Children’s understanding of probability and certainty: An intervention study

Peter BryantTerezinha NunesDeborah EvansLaura Gottardis

Maria-Emmanouela Terlektsi

Why probability and certainty?

What did we need to consider when designing the intervention?•The cognitive demands of the conceptual field of probability

•Concepts that are relevant also in other conceptual fields

•The technical skills that children must have to be good problem solvers

The designRandom assignment of children from the same class to different treatment groups

•Baseline: an unseen control group that received extra attention from the class teacher (better than business as usual because the same number of students with their own teacher)

•Certainty group: taught relevant technical skills (problem solving, inverse relations between operations, multiplicative reasoning)

•Probability group: taught technical and conceptual knowledge about probabilities

The probability teaching programme

The programme was designed to:• address the cognitive demands of the

conceptual field of probability

• address the required technical skills

• use research in other conceptual fields that make the same cognitive demands

The problem solving teaching programme

The programme was designed to:• promote technical skills related to problem solving

(reading problems and questions carefully)

• promote conceptual skills (additive and multiplicative reasoning)

• use diagrams and be as interesting as the probability lessons (playing games, throwing dice, recording results)

Randomness Sample SpaceSample space and quantification of

probabilities

Association between

variables (1 day)

Test 1 Test 2 Test 3 Test 4 Test 5

Summerbreak

The pre- and post-tests used tasks adapted from the literature

Test 1: pre-test measures for sample space and problems involving – provided a reliable measure of initial ability

Test 2: provided test-retest reliability

Test 3: pre-test measure for quantification of probabilities and more difficult problems about non-probabilistic situations

Tests 4 and 5: post-tests used to assess the programmes (problem solving and probability)

Randomness

The development of children’s understanding of randomness is related to their understanding of certainty

•consider how children use the word “random”

•compare random and certain outcomes

•consider the possibility of local patterns, over a restricted number of events

Randomness

• Randomness as a fair way of starting a game

• Fair and unfair ways:

– shuffling cards: Happy families game

– throwing dice: normal and loaded dice (recording outcomes)

• Possible, probable, impossible and improbable events

• Predictable and unpredictable sequences (recording and analysing sequences)

– some things are difficult to predict but not random

– local patterns do not allow for prediction over many trials

b c a

broccolicat apple

Predict the correct order of the figures

c p a

p ac

Predict the correct order of the figures

Sample space

The definition of an event and of all possible events

•an object as the junction of two (or more) dimensions leading to the concept of an event

•teaching children about Cartesian product problems

•considering how the same outcome may be obtained in different ways and defining a sample space with aggregation of outcomes

An object as the junction of two propertiesIdentifying all the possible objects in a matrix: Cartesian product

The difficulty of generating all possible combinationsThe need for a system

Aggregation in sample space

Quantification of probabilities

Understanding ratio and proportions•connecting sample space and quantification of probabilities

•quantification using frequencies and ratios is more easily understood

•comparing probabilities when the total number of cases is different

•evaluating chances when the ratio is presented in different ways

When repetition must be eliminated

Probabilities and changes in the sample spaceThe difficulty of comparing when the sample space is not the same

The need for ratios or proportions when the total is differentAnalysing ratios with concrete materialsUsing ratio notation to compare probabilitiesNoting inverse relations

Calculating ratios with a calculator, when you don’t “see” it

Results

• The groups did not differ at pre-test

• There were no school differences at pre-test

• Intervention conditions varied considerably between schools: separate spaces in one school, a large room with little interference between groups in the second, a smaller room with noise from one group interfering with the second in the third school

• Analyses use repeated measures (T4 and T5), a covariate (T1 for sample space and T3 for quantification)

Proportion of correct answers to sample space questions

The group effect was significant and the covariate was significant. The school effect was not significant and neither was the interaction between group and school. The probability group differed significantly from each of the other two.

1st pulled out

R R B

R

R

B

2nd pulled out

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

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

Both of these are equally likely

R

B

The Diagram What is most likely to happen?

Lecoutre’s (1996) problem: There are 3 chips in a bag, two red and one blue. You shake the bag and pull out two chips without looking. Complete the diagram to work out the possible combinations and answer the question.

Percent correct before and after connecting sample space and quantification of probabilities

Mean correct to quantification of probabilities questions

The group effect was significant and the covariate was significant. The school effect was not significant and neither was the interaction between group and school. The probability group differed significantly from the unseen group but the other group differences were not significant.

Conclusions• Good evidence of programme effectiveness, supporting

the notion that children can learn much about probability by the end of primary school

• Both groups show significant improvement in the specific domains taught when compared to the unseen group

• Some conceptual field specificity (sample space and quantification based on sample space)

• Some general conceptual and technical knowledge (proportional reasoning)