Moving on with Moving on with StatisticsStatistics

How to engage students with the How to engage students with the subject post 16subject post 16

Mark Kent, Mark Kent, Head of Mathematics and Head of Mathematics and

Computing/ICT FacultyComputing/ICT FacultyThe Sixth Form College Solihull The Sixth Form College Solihull

sallysallymark@kent25.freeserve.csallysallymark@kent25.freeserve.co.uko.uk

My experience in Statistics My experience in Statistics EducationEducation• Worked in Sixth Form Colleges for nearly 20 years, teaching Worked in Sixth Form Colleges for nearly 20 years, teaching

A Level and International Baccalaureate Mathematics and A Level and International Baccalaureate Mathematics and Statistics.Statistics.

• MSc with Sheffield Hallam University in Applied Statistics MSc with Sheffield Hallam University in Applied Statistics with Stats Education in 1997with Stats Education in 1997

• Led the Maths Department at Cadbury College Birmingham Led the Maths Department at Cadbury College Birmingham for 7 years – developed A level Statistics course growth to for 7 years – developed A level Statistics course growth to around 150 studentsaround 150 students

• Set up and led the first Specialist Maths Sixth Form College Set up and led the first Specialist Maths Sixth Form College project – a Maths/Stats outreach to Birmingham secondary project – a Maths/Stats outreach to Birmingham secondary students.students.

• Joined The Sixth Form College Solihull in January 2006 as Joined The Sixth Form College Solihull in January 2006 as Head of the Maths and Computing/ICT FacultyHead of the Maths and Computing/ICT Faculty

Statistics a Popular Option Statistics a Popular Option Post 16Post 16• AQA Statistics A level a practical Statistics course AQA Statistics A level a practical Statistics course

for students with a minimum of a grade C at GCSE – for students with a minimum of a grade C at GCSE – suitable for Intermediates!!suitable for Intermediates!!

• Hundreds of students take the course at Cadbury and Hundreds of students take the course at Cadbury and Solihull – about 75% with Intermediate B and C Solihull – about 75% with Intermediate B and C grades at GCSE.grades at GCSE.

• Pass rates at A level close to 100% over a seven year Pass rates at A level close to 100% over a seven year period. AS pass rates around 80%.period. AS pass rates around 80%.

• Student questionnaire responses and AS to A2 Student questionnaire responses and AS to A2 progression indicate both enthusiasm for the subject progression indicate both enthusiasm for the subject and often pleasant surprise with grades obtained!and often pleasant surprise with grades obtained!

Statistics isn’t Statistics isn’t Mathematics!Mathematics!

• Students who often struggle with Pure Students who often struggle with Pure Mathematics can do very well at Applied Mathematics can do very well at Applied Statistics.Statistics.

• It requires a different approach/mindset.It requires a different approach/mindset.

Statistics is the missing Statistics is the missing link!link!

• A large number of other AS/A2 subjects use A large number of other AS/A2 subjects use Statistics all the time e.g. Psychology, Statistics all the time e.g. Psychology, Biology, Geography, Economics.Biology, Geography, Economics.

• Many careers involve use of statistical Many careers involve use of statistical techniques.techniques.

A philosophy of Statistics A philosophy of Statistics EducationEducation• It should be in context.It should be in context.• Its delivery should begin with practical activity, Its delivery should begin with practical activity,

adding a theoretical structure later, or adding a theoretical structure later, or simultaneously.simultaneously.

• Use technology, but start with something concrete Use technology, but start with something concrete – stats education research indicates this is the best – stats education research indicates this is the best approach.approach.

• Avoid using abstract, artificial data.Avoid using abstract, artificial data.• Fit your activities to the interests of the students – Fit your activities to the interests of the students –

post 16 this is food, money, mobile phones etc.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 Don’t worry about things going wrong – they often

do but the students don’t mind!!do but the students don’t mind!!

The philosophy in The philosophy in practicepractice• Plan to incorporate a practical Plan to incorporate a practical

example into every topic area taught example into every topic area taught on the work-scheme.on the work-scheme.

• Practicals need not take the whole Practicals need not take the whole lesson – 10 minutes is often enough.lesson – 10 minutes is often enough.

• Avoid the urge to rush the teaching Avoid the urge to rush the teaching of a topic without concrete examples of a topic without concrete examples – it’s a false economy.– it’s a false economy.

The importance of the first The importance of the first lessonlesson

• Do Do something something memorable memorable – e.g. a – e.g. a smokers’ smokers’ fitness fitness experiment:experiment:

Smoker

Incr

ea

se in

pu

lse

ra

te

YesNo

100

90

80

70

60

50

40

30

Boxplot of Increase in pulse rate with exercise vs Smoker

Mars Bars and Memory – a Mars Bars and Memory – a good first lesson at AS levelgood first lesson at AS level

• Record how many random digits out of 10 Record how many random digits out of 10 students can remember at lesson start.students can remember at lesson start.

