Lesson 1 A Fair Comparison

Teacher Advice

Goals of Lesson

- To raise awareness of the battles women have had to compete in many Olympic events and to develop the focus of the case – Are women improving faster than men?

- To begin to consider how improvements in athletic performance can be compared.

- To identify what a ‘fair’ comparison of events might be.

- To understand how to work through an enquiry and what are the features of a high quality enquiry.

Features of the Lesson

- Pupils will become aware of past beliefs about women’s limitations and later debate about whether or not they may one day surpass men.

- Pupils will choose an event and begin to consider whether or not women have improved faster than men for their event through a basic comparison.

- Pupils will then consider how they might complete a ‘fair’ comparison of men’s and women’s improvements.

- Pupils will then consider how they might go about a complete formal enquiry into the Case question. To do this they will look at sample enquiry reports to identify the features that make up a high quality enquiry.

Summary

Part 1: Engaging pupils in the issue A brief discussion of some Olympic data to open up the issue and provide the focus for the investigation. (10 minutes).
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Part 2: An initial comparison Pupils begin to analyse the data for a specific event on which to focus for the remainder of the Case. They do this without teacher input as to how they might handle the data. (15 minutes).
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Part 3: A fair comparison Pupils reflect on their initial method of analysis with a focus on finding a ‘fair’ comparison of men’s and women’s improvements. (15 minutes).
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Part 4: The notion of a maths enquiry Pupils reflect on what is meant by an enquiry in maths and begin to consider the features of a good enquiry. (10 minutes).

Prerequisites

- Basic knowledge of Olympic Games.

Preparation

- A way of showing the Year of first participation for some Olympic events.

- Photocopies of Detailed list of performances in the six selected events, Enquiry process and Sample enquiries.

- Become familiar with background reading on women’s past participation, the debate on whether women have been improving faster than men, the different pieces of Teacher advice and how to use the 800 m data.

- Each pupil should have a LOG BOOK (small exercise book) in which to perform calculations, record notes, carry out rough drafts of work etc. Work which will be submitted can be transferred onto an appropriate medium later.

- Pupils will need a calculator regularly throughout the entire case.

Prior Learning for Lesson 1

Pupils should be familiar with the concept of absolute difference – the total change in a variable.

Part 1: Engaging pupils in the issue

Pupils are confronted with past thinking about the inadequacies of women and the much later debate about whether or not women are improving faster than men. (10 Minutes).

Activities | Teaching and Learning Notes
Show the table Year of first participation – some Olympic events and ask what the thinking might have been that meant women’s participation was limited to so few events. Key Issue: Women were regarded as physically inferior. | For decades women were considered incapable of competing in many events. Details are provided in the background article The changing role of women in the Olympic games. A brief summary is provided in: TL1: Biased reporting of the first women’s 800m. Use these to relate how prejudice and a 1928 reporter’s account allowed Olympic officials to ban women from this event for 32 more years. TL1: Biased reporting of the first women’s 800m What follows is a summary of the relevant points from the article, The changing role of women in the Olympic games. It is useful to read the whole article as it gives a number of interesting facts that may enrich the discussion. The founder of the modern Olympics, Baron Pierre de Courberton, did not believe women should participate in the Olympics and barred them from the first games. Women did begin to compete in some events, but not track and field until 1928. Women set up their own international athletics body in 1921 and ran women’s Olympics from 1922 until 1934. The men’s body was jealous of the success of the women’s games and reluctantly allowed the women to compete in 5 (only) track and field events in 1928. The men did not believe that the women were capable of the rigours of the 800m and, as detailed below, a leading sports writer falsely reported that the race was excessively grueling and this event was not run for women again until 1960. John Tunis, a prominent sportswriter of the day, portrayed the 800-metre event as follows: “Below us on the cinder path were 11 wretched women, 5 of whom dropped out before the finish, while 5 collapsed after reaching the tape”. Records, clearly indicate that only 9 women started, not 11, and all 9 finished the race. The winner set a world record. She and a few of the other competitors were understandably spent after racing at world-record pace. Some of them lay down beside the track, but none of them dropped out or collapsed from exhaustion. [This report] suited the purpose of the male-dominated administration, who prevented women from competing in any race longer than 200 metres in the Olympic Games for the next 32 years. It is interesting to compare a report on the men’s 800m in 1904: “at the finish of the 800-metre run, two men fell to the track, completely exhausted. One man was carried to his training quarters, helpless. Another was laid out on the grass and stimulants were used to bring him back to life.” (DeFranz, 1997). Print TL1 Close Box Some teachers have collected some clips of Olympic bloopers to provide an engaging initial context.
Open up the issue that more recently there has been debate about whether or not women are actually improving faster than men. Present this as the focus of an investigation that is to take the next three lessons. Key Question: Have women improved faster than men? | You have now fast forwarded to the 1990s. There has been considerable debate on this issue that is summarized in the teacher resource Women of the future are on track to run faster than men. This article should not be given to pupils at this point as it tends to pre-empt the investigation. At this stage pupils should come up with their own opinions. They can be shown the document later. Direct pupils’ attention to the focus of the case, “Olympics: Have women improved faster than men?” Have some pupils provide their gut reaction, with reasons if possible.

