10. PREPARING THE GROUND FOR 'A' LEVEL STATISTICS Jim Garbutt Burnholme School, York
10.1 Introduction
Many of the initiatives and courses on offer for statistics teachers are aimed at the post 16 age group. Perhaps this is a result of the number of qualified and eminent statisticians working with students of this age. However, to ensure a good supply of students in this age group there is a need to ensure good statistics teaching in our secondary and primary schools. A problem does exist here as many teachers feel out of their depth and adopt a safe mathematical approach to statistics. Hence the text book becomes the key to the teaching and the answers to problems are always known in advance and always work out quite nicely. Grubby data does not figure in the teaching. This is a product of the lack of in service training for teachers in these schools. There are several courses entitled 'Practical A level Statistics' but there are too few 'Practical Primary Statistics' or 'Practical Data Handling' courses. Perhaps the level of statistical knowledge is considered too elementary to warrant such inputs. However I can assure you that there is a need to provide inset at this level so that teachers can experience the excitement of handling data practically. This will increase their own and their students' motivation and enthusiasm.
10.2 Background
Originally I taught statistics to 16+, CSE and A level. I had no time for practicals because they 'hindered the completion of the syllabus'. Consequently I became reliant on the text book. Coursework was beginning to enter into the departmental discussions but I did not want to consider its value for many reasons that could be summarised as 'A fear of something new'.
About this time I became Head of BTEC/City and Guilds and had to consider coursework seriously as this was the sole form of assessment. The fear element of coursework was outweighed by the offer of an incentive 'B' allowance, what better form of motivation! Cross curricular approaches were adopted by some of the teachers in the team but I did not do this in statistics. The text books were still in full use and yet statistics is cross curricular by its very nature.
By now GCSE coursework was also in full flow and I was becoming disatisfied with and sceptical of some of the activities we passed off as coursework. I started to search for suitable professional development and enquired about the Post Graduate Diploma in Statistics and Statistics Education offered by Sheffield City Polytechnic (now Sheffield Hallam University). North Yorkshire County Council agreed to pay for me and so I commenced the course. The work was very challenging and gave me a plethora of ideas for the classroom. The course was without doubt the most career relevant one I have attended. I tentatively tried one or two of the more straight forward practicals and found that both myself and my groups enjoyed the work immensely. I found myself looking forward to practicals and investigations and wherever possible avoiding the text book. Students of all ages and abilities produced display work which was mounted on various walls and the whole process was very rewarding.
I am convinced that the confidence and ideas gained from the
PG Dip played an important part in my appointment as Head of
Mathematics at Burnholme School in York. However this move did
allow me to develop my Statistics teaching considerably.
Initially the developments were very small for the following
reasons:
I had a very tight budget and a great deal of work to do;
I did not want to alienate members of staff who were all new to me;
I wanted to show my passion for data handling without being messianic.
The approach did work and the staff proved to be just as enthusiastic as well.
10.3 Preparing the ground
Statistics is not mathematics and the text book mathematical
approach is neither the most efficient nor the most relevant
method for teaching this exciting and practical subject. My
approach to teaching statistics is based upon the following:
Always choose a practical alternative where one exists;
Critically analyse data sets;
Experiment and present results;
Displaying the results is important.
Statistics is cross curricular and this should be supported by the teaching.
The cross curricular nature of statistics is important for successful statistics teaching. I have often been asked by Science and Geography staff why we do not teach pie charts or bar charts etc? The problem is not that the ideas are not taught but that students compartmentalise their knowledge and tend not to transfer it between subjects. However this is hardly surprising given a compartmentalised curriculum. This prompted an examination of the need and provision of statistics across the curriculum using the model suggested by Rouncefield and Holmes (1990). The results were quite astounding and gave me the ideas for some of the activities described below.
The following activities have been developed and refined over a period of 3 years. I have used every one of them successfully in the classroom and on in service courses for teachers. Time is necessary for the development and there is a need to attach the correct degree of importance to the subject. In my case it helped to be a Head of Department with a Business Studies Headteacher. Statistics gained a lot of 'Street credibility' in the school.
10.4 The Activities
10.4.1 Parts of the body
The idea for this practical came while I was watching the television programme 'Quincy'. He found the remains of a thigh bone and within one hour knew that the victim was a 6'4" male weighing 240lbs with a passion for hamburgers!
