Transcript Slide 1

Children’s understanding of probability
A 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
Susan and Julie were cycling at the
same speed. 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?
• Susan: 3xJulie
Susan: 15x3: 45
• Susan: 6 circuits ahead
Susan: 15+6=21
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 events
If 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 probability
If 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/fil
es/P2.pdf
http://www.nuffieldfoundation.org/sites/default/fil
es/P4.pdf
• A study carried out for the DfE showed how
understanding relations predicts KS2 and KS3
mathematics results
Cognitive
Measures
(8 yrs)
5.5 years
IQ
3 years
Working
Memory
Arithmetic
Maths
Reasoning
8
9
School
achievement
(14 yrs)
School
achievement
(11 yrs)
Maths
Maths
Science
Science
English
English
10
11
12
Age in years
13
14
Key stage 2:
Mathematics
N=2488
.60
Key stage 2
.12
.46
Arithmetic
.53
Mathematical
Reasoning:
Year 4
.31
.48
Attention and
Memory
.64
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
Key stage 3:
Mathematics
N=1595
.61
Key stage 3
.11
.33
.47
Arithmetic
.51
Mathematical
Reasoning
Year 4
.56
Memory and
Attention
.62
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.
What is most likely to happen?
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
Thinking systematically and logically about
random events
R
R
1st pulled out
B
What is most likely to happen?
2nd pulled out
R
R
B
It is most likely that you would
pull out two red chips
R
It is most likely that you would
pull out a mixture, one red and
one blue
B
Both of these are equally likely
R
R
B
R
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 predicted
continued order
Older children predicted
progressive 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
Difficulty in understanding the independence of random events
 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;
Positive
recency
2. The next draw is more likely to
be a green ball than a blue one;
Negative
recency
3. The two colours are equally
likely.
Correct answer
Chiesi & Primi
Percentages for the different answers in Chiesi &
Primi’s task
Positive
recency
8yrs
Negative
recency
Correct
answer
0
10yrs
40
College
student
41
Percentages for the different answers in Chiesi &
Primi’s task
Positive
recency
Negative
recency
Correct
answer
8yrs
66
34
0
10yrs
30
30
40
College 16
student
43
41
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
HHHH
H
T
H
H
HHHT
HHTH
H
T
T
HHTT
H
T
H
H
T
HTHH
HTHT
HTTH
T
H
T
HTTT
THHH
H
THHT
THTH
T
H
H
H
T
THTT
T
T
T
H
H
T
T
H
16 equiprobable possible outcomes
T
TTHH
TTHT
TTTH
TTTT
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
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
What is the probability of 13?
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
What is the probability of 6?
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
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
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
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
Difficulty with quantification: a PISA question
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?”
The proportion of white in A is .33:
The proportion of white in B is .29
The white-black ratio is 1:2 in A and 1:2.5 in B
Only 27% of a large group of 15-year olds
got the right answer: worse than chance
PISA, 2003
level
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
These are the
cards in box 1
or
These are the
cards in box 2
box 2
box 1
.
It doesn’t matter
which box
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
These are
the cards in
box 2
These are the
cards in box 1
or
box 1
It doesn’t matter which box
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 fairness
seem a good start for working on their learning about
randomness
The importance of the sample space has been badly
underestimated 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 them
to 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