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Designing Effective Lessons on Probability: A Pilot Study Focused on the Illusion of Linearity

Title:

Designing Effective Lessons on Probability: A Pilot Study Focused on the Illusion of Linearity

Miszaniec, Jean-Marc (2016) Designing Effective Lessons on Probability: A Pilot Study Focused on the Illusion of Linearity. Masters thesis, Concordia University.

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Abstract

This thesis is a summary of a pilot study adopting the design experiment methodology investigating alternative approaches to the teaching of probability – the classical approach (called “Laplace approach” in our study) and the axiomatic approach (called “Properties approach”) – and alternative pedagogical treatments of these approaches: with or without explicit refutations of common misconceptions by the teacher. Four short video-lessons were designed, focused on Bernoulli trials: “Laplace-exposition”; “Laplace-refutation”; “Properties-exposition” and “Properties-refutation “. The primary misconception addressed in the study was the erroneous application of proportionality models: the misconception called “illusion of linearity” in the context of probability. The design and layout of the videos’ visuals and audio were informed by the principles of cognitive load theory in multimedia to ensure optimal learning potential. Each lesson was tried with one volunteer participant, a post-college student. Each participant listened to the video-lesson assigned to him or her, solved six related probability questions, responded to a questionnaire and was interviewed one-on-one by the researcher. Data in the form of participants’ written responses, transcripts of the interviews and researcher’s notes were used for a thorough qualitative analysis and to inform future iterations of the study.
Research suggests that the primary obstacles to probability education are misconceptions originating from cognitive biases and limitations. Often misconceptions in probability are attributed to intuitions. Intuitions are easily accessed and experience-based knowledge that can help or hinder individuals while problem solving. One question of this study was whether stating and refuting known misconceptions during a lecture-style lesson on Bernoulli trials (refutation treatment) promotes learning better than a lesson without the mention of misconceptions (exposition treatment).
This study also intended to illuminate the nature of probability misconceptions, as research suggests that attributing them to “intuitions” may be misleading. “Intuition”, especially, “robust intuition”, usually refers to a connected set of beliefs, but our observations suggest that students’ predictions about probability of events are based on something much less homogeneous and connected. Participants in the study did not have consistent and robust intuitions about Bernoulli trials but rather a weaker form of basic knowledge called phenomenological primitives. Their intuitions would have to be developed, perhaps through a frequentist approach to probability. Participants had a high distrust in their initial guesses – a symptom of the weakness or lack of “robust intuition” – and this played an important role throughout the problem solving process. The Properties approach seemed to have more success than the Laplace approach because of its computational simplicity and because participants may have naturally implemented the Properties approach when thinking of chance events. The approaches did not seem to alter the use of misconceptions. The refutation treatment did not have the expected outcome on learning. Participants subjected to it performed better than those subjected to the exposition treatment but this could be because they “knew” (were told) – not necessarily “understood” – that changes to the number of trials and the number of desired events affect the probability of the desired event in a non-proportional manner. Future iterations of this study would explore further the above conjectures about the nature of the basis of students’ predictions about probability and the appropriateness of different approaches to probability to build on this basis. Also, other common probability misconceptions would be explored to determine their cognitive origins in order to inform the design of instructional content.

Divisions:Concordia University > Faculty of Arts and Science > Mathematics and Statistics
Item Type:Thesis (Masters)
Authors:Miszaniec, Jean-Marc
Institution:Concordia University
Degree Name:M.T.M.
Program:Teaching of Mathematics
Date:3 May 2016
Thesis Supervisor(s):Sierpinska, Anna
Keywords:Mathematics education probability intuition obstacles lessons
ID Code:981221
Deposited By: JEAN-MARC MISZANIEC
Deposited On:17 Jun 2016 15:00
Last Modified:18 Jan 2018 17:52

References:

