Use of External Visual Representations in
probability problem solving
James E. Corter
Teachers College,
corter@exchange.tc.columbia.edu
Doris C. Zahner
Teachers College,
dwc14@columbia.edu
ABSTRACT
We investigate the use of external visual representations
in probability problem solving. Twenty-six students enrolled in an introductory
statistics course for social sciences graduate students (post-baccalaureate)
solved eight probability problems in a structured interview format. Results
show that students spontaneously use self-generated external visual
representations while solving probability problems. The types of visual
representations used include: reorganization of the given information,
pictures, novel schematic representations, trees, outcome listings, contingency
tables, and Venn diagrams. The frequency of use of each of these different
external visual representations depended on the type of probability problem
being solved. We interpret these findings as showing that problem solvers
attempt to select representations appropriate to the problem structure, and
that the appropriateness of the representation is determined by the problem’s
underlying schema.
Keywords: Statistics
education research; Probability problem solving; Visual representations; Trees;
Outcome listings; Venn diagrams
__________________________
Statistics Education Research
Journal, 6(1), 22-50, http://www.stat.auckland.ac.nz/serj
Ó International Association for Statistical Education
(IASE/ISI), May, 2007
REFERENCES
Batanero, C., Godino, J., & Roa, R. (2004). Training teachers to teach probability. Journal of Statistics Education, 12(1).
[Online: www.amstat.org/publications/jse/v12n1/batanero.html]
Chance, B., & Garfield, J. (2002). New approaches to gathering data on student learning for research in statistics education. Statistics Education Research Journal, 1(1), 38-41.
[Online: www.stat.auckland.ac.nz/~iase/serj/SERJ1(2).pdf]
De Bock, D., Verschaffel, L., Janssens, D., Van Dooren, W., & Claes, K. (2003). Do realistic contexts and graphical representations always have a beneficial impact on students’ performance? Negative evidence from a study on modeling non-linear geometry problems. Learning and Instruction, 13(4), 441-463.
Garfield, J., & Ahlgren, A. (1988). Difficulties in learning basic concepts in probability and statistics: Implications for research. Journal for Research in Mathematics Education, 19(1), 44-63.
Gelman, A., & Nolan, D. (2002). Teaching statistics: A bag
of tricks.
Gigerenzer, G. (1994). Why the distinction between
single-event probabilities and frequencies is important for psychology (and
vice versa). In G. Wright & P. Ayton (Eds.), Subjective Probability (pp. 129-161).
Ginsburg, H. (1997). Entering the child's mind: The clinical interview in
psychological research and practice.
Hall, V., Bailey, J., & Tillman, C. (1997). Can student-generated illustrations be worth ten thousand words? Journal of Educational Psychology, 89(4), 677-681.
Hampson, P., & Morris, P. (1990). Imagery and
working memory. In P. J. Hampson, D. E. Marks,
& J. T. E. Richardson (Eds.), Imagery: Current developments (pp.
78-102).
Hannafin, R. D., Burruss, J. D., & Little, C. (2001). Learning with dynamic geometry programs: Perspectives of teachers and learners. The Journal of Educational Research, 94(3), 132-144.
Hannafin, R. D., & Scott, B. (1998). Identifying critical learner traits in a dynamic computer-based geometry program. The Journal of Educational Research, 92(1), 3-12.
Hegarty, M., & Just, M. (1993). Constructing mental models of machines from text and diagrams. Journal of Memory and Language, 32(6), 717-742.
Hegarty, M., & Kozhevnikov, M. (1999). Types of visual-spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684-689.
Hollebrands, K. (2003). High school students’ understandings of geometric transformations in the context of a technological environment. Journal of Mathematical Behavior, 22(1), 55-72.
Kahneman, D., Slovic, P., & Tversky, A. (1982). Judgment under uncertainty: Heuristics and biases.
Kaufmann, G. (1990). Imagery effects on problem
solving. In P. J. Hampson, D. E. Marks, & J. T.
E. Richardson (Eds.), Imagery: Current developments (pp. 169-197).
Keeler, C. M., & Steinhorst, R. K. (2001). A new approach to learning probability in the first statistics course. Journal of Statistics Education, 9(3).
[Online: www.amstat.org/publications/jse/v9n3/keeler.html]
Kintsch, W., & Greeno, J.G. (1985). Understanding and solving word arithmetic problems. Psychological Review, 92(1), 109-29.
Koedinger, K. R., & Anderson, R. (1997). Intelligent tutoring goes to school in the big city. International Journal of Artificial Intelligence in Education, 8, 30-43.
Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6(1), 59-98.
Konold, C. (1995). Confessions of a coin flipper and would-be instructor. The American Statistician, 49(2), 203-209.
Konold, C. (1996). Representing probabilities with pipe diagrams. The Mathematics Teacher, 89(5), 378-384.
Konold, C., Pollatsek, A., Well, A. D., Lohmeier, J., & Lipson, A. (1993). Inconsistencies in students’ reasoning about probability. Journal for Research in Mathematics Education, 24, 392-414.
Kozhevnikov, M., Hegarty, M., & Mayer, R. (2002). Revising the visualizer-verbalizer dimension: Evidence for two types of visualizers. Cognition and Instruction, 20(1), 47-77.
Lehrer, R., & Schauble, L. (1998). Reasoning about structure and function: Children’s conceptions of gears. Journal of Research in Science Teaching, 35(1), 3-25.
