Children’s understanding of probability
Children’s understanding of probability
Dealing with uncertainty and probability
When events happen randomly, we cannot be certain about what will happen next.
But we can analyse and compare the probabilities of particular events logically and mathematically, provided that we know enough about all the possible things that could happen (the sample space)
Of course this kind of analysis makes a variety of intellectual demands on the people carrying it out.
The questions that we tried to answer in our report were: (a) what these demands are (b) how to help children learn to satisfy them.
Conclusions of the report on children’s understanding of probability
Learning about probability makes three main cognitive demands:Understanding the nature and the value of randomness and how it leads
to uncertaintyIdentifying and working out the sample space (all the possible outcomes)Quantifying and comparing probabilities: a proportional task There is a great deal of good research on children learning these aspects
of probability, but much of this was done in other (non-probability) contexts
This research should be the basis for devising new ways of teaching probability
1. The nature of randomness and randomisation
Randomness & randomisation: the problems and the solutions
Piaget & Inhelder claimed that young children can’t discriminate random from determined sequences
Problem of reversibility vs irreversibility
They worked with 5-13yr-olds on progressive randomisation
Younger children predictedcontinued order
Older children predictedprogressive mixing
Mixed vs ordered Children may do a great deal better than they do in the tilting
tray task if they are given randomisation tasks in more familiar contexts: like shuffling cards
It is probably the case that older children and adults get the idea of randomisation leading to mixed outcomes altogether too well
This is apparent in the “representativeness error” described by Kahneman & Tversky:
Many adults judge the order of the next six babies as more likely to be BGGBGB than BBBGGG, but the probability of both is the same . P=016 (1/64 )
The independence of successive events in random sequences
One hallmark of random sequences is the independence of successive events
Many adults either forget this or do not understand it when they make the very common negative recency error (lightning never strikes twice).
What about children?
A purple marbleA yellow marble
Each colour is just as likely
15 yellow marbles15 purple marbles
15 purple and 15 yellow balls in a bag Someone has already drawn four balls from the bag
(replacing the ball after each draw) and all four were purple. 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 purple ball than a yellow one; 2. The next draw is more likely to be a yellow ball than a purple 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
The social value of randomness
Randomisation is one way of ensuring fairness, as in a lottery or in other forms of selection (e.g. for children's games).
It is also a necessary part of starting some games (shuffling cards, throwing a coin) to ensure that one team or competitor does start with an unfair advantage.
We really need to study children’s understanding of randomeness in this sort of familiar context
Uses of randomisation: fairness
Paparistodemou et al worked with 5-8yr-old children on a computer microworld called Space Kid with a game about fair distribution
In this game the space kid is in peril of hitting a blue mine below him and a red mine above him
His up-down movements are determined by the number of times that a white ball moving unpredictably around hits a number of red and blue balls
Paparistodemou, Noss &Pratt
Paparistodemou, Noss &Pratt
Comments on the Paparistodemou et al. study
Making the connection between fairness and randomisation (‘unsteerable fairness’) is an excellent idea
There are other ways of achieving fairness: one is a controlled predictable procedure like sharing.
We need to compare tasks in which randomisation is a better way of achieving fairness than sharing (e.g. lotteries) and other tasks in which sharing is a better way than randomisation
Conclusions on randomness and randomisation
Young children may have some difficulties in distinguishing the nature of non-random, determined events from random, uncertain events
However, probably through informal experience, they do seem to take quite easily to the idea of a link between randomisation and fairness
This link should provide a good way of teaching children about randomisation and uncertainty
2. Sample space
HHHHHHTHHTHHHTTHTTTHTTHHTHTHTHHH
HHHTHHTTHTHTHTTTTTTTTTHTTHTTTHHT
The sample space of four successive tosses of a coin
Tree diagram to represent the sample space of four successive tosses of a coin
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T T
TH
H H
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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
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TTHT TTTH
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There are 3 chips in a bag, two red and one blue. You shake the bag and pull out one without looking. It could be red or blue. Then you pull out another one. What could the colours of the first and the second chip be? Work out the possible combinations and answer the question. What is most likely to happen? It is most likely that I
would pull out 2 red chipsIt is most likely that I would pull out 1 red and 1 blue chipBoth of these are equally likely
Lecoutre, 1996
There are 3 chips in a bag, two red and one blue. You shake the bag and pull out one without looking. It could be red or blue. Then you pull out another one. What could the colours of the first and the second chip be? Work out the possible combinations and answer the question. What is most likely to happen? It is most likely that I
would pull out 2 red chipsIt is most likely that I would pull out 1 red and 1 blue chipBoth of these are equally likely
1st pulled out
R
R
B
2nd pulled out
Lecoutre, 1996
There are 6 possible outcomes.One R and one W is twice as probable as two RsBut only around 50% of various groups of undergraduates gave the correct answer
Lecoutre, 1996
Difficulty in working out all the possible outcomes
R1-W R2 -W R1 -R2 R2 -R1 W -R1 W -R2
Other problems that could be removed by working on the sample space
Kahneman & Tversky’s representativeness error (BGBGGB more likely than BBBGGG) is actually a failure in inspecting the sample space or in wrongly aggregating the sample space.