• Students consume a mini mars bar and the Students consume a mini mars bar and the memory test is repeated every 5-10 minutes.memory test is repeated every 5-10 minutes.

• Results recorded and represented on box plots. Results recorded and represented on box plots. Memory peaks around 20 minutes after Memory peaks around 20 minutes after consumption.consumption.

Results of Mars Results of Mars ExperimentExperiment

Num

bre

of dig

its

rem

em

bere

d o

ut

of 10

2520151050

10

9

8

7

6

5

4

3

2

1

Boxplot of Memory Scores over time after mars bar consumption

Examples of Practicals in a Examples of Practicals in a teenage contextteenage contextMobile PhonesMobile 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 What does that mean and how can we test it – student

responses. responses. • Make use of simple equipment – could use rulers, calculators, Make use of simple equipment – could use rulers, calculators,

but best is a mobile phone – all students possess one!but best is a mobile phone – all students possess one!• Students time each other in pairs to text (not send) the word Students time each other in pairs to text (not send) the word

‘college’ with left and right hands separately.‘college’ with left and right hands separately.• Analysis depends on level – could use comparative box plots, Analysis depends on level – could use comparative box plots,

hypothesis test on differences (t test). Can also look at spread hypothesis test on differences (t test). Can also look at spread of data and breakdown into gender etc.of data and breakdown into gender etc.

ChocolateChocolate

Does Chocolate increase ability to concentrate?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.) Compare before and after (15 minutes approx.) consumption scores in remembering 10 consumption scores in remembering 10 random numbers.random numbers.

• Use AUTOGRAPH and Binomial Distribution Use AUTOGRAPH and Binomial Distribution – a sign test.– a sign test.

ScandalScandal

Modelling a Poisson DistributionModelling a Poisson Distribution• Distribute newspapers around the room– all have same brand Distribute newspapers around the room– all have same brand

of paper (Daily Mail, Guardian etc.) 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 Could also count number of pictures, number of

‘feelgood/happy’ stories.‘feelgood/happy’ stories.• Find mean and variance on calculators from class data – Find mean and variance on calculators from class data –

usually very close to each other. Fit a Poisson model – usually usually very close to each other. Fit a Poisson model – usually very good fit (don’t worry if it isn’t)very good fit (don’t worry if it isn’t)

What about the dry What about the dry topics?topics?• Make them as practical as possibleMake them as practical as possible

• Limit exposition at boardLimit exposition at board

• Use e.g. card-matching activities to get students Use e.g. card-matching activities to get students practicing without realisingpracticing without realising

• Avoid endless repetition of textbook exercises – use Avoid endless repetition of textbook exercises – use selectively.selectively.

• Mini whiteboards useful in testing understanding.Mini whiteboards useful in testing understanding.

• Variety essential.Variety essential.

Use of TechnologyUse of Technology Technology is a vital tool in doing statistics. It’s use in the classroom can Technology is a vital tool in doing statistics. It’s use in the classroom can

be extremely helpful: 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 MINITAB – a more powerful tool. Diagrams and analytical tools

superior to AUTOGRAPH and EXCEL.superior to AUTOGRAPH and EXCEL.• EXCEL – spreadsheets can be useful in a variety of ways e.g. in EXCEL – spreadsheets can be useful in a variety of ways e.g. in

simulations.simulations.• Graphics calculators – many statistical functions and graphing tools. Graphics calculators – many statistical functions and graphing tools.

Something all can get their hands on.Something all can get their hands on.• http://www.mathsnet.net/ – an excellent free website (for A Level) that http://www.mathsnet.net/ – an excellent free website (for A Level) that

has hundreds of applets and examples on Statistics and Mathematics for has hundreds of applets and examples on Statistics and Mathematics for insertion into lessons.insertion into lessons.

Avoid over-use of technology.Avoid over-use of technology. Try and use a concrete, practical example of something with students Try and use a concrete, practical example of something with students

first.first.

Monty Hall Demo on Monty Hall Demo on mathsnet.netmathsnet.net

Statistics in the Real Statistics in the Real WorldWorld• Try and plan at least one trip a year which illustrates Try and plan at least one trip a year which illustrates

work covered in classroom e.g. visit to Cadbury’s work covered in classroom e.g. visit to Cadbury’s factory to see quality control in action.factory to see quality control in action.

• Get a statistician to come in and talk about their Get a statistician to come in and talk about their work.work.

• Even better, stage a Statistics Conference at Even better, stage a Statistics Conference at school/college – one at Cadbury College for last two school/college – one at Cadbury College for last two years (Mobile Phones, Heart Disease). years (Mobile Phones, Heart Disease).

