The Chance and Probability Concepts Project
D. R. GREEN

What are children’s early ideas on probability and how do they develop?

This project, which is funded by the Social Science Research Council, began in November 1978 and is to run for three years. A main aim is to investigate just what concepts and intuitions concerning random processes are present in the minds of children of varying abilities across the 11-16 age range. The ability to list permutations, combinations and arrangements is also being investigated and an analysis of CSE and GCE O - Level mathematics scripts will be undertaken.

Over the past two decades the topic of 'Probability’ has been brought into the mathematics curriculum but it may be that this is more an empty gesture rather than a sound strategy. That the basic ideas behind probabilistic understanding are important for all pupils is not denied—indeed mathematical educationalists and psychologists point out that the education system has ignored probability and argue this must be rectified. For example, E. Fischbein (1975) writes: "The child is taught that explanation consists in specifying a cause; that a scientific prediction must be a certainty; that ambiguity and uncertainty are not acceptable in scientific reasoning, and so on. Even if all this is not explicitly stated it is implied in all that is taught in schools" (p. 71). The over-emphasis in school education on deterministic skills is apparent. However, it seems very unlikely that the kind of mechanical probability calculations asked for in most texts are of much help. These are often based on an a priori approach with the experimental side played down (so not to spoil the pure mathematics).

Although the findings of the Piagetian school have become well-known in the realm of number, probably few mathematics teachers are aware of "The Origin of the Idea of Chance in Children" (Piaget and Inhelder, 1975) which provides a useful starting point for consideration of probability intuitions in pupils.

There is the strong tendency of adults to assume that immature children possess the probabilistic intuitions and concepts which they themselves find so 'natural’. Experience may educate the teacher otherwise if he takes the trouble to analyse the statements made by his pupils. Nevertheless determining just what pupils are thinking is a hazardous business!

The first stage of the Chance and Probability Concepts Project was to devise a test comprising about thirty to forty questions and to interview individually, a cross-section of pupils, asking them about their answers to the written test. Preliminary findings from just three of those questions will now be presented.

Question 1

A small round counter is red on one side and green on the other side. The counter is tossed into the air. Which side (red or green) is more likely to be face up when the counter lands? (Explain)...

Typical Answers

A: Wendy (11 years, very bright) "It would more likely face the red side if you put it on the red side and throw it up"

B: Andrew (11 years, very bright) "Green - for go"

C: Dawn (12 years, low ability) "Red - because it spins round because it stays the same

D: David (11 years, low ability) "Same chance"

E: Jane (12 years, very bright) "There is a fair chance"

F: Nigel (12 years, average) "Equal"

G: Wendy (13 years, bright) "Nobody can say for sure

H: Teresa (14 years, average) "It has a 1/2 chance of getting on red or green"

I: Maria (14 years, above average) "They are both equally likely"

J: Jayne (14 years, average) "You don’t know, it could land on any side"

K: David (15 years, below average) "The green side is more likely"

L: Lionel (15 years, below average) "Well it would be a 50-50 chance"

OVERALL ~SUCCESS’: 9 out of 15

The description of the pupils’ abilities were provided by their mathematics teachers or the head teacher. It comes as something of a shock that a "very bright" boy (B) is satisfied with "Green for go" whereas a low ability" boy (D), who incidentally has the same mathematics teacher, recognises the symmetry of the situation.

Most eleven year olds tested were inclined to plump for one colour whereas most fifteen year olds trotted out some such phrase as "equally likely". Can we conclude that this demonstrates the availability of a concept to the older child but not to the younger? There is little justification for such a conclusion from this data. We would certainly expect such a developmental sequence but the influence of language may be ‘important. Perhaps for the younger pupils the question is phrased to demand a choice. Perhaps the key word "likely" triggers off a rote-learnt response in those older pupils who have been "taught probability", but who may not really know what it means. What does emerge from this very limited study is that the secondary school mathematics teacher cannot assume that the 'equally likely' nature of Heads/Tails is at all obvious to all pupils. Colour preference might be reasonably predicted in (say) six - year olds but possibly it persists much longer. The fierce interest in what is "fair", so pronounced in young children, doesn’t seem to help here! Of course this may be because "fairness’ is not mentioned in the wording of the question. (It would be interesting to see if describing the counter as "fair" makes any difference to the responses.)

The same question was given to a mixed ability class of 1st year pupils and a revised question, with "or is there no difference?" added, was given to two similar classes and the responses were as follows:
 
 
Original question
Revised question
"No difference"
16
17
"Red"
2
8
"Green"
7
13
Other
2
0
Totals
27
38

It would seem, therefore, that many pupils still show a preference for one side even when the equality possibility is made explicit.
 
