11. TEACHING STATISTICS THROUGH PRACTICAL WORK Mary Rouncefield Chester College, Cheyney Road, Chester. CH1 4BJ

11.1 Introduction

Statistics is clearly a useful and necessary tool required by large numbers of pupils for their project work in other subject areas, (geography, biology, social sciences etc). But what are these pupils learning in their mathematics or statistics lessons to back up all this practical statistical work? What can the mathematics teacher provide to develop the necessary skills? First we consider the processes required in any project:

Unfortunately, the mathematics teacher tends in many instances to concentrate on stage 4 of the process and children may be drilled on quite complex statistical calculations without understanding their purpose. (The standard deviation is an example of this.) Often little consideration is given to the quality of the data collected and the most appropriate method of selecting a sample for a particular purpose. I suspect that many children are going out with questionnaires containing badly phrased or leading questions which will give results which are inadequate, misleading or at worst entirely wrong. Should the mathematics teacher consider questionnaire design and ways of avoiding bias at the data collection stage?

How do pupils obtain their samples? Is it satisfactory to take the first ten people they meet in the school corridor or at the local shopping centre? Should the mathematics teacher consider sampling methods and help pupils to understand the need for a representative sample?

11.2 Practical Statistics

In 1982, a project entitled 'Practical Statistics' was sponsored by the Centre for Statistical Education at the University of Sheffield, under the leadership of the director there, Peter Holmes. Our brief was to develop and trial practical work for students and teachers working with existing post 16 syllabuses. While we were not asked to redesign the content of the typical sixth form statistics course, we have endeavoured to move away from the traditional approach of

 
 THEORY    EXAMPLE(S)     PRACTICE

                                                                        to

PRACTICAL ACTIVITY    REAL DATA   DISCUSSION MODEL & THEORY
 

The project materials can be found in Rouncefield and Holmes (1989), and are aimed at students over 16 but the method of working is equally suited to pupils aged under 16.

Although our practical work tends to be essentially 'teacher led', the teacher should encourage discussion at every stage, so that students develop the skills necessary to embark on their own statistical investigations. These materials aim to help those teachers who are worried about tackling statistics syllabuses which stipulate that project work is part of the final assessment. The practical tasks included in our materials build up both the skills and the confidence of teachers and pupils alike.

Practical work in any subject area provides pupils with a concrete understanding of reality, and this can be built upon and developed into an understanding of the theoretical ideas behind that topic. Pupils gain experience of real situations in which particular probability models apply. How many teachers working through past examination papers with pupils at the end of an A level course have been asked 'Please .... what KIND of question is this? What distribution is it?'

We have tried to keep the time required to complete the practical tasks to within a time limit of one hour, and many take considerably less time than that. Teachers and lecturers do find themselves under pressure to complete the lengthy syllabuses currently in force. But even so, time invested in practical work is time well spent, and pays dividends in better understanding of the work covered.

11.3 Classroom Practicals using the Binomial Distribution

The practical activities described here are by no means all new, but they have been tried and tested, by myself and teachers involved in the 'Practical Statistics' project. Practical experiments are used not only to illustrate the binomial distribution, but to make it relevant and important to the pupils. The aim is to take statistics and probability out of the text book and into the pupils' direct experience.

While most teachers of statistics teach the binomial distribution to pupils aged 16+, I would like to suggest that this topic can be introduced at a slightly younger age, to enliven lessons on probability. Providing pupils have some understanding of the multiplication and addition laws of probability, it is not a great step then to go on to develop the binomial distribution using tree diagrams. We start with a simple case and progress to more complicated ones.

My child claims not only to be able to tell the difference between brands of cola drink but also that he positively dislikes the cheaper supermarket brand. Is this prejudice based on fact?

Can people really tell the difference?

Mavis is given three cups of cola. Two cups contain one brand of cola. A third cup contains a different brand, but she is not told which one this is. Can Mavis distinguish between them and correctly identify the different brand?

She can on this one occasion. Can she really tell the difference though? Or did she guess?

