pp 1–15 | Cite as

Word problem solving and pictorial representations: insights from an exploratory study in kindergarten

  • Iliada EliaEmail author
Original Article


The aim of this study was to investigate how pictorial representations with different semiotic characteristics affect additive word problem solving by kindergartners. The focus of the study is on three categories of additive problems (change problems, combine problems and equalize problems) and on representational pictures with different semiotic characteristics: (a) pictures in which the problem quantities are represented in pictorial form, that is, as groups of illustrated objects (PP pictures), (b) pictures in which the quantities are represented partly in pictorial form and in symbolic form (PS pictures), and (c) pictures in which the quantities are represented in symbolic form (SS pictures). Data were collected from 63 kindergartners using a paper-and-pencil test. Results showed that the semiotic characteristics of representational pictures had a strong and significant effect on performance. Children’s performance was higher in the problems with PP pictures but declined in the problems with PS and SS pictures. However, the differences in children’s performance across the problems with different representational format varied between the problem categories and their mathematical structures. The semiotic characteristics of representational pictures had an important role in the establishment of close relations between children’s solutions in problems in different categories. Detailed analysis of children’s answers to the problems revealed a number of picture-related difficulties. Findings are discussed and directions for future research are drawn considering the methodological limitations of the study.


Word problems Representational pictures Kindergartners Addition Subtraction 



The study reported in this paper was supported by the University of Cyprus (start-up funding).


