With thanks to Tony Whelan from the LSE for some of the following. Tony is a highly experienced class teacher who has run classes in Maths, Statistics and OR.
- Provide background: In some sessions, it may be appropriate to discuss the theory and methods involved in a topic, at a fairly general level, and then to use that discussion as the basis for approaching the issues raised by homework exercises.
On an elementary statistics course, homework revealed that students had considerable difficulty with one important idea, namely that of an estimator. One successful class session involved spending half the time studying the relevant definitions and properties, with lots of examples of things that were, and that were not, estimators. This clarified the issues involved, and it was then possible to go back to the homework questions and clarify how the basic ideas applied in all of them.
- Read and contextualise the question(s): In most sessions it is fruitful to encourage students to read questions carefully and to absorb the information in the question. In many applied areas this can be motivated by the observation that, in the "real world", real problems require considerable effort and thought to decide what is important about them, and what mathematical approach(es) might be fruitful.
- Identify thought processes: In most sessions it is also fruitful to discuss the thought processes that students need to engage in while approaching how to solve a problem: at each stage, students need to be able to decide, "what should I do next"?
In an elementary statistics course, there are strategies for calculating probabilities using two results known as Bayes' Formula and the Total Probability Formula. It is often useful, at an appropriate stage, to (re-)display those results, in a different colour from the "solution", to remind students just why the next calculation is the appropriate one to carry out. Similarly, in explaining the Gaussian Elimination method of manipulating matrices, it can be useful to put coloured boxes around the key cells and blocks being used at various stages in the calculations.
- Use examples: It is frequently useful to motivate ideas and techniques by reference to realworld examples.
In an elementary statistics course, students meet the concept of "outliers", that is to say values in a set of data that seem a long way away from the bulk of the known data. In real-world situations, such anomalies can be due to, for instance, instrument errors. The discovery of the famous hole in the ozone layer, over Antarctica, illustrates both the importance and the difficulty of dealing with this problem in "real-world" situations: it was discovered using meteorological balloons, but then the question arose why meteorological satellites observing the same area earlier had not identified it first. It turned out that the computer programmes used to analyse the satellite data had been so written as to reject, as "outlier" instrument errors, true readings which ought to have revealed the ozone hole but were ignored until it was discovered a different way.
- Prepare and structure: Make sure that classes are well prepared, with a proper structure: some ideas about this can be found just above, and also in the section on 'Preparation and planning'.
- Explain, then summarise: Be prepared to repeat things, often from slightly different angles, and to summarise the ideas you are trying to get across, e.g. as bullet points.
- Observe your audience: Pay careful attention to whether students appear to be following what is being said: there are all sorts of clues that can help with this, involving body language and facial expressions as well as any explicit questions or interjections that they make.
- Encourage participation: Even when a class teacher is dominating the discussion (which will often be the case in problem-solving classes), s/he should make sure that students are encouraged to yell out if something is unclear, or wrong.
- Involve students: One other technique that helps to involve students, even when a class teacher is dominating the discussion, is from time to time to ask something like "Someone tell me what comes next". This approach can be varied by asking particular students something similar, but whatever detailed approach may be used, teachers need to be aware of the twin dangers of the "pushy" student, who likes to show off how much s/he knows, intimidating or discouraging others, and of the shy or nervous student, who needs to be encouraged to respond in such situations.
- Use follow-on exercises to check on understanding: Students can be told in advance that they will be given an exercise in class as a follow-on from, or as another example of, an exercise they have prepared. They could work on these in small groups with the groups reporting back.
- Give students enough time: If you give students work to do in the class as a follow-on exercise from the ones they have prepared, give them enough time to complete it, or at least to get sufficiently far through it to benefit from the subsequent explanation.