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Moving on with Statistics How to engage students with the subject post 16 Mark Kent, Head of Mathematics and Computing/ICT Faculty The Sixth Form College Solihull sallysallymark@kent25.freeserve.co.uk

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My experience in Statistics Education Worked in Sixth Form Colleges for nearly 20 years, teaching A Level and International Baccalaureate Mathematics and Statistics. Worked in Sixth Form Colleges for nearly 20 years, teaching A Level and International Baccalaureate Mathematics and Statistics. MSc with Sheffield Hallam University in Applied Statistics with Stats Education in 1997 MSc with Sheffield Hallam University in Applied Statistics with Stats Education in 1997 Led the Maths Department at Cadbury College Birmingham for 7 years – developed A level Statistics course growth to around 150 students Led the Maths Department at Cadbury College Birmingham for 7 years – developed A level Statistics course growth to around 150 students Set up and led the first Specialist Maths Sixth Form College project – a Maths/Stats outreach to Birmingham secondary students. Set up and led the first Specialist Maths Sixth Form College project – a Maths/Stats outreach to Birmingham secondary students. Joined The Sixth Form College Solihull in January 2006 as Head of the Maths and Computing/ICT Faculty Joined The Sixth Form College Solihull in January 2006 as Head of the Maths and Computing/ICT Faculty

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Statistics a Popular Option Post 16 AQA Statistics A level a practical Statistics course for students with a minimum of a grade C at GCSE – suitable for Intermediates!! AQA Statistics A level a practical Statistics course for students with a minimum of a grade C at GCSE – suitable for Intermediates!! Hundreds of students take the course at Cadbury and Solihull – about 75% with Intermediate B and C grades at GCSE. Hundreds of students take the course at Cadbury and Solihull – about 75% with Intermediate B and C grades at GCSE. Pass rates at A level close to 100% over a seven year period. AS pass rates around 80%. Pass rates at A level close to 100% over a seven year period. AS pass rates around 80%. Student questionnaire responses and AS to A2 progression indicate both enthusiasm for the subject and often pleasant surprise with grades obtained! Student questionnaire responses and AS to A2 progression indicate both enthusiasm for the subject and often pleasant surprise with grades obtained!

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Statistics isn’t Mathematics! Students who often struggle with Pure Mathematics can do very well at Applied Statistics. Students who often struggle with Pure Mathematics can do very well at Applied Statistics. It requires a different approach/mindset. It requires a different approach/mindset.

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Statistics is the missing link! A large number of other AS/A2 subjects use Statistics all the time e.g. Psychology, Biology, Geography, Economics. A large number of other AS/A2 subjects use Statistics all the time e.g. Psychology, Biology, Geography, Economics. Many careers involve use of statistical techniques. Many careers involve use of statistical techniques.

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A philosophy of Statistics Education It should be in context. It should be in context. Its delivery should begin with practical activity, adding a theoretical structure later, or simultaneously. Its delivery should begin with practical activity, adding a theoretical structure later, or simultaneously. Use technology, but start with something concrete – stats education research indicates this is the best approach. Use technology, but start with something concrete – stats education research indicates this is the best approach. Avoid using abstract, artificial data. Avoid using abstract, artificial data. Fit your activities to the interests of the students – post 16 this is food, money, mobile phones etc. Fit your activities to the interests of the students – post 16 this is food, money, mobile phones etc. Make maximum use of statistical experiments. Make maximum use of statistical experiments. Don’t worry about things going wrong – they often do but the students don’t mind!! Don’t worry about things going wrong – they often do but the students don’t mind!!

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The philosophy in practice Plan to incorporate a practical example into every topic area taught on the work- scheme. Plan to incorporate a practical example into every topic area taught on the work- scheme. Practicals need not take the whole lesson – 10 minutes is often enough. Practicals need not take the whole lesson – 10 minutes is often enough. Avoid the urge to rush the teaching of a topic without concrete examples – it’s a false economy. Avoid the urge to rush the teaching of a topic without concrete examples – it’s a false economy.

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The importance of the first lesson Do something memorable – e.g. a smokers’ fitness experiment: Do something memorable – e.g. a smokers’ fitness experiment:

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Mars Bars and Memory – a good first lesson at AS level Record how many random digits out of 10 students can remember at lesson start. Record how many random digits out of 10 students can remember at lesson start. Students consume a mini mars bar and the memory test is repeated every 5-10 minutes. Students consume a mini mars bar and the memory test is repeated every 5-10 minutes. Results recorded and represented on box plots. Memory peaks around 20 minutes after consumption. Results recorded and represented on box plots. Memory peaks around 20 minutes after consumption.