Part 2: An initial comparison

With the focus clear, pupils have a first attempt, deliberately without specific direction, at addressing the question "How can improvements in performance can be compared?" (15 minutes).

Activities | Teaching and Learning Notes
Pupils choose a specific event to focus on for the remainder of the Case and begin analysing the data by completing some basic comparisons of men’s and women’s improvements. Key Questions: Have women improved faster than men? How can we best answer this? What issues emerge as we make our first attempts? | Organise pupils into groups. Note that there are data for six events as well as the 800m which you will use as a model, it is preferable, but not essential that all six events are analysed, however for lesson 2 to work it is necessary that most of them are analysed. Tell pupils that the first key aim of this Case is to develop a detailed, justified answer to this question. Have women improved faster than men? Each group will choose one event to work with. The events which pupils may choose from include 100m track, 400m track, shot put, javelin, long jump and 100m freestyle. Give out Detailed list of performances in the six selected events; this is the data the pupils will use. Provide pupils with the opportunity to consider the data for their events and make a start on producing an answer to the Case question. After a short period, have groups provide their methods for how they have attempted to answer the Case question. Different groups are likely to vary in how quickly they recognize the issues of making a fair comparison and you will need to be prepared to respond flexibly to what they come up with. (See TL 2: They may see the need for percentages immediately) TL 2: They may see the need for percentages immediately It is tempting to follow the lesson plan letter for letter when it is written by someone else. However in lesson 1 some pupils have identified that percentages would be more useful in comparing performance than the absolute values. Sticking to the lesson plan without responding to pupils’ insights will risk curtailing motivation and inhibiting progress. Allow pupils the flexibility and freedom to govern their own course of analysis (with your guidance of course!). This may result in them deviating from the prescribed investigation and finding their own, perhaps surprising, interpretation of the data. Print TL2 Close Box If needed, direct pupils’ attention to the concept of absolute difference from first performance to most recent performance (which will be critiqued in the next stage) and model this with the 800m track event.
Teachers use the 800m data as a particular example different from that which the pupils investigate. If and how teachers use this data with pupils will depend on how much guidance pupils need. It may be used to guide particular individuals or a group of pupils through the preliminary investigation. | (See TL3: Basic comparison for 800m for further advice on this). It is to important avoid being too directive if pupils are seriously to engage with ideas about how they might handle the data. (See TL 4: The issue of independent thinking) TL4: The issue of independent thinking This Case presents pupils with the opportunity to develop their independent thinking. They are given a range of information is provided and an overall aim, but have a considerable amount of freedom in (a) selecting which data to use, and (b) deciding how to use this data to answer the question. In encouraging, responding to and managing this aspect of the case, it is important to avoid any suggestion of the ‘I have the answer, see if you can find it’ approach that at times characterises mathematics learning. This is important when introducing the Case, and is particularly crucial when modeling a basic comparison with the 800 m data. This aim of this is to raise the notion of an unfair comparison and time should be allowed for pupils to comment on, criticise and consider alternatives to the approach shown. Print TL4 Close Box TL3: Basic comparison for 800m When given time to compare the improvements in men’s and women’s performances, it is likely that many pupils will have calculated the difference between the first performance and the most recent performance (absolute difference) for men and women. There are other sensible answers which pupils may offer, some of which may be ‘fairer’ than the basic comparison below. The comparison below calculates absolute difference from first performance to most recent performance for men and women and can be used as a demonstration if/when pupils raise this as a method for comparing improvements. This analysis is intended to be identified as being ‘unfair’ and this provides a stimulus for the next activity. 800 m model
| First recorded time (s) | Most recent recorded time (s) | Difference (s)
Men | 131.0 | 104.65 | 21.93
Women | 136.8 | 114.87 | 26.35