If Mavis really cannot tell the difference, the probability of her guessing correctly is 1/3.

                                                                        1/3                    she guesses correctly
Mavis tries the cola and guesses 
                                                                         2/3                   she guesses incorrectly
 
 
The probability of her guessing correctly is quite high, then.

Suppose we let someone else try. George has a go.
 

 
What is the probability that they could both be guessing?
 

            MAVIS TRIES                            GEORGE TRIES

If two people can correctly distinguish between the brands of cola this is much more convincing. There is only a probability of 1/9 them both guessing correctly. So it does seem much more likely that their correct answers are based on a real difference.

Is this convincing enough?

How many people do we need to make a correct identification for us to be convinced?

Is this sufficient evidence?

The tree diagram is beginning to look rather complicated.

    MAVIS TRIES                                    GEORGE TRIES            ANNETTE TRIES

 
_{Can we work out the probabilities another way?  By listing all the possible outcomes?}
 

            
Here we can notice the symmetry of the arrangements for 1 and 2 correct identifications, and develop strategies for ensuring that all possible outcomes are listed. (2³ in total).

Three correct identifications is pretty convincing then. The probability of three people getting this result by guesswork is only 1 in 27. What if four people took the test and three of them were correct. Would we be as convinced by that result?

The tree diagram for this situation is beginning to become rather fearsome. Let's try the other method:
 
 
 

In this situation we would certainly be convinced by all 4 people giving correct answers. But what about 3? Three correct out of four is paradoxically less convincing than three correct out of only three.

By now our pupils are calculating binomial probabilities almost without realising it. As listing all possible outcomes perhaps becomes more difficult for larger numbers of trials, pupils will be fascinated by Pascal's triangle and enjoy using it.

11.4 Reading Someone's Mind

This is another experiment popular with all pupils. Again results can be analysed in the same way using the binomial model. Ordinary playing cards can be used.

If the experiment consists of the subject having to identify (or guess?) the suit of 10 cards, how many must they get right in order for us to believe that they can read the experimenter's mind?

The binomial probabilities for someone guessing correctly are:
 
 

The probabilities for 10, 9, 8 or 7 correct responses are all very close to zero. If a subject gets this many right it is extremely unlikely that they are guessing. These results all seem very convincing. Just how far down can we go?

Is an accumulated probability of 0.08 (8%) sufficiently impressive? In that case we can accept results down to 5 correct responses. If we insist on a probability of 0.05 we can only accept results of 6 correct or better.

The possibilities for experimentation and hypothesis testing can be opened up still further by introducing pupils to the sign test. This test can be used to answer any of the following questions (and any others which involve comparing 2 measurements for the same person).

The sign test is easy to use and pupils who can calculate binomial probabilities can progress on to this, providing the underlying logic is explained. These are the results of a group of girls who tried two tasks: one testing spatial ability and one testing verbal ability. In order to test whether girls are faster at the verbal task we need only count how many do have faster times, and then decide whether this is a convincing result.

 
The model to be used to analyse the results here is the binomial distribution with n = 10 and p = 1/2. The reader may like to check that 8 girls would need to have faster results on the verbal test for us to be convinced.

11.5 Conclusions

The essence of the practical approach is to:

start with a                     PROBLEM

collect                            DATA

                                      ANALYSE and DISCUSS

develop a                       MODEL

In this way pupils are involved actively in the learning process and will enjoy and remember what they have done. Much work done by older pupils on probability tends to consist of paper and pencil exercises (usually tree diagrams). If they are lucky, younger pupils may be allowed to experiment with dice, cards and coins. Hopefully these experiments will continue that progression and allow the pupil to make decisions on the basis of probabilistic arguments.

11.6 References

Rouncefield, M and Holmes, P (1989). Practical Statistics. Macmillan.

Hudson, B and Rouncefield, M. (1993) Handling Data, Years 10-11. Extension Focus book. Century Maths Series, Stanley Thornes.

The following paper makes many references to methods in the Scottish system of education and there may be items and jargon that are unfamiliar to teachers south of the Scottish border.

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