Abstract
Data from 154 interviews with students in grades 3 to 13 were analyzed to suggest a developmental progression of conceptual understanding associated with the sample space for two ordinary six-sided dice tossed simultaneously. The model was then considered in the light of responses to an extension task involving three six-sided dice with four sides painted black and two white. Forty-five of the interviews were longitudinal interviews of students 3 or 4 years after the original interviews, allowing for analysis of change in levels of understanding across time. This study suggests some of the intuitions and understandings that form the intermediate steps to a complete understanding of outcomes for two dice, as well as documenting the conceptual regression likely to occur when a more difficult task is encountered. Links to previous research, including well-known misconceptions, and educational implications of the model are among the discussion points.
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References
Amir, G.S. & Williams, J.S. (1999). Cultural influences on children’s probabilistic thinking. Journal of Mathematical Behavior, 18(1), 85–107.
Australian Education Council. (1991). A national statement on mathematics for Australian schools. Carlton, VIC: Author.
Australian Education Council. (1994). Mathematics: A curriculum profile for Australian schools. Melbourne: Curriculum Corporation.
Biggs, J.B. & Collis, K.F. (1982). Evaluating the quality of learning: The SOLO taxonomy. New York: Academic Press.
Bond, T.G. & Fox, C.M. (2001). Applying the rasch model: Fundamental measurement in the human sciences. Mahwah, NJ: Lawrence Erlbaum.
Collis, K.F. & Biggs, J.B. (1991). Developmental determinants of qualitative aspects of school learning. In G. Evans (Ed.), Learning and teaching cognitive skills (pp. 185–207). Hawthorn, VIC: Australian Council for Educational Research.
Department of Education and the Arts. (1993). Pack 5: Chance and data mathematics guidelines K-8. Hobart, Tas.: Curriculum Services Branch.
Fischbein, E. & Gazit, A. (1984). Does the teaching of probability improve probabilistic intuitions? An exploratory research study. Educational Studies in Mathematics, 15, 1–24.
Fischbein, E. & Schnarch, D. (1997). The evolution with age of probabilistic, intuitively based misconceptions. Journal for Research in Mathematical Education, 28, 96–105.
Fischbein, E., Nello, M.S. & Marino, M.S. (1991). Factors affecting probability judgements in children and adolescents. Educational Studies in Mathematics, 22, 523–549.
Green, D.R. (1982). Probability concepts in 11–16 year old pupils. Report of research sponsored by the Social Science Research Council. Loughborough: CAMET, University of Technology.
Jones, G.A. (1974). The performances of first, second and third grade children on five concepts of probability and the effects of grade, I.Q. and embodiments on their performances. Unpublished doctoral thesis. Bloomington: Indiana University.
Jones, G.A., Langrall, C.W., Thornton, C.A. & Mogill, A.T. (1997). A framework for assessing and nurturing young children’s thinking in probability. Educational Studies in Mathematics, 32, 101–125.
Kahneman, D. & Tversky, A. (1972). Subjective Probability: A judgement of representativeness. Cognitive Psychology, 3, 430–454.
Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6, 59–98.
Konold, C., Pollatsek, A., Well, A., Lohmeier, J. & Lipson, A. (1993). Inconsistencies in students’ reasoning about probability. Journal for Research in Mathematics Education, 24, 392–414.
Lecoutre, M-P. (1992). Cognitive models and problem spaces in “purely random” situations. Educational Studies in Mathematics, 23, 557–568.
Li, J. & Pereira-Mendoza, L. (2002). Misconceptions in probability. In B. Phillips (Ed.), Proceedings of the Sixth International Conference on Teaching Statistics, Cape Town, South Africa [CD-ROM]. Voorburg, The Netherlands: International Statistical Institute.
Miles, M.B. & Huberman, A.M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). Thousand Oaks, CA: Sage Publications.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA.
Page, D.A. (1959). Probability. In H.P. Fawcett, A.M. Hach, C.W. Junge, H.W. Syer, H. van Egen & P.S. Jones (Eds.), The growth of mathematical ideas: Grades K-12 - 24th NCTM Yearbook (pp. 229–271). Reston, VA: National Council of Teachers of Mathematics.
Piaget, J. & Inhelder, B. (1975). The origin of the idea of chance in children. (L. Leake, Jr., P. Burrell, & H.D. Fishbein, Trans.). New York: W.W. Norton and Company. (original work published 1951)
Saldanha, L. & Thompson, P. (2002). Conceptions of sample and their relationship to statistical inference. Educational Studies in Mathematics, 51, 257–270.
Shaughnessy, J.M. (1977). Misconceptions of probability: An experiment with a small-group activity-based, model building approach to introductory probability at the college level. Educational Studies in Mathematics, 8, 295–316.
Shaughnessy, J.M. (1992). Research in probability and statistics: Reflections and directions. In D.A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 465–494). New York: NCTM & Macmillan.
Shaughnessy, J.M., Watson, J., Moritz, J. & Reading C. (1999, April). School mathematics students’ acknowledgment of statistical variation. In C. Maher (Chair), There’s more to life than centers. Presession Research Symposium conducted at the 77th Annual National Council of Teachers of Mathematics Conference, San Francisco, CA.
Watson, J.M. (1994). Instruments to assess statistical concepts in the school curriculum. In National Organizing Committee (Ed.), Proceedings of the Fourth International Conference on Teaching Statistics, Volume 1 (pp. 73–80). Rabat, Morocco: National Institute of Statistics and Applied Economics.
Watson, J.M., Collis, K.F. & Moritz, J.B. (1997). The development of chance measurement. Mathematics Education Research Journal, 9, 60–82.
Watson, J.M. & Kelly, B.A. (2004). Expectation versus variation: Students’ decision making in a chance environment. Canadian Journal of Science, Mathematics and Technology Education, 4, 371–396.
Watson, J.M. & Kelly, B.A. (2006). Expectation versus variation: Students’ decision making in a sampling environment. Canadian Journal of Science, Mathematics and Technology Education, 6, 145–166.
Watson, J.M. & Moritz, J.B. (1998). Longitudinal development of chance measurement. Mathematics Education Research Journal, 10(2), 103–127.
Watson, J.M., Moritz, J.B. (2003). Fairness of dice: A longitudinal study of students’ beliefs and strategies for making judgments. Journal for Research in Mathematics Education, 34, 270–304.
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Watson, J.M., Kelly, B.A. Development of Student Understanding of Outcomes Involving Two or More Dice. Int J of Sci and Math Educ 7, 25–54 (2009). https://doi.org/10.1007/s10763-007-9071-1
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DOI: https://doi.org/10.1007/s10763-007-9071-1