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Developmental Theory and Probabilistic Thinking

This has proven to be a fruitful area of math education research and includes early (1951) work by Piaget and Inhelder.

Soen, Chan Wai (June, 1997). Intuitive Thinking and Probability. React Issue No. 1. Accessed 4/10/03: The following are brief quotes from the article:

Piaget and Inhelder (1975) were the first researchers to study the development of the idea of chance in children. According to them, the concept of probability as a formal set of ideas develops only during the formal operational stage, which occurs about twelve years of age. By that age, children can reason probabilistically about a variety of randomizing devices. In an experiment to demonstrate that children have an intuitive understanding of the law of large numbers and that intuitive thinking about chance events starts even before they are taught, they used a game with pointers which were stuck onto cards divided into various sectors and then spun. They found the children could predict that, in the long run, the pointer would fall onto every region marked on the card.

Other researchers have looked at the effect of instruction. Fischbein (1987) explored the foundation of intuitive thinking as precursors to learning probability in mathematics. He asserted that there were primary intuitions which were related to personal experiences that appeared prior to instruction and secondary intuitions which appeared through instructional influence. He found that intuitive ideas, whether primary or secondary, often resulted in fallacies.

Implications for Teaching

Researchers have suggested a number of implications for instruction to overcome students' difficulties in learning a subject like probability that tends to be influenced by erroneous intuitive thinking.

1. Garfield and Ahlgren (1988) contend that before the teaching of probability, students must have an understanding of ratio and proportion. Students must be able to function at the formal operational level. They must have the necessary skills in dealing with abstractions.

[[Comment from Moursund: I did the bold facing. If this is a correct conclusion, then it says that the teaching of probabilistic reasoning at the middle school level is a strong example of a mismatch between a student's developmental level and the math content being taught. Relatively few students are at a formal operations level while they are in middle school.]]

2. Teachers need to recognize and confront common errors in students' probabilistic reasoning.

To recognize them, researchers like Fischbein (1987) and Konold (1991) advocate the use of in-depth interviews. It is important to make students aware of how beliefs and conceptions can affect probabilistic judgments. Through interviews with a few of my students I found that predicting the occurrence of an event could mean asking whether an event was sure to happen, a result similar to the use of the outcome approach. All these interpretations I might not have found out had I used a pencil and paper test.

[[Comment from Moursund. It would be nice to have a self-assessment instrument or a simple instrument that does not require individual interviews.]]


Piaget, J. & Inhelder, B. (1975) The Origin of the Idea of Chance in Children. Routledge and Kegan Paul.

Piaget, J., & Inhelder, B. (1965). The Origin of the Idea of Chance in Children. (Original work 1951). Routledge & Kegan Paul, London.

[[Comment from Moursund. One of the two references given above has a wrong date. This book is referenced by a number of authors. I didn't find it listed on]]



Way, Jenni. (University of Western Sydney .) Laying foundations in chance and data. Accessed 4/10/03: Quoting from this article:

We all deal with situations involving the element of chance every day ; Will it rain today? What will I do if I miss the train? Is this a fair game? Much of our decision making involves trying to predict events and considering possible results of our actions. Whenever chance is involved, the unpredictability of the situation means that our normal logical thinking processes may not be entirely appropriate. For example, when you fill out Lotto entries, do you deliberately spread the numbers out? Why? Each number has the same chance of winning. The Lotto machine knows nothing of numerical order! (Actually, if you choose a string of consecutive numbers and win, you'd probably have the prize pool to yourself!). Our Australian culture includes some classic expressions that refer to our encounters with chance: 'Buckley's chance'; 'Murphy's law'; 'Against all odds'; 'Fat chance!'; 'In your dreams!', and of course some less polite ones.

Learning to reason 'correctly' using probability concepts, such as randomness and chance, is a skill that very few people have mastered, mainly because we have had very little assistance to do so. A fairly large body of research, for example Piaget and Inhelder (1965), Fischbein (1975), Green (1983), Jones et al. (1996), Way (1996), has shown that young children develop intuitive understandings of basic probability concepts without instruction. There is also evidence that many of these intuitions are misleading or incorrect, and that by adulthood, they develop into misconceptions that are extremely difficult to correct. However, studies such as Fischbein's and Jones's have established that children are responsive to appropriate instruction on probability concepts.

[[Comment from Moursund. The second paragraph seems particularly important to me.]]


Drier, H. S. (2000). The Probability Explorer: A research-based microworld to enhance children's intuitive understandings of chance and data. Focus on Learning Problems in Mathematics 22(3-4), 165-178. Accessed 4/10/03:

[[Comment from Moursund. A number of the articles suggest that hands on experience is essential in the instructional process. One can think of a MicroWorld as a virtual Manipulative. Thus, it provides an alternative to the traditional hands on work.]]


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