Students are required to measure parts of their bodies eg head circumference, thigh, forearm and compare these measurements with their height. The data can be analysed using scattergraphs and correlation coefficients and students might even try and obtain a linear model after interpreting the scattergraphs. This has allowed the students to extend themselves into the higher coursework levels. One interesting discussion occurred at the GCSE statistics moderators' meeting. A very able candidate had programmed a spreadsheet to correctly evaluate Spearman's Rank Correlation Coefficient for the various data collected. The whole piece was imaginative and an excellent piece of statistics. However the moderator said that the piece could only receive a low mark because the candidate had not completed any of the calculations by hand. The spreadsheet was simply multiplying the numbers. When I asked why, the answer was that unless the calculations were present the student had not demonstrated an understanding of the topic. This referred to the use of any statistical package. Packages can be used for coursework provided a hand written example exists of the graph or calculation! I did find this astounding because the examination is the medium for testing 'calculations'. Coursework should be freed from these ties. I am concerned by this mathematical attitude to statistics: 'If no hand calculation is present you have not demonstrated an understanding'.
10.4.2 Pulse Rates
This is another practical for bivariate data and based on material in Garbutt (1991a). Students investigate the factors that might affect pulse rates, for example height, weight, shoe size, eye colour.
10.4.3 Capture-Recapture
Capture-recapture is described in Garbutt (1991b) and is a hybrid of the activity in Davies (1993) book 4. This is an excellent publication with activities and photocopiable worksheets for all levels within Ma5. There are two volumes that take the form of teachers' guides. The idea arose from a conversation with Richard Dacosta, the Head of Science at Burnholme School, about a unit on Ecology. Counters are set up in a series of bags to represent a declining animal population. For instance there may be 80 counters in the first bag, 40 in the second, 20 in the third and 10 in the fourth bag. Each counter represents say 50 Elephants and each bag a different year. Students are asked to sample the number of Elephants in each year and write either a scientific report, a letter to their MP or a Newspaper report about their findings. After this a further bag is given out representing the population 10 years on. The bag contains about 60 counters to simulate a dramatic increase in the Elephant population. The students sample this population and write a follow up article. The piece of work links in with the ecology unit to add relevance to the simulation. Otherwise, a simulation can become too divorced from reality!
On one occasion I used this activity with two groups of teachers on an in service course. Their reaction was quite surprising and worrying. The first group had sampled the first four bags and had obtained almost perfect results of 8000, 4000, 1000 and 500 elephants respectively. I asked them one or two questions about the type of article they would choose to write, gave them the fifth bag and left them to get on while I worked with the second group. The second group were completely bemused. They had taken out ten counters from the first bag, marked the counters and replaced them in the bag. They had then taken out a second sample of ten counters but unfortunately had not chosen any marked ones. This does mess up the arithmetic somewhat. I asked them what the problem was and they replied most angrily 'it doesn't work'. I asked why and they drew my attention to the absence of marked counters in the second sample. I reminded them of the nature of the activity and suggested they choose a few more counters. However this was regarded as cheating because I had said 'choose about 10 counters' in my introduction. Having sorted that one out I returned to the first group expecting them to have finished. How wrong I was. That group was most confused. I asked what was wrong and they replied that there were about 5000 or 6000 elephants in the final sample. Well done I said, that's an almost perfect result. However they thought this was stupid because according to the pattern there should have been less than 500 elephants in the final sample. This was their prediction and it had to happen. They refused to consider the activity further and would only use it with children if the bags all followed the same pattern. 'What about re-population?' I asked, 'What about it?' they replied!. I have used this with students several times and they have found it a very exciting activity and some wonderful pieces of work have been completed. The teachers were of a variety of ages and most of them were teaching statistics up to 'A' level.
10.4.4 Radioactive Decay
See the worksheet in Morse (1991). Martin Morse, the Head of Lower School at Burnholme, introduced this activity to me. Again it is a slight hybrid of the original. Radioactive isotopes decay and although it is impossible to predict which atom will decay next it is possible to predict the amount of material decaying using Half Lives. Dice are used to simulate different rates of decay.
10.4.5 Dice Games
I use a variety of dice activities because they are simple to set up and can give rise to a wealth of exciting lessons. The three lessons outlined have all proved to be very popular with students, teachers and parents on 'Maths for Parents' evenings.
