### A Preliminary Genetic Decomposition of Probabilistic Independence

#### Abstract

*The purpose of this research is to construct a preliminary genetic decomposition delineating the mental constructions underlying probabilistic independence. This delineation will be considered within the framework of APOS theory. While the use of the term independence in probability is often conflated with causation, the technical definition relies instead upon an understanding of conditional probability. I hypothesize that the concept of independence is only fully available to students with a schema conception of probability. I offer additional hypotheses, supported by literature and anecdotal teaching experience, regarding students’ quantification of probability and construction of combinatorial reasoning.*

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Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Fuentes, S. R., Trigueros, M., & Weller, K. (2014). Genetic decomposition. In APOS theory: A framework for research and curriculum development in mathematics education (pp. 27–55). New York, NY: Springer.

D’Amelio, A. (2009). Undergraduate student difficulties with independent and mutually exclusive events concepts. The Montana Mathematics Enthusiast, 6(1&2), 47-56.

Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In Advanced mathematical thinking (pp. 95–126). Springer Netherlands.

English, L. D. (1991). Young children’s combinatorics strategies. Educational Studies in Mathematics, 22, 451–474.

Jones, G.A., Langrall, C.W. & Mooney, E.S. (2007). Research in probability: Responding to classroom realities. In The Second Handbook of Research on Mathematics, Ed. F.K. Lester, pp. 909–956. Reston, VA: National Council of Teachers of Mathematics (NCTM)

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(2), 101-125.

Jones, G.A., Langrall, C.W., Thornton, C.A. & Mogill, A.T. (1999). Students’ probabilistic thinking in instruction. Journal for Research in Mathematics Education, 30(5), 487–519.

Mathews, D. & Clark, J. (2003). Successful Students’ Conceptions of Mean, Standard Deviation and the Central Limit Theorem. Unpublished paper. Retrieved October 20, 2015, from http://www1.hollins.edu/faculty/clarkjm/stats1.pdf

Ollerton, R. L. (2015). A unifying framework for teaching probability event types. International Journal of Mathematics Education in Science and Technology, 46(5), 790-794.

Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children. (L. Leake, Jr., P. Burrell & H. D. Fischbein, Trans.). New York, NY: Norton. (Original work published 1951)

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 Mathematics, 8(3), 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, NY: Maxwell Macmillan.

Shaughnessy, J.M. (2003). Research on students’ understanding of probability. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 216–225). Reston, VA: NCTM.

Shin, J., & Steffe, L. P. (2009). Seventh graders’ use of additive and multiplicative reasoning for enumerative combinatorial problems. In S. L. Swars, D. W. Stinson, & S. Lemons-Smith (Eds.), Proceedings of the 31st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 170–177). Atlanta, GA: Georgia State University.

Steffe, L. P. Schemes of action and operation involving composite units. Learning and Individual Difference, 4(3), 259–309.

Tarr, J. E., & Jones, G. A. (1997). A framework for assessing middle school students’ thinking in conditional probability and independence. Mathematics Education Research Journal, 9(1), 39–59.

Tillema, E. (2011). Students’ combinatorial reasoning: The multiplication of binomials. In L. R. Wiest & T. Lamberg (Eds.), Proceedings of the 33rd Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 321–328). Reno, NV: University of Nevada, Reno.

Tillema, E. S. (2012). Relating one- and two-dimensional quantities: Students’ multiplicative reasoning in combinatorial and spatial contexts. In R. Mayes, & L. Hatfield (Eds.), Quantitative reasoning and mathematical modeling: A driver for STEM integrated education and teaching in context (pp. 113–126). Laramie: University of Wyoming.

Tversky, A., & Kahneman, D. (1971). Belief in the law of small numbers. Psychological Bulletin, 76, 105–110.

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