We learn mathematics more easily if we talk about examples

It¡¯s a typical situation at middle school, grammar school or university: you are preparing for a maths exam. Is it worth studying for it in a group, if afterwards it is your own performance that will be judged? And how should group learning work so that each individual subsequently performs better?

These are the questions being tackled by Anne Deiglmayr and Lennart Schalk. The two lecturers are carrying out research at the Chair for Research on Learning and Instruction led by Elsbeth Stern at ETH Zurich. They have already presented the initial results of their current project at conferences (for example at the external page 2013 Annual Meeting of the Cognitive Science Society). Their results are based on experiments that they are carrying out in ETH Zurich¡¯s Decision Science Laboratory with students from ETH, the University of Zurich and the Zurich University of Teacher Education.

In each of these experiments, three students form a team. In the first phase, they all study individually on a computer. They are given textual problems relating to the basic principles of probability theory, with the problems all being of the same kind but drawn from different areas of application. Afterwards, they compare notes on chat forums and work on the problems together.

With the help of pre- and post-testing, the researchers can work out how much progress each individual volunteer has made; they can use the chat records to analyse how the students interact with one another and how the group learning is related to their individual performances.

Focusing on individual performance

¡°As a researcher on learning, my focus is more so on individual than on group performance,¡± says Anne Deiglmayr. ¡°We are not studying group performance per se, but rather how individual learning improves if a person collaborates with others and learns as part of a group.¡±

Lennart Schalk gives an example: group work is generally focused on a product. For example, engineers from various disciplines contributed different expertise to designing the Mars Rover, a remote-controlled robot which is now exploring Mars and transmitting measurements and data back to scientists on Earth.

Similar situations can be created in a learning and teaching context, when resources and information are distributed between the members of a group. Psychologists call this type of learning situation a ¡°group puzzle¡±. In a traditional group puzzle, each individual first learns about one sub-topic on which he or she is the expert in the group.

Then the group pools its information so as to explore the overall subject on the basis of the sub-topics and use what they have learnt in order to solve problems. The different interdependencies that this creates increases the motivation for the members of the group to cooperate with one another; the people learning also save time by dividing up the work.

However, for teaching in secondary schools and universities, this traditional type of group puzzle is not usually the best approach. ¡°If the model of the traditional group puzzle is applied in the classroom, then in the end the students are not as confident with the topics or learning content that the others talked about as they are with those that they studied themselves,¡± says Deiglmayr. ¡°As a general principle, when groups are supposed to work together on the basis of different sources of expertise, they rarely share all of their knowledge but merely consolidate what they already knew.¡±

Many stories, one principle

This is why, in contrast to the traditional group puzzle, Deiglmayr and Schalk do not assign different topics or learning content to the various members of the group in their experiments. Instead, they all have to work on problems that demonstrate the same principles and concepts. What does differ between the students is the area of application from which the tasks are taken.

This means that they are all learning from different material and can benefit in the discussion from the different perspectives of the other students. The results from an initial Download study confirm the hypothesis that the authors¡¯ modified group puzzle promotes learning better than a traditional one.

For example, in the modified group puzzle, one member of the group works on the underlying principles of probability theory using the example of different-coloured bike helmets that are handed out on a cycling tour lasting several days. Another person uses the example of a chemist taking unlabelled samples out of a cabinet, while the third works on the example of equally matched ski jumpers competing against one another.

In the first example, a course leader hands out five different-coloured bike helmets to five participants each morning. Typical questions would be: ¡°I am always given my helmet first when all five are still there. My friend always comes after me. What is the probability that I will be given the red helmet on the first day and the yellow one on the second?¡± or ¡°What is the probability that my friend and I will receive the red and the yellow helmets on the first day, either way round?¡± or ¡°What is the probability that I will be given the red helmet on the first day and my friend the yellow one?¡±

People usually learn the underlying mathematical principles in secondary school, but some students will have forgotten them again by the time they go to university. The researchers only selected participants for their study who needed to refresh their knowledge.

Working out the basic principles for yourself

As they compare and contrast the examples they are given, the volunteers learn the underlying mathematical principles. By discussing their tasks and collaborating with one another, the volunteers realise that these principles apply in different contexts and recognise how they can use them themselves in others. ¡°This ability to transfer knowledge is important,¡± says Lennart Schalk, ¡°because no school can prepare students for every possible application of a scientific or mathematical principle or concept.¡±

With a view to teaching in practice, Anne Deiglmayr concludes: ¡°The aim of maths teaching should not be the endless learning by rote of formulae and typical examples, but rather to teach conceptual and transferable knowledge.¡± This kind of knowledge can be applied flexibly, Deiglmayr says. Her findings suggest that collaborative group learning encourages this, because people can then discuss and compare different approaches to solutions, principles and concepts.

¡°It can be very helpful if a teacher sets out a good structure for the collaborative learning in advance, for example by distributing teaching material and suggesting questions for discussion in more detail that will lead students to the underlying principles. But it is better if they still practise the actual processes, such as how to calculate the answers quickly, by themselves.¡±