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