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Results of Mars Experiment

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Examples of Practicals in a teenage context Mobile Phones Start with a question – are students ambidextrous? Start with a question – are students ambidextrous? What does that mean and how can we test it – student responses. What does that mean and how can we test it – student responses. Make use of simple equipment – could use rulers, calculators, but best is a mobile phone – all students possess one! Make use of simple equipment – could use rulers, calculators, but best is a mobile phone – all students possess one! Students time each other in pairs to text (not send) the word ‘college’ with left and right hands separately. Students time each other in pairs to text (not send) the word ‘college’ with left and right hands separately. Analysis depends on level – could use comparative box plots, hypothesis test on differences (t test). Can also look at spread of data and breakdown into gender etc. Analysis depends on level – could use comparative box plots, hypothesis test on differences (t test). Can also look at spread of data and breakdown into gender etc.

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Chocolate Does Chocolate increase ability to concentrate? A variation on the Mars Bars experiment. A variation on the Mars Bars experiment. Compare before and after (15 minutes approx.) consumption scores in remembering 10 random numbers. Compare before and after (15 minutes approx.) consumption scores in remembering 10 random numbers. Use AUTOGRAPH and Binomial Distribution – a sign test. Use AUTOGRAPH and Binomial Distribution – a sign test.

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Scandal Modelling a Poisson Distribution Distribute newspapers around the room– all have same brand of paper (Daily Mail, Guardian etc.) Distribute newspapers around the room– all have same brand of paper (Daily Mail, Guardian etc.) Ask to record the number of scandals in the first five pages. Ask to record the number of scandals in the first five pages. Data is messy – what is a scandal? Data is messy – what is a scandal? Could also count number of pictures, number of ‘feelgood/happy’ stories. Could also count number of pictures, number of ‘feelgood/happy’ stories. Find mean and variance on calculators from class data – usually very close to each other. Fit a Poisson model – usually very good fit (don’t worry if it isn’t) Find mean and variance on calculators from class data – usually very close to each other. Fit a Poisson model – usually very good fit (don’t worry if it isn’t)

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What about the dry topics? Make them as practical as possible Make them as practical as possible Limit exposition at board Limit exposition at board Use e.g. card-matching activities to get students practicing without realising Use e.g. card-matching activities to get students practicing without realising Avoid endless repetition of textbook exercises – use selectively. Avoid endless repetition of textbook exercises – use selectively. Mini whiteboards useful in testing understanding. Mini whiteboards useful in testing understanding. Variety essential. Variety essential.

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Use of Technology Technology is a vital tool in doing statistics. It’s use in the classroom can be extremely helpful: Technology is a vital tool in doing statistics. It’s use in the classroom can be extremely helpful: AUTOGRAPH – useful for recording, graphing, analysing data collected. AUTOGRAPH – useful for recording, graphing, analysing data collected. MINITAB – a more powerful tool. Diagrams and analytical tools superior to AUTOGRAPH and EXCEL. MINITAB – a more powerful tool. Diagrams and analytical tools superior to AUTOGRAPH and EXCEL. EXCEL – spreadsheets can be useful in a variety of ways e.g. in simulations. EXCEL – spreadsheets can be useful in a variety of ways e.g. in simulations. Graphics calculators – many statistical functions and graphing tools. Something all can get their hands on. Graphics calculators – many statistical functions and graphing tools. Something all can get their hands on. http://www.mathsnet.net/ – an excellent free website (for A Level) that has hundreds of applets and examples on Statistics and Mathematics for insertion into lessons. http://www.mathsnet.net/ – an excellent free website (for A Level) that has hundreds of applets and examples on Statistics and Mathematics for insertion into lessons. Avoid over-use of technology. Try and use a concrete, practical example of something with students first. Try and use a concrete, practical example of something with students first.

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Monty Hall Demo on mathsnet.net

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Statistics in the Real World Try and plan at least one trip a year which illustrates work covered in classroom e.g. visit to Cadbury’s factory to see quality control in action. Try and plan at least one trip a year which illustrates work covered in classroom e.g. visit to Cadbury’s factory to see quality control in action. Get a statistician to come in and talk about their work. Get a statistician to come in and talk about their work. Even better, stage a Statistics Conference at school/college – one at Cadbury College for last two years (Mobile Phones, Heart Disease). Even better, stage a Statistics Conference at school/college – one at Cadbury College for last two years (Mobile Phones, Heart Disease). Make it cross – curricular (involve Biologists, social scientists) and work together with other institutions (other schools, the RSS, Plymouth University etc.) Make it cross – curricular (involve Biologists, social scientists) and work together with other institutions (other schools, the RSS, Plymouth University etc.)

