Children’s understanding of probabilityA review prepared for

The Nuffield Foundation

Peter Bryant and Terezinha Nunes

This great book of the universe, which stands continually open to our gaze, cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed: the language of mathematics

Galileo -1564/1642

• Susan and Julie were cycling at the same speed in the Velodrome. Susan started first. When Julie had completed the circuit 3 times, Susan had completed it 9 times. How many times had Susan completed the circuit when Julie completed it 15 times?

Using mathematics to understand the world

• Quantities– how many times Julie had gone around the circuit– how many times Susan had gone around the

circuit• Relations between the quantities

– Susan had gone around 3 times the number that Julie had gone around

– Susan had gone around 6 times more than Julie

Using and testing mathematical models

• When Julie had completed the circuit 3 times, Susan had completed it 9 times. How many times had Susan completed the circuit when Julie completed it 15 times?

• Susan: 3xJulie Susan: 15x3: 45

• Susan: 6 circuits ahead Susan: 15+6=21

Susan and Julie were cycling at the same speed. Susan started first.

Quantitative reasoning

• Representing the world with numbers• Representing relations with numbers• Operating on the numerical representations• Following assumptions to their logical

conclusions• Testing the adequacy of the model

Two sorts of quantitative thinking

• Modelling non-random eventsIf you have £20 to distribute fairly to 5 people, you

know exactly how much each one will get

Two sorts of quantitative thinking

• Modelling with probabilityIf you toss a coin, we don’t know exactly what will

happen each time but we know:what is possible roughly what is most likely to happen if you toss the

coin a large number of times if a coin that we tossed lots of times is departing from

the expected pattern of probability

• At first glance, it seems pointless to model random events

• But in many probabilistic situations, we want to know whether something that looks like an association could be random

• This is crucial to scientific and statistical reasoning

• If you get a flu jab, are you less likely to have flu?

• What is common to both forms of quantitative reasoning– A way of thinking – Use of mathematical concepts

• What may be different– Specific concepts used

• What can be used in one form of thinking even though it was learned in the other?

Relations are crucial in problem solving

• A review prepared for the Nuffield Foundation on how children learn mathematics stressed the importance of relations

http://www.nuffieldfoundation.org/sites/default/files/P2.pdfhttp://www.nuffieldfoundation.org/sites/default/files/P4.pdf• A study carried out for the DfE showed how

understanding relations predicts KS2 and KS3 mathematics results

8 9 10 11 12 13 14Age in years

Cognitive Measures (8 yrs)

IQ

Working Memory

Arithmetic

Maths Reasoning

School achievement

(11 yrs)

Maths

Science

English

School achievement

(14 yrs)

Maths

Science

English

3 years

5.5 years

Mathematical Reasoning:

Year 4

.60

Key stage 2

Key stage 2:Mathematics

Attention and Memory.48Arithmetic .53

.12.46 .31

.64

N=2488

Key findings• Mathematical reasoning and arithmetic knowledge make separate contributions to the prediction of KS2 Mathematics results• Reasoning makes a much stronger contribution than arithmetic

MathematicalReasoning

Year 4

.61

Key stage 3

Key stage 3:Mathematics

Memory andAttention

.56Arithmetic .51

.62

.33.47

.11

N=1595

Key findings• Mathematical reasoning and arithmetic knowledge make separate contributions to the prediction of KS3 Mathematics results• Reasoning makes a much stronger contribution than arithmetic

• There are 3 chips in a bag, two red and one blue. You shake the bag and pull out two chips without looking.

It is most likely that you would pull out two red chips

It is most likely that you would pull out a mixture, one red and one blue

Both of these are equally likely

What is most likely to happen?

1st pulled outR R B

R

R

B

2nd pulled out

It is most likely that you would pull out two red chips

It is most likely that you would pull out a mixture, one red and one blue

Both of these are equally likely

R

B

What is most likely to happen?

R

B

R

R

Thinking systematically and logically about random events

Thinking about random events

• Identifying quantities and relations

• Using concepts that are specific to understanding random events

• Questions so far?

Understanding probability

Randomness produces uncertainty: we can’t make precise predictions about uncertain situations

Nevertheless we can, and often do, think about uncertainty rationally and we can analyze uncertain contexts logically.

If we know what all the possible outcomes are, we can work out the probability of specific outcomes, and this is tremendously useful.

This raises the question: is it possible to show young children how to work out and to understand probability?

Three crucial areas in the understanding of probability

The nature of randomness and randomising

The need to work out and organise the sample space

The quantification of probability

1. The nature of randomness : understanding randomising & the independence of separate events

Randomness & randomising Most previous research on children’s understanding

of randomness has concentrated on their reactions to various kinds of randomising (usually in quite unfamiliar contexts) e.g. research by Piaget & Inhelder).

Piaget & Inhelder worked with 5-13yr-olds on progressive randomisation

Younger children predictedcontinued order

Older children predictedprogressive mixing

Randomness & randomising Most previous research on children’s understanding of

randomness has concentrated on their reactions to various kinds of randomising (usually in quite unfamiliar contexts) e.g. research by Piaget & Inhelder).

