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Diagrams Affect Choice of Strategy in Probability Problem Solving

  • Chenmu XingEmail author
  • James E. Corter
  • Doris Zahner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9781)

Abstract

We investigated whether diagrams influence strategy choice and success in solving elementary combinatorics problems. Generic diagrams (trees or two-way tables) were provided to solvers as aids. Participants’ coded solution strategies revealed that problem solvers tended to utilize mathematical structures and solutions that easily mapped to the diagrams’ visuospatial relations. For example, when provided with an unlabeled N-by-N table, solvers tended to proceed by defining an equally-likely outcome space (an “outcome-search” solution); when provided with a binary tree, solvers tended to adopt a “sequential” solution defining stage-wise simple or conditional probabilities; when provided with an N-ary tree cuing equally-likely outcomes, choices were split between the two solution types. Furthermore, the tree and the table showed different patterns of characteristic errors, and perhaps for this reason, the tree led to higher accuracy for one problem that involved sequential sampling without replacement, while the table was best for the other problem, involving independent events. The results support arguments that the content and structure of diagrams must be congruent to that of the problem at hand and be easily and accurately perceived to be effective, and demonstrate that diagrams can influence strategy choice in problem solving.

Keywords

Probability problem solving Diagram congruence Diagram design 

References

  1. 1.
    Hegarty, M., Kozhevnikov, M.: Types of visual–spatial representations and mathematical problem solving. J. Educ. Psychol. 91(4), 684–689 (1999)CrossRefGoogle Scholar
  2. 2.
    Heiser, J., Tversky, B.: Arrows in comprehending and producing mechanical diagrams. Cogn. Sci. 30, 581–592 (2006)CrossRefGoogle Scholar
  3. 3.
    Manalo, E., Uesaka, Y.: Quantity and quality of diagrams used in math word problem solving: a comparison between New Zealand and Japanese students. In: Paper Presented at the New Zealand Association for Research in Education (NZARE) National Conference, Rotorua, New Zealand, December 2006Google Scholar
  4. 4.
    Zahner, D., Corter, J.E.: The process of probability problem solving: use of external visual representations. Math. Thinking Learn. 12(2), 177–204 (2010)CrossRefGoogle Scholar
  5. 5.
    Novick, L.R., Catley, K.M.: Reasoning about evolution’s grand patterns: college students’ understanding of the tree of life. Am. Educ. Res. J. 50, 138–177 (2013)CrossRefGoogle Scholar
  6. 6.
    Gattis, M., Holyoak, K.J.: Mapping conceptual to spatial relations in visual reasoning. J. Exp. Psychol. Learn. Mem. Cogn. 22(1), 231–239 (1996)CrossRefGoogle Scholar
  7. 7.
    Simkin, D., Hastie, R.: An information-processing analysis of graph perception. J. Am. Stat. Assoc. 82(398), 454–465 (1987)CrossRefGoogle Scholar
  8. 8.
    Tversky, B., Corter, J.E., Gao, J., Tanaka, Y., Nickerson, J.: People, place, and time: inferences from diagrams. In: Proceedings of the 35th Annual Conference of the Cognitive Science Society, pp. 3593–3597. Cognitive Science Society, Austin (2013)Google Scholar
  9. 9.
    Gick, M.L., Holyoak, K.J.: Schema induction and analogical transfer. Cogn. Psychol. 15, 1–38 (1983)CrossRefGoogle Scholar
  10. 10.
    Novick, L.R.: Representational transfer in problem solving. Am. Psychol. Soc. 1(2), 128–132 (1990)Google Scholar
  11. 11.
    Mason, D.L., Corter, J.E., Tversky, B., Nickerson, J.V.: Structure, space and time: some ways that diagrams affect inferences in a planning task. In: Cox, P., Plimmer, B., Rodgers, P. (eds.) Diagrams 2012. LNCS, vol. 7352, pp. 277–290. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  12. 12.
    Braithwaite, D.W., Goldstone, R.L.: Flexibility in data interpretation: effects of representational format. Front. Psychol. 4, 1–16 (2013)CrossRefGoogle Scholar
  13. 13.
    Zacks, J., Tversky, B.: Bars and lines: a study of graphic communication. Mem. Cogn. 27(6), 1073–1079 (1999)CrossRefGoogle Scholar