  1. Bodin, A., Coutourier, R., & Gras, R. (2000). CHIC: Classification hiérarchique implicative et cohésive-Version sous WindowsCHIC 1.2. Rennes: Association pour la Recherche en Didactique des Mathématiques.Google Scholar
  2. Briars, D. J., & Larkin, J. H. (1984). An integrated model of skill in solving elementary problems. Cognition and Instruction,1, 245–296.CrossRefGoogle Scholar
  3. Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: a study of kindergarten children’s problem-solving processes. Journal for Research in Mathematics Education,24, 428–441.CrossRefGoogle Scholar
  4. Carpenter, T. P., & Moser, J. M. (1982). The development of addition and subtraction problem-solving skills. In T. P. Carpenter, J. M. Moser, & T. A. Romberg (Eds.), Addition and subtraction: A cognitive perspective (pp. 9–24). Hillsdale: Lawrence Erlbaurn Associates.Google Scholar
  5. Cyprus Ministry of Education and Culture. (2010). Program of studies in mathematics. Nicosia: Cyprus Ministry of Education and Culture.Google Scholar
  6. Daroczy, G., Wolska, M., Meurers, W. D., & Nuerk, H. C. (2015). Word problems: a review of linguistic and numerical factors contributing to their difficulty. Frontiers in Psychology,6, 348.CrossRefGoogle Scholar
  7. De Corte, E., & Verschaffel, L. (1985). Beginning first graders’ initial representation of arithmetic word problems. Journal of Mathematical Behavior,4, 3–21.Google Scholar
  8. De Corte, E., & Verschaffel, L. (1987). The effect of semantic structure on first graders’ solution strategies of elementary addition and subtraction word problems. Journal for Research in Mathematics Education,18, 363–381.CrossRefGoogle Scholar
  9. De Corte, E., & Verschaffel, L. (1988). Computer simulation as a tool in research on problem solving in subject-matter domains. The International Journal of Educational Research,12, 49–69.Google Scholar
  10. De Corte, E., Verschaffel, L., & De Win, L. (1985). The influence of rewording verbal problems on children’s problem representations and solution. Journal of Educational Psychology,77, 460–470.CrossRefGoogle Scholar
  11. Dewolf, T., Van Dooren, W., Ev Cimen, E., & Verschaffel, L. (2014). The impact of illustrations and warnings on solving mathematical word problems realistically. The Journal of Experimental Education,82(1), 103–120.CrossRefGoogle Scholar
  12. Elia, I., Gagatsis, A., & Demetriou, A. (2007). The effects of different modes of representation on the solution of one-step additive problems. Learning and Instruction,17, 658–672.CrossRefGoogle Scholar
  13. Elia, I., & Philippou, G. (2004). The functions of pictures in problem solving. In M. Johnsen Høines & A. Berit Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 327–334). Bergen: PME.Google Scholar
  14. Elia, I., & Van den Heuvel-Panhuizen, M. (2015). Mapping kindergartners’ number competence. In X. Sun, B. Kaur & J. Novotná (Eds.), Proceedings of the twenty-third ICMI Study: Primary mathematics study on whole numbers (pp. 177–185). Macau: University of MacauGoogle Scholar
  15. Ginsburg, H. P., Klein, A., & Starkey, P. (1998). The development of children’s mathematical thinking: Connecting research with practice. In I. Sigel & A. Renninger (Eds.), Handbook of child psychology (5th ed., Vol. 4, pp. 401–476). New York: Wiley.Google Scholar
  16. Lahanier-Reuter, D. (2008). Didactics of mathematics and implicative statistical analysis. In R. Gras, E. Suzuki, F. Guillet, & F. Spagnolo (Eds.), Studies in computational intelligence 127: Statistical implicative analysis (pp. 277–298). Heidelberg: Springer-Verlag.CrossRefGoogle Scholar
  17. Li, Y., Zhang, M., Chen, Y., Deng, Z., Zhu, X., & Yan, S. (2018). Children’s non-symbolic and symbolic numerical representations and their associations with mathematical ability. Frontiers in Psychology,9, 1035.CrossRefGoogle Scholar
  18. Li, Y., Zhang, M., Chen, Y., Zhu, X., Deng, Z., & Yan, S. (2017). Children’s non-symbolic, symbolic addition and their mapping capacity at 4–7 years old. Frontiers in Psychology,8, 1203.CrossRefGoogle Scholar
  19. Libertus, M. E., Feigenson, L., & Halberda, J. (2011). Preschool acuity of the approximate number system correlates with school math ability. Developmental Science,14, 1292–1300.CrossRefGoogle Scholar
  20. Lyons, I. M., Bugden, S., Zheng, S., De Jesus, S., & Ansari, D. (2018). Symbolic number skills predict growth in non-symbolic number skills in kindergarteners. Developmental Psychology,54(3), 440–457.CrossRefGoogle Scholar
  21. Matalliotaki, E. (2012). Resolution of division problems by young children: What are children capable of and under which conditions? European Early Childhood Education Research Journal,20(2), 283–299.CrossRefGoogle Scholar
  22. Mayer, R. E. (2003). The promise of multimedia learning: using the same instructional design methods across different media. Learning and Instruction,13, 125–139.CrossRefGoogle Scholar
  23. Múñez, D., Orrantia, J., & Rosales, J. (2013). The effect of external representations on compare word problems: Supporting mental model construction. The Journal of Experimental Education,81(3), 337–355.CrossRefGoogle Scholar
  24. Nesher, P., Greeno, J. G., & Riley, M. S. (1982). The development of semantic categories for addition and subtraction. Educational Studies in Mathematics,13, 373–394.CrossRefGoogle Scholar
  25. Nistal, A. A., Van Dooren, W., Clarebout, G., Elen, J., & Verschaffel, L. (2009). Conceptualising, investigating and stimulating representational flexibility in mathematical problem solving and learning: a critical review. ZDM - The International Journal on Mathematics Education,41(5), 627–636.CrossRefGoogle Scholar
  26. Riley, M. S., & Greeno, J. G. (1988). Developmental analysis of understanding language about quantities and of solving problems. Cognition and Instruction,5(1), 49–101.CrossRefGoogle Scholar
  27. Schnotz, W., & Bannert, M. (2003). Construction and interference in learning from multiple representation. Learning and Instruction,13, 141–156.CrossRefGoogle Scholar
  28. Sophian, C., & Vong, K. I. (1995). The parts and wholes of arithmetic story problems: developing knowledge in the preschool years. Cognition and Instruction,13(3), 469–477.CrossRefGoogle Scholar
  29. Sweller, J. (1994). Cognitive load theory, learning difficulty, and instructional design. Learning and Instruction,4, 295–312.CrossRefGoogle Scholar
  30. Van den Heuvel-Panhuizen, M. (1996). Assessment and realistic mathematics education. Utrecht: CD-β Press, Center for Science and Mathematics Education.Google Scholar
  31. Verschaffel, L., & De Corte, E. (1997). 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: Psychology Press.Google Scholar

Copyright information

© FIZ Karlsruhe 2019

Authors and Affiliations

  1. 1.Department of EducationUniversity of CyprusNicosiaCyprus

Personalised recommendations