Building a Connection between Experimental and Theoretical Aspects of Probability

Seth Ireland; Jane Watson

doi:10.29333/iejme/244

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Seth Ireland, Jane Watson

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Abstract

This paper addresses a question identified by Graham Jones: what are the connections made by students in the middle years of schooling between classical and frequentist orientations to probability? It does so based on two extended lessons with a class of Grade 5/6 students and in-depth interviews with eight students from the class. The Model 1 version of the software TinkerPlots was used in both settings to simulate increasingly large samples of random events. The aim was to document the students’ understanding of probability on a continuum from experimental to theoretical, including consideration of the interaction of manipulatives, the simulator, and the law of large numbers. A cognitive developmental model was used to assess students’ understanding and recommendations are made for classroom interventions.

Keywords

Experimental estimates of probability
Theoretical probability
simulation
law of large numbers
TinkerPlots

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, 2009, Volume 4, Issue 3, 339-370

https://doi.org/10.29333/iejme/244

Publication date: 12 Dec 2009

Article Views: 4003

Article Downloads: 2921

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References

Abrahamson, D. & Cendak, R. M. (2006). The odds of understanding the law of large numbers: A design for grounding intuitive probability in combinatorial analysis. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the Thirtieth Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 1–8). Prague: Charles University. Retrieved June 5, 2009, from http://gse.berkeley.edu/faculty/dabrahamson/publications/Abrahamson-Cendak_PME30.pdf.
Abrahamson, D., Janusz, R., & Wilensky, U. (2006). There once was a 9-Block ... – A middle-school design for probability and statistics. Journal of Statistics Education, 14(1). Retrieved June 5, 2009, from http://www.amstat.org/publications/jse/v14n1/abrahamson.html.
Abrahamson, D., & Wilensky, U. (n. d.). Connected Mathematics: ProbLab. The Center for Connected Learning and Computer-Based Modeling. Retrieved June 5, 2009, from http://ccl.northwestern.edu/ProbLab/.
Abrahamson, D., & Wilensky, U. (2007). Learning axes and bridging tools in a technology-based design for statistics. International Journal of Computers for Mathematics Learning, 12(1), 23–55. Retrieved June 5, 2009, from http://ccl.northwestern.edu/papers/2007/04-learningaxes.pdf.
Basson, A., Krantz, S., & Thornton, B. (2006). A new kind of instructional mathematics laboratory. Primus, 16(4), 332– 348.
Ben-Zvi, D., Gil, E., & Apel, N. (2007). What is hidden beyond the data? Helping young students to reason and argue about some wider universe. In D. Pratt & J. Ainley (Eds.), Reasoning about Informal Inferential Statistical Reasoning: A collection of current research studies. Proceedings SRTL-5. Retrieved June 5,, 2009, from http://srtl.stat.auckland.ac.nz/srtl5/view_presentation/presentation_id=49.
Biggs, J. B. & Collis, K. F. (1982). Evaluating the quality of learning: The SOLO taxonomy (structure of the observed learning outcome). New York: Academic Press.
Biggs, J. B. & Collis, K. F. (1991). Multimodal learning and the quality of intelligent behaviour. In H. A. H. Rowe (Ed.), Intelligence: Reconceptualisation and measurement (pp. 57–76). Hillsdale, NJ: Lawrence Erlbaum Associates.
Blythe, T. (1998). The teaching for understanding guide. San Francisco: Jossey-Bass.
Borovcnik, M. & Bentz, H.-J. (1991). Empirical research in understanding probability. In R. Kapadia & M. Borovcnik (Eds.), Chance encounters : Probability in education (pp. 73–105). Dordrecht: Kluwer.
Clements, D. (1999). Concrete manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1(1), 45–60.
Department of Education. (2008). The Tasmanian Curriculum: Mathematics-numeracy. K–10 syllabus and support materials. Hobart: Author. Retrieved June 5, 2009, from http://www.education.tas.gov.au/curriculum/standards/maths.
Erickson, T. (2006). Using simulation to learn about inference. In A. Rossman & B. Chance (Eds.), Proc. Seventh Intern. Conf. Teaching Statistics: Working Cooperatively in Statistics Education, Salvador, Brazil. Voorburg: International Statistical Institute. [CDROM].
Fischbein, E. (1982). Intuition and proof. For the Learning of Mathematics, 3(2), 9–19.
Ireland, S. (2007). Making connections between concrete objects and abstract concepts: Developing students’ understanding of the connection between observed experimental probability outcomes and the associated theoretical probability, aided by computer simulation software. Unpublished Honours dissertation. Hobart: Education Faculty, University of Tasmania.
James, G. & James, R. (1959). Mathematics dictionary. Princeton, NJ: D. Van Nostrand Company Inc.
Jones, G. (2005). Reflections. In G. Jones (Ed.), Exploring probability in school: Challenges for teaching and learning (pp. 367–372). New York: Springer.
Konold, C. (2006). Handling complexity in the design of educational software tools. In A. Rossman & B. Chance (Eds.), Proceedings of the Seventh International Conference on Teaching Statistics: Working Cooperatively in Statistics Education, Salvador, Brazil. [CDROM] Voorburg: International Association for Statistical Education and the International Statistical Institute.
Konold, C., Harradine, A., & Kazak, S. (2007). Understanding distributions by modelling them. International Journal of Computers for Mathematical Learning, 12, 217–230.
Konold, C. & Kazak, S. (2008). Reconnecting data and chance. Technology Innovations in Statistics Education, 2, Article 1. Retrieved June 5, 2009, from http://repositories.cdlib.org/uclastat/cts/tise/vol2/iss1/art1.
Konold, C. & Lehrer, R. (2008). Technology and mathematics education: An essay in honor of Jim Kaput. In L.D. English (Ed.), Handbook of international research in mathematics education (2nd ed.) (pp. 49–71). New York: Routledge.
Konold, C. & Miller, C. D. (1993). ProbSim. [Computer software] Amherst, MA: University of Massachusetts.
Konold, C. & Miller, C. D. (2005). TinkerPlots: Dynamic data exploration. [Computer software] Emeryville, CA: Key Curriculum Press.
Malhotra, N., Hall, J., Shaw, M., & Crisp, M. (1996). Marketing research: An applied orientation. Sydney: Prentice Hall Australia.
Mills, J. (2002). Using computer simulation methods to teach statistics: A review of the literature. Journal of Statistics Education, 10(1). Retrieved June 5, 2009, from http://www.amstat.org/publications/jse/v10n1/mills.html.
Paparistodemou, E. & Meletiou-Mavrotheris, M. (2008). Developing young students’ informal inference skills in data analysis. Statistics Education Research Journal, 7(2), 83–106. Retrieved June 5, 2009, from http://www.stat.auckland.ac.nz/~iase/serj/SERJ7(2)_Paparistodemou.pdf.
Piaget, J. (1955). The construction of reality in the child. London: Routledge & Kegan Paul.
Pratt, D. & Noss, R. (2002). The microevolution of mathematical knowledge: The case of randomness. Journal of Learning Science, 11, 453–488.
Prediger, S. (2008). Do you want me to do it with probability or with my normal thinking? Horizontal and vertical views on the formation of stochastic conceptions. International Electronic Journal of Mathematics Education, 3(3), 126–154. Retrieved June 5, ,2009, from http://www.iejme.com/032008/ab1.htm.
Rider, R. & Stohl Lee, H. (2006). Differences in students’ use of computer simulation tools and reasoning about empirical data and theoretical distributions. In A. Rossman & B. Chance (Eds.), Proc. Seventh Intern. Conf. Teaching Statistics: Working Cooperatively in Statistics Education, Salvador, Brazil. Voorburg: International Statistical Institute. [CDROM].
Sarama, J. & Clements, D. (1998). Using computers for algebraic thinking. Teaching Children Mathematics, 5(3), 186– 190.
Steen, L. (2007). Every teacher is a teacher of mathematics. Principal Leadership, 7(5), 16–20.
Stein, M. & Bovalino, J. (2001). Manipulatives: One piece of the puzzle. Mathematics Teaching in the Middle School, 6(6), 356–359.
Stohl, H., Rider, R., & Tarr, J. (2004). Making connections between empirical and theoretical probability: Students’ generation and analysis of data in a technological environment. Retrieved June 5, 2009, from http://www.probexplorer.com/Articles/LeeRiderTarrConnectE&T.pdf
Stohl, H. & Tarr, J. (2002). Developing notions of inference using probability simulation tools. Journal of Mathematical Behaviour, 21, 319–337.
Van de Walle, J. (2004). Elementary and middle school mathematics: Teaching developmentally (5th ed.). Boston: Pearson Education, Inc.
Watson, J. M. & Callingham, R. A. (2003). Statistical literacy: A complex hierarchical construct. Statistics Education Research Journal, 2(2), 3–46. Retrieved June 5, 2009, from http://www.stat.auckland.ac.nz/~iase/serj/SERJ2(2)_Watson_Callingham.pdf.
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. & 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.
Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematics education. In I. Harel & S. Papert (Eds.), Constructionism (pp. 193–204). Norwood, NJ: Ablex Publishing Corporation. Retrieved June 5, 2009, from http://ccl.northwestern.edu/papers/concrete/.
Wordnet. (n. d.). Computer simulation. Wordnet: A lexical database for the English language. Retrieved June 1, 2007, from http://wordnetweb.princeton.edu/perl/webwn?s=computer%20simulation.

