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

Supported by The Nuffield Foundation

There is general agreement today that students should leave school with a good grasp of all the basic mathematical ideas about how to deal with uncertainty.

As adults, they will have to make important decisions about the probability of some events taking place and about the risk of a wide range of dangers. The understanding of uncertainty also plays a central role in modern science. Many scientific discoveries of great importance would have been impossible if scientists had only conceived of the world in terms of certainty.

Our project investigates the connections between primary school children’s thinking about probabilistic events and mathematical reasoning that does not involve ideas about randomness, which we can call simply quantitative problem solving. Our hypothesis is that problems that involve randomness and problems that do not involve randomness make some common demands on children’s reasoning: children need to think clearly about both kinds of problem, be systematic about using the information, and think logically about the connections between the different pieces of information in the problem. But thinking about probability involves also specific concepts beyond this: children must have some understanding of randomness, of sample space and of how to quantify probabilities.

We have two aims that will help us test this hypothesis. One aim is to promote children’s understanding of randomness and probability and test whether this improves their ability to solve mathematical problems in general as well as in situations that involve uncertainty. The second aim is to promote children’s understanding of numerical operations in a context where they can be certain of the outcomes and to test whether children who improve their quantitative problem solving ability in this context also improve their understanding of probability.

In the first phase of this project, we worked with three groups of  9- to 11-year-old children. All of them were given the same pre-test, which measures their ability to solve probabilistic problems and also their quantitative problem solving skills when the questions do not involve randomness. The children in the probability learning group were taught how to solve problems that involve randomness and how to compare probabilities in situations where different outcomes are possible. They were invited to think about what might happen and how likely it is that certain events would happen.  The children in the second group worked on quantitative problems that involved relative numbers and more than two quantities and relations between quantities. None of these problems involved random events. The third group, a control group, was given no extra teaching between the pre- and the post-tests, but did have a chance to learn about the ideas included in the teaching programmes after the study had been completed.

The teaching programmes were developed on the basis of the literature syntheses carried out at the request of the Nuffield Foundation which can be found below. Click on the titles to access them.

During this first phase, we found that:

  • The children in the probability group made significantly more progress in understanding randomness, sample space and quantification of probabilities than each of the other two groups; their progress was stable two months after the teaching had ended and they continued to show a significant advantage over each of the other two groups. Download the results of the intervention about probability.
  • They were able to learn how to use these ideas in order to think about associations between events in a single teaching session and out-performed each of the other groups in a measure of the understanding of association between measures. Download the results of the intervention on correlations.
  • The children in the quantitative problem solving group performed significantly better than the control group in problem solving measures.  The children in the probability group performed as well as those who had been taught about quantitative problem solving in the problem solving measures. Download the results of the intervention on problem solving.

The second phase of the project assessed whether teachers would be able to implement the programme in their classrooms and replicate the results of our successful teaching of probability and quantitative problem solving. Teachers working with years 5 or 6 pupils were invited to a professional development day in order to learn about the teaching programmes and then to test them in their classroom. Some teachers were representing a cluster of schools and besides implementing the programme themselves they invited others to attend a twilight session which we offered to the teachers in the cluster. Other teachers were unable to attend and we offered them a one-to-one session about the programme.

The teachers chose which programme they wanted to implement in their classroom, either the probability or the quantitative problem solving programme. Their children were assessed by us before they had started the programme (pre-test) and at the end of the year (post-test), after the teachers had implemented the programme. In this phase, there was no unseen comparison group, all the children in the study had participated either in the problem solving or the probability teaching programme.

During this phase, we found that:

  • Teachers were able to implement the programme in their classrooms. A total of 25 schools participated in this teacher-led trial of the programmes; 810 pupils completed both the pre-test and the post-test. The type of training that the teachers received before implementing the programme did not affect the results. We think that this was due to the teachers’ conscientious use of the problems in the order in which they appear in the programme and their skilful management of the discussions once the pupils had reached their answers.
  • The children in the probability group made significantly more progress that the children in the problem solving group when the questions in the assessment referred to randomness, sample space, quantification of probabilities or association between variables. The programme was effective for pupils who at pre-test performed below the mean as well as for those who at pre-test performed above the mean. You can download a description of the results by clicking here.
  • The children in the quantitative problem solving made significantly more progress in problem solving than the probability group. This was true of pupils who at pre-test performed below the mean as well as those who at pre-test performed above the mean. You can download a description of the results by clicking here.
  • If you would like to find out about each of the programmes, you can download the teacher handbook for the probability and the problem solving teaching.
  • Download the Handbook for the Probability Programme by clicking here.
  • Download the Handbook for the Problem Solving Programme by clicking here.

Research team

  • Peter Bryant
  • Terezinha Nunes
  • Deborah Evans
  • Laura Gottardis
  • Maria-Emmanouela Terlektsi

Page last modified: July 3, 2014

Certainty

Probability