But randomising is actually a socially useful activity (e.g. shuffling cards, lotteries), and is familiar to most young children.

It’s possible that teaching children about randomness through useful randomising would be successful.

Independence of separate events The probability of a particular outcome at any point

in a random sequence is unaffected by what has happened before in the sequence: if I’ve throw 5 heads in succession, the probability of another head on the next throw is still 0.5.

This independence seems hard for many children, and some adults, to understand

15 green and 15 blue balls in a bag Someone has already drawn four balls from the bag

(replacing the ball after each draw) and all four were blue. This person is going to make another draw. What is likely to happen on his next draw?

1. The next draw is more likely to be a blue ball than a green one;

2. The next draw is more likely to be a green ball than a blue one;

3. The two colours are equally likely.

Positive recency

Negative recency

Correct answerChiesi & Primi

Difficulty in understanding the independence of random events

Percentages for the different answers in Chiesi & Primi’s task

Positive Negative Correct recency recency answer8yrs 0

10yrs 40

College 41student

Percentages for the different answers in Chiesi & Primi’s task

Positive Negative Correct recency recency answer

8yrs 66 34 0

10yrs 30 30 40

College 16 43 41student

Randomness and fairness Randomness has a positive and useful side. It is a valuable way

of ensuring fairness in some situations.

This is particularly true in games and lotteries e.g tossing a coin to decide which side bats first

In games, there are some ways of randomising that are better than others: eeny-meeny-miny-mo is a more predictable and therefore less random procedure than shuffling cards or having to throw a six in order to start in a game like Ludo.

Conclusions and a possible solution : randomness ensures fairness in games

Children have to learn about determined, reversible cause-effect sequences

Initially their view of random, uncertain events is that they will be as predictable and reversible as determined events

Children may learn well about randomising if they are given randomisation tasks in more familiar contexts where randomising is a way of ensuring fairness: like shuffling cards and tossing dice

2. The need to work out and organise (aggregate) the sample

space

Tree diagram to represent the sample space of four successive tosses of a coin

H

H

H

T T

TH

H H

H

T

T T

T

H

H

H

T T

TH

H H

H

T

T T

T

T

H

HHHH

HHHTHHTH

HHTT

HTHH

HTHTHTTH

HTTT

THHH

THHTTHTH

THTT

TTHH

TTHT TTTH

TTTT16 equiprobable possible outcomes

Two-dice problem

Fischbein & Gazit asked children about the problem of two dice adding up to particular totals

What is the probability of: (a) 6? (b) 13?(c) a number bigger than 9?

The sample space for the two-dice problem

1 2 3 4 5 61 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12

What is the probability of 13?

1 2 3 4 5 61 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12

What is the probability of 6?

1 2 3 4 5 61 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12

5 ways of getting 6

probability of 6 =5/36

p=.139

What is the probability of bigger than 9?

1 2 3 4 5 61 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12

6 ways of getting >9

p=.167

Percent correct in two-dice task

10 years 12 years

p of 13 38 78

p of 6 0 51

p of >9 0 28

Conclusions and a possible solution :

Teaching about need for working out sample space as first step, again in the context of games

Use of diagrams – particularly tree diagrams

This teaching must involve some categorization

3. The quantification of probability

Box A contains 3 marbles of which 1 is white and 2 are black. Box B contains 7 marbles of which 2 are white and 5 black. You have to draw a marble from one of the boxes with your

eyes covered. From which box should you draw if you want a white marble?”

PISA, 2003

The proportion of white in A is .33: The proportion of white in B is .29

Only 27% of a large group of 15-year olds got the right answer: worse than chance level

Difficulty with quantification: a PISA question

The white-black ratio is 1:2 in A and 1:2.5 in B

The cards are shuffled several times and then put into the box where they belong.

Tick box 1 or box 2 or tick the It doesn’t matter which box

or

.box 1 box 2

It doesn’t matter which box

These are the cards in box 1

These are the cards in box 2

This is a problem which does not need a proportional solution

In this quite easy comparison, the child can solve the problem just by directly comparing the number of squares in the two sets

Tick box 1 or box 2 or tick the It doesn’t matter which box

or

box 1 box 2It doesn’t matter which box

These are the cards in box 1

These are the cards in box 2

This is a problem which does need some kind of a proportional solution

Because the number of both kinds of card differed between the two boxes. This makes the problem a genuinely proportional one.

There is overwhelming evidence that this kind of comparison is difficult for children

Piaget & Inhelder report that the children who do solve such problems reach their solution by calculating ratios, not fractions.

Conclusions

Children’s understanding of and interest in fairnessseem a good start for working on their learning about randomness

The importance of the sample space has been badlyunderestimated in existing research. We think that it can be taught with the help of diagrams and concrete material.

Children are more successful at solving proportional problems using ratios than using fractions. This gives us an important lead into how to teach themto quantify and compare probabilities

Our central idea is that we can teach these three elements separately, but in a cumulative way

Children’s understanding of probability.A report prepared for

The Nuffield Foundation

http://www.nuffieldfoundation.org/sites/ default/files/files/Nuffield_CuP_FULL_REPORTv_FINAL.pdf