Print TL3 Close Box TL5: Key features in the improvements of selected events This resource is not intended to be given to pupils. The events below were chosen to give a mix of results to the question have women improved faster than men? In some cases the answer is clearly yes, in others it is less clear. Note that the question Are women still improving faster than men? Is a different question very relevant to lesson three when examining scatter plots of results. In all cases below the gold medal performances are compared over the same time period –which means the time that the women have been competing. Students are likely to begin by comparing the improvements using the total time the men have been competing. This is fine as an initial suggestion, but for several reasons it is not a fair comparison and this should emerge during lesson 1. For a discussion of the improvements the men have made since their first competition and the many unusual features of the 1896 Olympics see TL7 1896 –A major outlier. 100m track since 1928 If you compare the period when women competed: 1928-2008, then the women have improved by only slightly more than the men: 11.6% versus 10.3%, if you compare the % improvement in the best result since 1928, then the women’s improvement (to the 1988 result) is now 13.6% - the men’s doesn’t change as 2008 was also the best ever result. However there are some outlier results that are affecting the interpretation of these figures: the 1928 men’s result could be regarded as slow (1932 was 0.5 seconds faster, there is no recorded reason for this) and in 2008 Usain Bolt took 0.16 seconds off the previous fastest. Also the 1988 performances by Florence Griffith Joyner were by a long way the fastest ever by a women and there have been numerous suspicions of drug use –but nothing was established before (or after) she died at age 38. If you compared the improvements from 1932 to 2004 then the women improved 8.2% and the men 4.4%.
| % Improvement since 1928 | % Improvement from 1928 to best result
Men | 10.3% | 10.3%
Women | 11.6% | 13.6%
400m track since 1964 Again the women have improved slightly more than the men: 4.6% to 2.4% if we take changes from 1964 (when women first ran this event) to 2008. If we take the percentage improvements from 1964 to the best performance (both in 1996) then the women lead 7.2% to 3.6%. Note that the women have shown no real improvement since 1976 and the men since 1968. The 1968 result is considered to have been helped by the thinner air of Mexico City.
| % Improvement since 1964 | % Improvement from 1928 to best result
Men | 2.4% | 3.6%
Women | 4.6% | 7.2%
800m track since 1928 (teacher’s model event) Here the women have improved much faster than the men -16.0% to 6.4% to the 2008 result and 17.1% to 8.2% if we take improvement to the fastest ever gold medal performances. The women did not run 800m again until 1960, but if we compare improvement since 1960, the women have still improved at least twice as much as the men.
| % Improvement since 1928 | % Improvement from 1928 to best result
Men | 6.4% | 8.2%
Women | 16.0% | 17.1%
Long jump since 1948 This is another event where the women have improved much more than the men: 23.6% to 9.1% to the 2008 results and 29.8% to 14.0% if we measure improvement to the best results (1988 and 1968 for women and men). The 1968 leap of Bob Beamon is a huge outlier, he added 55cm to the world record in a performance helped by high altitude and a tailwind.
| % Improvement since 1948 | % Improvement from 1928 to best result
Men | 9.1% | 14.0%
Women | 23.6% | 29.8%
Shot put since 1948 The difference in weights is greater here-7.26Kg to 4Kg but as with javelin the women have improved more -49.7% to 25.6% to 2008 and 63.05% to 31.3% if we measure to the best results (1980 and 1988). However neither men nor women have shown any sustained improvement since 1972. The shot putt is an event plagued by use of steroids and these were largely undetected in the 1970s and 80s.
| % Improvement since 1948 | % Improvement from 1928 to best result
Men | 25.6% | 31.3%
Women | 49.5% | 63.0%
Javelin since 1932 The women throw a lighter javelin (600g versus 800g) but they have improved much more than the men: 63.5% to 24.6% to 2008 and 71% to 30% to the best performances (1988 and 1976). Note that in 1986 (men) and 1991 (women) the javelin’s centre of mass was altered in ways that made it travel a shorter distance and also land at a better angle so it would stick in more and make the throw easier to measure. Results dropped in the Olympics after these changes More reliable measurement was one reason for this change in the javelin, but the other was that the men were starting to land the javelin in the running lanes at the other end of the ground and this was risky for runners! Both men and women have improved much more in this sport than, say sprinting, one reason is better technique, but there have also been several improvements to javelins that have made them travel better.
| % Improvement since 1932 | % Improvement from 1928 to best result
Men | 24.9% | 30.1%
Women | 63.5% | 71.0%
100m freestyle swimming since 1912 In swimming, the improvement in both men’s and women’s performance has been greater and much more regular than in track and field –the most recent Olympics has been the fastest in all but one of the 26 swimming events. Women did not compete in many swimming events until 1968, but have been swimming this event since 1912. Since that time they have improved 35.6% to the men’s 25.6%. If we take improvement since 1968, when the number of events for women rose sharply then the women have beaten the men by a small margin 11.5% to 9.6%; these latter numbers are very similar to the numbers in all other swimming events where men and women have improved by 10-15% and usually, but not always, the women have improved by 1-2 percentage points more. Swimming results follow much smoother patterns with far fewer outlier results that track and field.
| % Improvement since 1912 | % Improvement from 1928 to best result
Men | 25.6% | 25.6%
Women | 35.4% | 35.4%
Print TL5 Close Box After this discussion, allow groups time to complete these initial calculations for their event.