How to cite this article

APA

Ireland, S., & Watson, J. (2009). Building a Connection between Experimental and Theoretical Aspects of Probability. International Electronic Journal of Mathematics Education, 4(3), 339-370. https://doi.org/10.29333/iejme/244

Vancouver

Ireland S, Watson J. Building a Connection between Experimental and Theoretical Aspects of Probability. INT ELECT J MATH ED. 2009;4(3):339-70. https://doi.org/10.29333/iejme/244

AMA

Ireland S, Watson J. Building a Connection between Experimental and Theoretical Aspects of Probability. INT ELECT J MATH ED. 2009;4(3), 339-370. https://doi.org/10.29333/iejme/244

Chicago

Ireland, Seth, and Jane Watson. "Building a Connection between Experimental and Theoretical Aspects of Probability". International Electronic Journal of Mathematics Education 2009 4 no. 3 (2009): 339-370. https://doi.org/10.29333/iejme/244

Harvard

Ireland, S., and Watson, J. (2009). Building a Connection between Experimental and Theoretical Aspects of Probability. International Electronic Journal of Mathematics Education, 4(3), pp. 339-370. https://doi.org/10.29333/iejme/244

MLA

Ireland, Seth et al. "Building a Connection between Experimental and Theoretical Aspects of Probability". International Electronic Journal of Mathematics Education, vol. 4, no. 3, 2009, pp. 339-370. https://doi.org/10.29333/iejme/244