianna Szendrei, Malcolm Swan and Lieven Verschaffel for their constructive comments on an earlier draft of this paper. Parts of the present study have been presented at a conference (Csíkos, Szitányi & Kelemen, 2009). Also, part of the data have been published in Hungarian in Magyar Pedagógia (Csíkos, Szitányi & Kelemen, 2010).

## References

- Báthory, Z. (2003). Rendszerszintű pedagógiai felmérések. [System-level educational surveys].
*Iskolakultúra, 13*(8), 3–19.Google Scholar - Bell, P. (2004). On the theoretical breadth of design-based research in education.
*Educational Psychologist, 39*, 243–253.CrossRefGoogle Scholar - Berends, I. E., & van Lieshout, E. C. D. M. (2009). The effect of illustrations in arithmetic problem-solving: Effects of increased cognitive load.
*Learning and Instruction, 19*, 345–353.CrossRefGoogle Scholar - Boaler, J. (1994). When do girls prefer football to fashion? A analysis of female underachievement in relation to ‘realistic’ mathematics context.
*British Educational Research Journal, 20*, 551–564.CrossRefGoogle Scholar - Brown, A. L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings.
*The Journal of the Learning Sciences, 2*, 141–178.CrossRefGoogle Scholar - Cai, J., & Lester, F. K. (2005). Solution representations and pedagogical representations in Chinese and U.S. classrooms.
*Journal of Mathematical Behavior, 24*, 221–237.CrossRefGoogle Scholar - Chapman, O. (2006). Classroom practices for context of mathematics word problems.
*Educational Studies in Mathematics, 62*, 211–230.CrossRefGoogle Scholar - Clark, G. (1989). Screeining and identifying students talented in the visual arts. Clark’s Drawing Abilities test.
*Gifted Child Quaterly, 33*, 98–106.CrossRefGoogle Scholar - Cohen, J. (1969).
*Statistical power analysis for the behavioral sciences*. New York: Academic.Google Scholar - Cooper, B., & Harries, T. (2002). Children’s responses to contrasting ‘realistic’ mathematics problems: Just how realistic are children ready to be?
*Educational Studies in Mathematics, 49*, 1–23.CrossRefGoogle Scholar - Csíkos, C. (2005).
*A metacogniton-based training program in grade 4 in the fields of mathematics and reading*. Paper presented at the 11th European Conference for Research on Learning and Instruction, Nicosia, Cyprus, 23–27, August, 2005.Google Scholar - Csíkos, C., Szitányi, J., & Kelemen, R. (2009).
*Promoting 3rd grade children’s mathematical problem solving through learning about the role of visual representations*. Paper presented at the 13th European Conference for Research on Learning and Instruction held in Amsterdam, The Netherlands, August 25—August 29.Google Scholar - Csíkos, C., Szitányi, J., & Kelemen, R. (2010). Vizuális reprezentációk szerepe a matematikai problémamegoldásban. Egy 3. osztályos tanulók körében végzett fejlesztő kísérlet eredményei. [The role of visual representations in mathematical problem solving: Results of an intervention program with 3rd grade students].
*Magyar Pedagógia, 110*, 149–166.Google Scholar - Diezmann, C. M. (2005). Primary students’ knowledge of the properties of spatially-oriented diagrams. In H. L. Chick & J. L. Vincent (Eds.),
*Proceedings of the International Group for the Psychology of Mathematics Education*(pp. 281–288). Melbourne: PME.Google Scholar - Elia, I., Gagatsis, A., & Demetriou, A. (2007). The effects of different mode of representation on the solution of one-step additive problems.
*Learning and Instruction, 17*, 658–672.CrossRefGoogle Scholar - English, L. D. (1996). Children’ construction of mathematical knowledge in solving novel isomorphic problems in concrete and written form.
*Journal of Mathematical Behavior, 15*, 81–112.CrossRefGoogle Scholar - English, L. D. (1997). The development of fifth-grade children’s problem solving abilities.
*Educational Studies in Mathematics, 34*, 183–217.CrossRefGoogle Scholar - English, L. D., & Halford, G. S. (1995).