Part 3: A fair comparison

The teacher uses pupil’s initial responses to build meaning for what a ‘fair’ comparison of events might be. (15 minutes).

Activities | Teaching and Learning Notes
Encourage pupils to identify reasons why comparing absolute difference from first performance to most recent performance is not a fair comparison. (See TL6 and TL7 for issues to discuss.) Key Question: · “Is this a fair comparison?” | For background details of why these six events were chosen and how they differ in terms of relative improvement, See TL5: Key features in the improvements of selected events. Set pupils the challenge of identifying how they may complete a ‘fair’ comparison of improvements in their event and allow them to discuss this with their group. One issue is that the comparison should be comparing like with like. (See TL 6: Comparing Like with Like). TL6: Comparing like with like To complete an accurate and fair comparison between men’s and women’s improvements, as many variables as possible should be held constant. I f you compare men and women over the whole time that each of them have competed, then there appears to be no event where the women have improved more than men. However this is not a fair comparison, one reason is that it gives the men more time to improve and so the women may have improved faster. There are several possible options for completing a ‘fair comparison’ which pupils may offer as suggestions or may be suggested by the teacher if pupils are unable to provide suggestions Improvement from a common starting year (women’s first performance or other significant year) to most recent performance Improvement from a common starting year (women’s first performance or other significant year) to the best ever Olympic performance Improvement from the first performance of each sex over a common time frame There are other possibilities, often more sophisticated, such as calculating relative rates of improvement (though we do not suggest introducing these) and there are clearly pros and cons to each method of comparison. Pupils should choose which comparison method they believe is the most fair, while stronger pupils/classes may be encouraged to justify why they believe the method they chose was the fairest. 800m model The comparison below compares the improvement in men and women over the time women have been competing in the 800m and can be used as a demonstration if/when pupils raise this as a method for comparing improvements.
| 1928 performance (s) | 2008 performance (s) | Difference (s)
Men | 111.8 | 104.7 | 7.1
Women | 136.8 | 114.9 | 21.9
For teacher background, the table below shows the difference in comparisons when you move from comparing over the whole Olympic period to using the men’s improvement over the same period as women.
Event | Women’s improvement from when they first competed | Men’s improvement from when they first competed | Men’s improvement from the same year as the women
100m track | 1.42sec from 1928 | 2.31sec from 1896 | 1.11sec
400m track | 2.38sec from 1964 | 10.2sec from1896 | 1.11sec
800m track | 21.9sec from 1928 | 26.4sec from1896 | 7.2sec
Long jump | 1.34m from 1948 | 2.2m from 1896 | 0.72m
Shot Putt | 6.81m from 1948 | 10.29m from1896 | 4.39m
Javelin | 27.74m from 1932 | 35.74m from1908 | 17.86m
100m freestyle | 29.1sec from 1912 | 35sec from1896 | 16.2sec