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A Trip to the Grave! Find the nearest large church graveyard and divide the land into areas e.g. older graves, recent, child/family burials (if distinct areas in yard). Find the nearest large church graveyard and divide the land into areas e.g. older graves, recent, child/family burials (if distinct areas in yard). Assign groups of students to each area. Assign groups of students to each area. Students start at a random point and use dice and coins to decide movement (left, right, forward or back then number of paces) Students start at a random point and use dice and coins to decide movement (left, right, forward or back then number of paces) Record details on nearest grave – age at death, gender, occupation (if there), size of headstone, date buried etc. Record details on nearest grave – age at death, gender, occupation (if there), size of headstone, date buried etc. Pool data on return and use as a basis of class and project work. Pool data on return and use as a basis of class and project work.

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Dicey Statistics! Give each child a die and explain that you are interested in finding out the most likely number of throws up to and including the first six – ask them to guess this first. Give each child a die and explain that you are interested in finding out the most likely number of throws up to and including the first six – ask them to guess this first. Carry out the experiment once and tally results. Repeat for greater sample size. Carry out the experiment once and tally results. Repeat for greater sample size. Students are usually surprised by the result. Students are usually surprised by the result. Many people’s instincts about probability are based on false ideas. Many people’s instincts about probability are based on false ideas.

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Dice Experiment Theory A Geometric Distribution A Geometric Distribution Let X = the number of throws up to and including the first six Let X = the number of throws up to and including the first six X ~ G (1/6) P(X = 1) = 1/6 P(X = 2) = 1/6 * 5/6 P(X = 3) = 1/6 * (5/6)* (5/6) etc. Mean 1/(1/6) = 6 Mean 1/(1/6) = 6

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XProbabilityPredicted number in sample of 60 10.16666666710 20.1388888898 30.1157407417 40.0964506176 50.0803755145 60.0669795954 70.0558163293 80.0465136083 90.038761342 100.0323011172 110.0269175972 120.0224313311 130.0186927761 140.0155773131 150.0129810941 160.0108175791 170.0090146491 180.0075122070 X = 1 or 2 are the most likely scores

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A final practical example – works with year 7 upwards A final practical example – works with year 7 upwards Poverty Distribute squares of CDM bar around class – do it unevenly. Distribute squares of CDM bar around class – do it unevenly. Obtain class reaction – unfair? Obtain class reaction – unfair? Calculate mean – arrange it to be 2 if possible. Calculate mean – arrange it to be 2 if possible. Discussion on averages to ensue – which is best one. Discussion on averages to ensue – which is best one. Give say 6 squares of chocolate to the boy/girl with most – mean increases! Link to Aid to developing countries and question of corruption. Give say 6 squares of chocolate to the boy/girl with most – mean increases! Link to Aid to developing countries and question of corruption. Students decide how to redistribute squares so median is 2 and then mode is 2. Students decide how to redistribute squares so median is 2 and then mode is 2.

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Don’t be afraid to experiment! The course will be a success if you: Build in practical activities into work-schemes in EACH topic and encourage staff to try new ideas. Build in practical activities into work-schemes in EACH topic and encourage staff to try new ideas. Observe each other teaching – be supportive. Team-teaching can also help. Observe each other teaching – be supportive. Team-teaching can also help. Continue to produce a bank of low and hi-tech activities (card- matching, dominoes, data-sets on MINITAB and AUTOGRAPH etc.) and incorporate into your work-schemes. Continue to produce a bank of low and hi-tech activities (card- matching, dominoes, data-sets on MINITAB and AUTOGRAPH etc.) and incorporate into your work-schemes. This will take time (a two –year project) – involve the whole department, use Standards Unit materials and templates to help. This will take time (a two –year project) – involve the whole department, use Standards Unit materials and templates to help. Plan variety into individual lessons and the organisation of each term. Plan variety into individual lessons and the organisation of each term. Have at least one field trip/visit per year Have at least one field trip/visit per year Embed practical application/context into each topic – how is it used in the real world? Embed practical application/context into each topic – how is it used in the real world? Allow for the use of different student learning styles. Allow for the use of different student learning styles. Have regular reviews after each topic and build in at least 4 weeks of revision before each set of exams if possible. Have regular reviews after each topic and build in at least 4 weeks of revision before each set of exams if possible. Show enthusiasm for the subject! Show enthusiasm for the subject!

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In Closing ….. Statistics is a fascinating subject pre and post 16. Statistics is a fascinating subject pre and post 16. Students really enjoy using it to deal with real-world problems. Students really enjoy using it to deal with real-world problems. Intermediate C grade students up to A* students can do VERY well on an A Level Statistics course (AQA only truly applied course available, but MEI now has AS only course). Intermediate C grade students up to A* students can do VERY well on an A Level Statistics course (AQA only truly applied course available, but MEI now has AS only course). There is a desperate need world-wide for statisticians, and excellent career prospects for those who study it in higher education. There is a desperate need world-wide for statisticians, and excellent career prospects for those who study it in higher education. It is a really rewarding subject to teach. It is a really rewarding subject to teach.

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