(i) People often think that a 6 is the most difficult number to throw on a die. This results from the extra attention some games draw to the 6. Eg Throw a six to start. I am sure we have all felt the frustration of being stuck at the start while someone else races off to win the game and all because of that blasted 6 which is so hard to get?
Students are required to design an experiment to test if a six is in fact the most difficult number to throw. A more detailed explanation of this can be found in Garbutt (1991a), Schools Council (1980) and Green (1983).
(ii) A development from this is to introduce different polyhedra dice and ask the students to choose which dice they would use for a particular game. Dice are tested against each other and possibility space diagrams are used. However don't be surprised at the number of people who will still choose the die with the best colour despite any contradictory evidence.
(iii) Chinese Dice. This is based on material in Garbutt (1990), Rouncefield and Green (1989) and Garbutt (1991a). This is my favourite dice game because it is illogical. I introduce the dice by challenging students to beat me and saying 'But you have no chance'. Students always rise to the bait and playing for 10p is an added incentive. Of course I always win (my Roller proves this!) but this fun activity gives rise to some very interesting and serious work on data collection and probability. One teacher was so taken by this game that he purchased the dice and now refers to me as 'The Chinese Dice Man'. I have also had parents request the address for the dice so that they can purchase sets as presents. This reflects the intrigue and interest that exists in the wonderful world of statistics.
10.4.6 Dime Kits
There are two Dime Probability Kits that come complete with teachers' guides and probability paper. The kits comprise 24 sealed shakers containing balls, dice or counters plus a number of experiments. They are inexpensive and provide an excellent resource for National Curriculum coursework and discussion of probability at higher levels.
10.4.7 Identifying Newspapers
See the worksheet in Garbutt (1992). I was watching a programme about trying to establish if a newly discovered work was in fact written by William Shakespeare. I found the whole process intriguing and wondered if something similar could be done in the classroom. I asked the English department if they had a class set of 'undiscovered works by Shakespeare'. They said that they were out of them at the moment due to cut backs in the LMS budget or some equally lame excuse. However they did suggest I used newspapers as a suitable alternative. Five known articles from newspapers are used to try and identify 5 unknown articles. The interesting aspect to this practical is that the statistical results can be inconclusive unless style is also included. This highlights the fact that statistics is one of the tools of analysis and not the only one. Commonsense often has a role to play but is not often used.
10.4.8 Cost of Living
See the worksheet in Garbutt (1991c), Bass and Farham (1971) and Garbutt (1990). An old series of text books produced this lovely piece of statistical investigation. I was intending to throw them out. However, I glanced through the mound and found a price list from about 1970. In a flash of inspiration (perhaps the only one I have ever had!) I had the idea for 'Cost of Living'. Students collect the up to date prices of as many items in the list as is possible. This information is used to predict future prices. A few years ago a group of 'low ability and difficult students', the two often seem to go together don't they, completed a beautiful piece of display based upon this. The graph for the 1970 prices was quite small, the 1988 graph was much larger, but the 2010 graph was so large that we had to run it up the wall and along the ceiling. Another group constructed 3-d shopping baskets representing the price rises. Although the scales were not strictly correct, it was their imagination that created the ideas. The motivation of these young people on this project was superb and they gained much self esteem from having other teachers and friends comment on their display. I would urge you to display work no matter what the quality. Some people put in a lot of effort to produce what may look an ordinary piece of work and it is demotivating if only the 'pretty' pieces get onto the wall. Kids get a real buzz from seeing their work displayed and they are motivated by the practical statistics classroom.
10.4.9 The Plague
See the worksheets in Garbutt (1990, 1991d). This is a simulation based upon people catching the plague. Different graph papers, triangular, square and hexagonal, can represent different rates of infection. The activity can be used with a wide range of abilities and some very interesting work on sequences can develop. Extensions can occur by allowing immunity eg every third person is immune, or throw a die to decide immunity. A dramatic opening to the simulation can be created by having the group(s) in the hall and a person scream and collapse when they are infected. The work can link into PSE and Health Education and AIDS etc.
10.4.10 Which Ball is Best
Quite simply the students are given about 5 different types of ball and have to answer the question 'Which Ball is Best'? Video can be used to record students 'selling' the best ball. Incidentally balls bounce differently according to the surface they bounce on. So the question posed in this piece of work does in fact raise many more questions for the students to investigate and answer. Many different and ingenious experiments have been designed by my groups. If you like a noisy and bouncy start or end to the day then this is the activity for you. I always enjoy this lesson!