Logie, R., & Baddeley, A. (1990) Imagery
and working memory. In P. J. Hampson, D. E.
Marks, & J. T. E. Richardson (Eds.), Imagery: Current Developments
(pp. 103-128).
Lowrie, T., & Kay, R. (2001). Relationship between visual and nonvisual solution methods and difficulty in elementary mathematics. The Journal of Educational Research, 94(4), 248-255.
Mayer, R. (1989). Systemic thinking fostered by illustrations in scientific text. Journal of Educational Psychology, 81(2), 240-246.
Mayer, R. (1992). Mathematical problem solving:
Thinking as based on domain specific knowledge. In R. E.
Mayer (Ed.), Thinking, problem solving, cognition (pp. 455-489).
Mayer, R. (2001). Multimedia
learning.
Mayer, R., & Anderson, R. (1991). Animations need narrations: An experimental test of a dual-coding hypothesis. Journal of Educational Psychology, 83(4), 484-490.
Mayer, R., & Anderson, R. (1992). The instructive animation: Helping students build connections between words and pictures in multimedia learning. Journal of Educational Psychology, 84(4), 444-452.
Mayer, R., & Gallini, J. (1990). When is an illustration worth ten thousand words? Journal of Educational Psychology, 82(4), 715-726.
Mayer, R., Mautone, P., & Prothero, W. (2002). Pictorial aids for learning by doing in a multimedia geology simulation game. Journal of Educational Psychology, 91(4), 171-185.
Molitor, S., Ballstaedt, S. P., & Mandl, H. (1989). Problems in knowledge acquisition from text and pictures. In H. Mandl & J. Levin (Eds.), Knowledge acquisition from text and pictures (pp. 3-35). North-Holland: Elsevier Science Publishers.
Nemirovsky, R. (1994). On ways of symbolizing: The case of Laura and the velocity sign. Journal of Mathematical Behavior, 13(4), 389-422.
Novick, L. (1990). Representational transfer in problem solving. Psychological Science, 1(2), 128-132.
Novick, L. (2001). Spatial
diagrams: Key instruments in the toolbox for thought. In D. Medin
(Ed.), The psychology of learning and
motivation: Advances in research and theory, Volume 40 (pp. 279-325).
Novick, L., & Hmelo, C. (1994). Transferring symbolic representations across nonisomorphic problems. Journal of Experimental Psychology: Learning, Memory, and Cognition, 20(6), 1296–1321.
Novick, L.R., & Hurley, S. M. (2001). To matrix, network or hierarchy: That is the question. Cognitive Psychology, 42(2), 158-216.
O’Connell, A. A. (1993). A
classification of student errors in probability problem-solving. Unpublished doctoral dissertation, Teachers College,
O’Connell, A. A. (1999). Understanding the nature of errors in probability problem solving. Educational Research and Evaluation, 5(1), 1-21.
O’Connell,
A., & Corter, J. E. (1993, April). Student misconceptions in
probability problem-solving. Paper presented at annual meeting of the
American Educational Research Association,
Penner, D. E., Giles, N. D., Lehrer, R., & Schauble, L. (1996). Building functional models: Designing an elbow. Journal of Research in Science Teaching, 34(2), 125-143.
Pollatsek, A., Well, A., Konold, C., Hardiman, P., and Cobb, G. (1987). Understanding conditional probabilities. Organizational Behavior and Human Decision Processes, 40(2), 255-269.
Presmeg, N. C. (1986). Visualization in high school mathematics. For the Learning of Mathematics – An International Journal of Mathematics Education, 6(3), 42-46.
Qin, Y., & Simon, H. (1995).
Imagery and mental models in problem solving. In B. Chandrasekaran, J. Glasgow, &
N. H. Narayanan (Eds.), Diagrammatic reasoning (pp. 403-434).
Reusser, K. (1996). From cognitive modeling to
the design of pedagogical tools. In S. Vosniadou, E. De Corte, R. Glaser & H. Mandl (Eds.), International perspectives on the design
of technology–supported learning environments (pp. 81-103).
Riley, M., Greeno,
J., & Heller, J. (1983). Development of children’s
problem-solving ability. In H. Ginsburg (Ed.), The
development of mathematical thinking.
Russell, W. E. (2000). The use of
visual devices in probability problem solving. (Doctoral
Dissertation,
Santos-Trigo, M. (1996). An exploration of strategies used by students to solve problems with multiple ways of solution. Journal of Mathematical Behavior, 15(3), 263-284.
Scaife, M., &
Schwartz, D. L., & Black, J. B. (1996). Shuttling between depictive models and abstract rules: Induction and fallback. Cognitive Science, 20(4), 457-497.
Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction. Cognition and Instruction, 22(2), 129-184.
Sedlmeier, P. (2000). How to improve statistical thinking: Choose the task representation wisely and learn by doing. Instructional Science, 28(3), 227-262.
Sedlmeier, P., & Gigerenzer, G. (2001). Teaching Bayesian reasoning in less than two hours. Journal of Experimental Psychology: General, 130(3), 380-400.
Tversky, B. (2001). Spatial schemas in depictions. In M. Gattis
(Ed.), Spatial schemas and abstract thought
(pp. 79-112).
Zahner, D. C., & Corter, J. E. (2002,
April). Clinical interviewing to uncover the cognitive
processes of probability problem solvers. Paper presented at the 2002
Annual Meeting of the American Educational Research Association,
James E. Corter
Doris C. Zahner
Department
of Human Development
Teachers College,