14. TEACHING STATISTICS THROUGH GAMES Steven G. Gilmour University of Reading
14.1 Introduction
To understand the motivation behind this paper it is necessary to ask: 'Why do we teach statistics?' Statistics is not a branch of mathematics and the answer to the above question is simpler than the answer to the question of why we teach mathematics. We teach statistical methods because they are useful for solving problems in other disciplines. The range of other disciplines which use statistics will include most of the subjects A-level students are taking or have taken up to GCSE.
Given that statistics is all about solving problems, the flavour of statistics will be best acquired through problem based learning. This is well recognised and most statistics teachers put examples from a variety of fields of application into their courses. However, these examples often look to the students like they have been put in as an afterthought to try to pretend that the statistics is relevant.
In order to fully understand the statistical methods, students must solve real problems themselves and see the benefits of using statistics to do so. This is already fairly common practice with some elements of statistics - many A-level students, and some pre-A-level students, design, carry out and analyse surveys and questionnaires. However, although it is desirable, it is not so easy to do the same with the great number of techniques which are particularly relevant to the experimental sciences. At Reading we have developed and used over many years games which simulate real-life experimental situations; the early work was described by Mead and Stern (1973) and Mead (1974).
Clearly this type of learning differs from traditional lecture based courses and will usually replace some, rather than all, lectures. The students are put in the position of an experimenter and the teacher acts almost as a statistical consultant, but giving more theory in the lectures. By motivating the use of techniques through the games, the students should see the relevance of the theory and the mathematics which are presented in lectures. It becomes clear to the students that statistics is not a branch of mathematics, but that the mathematics is necessary for solving real problems.
14.2 Examples
Three examples are presented which are used in courses at Reading given to students taking degrees in the Faculty of Agriculture and the School of Biological Sciences. A majority of these students come to university without A-level Mathematics, but have done a 20 hour mathematics course. The second and third games are also used for students taking the MSc course in the Department of Applied Statistics; of course, the lectures around the games differ greatly for the different groups of students.
14.2.1 Curves
Before discussing regression in lectures, this game is used to motivate simple linear regression, illustrate the method of least squares and, most importantly, get the students to use regression methods sensibly, even without knowing the theory. The experiment is as follows. Air with varying concentrations of CO2 was passed over wheat leaves at a temperature of 35oC and the uptake of CO2 by the leaves was measured. The objective is to predict how much CO2 will be taken up if 100ppm is applied; or 150ppm; or 200ppm.
The experimenter's first step should be to plot the data and most students will do this without prompting (Figure 1). If they are then prompted to make predictions some will do so immediately, others will draw a straight line first. Having fitted a line by eye they are then asked to write down the equation of the straight line. Having thus motivated the students, who should now see the point of fitting lines, the teacher can move on to discuss the method of least squares to find the best fitting line. The necessary mathematics is then seen in context and appears less dry.
Figure 14.1:Data on the relationship between CO2 uptake (y) and CO2 concentration in air (x).
14.2.2 Tomato
The objective of this game is to motivate the concepts which are not part of the A-level Statistics syllabus, they are covered in at least some A-level Biology courses, although probably not with these names.
The objective of the experiment is to find the conditions which maximise the yield of tomatoes. In particular:
A glasshouse is available, which contains 12 plots, 6 north facing and 6 south facing. Two years of experimentation are to be done.
The students are then left to design the experiments on their own. This involves assigning a variety to each plot and deciding whether or not to use supplementary heating and/or lighting. Experience suggests that lack of knowledge of statistical design of experiments is not a handicap. In fact, it is often the students who have done courses on design before who struggle most. To answer the questions of interest, we should use the factorial set of eight treatments defined by all combinations of the two varieties, heat/no heat and light/no light. Four of the eight treatments should be repeated and then the treatments should be allocated to the plots, trying to balance them between the north and south sides.
Having designed an experiment, the students then obtain simulated data, either by inputting their design into a computer program, which generates the data, or by picking a slip of paper from an envelope, which contains previously generated random yields. The data acquired can be used to discuss the analysis of variance; alternatively, the analysis can be left as an exercise for the students.
14.2.3 Chick
This game is similar to TOMATO, but involves investigating the effect of adding copper to maize or wheat in the diet of chicks. Like TOMATO, this game is based on a real experiment carried out by researchers at the University of Reading. The statistics can get very complicated, but the game is used by agriculture students who can produce reasonable solutions without knowing all of the details. This game was described by Pollock, Ross-Parker and Mead (1979).
14.3 Conclusions
The major benefit of this approach to learning is that students get much more involved and will make discoveries by trial and error, rather than having to be told everything. The problems they are working on should be fairly realistic, so that they want to solve them.
A second benefit is that, because the objectives are clear, the students always know what they are trying to do. Therefore, although they can still make mistakes, it is more difficult to get completely lost. The answers should make sense even to the students who do not follow all of the mathematics.
Finally, the mathematics itself becomes more relevant because the students can see why it is being used. This can also help to arouse interest in the other parts of the mathematics syllabus, such as differentiation and integration.
Dangers arise from the fact that this approach to learning is necessarily multi-disciplinary. The students can become confused and ask 'Why are we doing Biology in the Maths class?' A related danger is that the mathematics, on which the students will eventually be tested, may seem unnecessary.
Our experience of using these games is that they are largely beneficial. Although the examples presented here may not all be relevant, similar ideas could undoubtedly be used for parts of the A-level Statistics syllabus. Work on this type of teaching material is now continuing under the Teaching and Learning Technology Programme (TLTP) Statistics Consortium.
14.4 References
Mead, R. (1974) The use of computer simulation games in the teaching of statistics to agriculturalists. International Journal of Mathematical Education in Science and Technology, 5, 705-712.
Mead, R. and Stern, R.D. (1973) The use of a computer in the teaching of statistics. Journal of the Royal Statistical Society, Series A, 136, 191-225.
Pollock, K.H., Ross-Parker, H.M. and Mead, R. (1979) A sequence of games useful in teaching experimental design to agriculture students. The American Statistician, 33, 70-76.