 

Question 2

The roof of a small garden shed has 16 square tiles as in the picture

.
 

 

 

   

 
 
 
     

 

 

     

 

 

     

 

It begins to snow. After a little while a total of 16 snowflakes have fluttered down Onto the roof. Put a x for each snowflake to show where you think they would land on the roof.

Explain your answer

Wendy -—-a random distribution (Fig. 1)

Andrew----one per square (Fig. 2)

Dawn —-all round the edge (Fig. 3)

David —-"one per square"

Nigel -— "about one in each square"

Wendy — "it is almost impossible to say

OVERALL SUCCESS’: 2 or 3 out of 9

It was apparent that pupils felt compelled to put the snowflakes neatly one-per-square when asked to draw them, but that may reflect a desire for a symmetrical pattern rather than an ignorance of randomness.

The question was then changed to a multiple choice format and class-tested. The results were as follows:
 
 
Responses (%)
Year
Number tested
Fig1
Fig 2
Fig 3
other
1
65
46
40
11
3
2
43
30
53
14
2
3
50
12
54
26
8
4
88
28
64
6
2
5
42
33
55
7
5

 

It is difficult to be sure what to conclude from these figures. The preference for the random’ distribution (Fig. 1) is most marked in the first year pupils (before the mathematicians have got at them’?). Certainly there is food for thought here!

Closer to the typical school curriculum was the following:

Question 3

(a) Two bags have each got some black balls and some white balls in them.

Bag S: 1 white and 2 black Bag T: 5 white and 2 black

John has to choose a bag and pick out one ball without looking. If he picks a white ball he will win a prize. Should John choose Bag S or Bag T? Which bag gives him a better chance of picking a white ball, or are they the same?

Other parts of the question were similarly worded, but without actual pictures, bag contents being as follows:

(b)  Bag U: 2 white and 2 black
      Bag V: 4 white and 4 black

(c)  Bag W: 3 white and 1 black
     Bag X: 3 white and 2 black

(d) Bag Y: 12 white and 4 black
     Bag Z: 20 white and 10 black

The percentages of correct responses were as follows:
Year
3(a)
3(b)
3(c)
3(d)
1
100
54
74
60
2
*
65
81
42
3
*
62
82
70
4
95
66
86
70
5
98
71
93
83

 

Parts (a) and (c) are both 1 variable’ problems because the number of only one of the colours varies. Piaget and Inhelder report that responses to such questions are systematically correct at stage Ila (7- 10 years). Parts (b) and (d) are '2 variable’ problems because the numbers of both colours vary. A comparison of two ratios is required here - i.e. a double comparison which Piaget and Inhelder say really requires the stage of formal operations (stage III) although non-systematic empirical methods at stage lIb (9 - 12 years) will sometimes yield correct answers.

The very high percentage of correct scores on (a) and (c) and the lower rate in (b) are, therefore, all expected, but certainly results for (d) are surprisingly high. When the explanations given for the proffered answers are examined one can see that the pupils are operating rather more intuitively’ than logically’.

For part (b) typical reasons given for choosing one particular bag were:

A: "U because there are less white and blacks than in bag V"

C: "U because there are less to choose from"

G: "because he only has a 2—1 chance in bag U but 4—1 in bag V".

Pupils A and C prefer the simpler case feeling that smaller numbers give an advantage to white (but not to black !). Pupil G ignores the presence of the black balls altogether.

For part (c) typical responses were:

A: "W because there are more white balls"

C: "W because not as many black balls"

G: "W because he has less chance of getting a black ball in bag W"

J: "W because the majority of the balls are white"

For part (d) typical responses were: A: "Y because there are less white balls than Z has"

C: "Y because there are less blacks"

E: "Y because there is a bigger proportion of white to black balls"

G: "Y because in bag Z there is half as many black balls as there is white. In bag Y there is less than half as many

J: "Y because there are lots of white balls and only a few black"

This is not the time to go further in either describing test results or interpreting the findings, as the project has only just begun. It is hoped that the above will suggest to mathematics teachers that considerable care must be exercised when contemplating the teaching of probability. Further reports will be published in due course and interested readers may care to write to the author with their comments. Offers of help from schools in the Midlands area or even further afield, with test trials, would be much appreciated.

The author gratefully acknowledges the co-operation of several schools, who must remain unnamed.

References

Fischbein, E. (1975) The Intuitive Sources of Probabilistic Thinking in Children, D. Reidel.

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

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