A better approach?
Having considered some of the problems involved in teaching probability, it is time to present the main features of an alternative (and, we believe, better) approach .
Our approach is to teach probability through experimentation, and to use mathematical models to solve contextualised problems. We do not make a big issue about whether probabilities are ‘known’ or ‘unknown’. In real life no probabilities are ever known, they are only assumed with more or less justification.
We regard experimentation as vital to understanding the role of chance and unpredictability. Ideally students should carry out experiments themselves using randomising devices. Our preference is for spinners, either where probabilities are obvious or where they are deliberately concealed – these better reflect real life where nicely balanced situations are rare. We prefer to avoid dice – the choice of outcomes is too restrictive, and numbers may have emotional connotations, quite apart from the practical problems of throwing them in a class.Having acquainted themselves with spinners as randomising devices in their own right, students can start using these as a way to model real situations – spinners can be labelled with specific outcomes, and students then simply count the number of times outcomes occur in a given number of trials. The idea is to use whole numbers initially, and bring in proportions, fractions and probability rules later. We also exploit the strong motivating role of playing competitive non-gambling games of chance, encouraging engagement by clearly making some outcomes more desirable than others.
We can summarise our approach as a series of 11 steps:
- Start with a problem, expressed as an appropriately simplified mathematical model or game.
- Use a spinner as a tool for investigating the model.
- Do experiments in small groups to promote discussion, recording the results of the spins either on paper or using physical objects, such as plastic cubes.
- Tally the data, then record it on a frequency tree.
- Discuss the narratives represented by sets of branches on the tree, emphasising that these are mutually exclusive, and that together they include all of the possible outcomes – the sample space for the experiment.
- Ask questions about the proportion of times specific events occur, and whether and why any results are surprising.
- Average the data from all groups, observing that this ‘smooths’ the data, so that trends in the data can be seen more clearly.
- If possible, conduct large numbers of trials using a computer simulation, helping students to understand that the more experimental results they have, the nearer the data approaches the results they would expect if they could conduct an infinite number of trials.
- Construct the expected frequency tree, discussing the proportion of times you expect each outcome/event to occur. Compare with the experimental data – the class average may well be very close to the expected results.
- Compare representations of data in frequency trees, contingency (2-way) tables, and Venn diagrams.
- Decide what fraction of times you expect each outcome on the spinner, and use these fractions as probabilities on tree branches to arrive finally at the probability tree.
Note that probability is the final step in our teaching method. The preceding steps provide scaffolding for students to build up their experience of what happens experimentally, and provide many opportunities to discuss what different representations tell us, before they are faced with the abstract probability tree. Note too that we do not ask students to predict what will happen before they have had a chance to see what happens experimentally. We do not want them to form unfounded opinions that will then blind them to the evidence.
We will work through these approaches in more detail in the remainder of the course. You may find it useful at this stage to share some initial reflections on these points:
- How do our approaches compare to your existing methods for teaching probability?
- Do you think that our approaches could work with your students?