*Mathematics education: Models and processes*. Mahwah, NJ: Erlbaum.Google Scholar - Gagatsis, A., & Elia, I. (2004). The effects of different modes of representations on mathematical problem solving. In M. Johnsen-Hoines & A. Berit-Fuglestad (Eds.),
*Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education: Vol. 2*(pp. 447–454). Bergen, Norway: Bergen University College.Google Scholar - Geary, D. C. (1999). Sex differences in mathematical abilities: Commentary on the math-fact retrieval hypothesis.
*Contemporary Educational Psychology, 24*, 267–274.CrossRefGoogle Scholar - Goldin, G. A., & Kaput, J. (1996). A joint perspective on the idea of representation in learning and doing mathematics. In L. Steffe, P. Nesher, P. Cobb, G. Goldin, & B. Greer (Eds.),
*Theories of mathematical learning*(pp. 397–430). Hillsdale: Erlbaum.Google Scholar - Gravemeijer, K., & Cobb, P. (2006). Design research from a learning design perspective. In J. van den Akker, K. Gravemeijer, S. McKenney, & N. Nieveen (Eds.),
*Educational design research*(pp. 17–51). London: Routledge.Google Scholar - Halpern, D. F., & LaMay, M. L. (2000). The smarter sex: A critical review of sex differences in intelligence.
*Educational Psychology Review, 12*, 229–246.CrossRefGoogle Scholar - Hegarty, M., & Kozhevnikov, M. (1999). Types of visual-spatial representations and mathematical problem solving.
*Journal of Educational Psychology, 91*, 684–689.CrossRefGoogle Scholar - Hegarty, M., Mayer, R. E., & Monk, C. A. (1995). Comprehension of arithmetic word problems: A comparison of successful and unsuccessful problem solvers.
*Journal of Educational Psychology, 87*, 18–32.CrossRefGoogle Scholar - Jonassen, D. H. (2003). Designing research-based instruction for story problems.
*Educational Psychology Review, 15*, 267–296.CrossRefGoogle Scholar - Kárpáti, A. (2001).
*Firkák, formák, figurák. A vizuális nyelv fejlődése a kisgyermekkortól a serdülőkorig. [Scribbles shapes and figures. The visual language of children]*. Budapest Hungary: Dialóg Campus Kiadó.Google Scholar - Kozhevnikov, M., Hegarty, M., & Mayer, R. E. (2002). Revising the visualizer–verbalizer dimension: Evidence for two types of visualizers.
*Cognition and Instruction, 20*, 47–77.CrossRefGoogle Scholar - Kramarski, B., Mevarech, Z. R., & Arami, M. (2002). The effects of metacognitive instruction on solving mathematical authentic tasks.
*Educational Studies in Mathematics, 49*, 225–250.CrossRefGoogle Scholar - Kramarski, B., Mevarech, Z. R., & Lieberman, A. (2001). Effects of multilevel versus unilevel metacognitive training on mathematical reasoning.
*The Journal of Educational Research, 94*, 292–300.CrossRefGoogle Scholar - Lave, J. (1992). Word problems: A microcosm of theories of learning. In P. Light & G. Butterworth (Eds.),
*Context and cognition. Ways of learning and knowing*(pp. 74–92). Hillsdale, NJ: Erlbaum.Google Scholar - Mayer, R. E., & Hegarty, M. (1996). The process of understanding mathematical problems. In R. J. Sternberg & T. Ben-Zeev (Eds.),
*The nature of mathematical thinking*(pp. 29–53). Mahwah, NJ: Erlbaum.Google Scholar - Mevarech, Z. R., & Kramarski, B. (1997). IMPROVE: A multidimensional method for teaching mathematics in heterogeneous classrooms.
*American Educational Research Journal, 2*, 365–394.Google Scholar - Morris, S. B. (2005).
*Effect size estimation from prestest–posttest–control designs with heterogeneous variances.*Paper presented at the 20th Annual Conference of the Society for Industrial and Organizational Psychology, Los Angeles, CA.Google Scholar - Mwangi, W., & Sweller, J. (1998). Learning to solve compare word problems: The effect of example format and generating self-explanations.
*Cognition and Instruction, 16*, 173–199.CrossRefGoogle Scholar - Nemzeti Alaptanterv (2007) [National Core Curriculum] Available from: http://www.okm.gov.hu/letolt/kozokt/nat_070926.pdf.