  14. 14.
    Markman, A.B.: Knowledge Representation. Lawrence Erlbaum Associates, Mahwah (1999)Google Scholar
  15. 15.
    Novick, L.R., Hurley, S.M.: To matrix, network, or hierarchy: that is the question. Cogn. Psychol. 42(2), 158–216 (2001)CrossRefGoogle Scholar
  16. 16.
    Gattis, M.: Mapping relational structure in spatial reasoning. Cogn. Sci. 28, 589–610 (2004)CrossRefGoogle Scholar
  17. 17.
    Tversky, B., Kugelmass, S., Winter, A.: Cross-cultural and developmental trends in graphic productions. Cogn. Psychol. 23(4), 515–557 (1991)CrossRefGoogle Scholar
  18. 18.
    Shah, P., Mayer, R.E., Hegarty, M.: Graphs as aids to knowledge construction: Signaling techniques for guiding the process of graph comprehension. J. Educ. Psychol. 91(4), 680–702 (1999)CrossRefGoogle Scholar
  19. 19.
    Tversky, B., Zacks, J., Lee, P., Heiser, J.: Lines, blobs, crosses and arrows: diagrammatic communication with schematic figures. In: Anderson, M., Cheng, P., Haarslev, V. (eds.) Diagrams 2000. LNCS (LNAI), vol. 1889, pp. 221–230. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  20. 20.
    Russell, W.E.: The use of visual devices in probability problem solving. Doctoral dissertation, Columbia University, Dissertation Abstracts International, 61, 1333 (2000)Google Scholar
  21. 21.
    Corter, J.E., Zahner, D.C.: Use of external visual representations in probability problem solving. Stat. Educ. Res. J. 6(1), 22–50 (2007)Google Scholar
  22. 22.
    Bobek, E.J., Corter, J.E.: Effects of problem difficulty and student expertise on the utility of provided diagrams in probability problem solving. In: Ohlsson, S., Catrambone, R. (eds.) Proceedings of the 32nd Annual Conference of the Cognitive Science Society, pp. 2650–2655. Cognitive Science Society, Austin (2010)Google Scholar
  23. 23.
    Ainsworth, S.: The functions of multiple representations. Comput. Educ. 33(2), 131–152 (1999)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Novick, L.R., Hurley, S.M., Francis, M.: Evidence for abstract, schematic knowledge of three spatial diagram representations. Mem. Cogn. 27(2), 288–308 (1999)CrossRefGoogle Scholar
  25. 25.
    Novick, L.R., Hmelo, C.E.: Transferring symbolic representations across nonisomorphic problems. J. Exp. Psychol. Learn. Mem. Cogn. 20(6), 1296–1321 (1994)CrossRefGoogle Scholar
  26. 26.
    Gugga, S.S., Corter, J.E.: Effects of temporal and causal schemas on probability problem solving. In: Bello, P., Guarini, M., Scassellati, B. (eds.) Proceedings of the 36th Annual Conference of the Cognitive Science Society, pp. 2650–2655. Cognitive Science Society, Austin (2014)Google Scholar
  27. 27.
    Tversky, B., Morrison, J.B., Betrancourt, M.: Animation: can it facilitate? Int. J. Hum. Comput. Stud. 57(4), 247–262 (2002)CrossRefGoogle Scholar
  28. 28.
    Larkin, J.H., Simon, H.A.: Why a diagram is (sometimes) worth ten thousand words. Cogn. Sci. 11, 65–99 (1987)CrossRefGoogle Scholar
  29. 29.
    Gentner, D., Markman, A.B.: Structure mapping in analogy and similarity. Am. Psychol. 52(1), 45–56 (1997)CrossRefGoogle Scholar
  30. 30.
    Tversky, B.: Visualizations of thought. Top. Cogn. Sci. 3, 499–535 (2011)CrossRefGoogle Scholar
  31. 31.
    Tversky, B., Corter, J.E., Yu, L., Mason, D.L., Nickerson, J.V.: Representing category and continuum: visualizing thought. In: Cox, P., Plimmer, B., Rodgers, P. (eds.) Diagrams 2012. LNCS, vol. 7352, pp. 23–34. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  32. 32.
    Bauer, M.I., Johnson-Laird, P.N.: How diagrams can improve reasoning. Psychol. Sci. 6, 372–378 (1993)CrossRefGoogle Scholar
  33. 33.
    Glenberg, A.M., Langston, W.E.: Comprehension of illustrated text: pictures help to build mental models. J. Mem. Lang. 31, 129–151 (1992)CrossRefGoogle Scholar
  34. 34.
    Pinker, S.: A theory of graph comprehension. In: Freedle, R. (ed.) Artificial Intelligence and the Future of Testing, pp. 73–126. Erlbaum, Hillsdale (1990)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Teachers College, Columbia UniversityNew YorkUSA
  2. 2.Council for Aid to EducationNew YorkUSA

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