EXAMINATION OF DIAGRAMMATIC REPRESENTATION AND VERBAL PROBLEM-SOLVING REPRESENTATIONS OF PRIMARY SCHOOL STUDENTS
This study aimed to examine the diagrammatic representation skills and problem-solving performances of students according to their problem-solving representations. A cross-sectional survey design using quantitative methods was used in this study. The sample consisted of 31 second-grade and 41 third-grade students from a public primary school in Turkey. The Diagrammatic Representation Test and Mathematical Operations Test were used in this study. The data were analyzed with descriptive statistical analysis, the chi-square test, the independent samples t-test, discriminant analysis and logistic regression analysis. The findings indicated that while the preferred types of representations for solving verbal problems and problem-solving performance did not vary significantly based on grade level, scores obtained from the diagrammatic representation test exhibited significant differences. It was observed that students' problem-solving performance and diagrammatic skills could predict their preferred types of representations for solving verbal problems. Consequently, students who possess knowledge regarding effective representation preferences, as well as the ability to construct and utilize them, are more likely to generate appropriate and high-quality representations, leading to accurate problem-solving outcomes. This, in turn, enhances their performance in diagrammatic representation tasks.
Keywords: Diagrammatic representation, pictorial representation, schematic representation, symbolic representation, verbal problem solving.
Acevedo Nistal, A., Clarebout, G., Elen, J., Van Dooren,W., & 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, 627-636. https://doi.org/10.1007/s11858-009-0189-1
Baltacı, A. (2018). Nitel araştırmalarda örnekleme yöntemleri ve örnek hacmi sorunsalı üzerine kavramsal bir inceleme [A conceptual review of sampling methods and sample size problems in qualitative research]. Bitlis Eren University Social Science Journal, 7(1), 231-274.
Beckmann, S. (2004). Solving algebra and other story problems with simple diagrams: A method demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14, 42–46.
Blazhenkova, O., & Kozhevnikov, M. (2009). The new object‐spatial‐verbal cognitive style model: Theory and measurement. Applied Cognitive Psychology: The Official Journal of the Society for Applied Research in Memory and Cognition, 23(5), 638-663. https://doi.org/10.1002/acp.1473
Boonen, A. J. H., van der Schoot, M., Van Wesel, F., De Vries, M. H. & Jolles, J. (2013). What underlies successful word problem solving? A path analysis in sixth grade students. Contemporary Educational Psychology, 38(3), 271–279. https://doi.org/10.1016/j.cedpsych.2013.05.001
Boonen, A. J. H., Van Wesel, F., Jolles, J., & Van Der Schoot, M. (2014). The role of visual representation type, spatial ability, and reading comprehension in word problem solving: An item-level analysis in elementary school children. International Journal of Educational Research, 68, 15–26. https://doi.org/10.1016/j.ijer.2014.08.001
Booth, J. L., & Koedinger, K. R. (2012). Are diagrams always helpful tools? Developmental and individual differences in the effect of presentation format on student problem solving. British Journal of Educational Psychology, 82(3), 492-511. https://doi.org/10.1111/j.2044-8279.2011.02041.x
Cheng, P. C. (2004, March). Why diagrams are (sometimes) six times easier than words: benefits beyond locational indexing. In International Conference on Theory and Application of Diagrams (pp. 242-254). Springer, Berlin, Heidelberg.
Cooper, J. L., Sidney, P. G., & Alibali, M. W. (2018). Who benefits from diagrams and illustrations in math problems? Ability and attitudes matter. Applied Cognitive Psychology, 32(1), 24-38. https://doi.org/10.1002/acp.3371
Davenport, J. L., Yaron, D., Klahr, D., & Koedinger, K. (2008). When do diagrams enhance learning? A framework for designing relevant representations. In Proceedings of the 8th international conference on International conference for the learning sciences - Volume 1 (ICLS'08). International Society of the Learning Sciences, 191–198. https://doi.org/10.5555/1599812.1599834
Diezmann, C., & Lowrie, T. (2009). Primary students' spatial visualization and spatial orientation: an evidence base for instruction. In Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (pp. 417-424). PME, Greece.