- Opfer, J. E., & Siegler, R. S. (2007). Representational change and children’s numerical estimation.
*Cognitive Psychology, 55*, 169–195.CrossRefGoogle Scholar - Palm, T. (2008). Impact of authenticity on sense making in word problem solving.
*Educational Studies in Mathematics, 67*, 37–58.CrossRefGoogle Scholar - Pantziara, M., Gagatsis, A., & Elia, I. (2009). Using diagrams as tools for the solution of non-routine mathematical problems.
*Educational Studies in Mathematics, 72*, 39–60.CrossRefGoogle Scholar - Presmeg, N. (1986). Visualisation in high school mathematics.
*For the Learning of Mathematics, 6*, 42–46.Google Scholar - Sáenz-Ludlow, A., & Walgamuth, C. (1998). Third-graders interpretations of equality and the equal symbol.
*Educational Studies in Mathematics, 35*, 153–187.CrossRefGoogle Scholar - Schneider, M., Heine, A., Thaler, V., Tornbeyns, J., De Smedt, B., Verschaffel, L., et al. (2008). A validation of eye movements as a measure of elementary school children’s developing number sense.
*Cognitive Development, 23*, 409–422.CrossRefGoogle Scholar - Selter, C. (1998). Building on children’s mathematics—a teaching experiment in grade three.
*Educational Studies in Mathematics, 36*, 1–27.CrossRefGoogle Scholar - Siegler, R. S., & Opfer, J. E. (2003). The development of numerical estimation: Evidence for multiple representations of numerical quantity.
*Psychological Science, 14*, 237–243.CrossRefGoogle Scholar - Uesaka, Y., Manalo, E., & Ichikawa, S. (2007). What kinds of perceptions and daily learning behaviors promote students’ use of diagrams in mathematics problem solving?
*Learning and Instruction, 17*, 322–335.CrossRefGoogle Scholar - Van Dooren, W., Verschaffel, L., & Onghena, P. (2003). Pre-service teachers’ preferred strategies for solving arithmetic and algebra word problems.
*Journal of Mathematics Teacher Education, 6*, 27–52.CrossRefGoogle Scholar - Van Meter, P., Aleksic, M., Schwartz, A., & Garner, J. (2006). Learner-generated drawing as a strategy for learning from content area text.
*Contemporary Edcuational Pschology, 31*, 142–166.CrossRefGoogle Scholar - Van Meter, P., & Garner, J. (2005). The promise and practice of learner-generated drawing: Literature review and synthesis.
*Educational Psychology Review, 17*, 285–325.CrossRefGoogle Scholar - Verschaffel, L., & De Corte, E. (1997a). Word problems: A vehicle for promoting authentic mathematical understanding and problem solving in the primary school? In T. Nunes & P. Bryant (Eds.),
*Learning and teaching mathematics: An international perspective*(pp. 69–97). Hove, UK: Psychology Press.Google Scholar - Verschaffel, L., & De Corte, E. (1997b). Teaching mathematical modeling and problem-solving in the elementary school. A teaching experiment with fifth graders.
*Journal for Research in Mathematics Education, 28*, 577–601.CrossRefGoogle Scholar - Verschaffel, L., De Corte, E., & Lasure, S. (1994). Realistic considerations in mathematical modelling of school arithmetic word problems.
*Learning and Instruction, 7*, 339–359.CrossRefGoogle Scholar - Verschaffel, L., De Corte, E., Lasure, S., Van Vaerenbergh, G., Bogaerts, H., & Ratinckx, E. (1999). Design and evaluation of a learning environment for mathematical modeling and problem solving in upper elementary school children.
*Mathematical Thinking and Learning, 1*, 195–229.CrossRefGoogle Scholar - Verschaffel, L., Greer, B., & De Corte, E. (2000).
*Making sense of word problems*. Lisse, The Netherlands: Swets & Zeitlinger.Google Scholar - Verschaffel, L., Greer, B. & Torbeyns, J. (2006). Numerical thinking. In Boero, P. & Gutiérrez, A. (Eds.),
*Handbook of research on the psychology of mathematics education: Past, present and future*(pp. 51–82). Rotterdam, the Netherlands: Sense Publishers.Google Scholar