Bachelard, G. (1938/1983). La formation de l'esprit scientifique. Paris: PUF.
Borovcnik, M., Bentz, H.-J., & Kapadia, R. (1991). A probabilistic perspective. In R. Kapadia, & M. Borovcnik, Chance encounters: Probability in education (pp. 27-71). Dortrecht: Kluwer.
Brousseau, G. (1997). Theory of didactical situations in mathematics. Dortrecht: Kluwer.
Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18 (1), 32-42.
Chennu, S. (2013). Expectation and attention in hierarchical auditory prediction. The Journal of Neurosciene, 33(27), 111940-11205.
Chiasson, & McMillan. (2008). Astronomy Today: The Solar System (6e). San Francisco: Pearson Education Inc.
Cobb, P., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9-13.
d'Alembert, J. (1754). Croix ou pile. In D. Diderot, & J. d'Alembert, Encyclopédie (Vol. 4, p. 512). Paris. Retrieved from http://artflsrv02.uchicago.edu/cgi-bin/philologic/getobject.pl?c.3:1169:29.encyclopedie0513.5344355.5344361.5344364
De Bock, D., Van Dooren, W., Janssens, D., & Verschaffel, L. (2002). Use of linear reasoning: An in-depth study of the nature and the irresistibility of secondary school students' errors. Educational studies in mathematics, 311-334.
Diderot, D. (n.d.). Jouer. Retrieved March 14, 2016, from Encyclopédie ou Dictionnaire Raisonné des Arts et Métiers (1751-1772; page 8:884): http://artflsrv02.uchicago.edu/cgi-bin/philologic/getobject.pl?c.7:2770:1.encyclopedie0513
Diderot, D., & d'Alembert, J. (1751-1772). L'Encyclopédie ou Dictionnaire raisonné des arts et métiers. Paris.
diSessa, A. (1983). Phenomenology and the Evolution of Intuition. In D. Genter, & A. Stevens, Mental Models (pp. 15-33). Hillsdale, NJ: Lawrence Erlbaum Associates.
Fine, T. (1973). Theories of probability. An examination of foundations. New York and London: Academic Press.
Fischbein, E. (1975). The intuitive Sources of Probabilistic Thinking in Children. Dordrecht, Holland: D. Reidel Pulishing Company.
Fischbein, E. (1999). Intuitions and schemata in mathematical reasoning. Educational Studies in Mathematics, 38, 11-50.
Fischbein, E., & Schnarch, D. (1997). The evolution with age of probabilistic intuitively based misconceptions. Journal for Research in Mathematics Education 28(1), 96-105.
Freudenthal, H. (1973). Mathematics as an educational task. Dortrecht: D. Reidel.
Gardner, M. (1987). The Second Scientific American Book of Mathematical Puzzles and Diversions. University of Chicago Press.
Gorroochurn, P. (2011). Errors of probability in a historical context. The American Statistician (65), 246-255.
Greer, B. (2001). Understanding probabilistic thinking: the legacy of Efraim Fischbein. Educational Studies in Mathematics.
Guenther, W. (1968). Concepts of Probability. New York: Mcgram-Hill Boo Company.
Hoffrage, U., & Gigerenzer, G. (1995). How to improve bayesian reasoning without instruction: frequency formats. Psychological Review 102(4), 684-704.
Jaynes, E. (1973). The well posed problem. Foundation of Physics, 477-493.
Kahneman, D. (2003). A perspective on judgment and choice: maps of bounded rationality. The American Economic Review 58(9), 697-720.
Kahneman, D. (2011). Thinking fast and slow. New York, New York: Ferrar, Strauss and Giroux.
Kahneman, D., & Frederic, S. (2002). Heuristics of Intuitive Judgment published in . . In T. Gilovich, D. D. Griffin, & D. Kahneman. New York.: Cambridge University Press.
Kahneman, D., & Tversky, A. (1972). Subjective probability: A judgment of representativeness. Cognitive Psychology, 3, 430-454.
Kahneman, D., Slovic, P., & Tversky, A. (1982). Judgement under Uncertainty: Heuristics and Biases. Cambridge: University Press.
