Abstract
The concept of probability plays a vital role in mathematics and scientific research, as well as in our everyday lives. It has also become one of the fastest growing segments of high school and college curricula, yet learning probability within school contexts has proved more difficult than many in education realize.
This chapter is in two broad parts. The first part synthesizes a discussion of randomness and probability that is situated at the nexus of bodies of literature concerned with the ontology of stochastic events and epistemology of probabilistic ideas held by people. Our synthesis foregrounds philosophical, mathematical, and psychological debates about the meaning of randomness and probability that highlight their deeply problematic nature, and therefore raises the equally problematic question of how instruction might support students’ understanding of them. We propose an approach to the design of probability instruction that focuses on the development of coherent meanings of randomness and probability—that is, schemes composed of imagery and conceptual operations that stand to support students’ coherent thinking and reasoning about situations that we see as entailing randomness and probability. The second part of the chapter reports on aspects of a sequence of classroom teaching experiments in high school that employed such an instructional approach. We draw on evidence from these experiments to highlight challenges in learning and teaching stochastic conceptions of probability. Our students’ challenges centered on re-construing given situations as idealized random experiments involving the conceptualization of an unambiguous and essentially repeatable trial, as a basis for conceiving of the probability of an event as its anticipated long-run relative frequency.
Research reported in this chapter was supported by National Science Foundation Grant No. REC-9811879. Any conclusions and recommendations stated here are those of the authors and do not necessarily reflect official positions of NSF. We wish to acknowledge the contribution of Patrick W. Thompson—dissertation advisor to both authors, and principal investigator of the research project on which this chapter is based.
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Notes
- 1.
Because of his background as a physicist, von Mises was more concerned with the link between probability theory and natural phenomena. von Mises regarded probability as “a scientific theory of the same kind as any other branch of the exact natural science” (ibid., p. 7).
- 2.
A probability curriculum starting with the axiomatic approach to probability may prevent students from developing an understanding of probability that is rooted in their experience. It is like teaching the concept of circle as a set of points (x,y) that satisfies the condition (x−a)2+(y−b)2=c 2, \(a,b,c,\in \mathbb{R}\), c≥0. If this definition precedes the introduction of the concept of distance and measurement of length, students may experience difficulties in visualizing the image of a circle as a set of points having the same distance from a fixed point in a two-dimensional surface.
- 3.
The failure of differentiating between possible and necessary as a developmental constraint was confirmed by later studies of Kahneman and Tversky (1982) and Konold (1989). They found that even when people overcame the developmental constraint, they might still fail, or rather refuse, to distinguish uncertain events from necessary ones due to a deterministic world view.
- 4.
We note that there are two other potential sources of deviation from the standard solution for the taxi problem. If a subject conceptualizes the problem as P(Taxi is Blue|Witness says “blue”), then to give the standard solution, the subject must take the witness’ accuracy rate as his or her propensity to say “Blue,” and the subject might see no compelling reason to assume this because there is no information on the witness’ propensity to say “I’m unsure”. Second, if the subject takes it as a fact that the witness said “Blue”, then P(Witness says “blue”) is 1, and hence P(Taxi is Blue|Witness says “blue”) is equal to P(Taxi is Blue).
- 5.
The term “probabilistic revolution” (Krüger 1987) broadly suggests a shift in world view within the scientific community from 1800 through 1930, from a deterministic reality where everything in the world is connected by necessity in the form of cause–effect relationships, to one in which uncertainty and probability have become central and indispensable.
- 6.
Consider the example given by Gigerenzer (ibid., p. 17): A scenario in probability format: The probability that a person has colon cancer is 0.3 %. If a person has colon cancer, the probability that the test is positive is 50 %; if not, the probability that the test is positive is 3 %. What is the probability that a person who tests positive actually has colon cancer? In natural frequency format, 30 out of every 10,000 people have colon cancer. Of these 30 people, 15 will test positive. Of the remaining 9,970 people, 300 will still test positive. To solve the problem in probability format, one uses Bayes’ theorem: (0.3 % × 50 %)/(0.3 % × 50 % + 99.7 % × 3 %). In the natural frequency format, 15/(300+15) will suffice. However, to justify substituting numbers in place of the percentages requires understanding that the underlying proportional relationships will remain constant no matter the population’s size.
- 7.
We relate an anecdote in anticipation of Bayesians’ objections to this statement: In a recent conversation with a prominent Bayesian probabilist, we asked “Suppose, at a given moment, and for whatever reason, you judge that the probability of an event is 0.4. At that moment, what does “0.4” mean to you?” He responded that it meant that he anticipates that this event will occur 40 % of the time.
- 8.
The authors were members of the research team that conducted the project.
- 9.
- 10.
Adapted from Konold (2002).
- 11.
The following symbol key is used for discussion protocols: “Instr” denotes an utterance made by the instructor; “[…]” denotes text that is not included in the presented excerpt; “—” denotes that the word or statement immediately preceding it was abruptly halted in speech. Underlined words and statements denote that they were emphasized in speech. Statements enclosed in parentheses generally provide contextual information not captured by utterances, such as participants’ nonverbal actions or the duration of pauses or silences in a discussion.
- 12.
The distinction between sampling with and without replacement had been established in classroom discussions prior to this point in the teaching experiment.
- 13.
“Int” denotes an utterance made by the interviewer.
- 14.
Our second teaching experiment consisted of 22 lessons, and eventually addressed the concepts of conditional probability, independence, and Bayes Theorem.
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Saldanha, L., Liu, Y. (2014). Challenges of Developing Coherent Probabilistic Reasoning: Rethinking Randomness and Probability from a Stochastic Perspective. In: Chernoff, E., Sriraman, B. (eds) Probabilistic Thinking. Advances in Mathematics Education. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7155-0_20
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