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 uncertainty  Identifying 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 predicted continued order Older children predicted progressive 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?

15 15 purple marbles yellow marbles A purple marble A yellow marble Each colour is just as likely

Difficulty in understanding the independence of  random events 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 answer

Chiesi & Primi

Percentages for the different answers in Chiesi & Primi’s task 8yrs

Positive Negative Correct recency recency answer

0 10yrs College student 40 41

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

Positive Negative Correct recency recency answer 8yrs

66

10yrs

30

College student

16 34 30 43 0 40 41

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

The sample space of four successive tosses of a coin HHHH HHTH HTHH HTTH TTTH TTHH HHHT HHTT HTHT HTTT TTTT TTHT THTH THHH THTT THHT

Tree diagram to represent the sample space of four successive tosses of a coin HHHH H H T H HHHT HHTH H T T HHTT H T H H HTHH T T H T HTHT HTTH HTTT THHH H H T H THHT THTH H T T THTT T TTHH T H H T H T TTHT TTTH TTTT T

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 chips It is most likely that I would pull out 1 red and 1 blue chip Both 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?

1 st out pulled R R B 2nd pulled out It is most likely that I would pull out 2 red chips It is most likely that I would pull out 1 red and 1 blue chip Both of these are equally likely Lecoutre, 1996

Difficulty in working out all the possible outcomes  R1-W  R2 -W  R1 -R2  R2 -R1  W -R1  W -R2 There are 6 possible outcomes.

One R and one W is twice as probable as two Rs

But only around 50% of various groups of undergraduates gave the correct answer Lecoutre, 1996

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

Comparing probabilities

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 calculation is a proportional one: the proportion of white in A is.33: the proportion of white in B is .29

 Only 27% of a large group of German 15 year olds got the right answer: worse than chance level PISA, 2003

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 These are the cards in box 1 or These are the cards in box 2 .

box 1 box 2

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 1 box 2

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 1 It doesn’t matter which box box 2

Tick box 1

or

box 2

or

tick the

It doesn’t matter which box

These are the cards in box 1 These are the cards in box 2 box 1

It doesn’t matter which box

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 look Of caution under stars that very well Might not have missed me when they shot and fell?

It was a risk I had to take – and took.

Robert Frost