Lamprianou, I., & Lamprianou, T. (2002). The nature of pupils' probabilistic thinking in primary school pupils in Cyprus,. Proceedings of the 26th Conference of the Interna-tional Group for the Psychology of Mathematics Education, Vol. 3, (pp. 273-280). Norwich, UK.
Laplace, P. (1814). Théorie analytique des probabilités. Seconde édition. Paris: Ve. Courcier. Retrieved from https://archive.org/details/thorieanalytiqu01laplgoog
Leinhardt, G., Zaslavsky, O., & Stein, M. (1990). Functions, graphs, and graphing: Tasks, learning, and teaching . Review of Educational Research 60(1), 1-64.
Leron, U., & Hazzan, O. (2009). Intuitive vs. analytical thinking: four perspectives . Educational studies in Mathematics (71), 263-278.
Maistrov, L. (1974). Probability theory: A historical sketch. New York: Academic Press.
Maplesoft Group, Inc. (2015, May 26). Home page. Retrieved from Maplesoft Group: http://www.maplesoftgroup.com/en
Marinoff, L. (1994). A resolution of Bertrand's Paradox. Philosophy of Science, 61(1), 1-24.
Miller, G. (1956). The magical number seven, plus or minus two: Some limits on our capacity for prcessing information. The Psychological Review, 81-97.
Muller, D. (2008). Designing effective multimedia for physics learning. Australia: University of Sydney.
Myers, D. (2010). Psychology. New York, NY: Worth Publishers.
Ore, O. (1960). Pascal and the intuition of probability theory. American Mathematical Monthly, 67(5), 409-419.
Piaget, J., & Inhelder, B. (1951). The Origin of the Idea of Chance in Children. Routledge, London.: Norton, W.W. &Company, Inc.
Pollock, E., Chandler, P., & Sweller, J. (2002). Assimilating complex information. Learning and Instruction, 61–86.
Popkin, R. H. (1999). The Columbia history of western philosophy. New York: Columbia University Press.
Ritson, R. (1998). The development of primary school children's understanding of probability. Queen's University, Belfast: Unpublished thesis.
Shaughnessy, M. J. (1977). An Experiment with a Small-Group, Activity-Based, Model Building Approach to Introductory Probability at the College Level. Educational Studies in Mathematics, 295-316.
Shiv, B., & Fedorkikhin, A. (1999). Heart and mind in conflict: the interplay of affect and cognition in consumer decision making. The journal of consumer research, 26(3), 278-293.
Sierpinska, A. (1990). Some remarks on understanding in mathematics. For the Learning of Mathematics, 10(3), 24-36.
Sierpinska, A. (1994). Understanding in mathematics. London: Falmer.
Sierpinska, A. (2005). On practical and theoretical thinking and other false dichotomies in mathematics education. In M. Hoffmann, J. Lenhard, & F. Seeger, Activity and sign: Grounding mathematics education (pp. 117-136). New York: Springer.
Sierpinska, A., Bobos, G., & Pruncut, A. (2011). Teaching absolute value inequalities to mature students. Educational Studies in Mathematics, 78(3), 275-305.
Steinbring, H. (1991). The theoretical nature of probability in the classroom. In R. Kapadia, & M. Borovcnik, Chance encounters in probability (pp. 135-167). Springer.
Van Dooren, W., De Bock, D., Depaepe, F., Janssens, D., & Verschaffel, L. (2003). The illusion of linearity: Expanding the evidence towards probabilistic reasoning. Educational Studies in Mathematics, 50, 113-138.
van Frassen, B. (1984). Belief and Will. Journal of Philosophy, 235-256.
Winkler, P. (2013, February 16). Retrieved from Simon's Foundation- Advancing Research in Basic Science and Mathematics: http://www.simonsfoundation.org/multimedia/simons-science-series/probability-intuition/
Winter, H. (1983). Zur problematik des beweisbedurfnisses. Journal fur Mathematik-didaktik , 59-95.
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