Counting as a Foundation for Learning to Reason About Probability
Based on findings from long-term and cross-sectional studies in a variety of contexts and across a variety of ages, we have found that in the activity of problem solving on strands of counting and probability tasks, students exhibit unique and rich repres
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Abstract Based on findings from long-term and cross-sectional studies in a variety of contexts and across a variety of ages, we have found that in the activity of problem solving on strands of counting and probability tasks, students exhibit unique and rich representations of counting heuristics as they work to make sense of the requirements of the tasks. Through the process of sense making and providing justifications for their solutions to the problems, students’ representations of the counting schemes become increasingly more sophisticated and show understanding of basic combinatorial and probabilistic reasoning.
1 Introduction The Roman philosopher, Cicero, remarked: “Probability is the very guide of life.” In today’s modern world where informed citizens need to understand the language of statistics and probability to absorb the information presented in news, media, health reports, sales and advertising, Cicero’s vision still applies. In 2000, the National Council of Teachers of Mathematics (NCTM) placed increased emphasis on probability and statistics learning in the K-12 curriculum by including these subjects as one of the five major content standards and recommending these as major content areas. The NCTM indicated that learning to reason probabilistically is not knowledge that learners intuitively develop. They point to the importance of including these subjects in the mainstream curriculum. Specifically, they indicate: “The kind of reasoning used in probability and statistics is not always intuitive, and so students will not necessarily develop it if it is not included in the curriculum” (NCTM 2000, p. 48). Several studies have shown that a strong foundation in counting methods is an important prerequisite for fundamental probabilistic reasoning (Alston and Maher C.A. Maher (B) Rutgers University, 10 Seminary Place, Room 231, New Brunswick, NJ 08901, USA e-mail: [email protected] A. Ahluwalia Brookdale Community College, 765 Newman Springs Road, Lincroft, NJ 07738, USA E.J. Chernoff, B. Sriraman (eds.), Probabilistic Thinking, Advances in Mathematics Education, DOI 10.1007/978-94-007-7155-0_30, © Springer Science+Business Media Dordrecht 2014
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2003; Benko 2006; Benko and Maher 2006; Kiczek et al. 2001; Francisco and Maher 2005; Maher 1998; Maher and Muter 2010; Maher et al. 2010; Shay 2008). The Rutgers Longitudinal Study, initiated in 1987, engaged students in working on a strand of counting and probability tasks in which the students were challenged to build and justify their solutions to the problems (Maher et al. 2010). The research has shown that under particular conditions in which students are invited to collaborate, share, revise and revisit ideas, sophisticated counting schemes are built (Maher 2005; Maher and Martino 1997). The counting schemes are represented in a variety of ways and, over time, are elaborated and expressed in more abstract forms, becoming increasingly more sophisticated and formal (Davis and Maher 1997; Maher and Weber 2010).
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