Van Dooren et al. demonstrated that most adolescents judge as correct the statement I roll a die 12 times. My chance of getting at least two 6s in these 12 throws is 3times as great as my chance of getting at least two 6s if I roll the die 4 times. They would probably not make this “linear” error if they worked out the sample space for 4 and 12 throws, since this would show that there is no linear relationship between the number of throws and the possibility of two sixes.
Aggregating
Counting out the number of possible alternatives is often not enough.
It is often necessary to form these alternatives into categories
This causes a lot of difficulties, particularly when the basic alternatives are equiprobable but category membership is not.
Abrahamson’s scoop task
Abrahamson gave 12-year old children big box of green and blue balls and a 4-ball scoop
And then asked them about the probability of the outcomes of a scoop
His questions were not about single outcomes but about categories of outcomes e.g. how likely is it to be 3G and 1B?
So he was asking them to aggregate
The sample space: 16 possible scoops
Abrahamson’s results This is a very difficult task for 12-year olds. Their commonest mistake is to say that the 5 aggregated categories are
equiprobable. The children’s justifications seem to show that the cause of their
difficulty is having to deal with two levels of data – the 16 individual equiprobable outcomes and the 5 non-equiprobable categories – at the same time and yet keep them separate.
We need much more research (on adults as well as children) on this possibility
A confusion between levels of aggregation probably also causes the ‘representativeness’ mistake
Back to the future
The topic of sample space raises a cognitive question , which is “how systematically can children think about the future”?
There is no data at all on children’s ability to list all the possible events in a particular context.
There is research on children’s counterfactual reasoning about alternative possibilities (what would have happened if Napoleon hadn’t had indigestion at Waterloo?) but this is post-hoc and about deterministic chains of events
3. The quantification of probability
The difference between extensive and intensive quantities
Some quantities, like weight and volume, are extensive: if you add one to another the quantity increases: if I have a kilo of apples and put another kilo into it, the basket is now heavier
Other quantities are intensive: they do not obey the same rules of addition: if two pieces of wood have the same density, and I join them together, the new object has the same density as its two parts
Intensive quantities are based on proportions Probability is an intensive quantity: the proportion of a
particular event to all the possible events
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 calculation is a proportional one: the proportion of white in A is.33: the proportion of white in B is .29Only 27% of a large group of German 15-year olds got the right answer: worse than chance level
Comparing probabilities
The cards are shuffled several times and then put into the box where they belong.
Tick the box where there’s a better chance of picking a circle, or tick the It doesn’t matter which box if you think that you’ve the same chance of picking a circle in one box as in the other
.
box 1 box 2
These are the cards in box 1
These are the cards in box 2
.It doesn’t matter which box
or
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
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
Tick box 1 or box 2 or tick the It doesn’t matter which box
box 1 box 2It doesn’t matter which box
These are the cards in box 1
These are the cards in box 2
Problems which did not need a proportional solution
An example is the trial in which there were the same number of circles (6 in both) in both boxes, but an unequal number of squares (5 in one and 6 in the other)
In this comparison, the child could solve the problem just by directly comparing the number of squares in the two sets
Piaget &
Problems which did need a proportional solution
In other trials the number of circles and the number of squares both differed between the two boxes. This made the problem a genuinely proportional one.
These were much the harder of the two kinds of problem in our pre-test.
Piaget & Inhelder report that the childen who did solve such problems reached their solution by calculating ratios, not fractions.
This chimes with our work on other intensive quantities
General conclusions
Children’s understanding of and interest in fairness seem a good start for working on their learning about randomness
The importance of analysing the sample space first has been badly underestimated in past research. We urgently need research on teaching children how to do this
Their preference for ratios over fractions gives us an important lead into how to teach them to quantify probabilities.
The relationship between these three kinds of knowledge needs investigation: this should combine intervention studies with longitudinal research
Bravado Have I not walked without an upward lookOf caution under stars that very wellMight not have missed me when they shot and fell?It was a risk I had to take – and took.
Robert Frost