Ertuna, L., & Uçar, Z. T. (2021). An investigation of elementary school 4-7th grade students' ability to link equivalent fractions' symbolic and graphical representations. Sakarya University Journal of Education, 11(3), 613-631. https://doi.org/10.19126/suje.992377
Fraenkel, J. R., & N. E. Wallen. (2003). How to design and evaluate research in education. New York: McGraw Hill.
Frick, A., & Newcombe, N. S. (2015). Young children's perception of diagrammatic representations. Spatial Cognition & Computation, 15(4), 227-245. https://doi.org/10.1080/13875868.2015.1046988
Galindo-Morales, E. (1994). Visualization in the calculus class: Relationship between cognitive style, gender, and use of technology (Doctoral dissertation), The Ohio State University.
Gültekin, S. B., & Altun, T. (2022). Investigating the Impact of Activities Based on Scientific Process Skills on 4th Grade Students' Problem-Solving Skills. International Electronic Journal of Elementary Education, 14(4), 491-500. https://doi.org/10.26822/iiejee.2022.258
Hatisaru, V. (2020). Exploring evidence of mathematical tasks and representations in the drawings of middle school students. International Electronic Journal of Mathematics Education, 15(3), em0609. https://doi.org/10.29333/iejme/8482
Hegarty, M., & Kozhevnikov, M. (1999). Types of visual-spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684-689.
Johnson-Laird, P. N. (1983). A computational analysis of consciousness. Cognition & Brain Theory, 6(4), 499–508.
Kalayc, S. (2005). Multvarate Statstcal Technques wth SPSS. Asl Publsher, Ankara, Turkey
Kozhevnikov, M., Kosslyn, S., & Shephard, J. (2005). Spatial versus object visualizers: A new characterization of visual cognitive style. Memory & cognition, 33(4), 710-726. https://doi.org/10.3758/BF03195337
Krutetskii V. A. (1976). The psychology of mathematical abilities in schoolchildren, University of Chicago Press, Chicago.
Lowrie, T. (2020). The utility of diagrams in elementary problem solving. Cognitive Development, 55, 1-12. https://doi.org/10.1016/j.cogdev.2020.100921
Lowrie, T., & Clements, M. K. (2001). Visual and nonvisual processes in Grade 6 students' mathematical problem solving. Journal of Research in Childhood Education, 16(1), 77-93. https://doi.org/10.1080/02568540109594976
Lowrie, T., & Kay, R. (2001). Relationship between visual and nonvisual solution methods and difficulty in elementary mathematics. Journal of Educational Research, 94(4), 94(4), 248–255. https://doi.org/10.1080/00220670109598758
Mayer, R. (2005). Cognitive theory of multimedia learning. In R. Mayer (Ed.), The Cambridge handbook of multimedia learning (pp. 31–48). Cambridge: Cambridge University Press.
Mayer, R. E. (1989). Models for understanding. Review of educational research, 59(1), 43-64. https://doi.org/10.3102/00346543059001043
Mayer, R. E., & Massa, L. J. (2003). Three facets of visual and verbal learners: Cognitive ability, cognitive style, and learning preference. Journal of educational psychology, 95(4), 833. https://doi.org/10.1037/0022-06220.127.116.113
Metin, M. (2014). Eğitimde bilimsel araştırma yöntemleri [Scientific research methods in education]. Ankara: Pegem Academy Publications.
Meyer, J. (2000). Performance with tables and graphs: Effects of training and a visual searchmodel. Ergonomics, 43, 1840‑1865. https://doi.org/10.1080/00140130050174509
Murata, A. (2004). Paths to learning ten-structured understanding of teen sums: Addition solution methods of Japanese grade 1 students. Cognition and Instruction, 22, 185–218. https://doi.org/10.1207/s1532690xci2202_2
Murayama, K. (2003). Learning strategy use and short- and long-term perceived utility. Japanese Journal of Educational Psychology, 51, 130–140. https://doi.org/10.5926/jjep1953.51.2_130
National Center for Education Statistics (2003). Teaching mathematics in seven countries: Results from the TIMSS 1999 video study. Washington, DC: Author.
National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: Author.
Presmeg, N. C. (2006, July). A semiotic view of the role of imagery and inscriptions in mathematics teaching and learning. In Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 19-34).
