
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: http://eduweb.nie.edu.sg/REACTOld/1997/1/1.html.
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 indepth
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 selfassessment 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 Amazon.com.]]
Way, Jenni. (University of Western Sydney .) Laying
foundations in chance and data. Accessed 4/10/03: http://hsc.csu.edu.au/pta/mansw/reflections/vol22no1_97way.htm.
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
researchbased microworld to enhance children's intuitive
understandings of chance and data. Focus on Learning
Problems in Mathematics 22(34), 165178. Accessed 4/10/03:
http://216.239.53.100/search?q=cache:VpemU0h6
jIC:www.probexplorer.com/Articles/Drier2000F
ocus.pdf+Inhelder+probabilistic+
thinking&hl=en&ie=UTF8
[[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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