Print TL6 Close Box Another issue is that many of the 1896 Olympics events were run in ways very different to later years and the results are major outliers. Pupils may notice this themselves or you may need to direct their attention to this. Reasons for this are given in: TL7: 1896 - A major outlier. TL7: 1896 - A major outlier If the men’s results are taken from 1896, they show much greater improvement that if they are taken from 1900. We label the 1896 data as outliers that do not provide a fair comparison. Today’s pupils have grown up with the Olympics being the elite competition. Things were very different in 1896. There were very few competitors (only 64 in all the athletics events from only 10 counties), moreover these were far from the best in their countries –the American team of 10 athletes had only one who was a national champion in his event. There were only 9 competitors from 5 countries in the long jump and 7 athletes from 4 countries in the shot putt; in other words the event in no way brought together the best athletes in the world. The track events were run on a 200m track (see photos below) that had very tight bends; this meant that the runners in every event longer than 100m had to slow down on the bends much more than runners on a 400m track. In swimming, the events were not held in a pool, but in a roughly measured course in the open ocean, moreover the modern freestyle swimming stroke was not developed until 1902 and a much slower stroke was used in 1896. In shot putt they threw from a square not a circle. The following table compares the men’s improvement from 1896 with the next time the event was run (1900 in all bar 100 freestyle when it was 1904).
Event | Men’s improvement from 1896 | Men’s improvement from the next year of the event
100m track | 2.31sec | 1.31 sec
400m track | 10.2sec | 4.8 sec
800m track | 26.4sec | 16.6 sec
Long jump | 2.2m | 1.35 m
Shot Put | 10.29m | 7.41 m
100m freestyle | 35sec | 15.6 sec
Print TL7 Close Box
After a short period, get pupils to provide possible techniques for completing a fair comparison. See TL6 for possible options. | Allow pupils to select the method they would like to use to complete an enquiry into the Case question. Reflect briefly with the class about the key points they learnt through identifying a ‘fair’ comparison. See TL6 for key points.

Part 4: The notion of a Maths Enquiry

Pupils focus on what will be expected of them in completing a good enquiry by building an understanding of how to work through an enquiry and the components of a high quality enquiry. (10 minutes).

Activities | Teaching and Learning Notes
As a guide for pupils for working through an enquiry and how they should present their work, provide pupils with the ‘enquiry process’ handout describing the elements of an enquiry and discuss. Provide pupils with three sample enquiries; one of which is high quality, one is of medium quality and one is of low quality. Key Questions For Homework: · What are the components of an Enquiry report? · Put the three example enquiry reports in order from best to worst, and explain why you ranked them in this way. | Possible solutions shown in Enquiry process solutions. Do not give this to pupils at this stage. You can use this to help shape the discussion of the homework at the start of Lesson 2. These sample enquiries answer the question using the calculations pupils have just completed. For the homework task, it is not intended that pupils complete a full enquiry answering the same question or using the calculations they have just completed. However by lesson 2 they do need to have completed what they regard as a fair comparison for their event.

Homework:

For homework pupils are asked to put the sample enquiries in order and identify what makes a high quality enquiry and record these features in the final column on their ‘enquiry process’ handout. If necessary they finish their calculations for their event.