Rellensmann, J., Schukajlow, S., & Leopold, C. (2017). Make a drawing. Effects of strategic knowledge, drawing accuracy, and type of drawing on students’ mathematical modelling performance. Educational Studies in Mathematics, 95(1), 53-78. https://doi.org/10.1007/s10649-016-9736-1
Sevimli, E. (2013). Bilgisayar cebiri sistemi destekli öğretimin farklı düşünme yapısındaki öğrencilerin integral konusundaki temsil dönüşüm süreçlerine etkisi [The effect of a computer algebra system supported teaching on processes of representational transition inintegral topics of students with different types of thinking]. (Unpublished Doctoral dissertation). Marmara University, Turkey.
Stenning, K., & Oberlander, J. (1995). A cognitive theory of graphical and linguistic reasoning: Logic and implementation. Cognitive science, 19(1), 97-140. https://doi.org/10.1016/0364-0213(95)90005-5
Stylianou, D. (2010). Teachers’ conceptions of representation in middle school mathematics. Journal of Mathematics Teacher Education, 13(4), 325-434. https://doi.org/10.1007/s10857-0109143-y
Surya, E., Sabandar, J., Kusumah, Y. S., & Darhim, D. (2013). Improving of junior high school visual thinking representation ability ın mathematical problem solving by Ctl. Journal. Math. Edu., 1(4). https://doi.org/10.22342/jme.4.1.568.113-126
Suwarsono, S. (1982). Visual imagery in the mathematical thinking of seventh grade students (Unpublished Doctoral dissertation). Monash University.
Taşova, H. İ. (2011). Matematik öğretmen adaylarının modelleme etkinlikleri ve performansı sürecinde düşünme ve görselleme becerilerinin incelenmesi [Investigating thinking and visualisation skills of preservice mathematics teachers in modelling activities and performance]. (Unpublished Doctoral dissertation). Marmara University, Turkey.
Tian, F., Hou, Y., Zhu, W., Dietrich, A., Zhang, Q., Yang, W., ... & Cao, G. (2017). Getting the joke: insight during humor comprehension–evidence from an fMRI study. Frontiers in psychology, 8, 1835. https://doi.org/10.3389/fpsyg.2017.01835
Tytler, R., Prain, V., Kirk, M. et al. (2023) Characterising a representation construction pedagogy for ıntegrating science and mathematics in the primary school. Int J of Sci and Math Educ 21, 1153–1175. https://doi.org/10.1007/s10763-022-10284-4
Uesaka, Y., & Manalo, E. (2012). Task‐related factors that influence the spontaneous use of diagrams in math word problems. Applied Cognitive Psychology, 26(2), 251-260. https://doi.org/10.1002/acp.1816
Uesaka, Y., Manalo, E., & Ichikawa, S. I. (2010, August). The effects of perception of efficacy and diagram construction skills on students’ spontaneous use of diagrams when solving math word problems. In International Conference on Theory and Application of Diagrams (pp. 197-211). Springer, Berlin, Heidelberg.
Van Garderen, D. (2007). Teaching students with LD to use diagrams to solve mathematical word problems. Journal of learning disabilities, 40(6), 540-553. https://doi.org/10.1177/00222194070400060501
Van Garderen, D., & Montague, M. (2003). Visual‐spatial representation, mathematical problem solving, and students of varying abilities. Learning Disabilities Research & Practice, 18(4), 246-254. https://doi.org/10.1111/1540-5826.00079
Van Garderen, D., Scheuermann, A., & Jackson, C. (2012). Developing representational ability in mathematics for students with learning disabilities: A content analysis of grades 6 and 7 textbooks. In Learning Disability Quarterly, 35(1), 24-38. https://doi.org/10.1177/0731948711429726
Van Garderen, D., Scheuermann, A., & Jackson, C. (2013). Examining how students with diverse abilities use diagrams to solve mathematics word problems. Learning Disability Disability Quarterly, 36(3), 145–160. https://doi.org/10.1177/0731948712438558
Zahner, D., & Corter, J. E. (2010). The process of probability problem solving: Use of external visual representations. Mathematical Thinking and Learning, 12(2), 177-204. https://doi.org/10.1080/10986061003654240
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