10.4.11 Databases
Many activities have been developed for databases. The Centre for Statistical Education, in the Department of Probability and Statistics at the University of Sheffield, has had a two year project that has developed materials. I have seen some of the draft activities, and as would be expected, they are excellent. The project officer is Mike Hammond. Contact the centre for further details. An activity developed by Lesley Lambert, the Deputy Headteacher at Burnholme School, that I have used successfully, is to create databases for shapes and solids and then play spot the shape. This has obvious links with other areas of the National Curriculum. Plastic shapes and solids are easily available and in 1993 cost about ,20 for a complete set.
10.4.12 Road Accidents
See the worksheet in Garbutt (1993). It is important to introduce statistics using real data and if possible data that are relevant to the students. This will capture their imagination and is more likely to precipitate discussion. One morning a large envelope appeared on my desk containing the North Yorkshire County Council traffic survey report (Moore, 1991). This was a wonderful stroke of luck as I was just about to commence a data handling section of the scheme of work. I prepared a worksheet that focuses on analysing the charts and density measures, as opposed to looking at the raw data. I contacted NYCC and asked for a class set of the reports. They agreed to send them through and have since visited me to see how the next edition can be best introduced to schools. Initially I spent a full lesson just looking at the report and asking questions such as 'Is the A1T 10 times as dangerous as the A629 as can be implied from the first column?'
At the moment I am developing worksheets for each level of the National Curriculum matched to this report as a way of encouraging other teachers in North Yorkshire to use real data.
A group of year 8 students developed this into a survey of a local road that was causing some anxiety. Phrases such as 'Get real sir, it's mad alley crossing that road' spring to mind. The group even built an electronic noise meter and went out on a very cold November morning to collect the data. I remember the day well as I had to lend my coat to one of the girls. So who says statistics doesn't motivate kids? Believe me it does, I am always witnessing it!
10.5 Conclusion
I believe all of these activities prepare the ground for 'A' level statistics because they prompt students to use real data, draw inferences, use their knowledge and judgement and to realise that there is not always just one correct answer. However, more importantly, I try to get all of the students to think statistically and not simply apply formula and techniques to get a number that matches the one in the back of the book.
I have a passion for statistics (can we drop data handling please) and I love working in this field. I get a real buzz from the enjoyment of the students in the classroom. I often feel I am working in isolation and would love to develop a 'share a practical' network with teachers around the country. So if some of my passion has rubbed off to the extent that you do develop something, please let me know or send it in to Teaching Statistics. That way we can all work towards improving the teaching of this wonderfully practical subject.
10.6 References
Bass, D. and Farnham, A. (1971). Action Mathematics 2. Cassell.
Davies, G. (1993). Practical Data Handling, Books A and B. Hodder and Stoughton.
Garbutt, J. (1990). Take Five. Teaching Statistics, 11(1), 2-14.
Garbutt, J. (1991a) Take Five More (Pulse Rates). Teaching Statistics, 13(3), 82-84.
Garbutt, J. (1991b). Take the Money and Run to Peking - Quick! Making Progress, Spring 1991. Stanley Thornes Press.
Garbutt, J. (1991c). Cost Of Living. Making Progress, Spring 1991. Stanley Thornes Press.
Garbutt, J. (1991d) The Plague. Making Progress, Autumn 1991. Stanley Thornes Press.
Garbutt, J. (1992). Identifying Newspapers. Making Progress, Spring 1992. Stanley Thornes Press.
Garbutt, J. (1993). Road Accidents and Casualties. Making Progress, Spring 1993. Stanley Thornes Press.
Green, D. (1983). Shaking A Six. Mathematics in School, 25 (1), 29-32.
Moore, M.O. (1991) (Editor). Road Accidents and Casualties, 1991. North Yorkshire County Council Surveyor's report.
Morse, M. (1991). Radioactive Decay. Making Progress, Spring 1991. Stanley Thornes Press.
Rouncefield, M. and Green, D.R. (1989) - Condorcet's Paradox - Teaching Statistics 11 (2) 46-49.
Rouncefield, M. and Holmes, P (1990). From Cooperation to Coordination (A file for coordinating school statistics teaching.) Centre for Statistical Education, University of Sheffield.
Schools Council (1980). Statistics in Your World; Shaking a Six. Foulsham Educational.