• Make it cross – curricular (involve Biologists, social Make it cross – curricular (involve Biologists, social scientists) and work together with other institutions scientists) and work together with other institutions (other schools, the RSS, Plymouth University etc.)(other schools, the RSS, Plymouth University etc.)

A Trip to the Grave!A Trip to the Grave!

• Find the nearest large church graveyard and divide the land Find the nearest large church graveyard and divide the land into areas e.g. older graves, recent, child/family burials (if into areas e.g. older graves, recent, child/family burials (if distinct areas in yard).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 Students start at a random point and use dice and coins to

decide movement (left, right, forward or back then number of decide movement (left, right, forward or back then number of paces)paces)

• Record details on nearest grave – age at death, gender, Record details on nearest grave – age at death, gender, occupation (if there), size of headstone, date buried etc.occupation (if there), size of headstone, date buried etc.

• Pool data on return and use as a basis of class and project Pool data on return and use as a basis of class and project work.work.

Dicey Statistics!Dicey Statistics!

• Give each child a die and explain that you are Give each child a die and explain that you are interested in finding out the most likely number of interested in finding out the most likely number of throws up to and including the first six – ask them to throws up to and including the first six – ask them to guess this first.guess this first.

• Carry out the experiment once and tally results. Carry out the experiment once and tally results. Repeat for greater sample size.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 Many people’s instincts about probability are based

on false ideas.on false ideas.

Dice Experiment TheoryDice Experiment Theory

• A Geometric DistributionA Geometric Distribution• Let X = the number of throws up to and including Let X = the number of throws up to and including

the first sixthe first six

X ~ G (1/6)X ~ G (1/6)P(X = 1) = 1/6P(X = 1) = 1/6P(X = 2) = 1/6 * 5/6 P(X = 2) = 1/6 * 5/6 P(X = 3) = 1/6 * (5/6)* (5/6) etc.P(X = 3) = 1/6 * (5/6)* (5/6) etc.

• Mean 1/(1/6) = 6Mean 1/(1/6) = 6

X Probability Predicted number in sample of 60

1 0.166666667 10

2 0.138888889 8

3 0.115740741 7

4 0.096450617 6

5 0.080375514 5

6 0.066979595 4

7 0.055816329 3

8 0.046513608 3

9 0.03876134 2

10 0.032301117 2

11 0.026917597 2

12 0.022431331 1

13 0.018692776 1

14 0.015577313 1

15 0.012981094 1

16 0.010817579 1

17 0.009014649 1

18 0.007512207 0

X = 1 or X = 1 or 2 are the 2 are the most most likely likely scoresscores

A final practical example – A final practical example – works with year 7 upwardsworks with year 7 upwardsPovertyPoverty• 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 – Give say 6 squares of chocolate to the boy/girl with most –

mean increases! Link to Aid to developing countries and mean increases! Link to Aid to developing countries and question of corruption.question of corruption.

• Students decide how to redistribute squares so median is 2 and Students decide how to redistribute squares so median is 2 and then mode is 2.then mode is 2.

Don’t be afraid to Don’t be afraid to experiment!experiment!The course will be a success if youThe course will be a success if you::• Build in practical activities into work-schemes in EACH topic and encourage Build in practical activities into work-schemes in EACH topic and encourage

staff to try new ideas.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, Continue to produce a bank of low and hi-tech activities (card-matching,

dominoes, data-sets on MINITAB and AUTOGRAPH etc.) and incorporate dominoes, data-sets on MINITAB and AUTOGRAPH etc.) and incorporate into your work-schemes.into your work-schemes.

This will take time (a two –year project) – involve the whole department, use This will take time (a two –year project) – involve the whole department, use Standards Unit materials and templates to help.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 yearHave at least one field trip/visit per year• Embed practical application/context into each topic – how is it used in the Embed practical application/context into each topic – how is it used in the

real world?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 Have regular reviews after each topic and build in at least 4 weeks of revision

before each set of exams if possible.before each set of exams if possible.• Show enthusiasm for the subject!Show enthusiasm for the subject!

In Closing …..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 Students really enjoy using it to deal with real-world

problems.problems.• Intermediate C grade students up to A* students can Intermediate C grade students up to A* students can

do VERY well on an A Level Statistics course (AQA do VERY well on an A Level Statistics course (AQA only truly applied course available, but MEI now has only truly applied course available, but MEI now has AS only course).AS only course).

• There is a desperate need world-wide for statisticians, There is a desperate need world-wide for statisticians, and excellent career prospects for those who study it and excellent career prospects for those who study it in higher education.in higher education.

• It is a really rewarding subject to teach.It is a really rewarding subject to teach.