Transcript Moving on with Statistics

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
[email protected]
My experience in Statistics
Education
• Worked in Sixth Form Colleges for nearly 20 years, teaching A
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Level and International Baccalaureate Mathematics and
Statistics.
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
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
Statistics a Popular Option
Post 16
• AQA Statistics A level a practical Statistics course for
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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.
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!
Statistics isn’t Mathematics!
• Students who often struggle with Pure
Mathematics can do very well at Applied
Statistics.
• It requires a different approach/mindset.
Statistics is the missing link!
• 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.
A philosophy of Statistics
Education
• It should be in context.
• Its delivery should begin with practical activity, adding a
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theoretical structure later, or simultaneously.
Use technology, but start with something concrete –
stats education research indicates this is the best
approach.
Avoid using abstract, artificial data.
Fit your activities to the interests of the students – post
16 this is food, money, mobile phones etc.
Make maximum use of statistical experiments.
Don’t worry about things going wrong – they often do
but the students don’t mind!!
The philosophy in practice
• Plan to incorporate a practical example
into every topic area taught on the workscheme.
• 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.
The importance of the first
lesson
Boxplot of Increase in pulse rate with exercise vs Smoker
• Do
90
Increase in pulse rate
something
memorable
– e.g. a
smokers’
fitness
experiment:
100
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No
Yes
Smoker
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.
• 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 of Mars Experiment
Boxplot of Memory Scores over time after mars bar consumption
Numbre of digits remembered out of 10
10
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Examples of Practicals in a
teenage context
Mobile Phones
• Start with a question – are students ambidextrous?
• 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!
• 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.
Chocolate
Does Chocolate increase ability to concentrate?
• A variation on the Mars Bars experiment.
• Compare before and after (15 minutes approx.)
consumption scores in remembering 10
random numbers.
• Use AUTOGRAPH and Binomial Distribution
– a sign test.
Scandal
Modelling a Poisson Distribution
• 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.
• Data is messy – what is a scandal?
• 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)
What about the dry topics?
• Make them as practical as possible
• Limit exposition at board
• Use e.g. card-matching activities to get students
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practicing without realising
Avoid endless repetition of textbook exercises – use
selectively.
Mini whiteboards useful in testing understanding.
Variety essential.
Use of Technology
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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.
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.
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.
Avoid over-use of technology.
Try and use a concrete, practical example of something with
students first.
Monty Hall Demo on
mathsnet.net
Statistics in the Real World
• Try and plan at least one trip a year which illustrates
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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.
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.)
A Trip to the Grave!
• Find the nearest large church graveyard and divide the land
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into areas e.g. older graves, recent, child/family burials (if
distinct areas in yard).
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)
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.
Dicey Statistics!
• Give each child a die and explain that you are
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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.
Students are usually surprised by the result.
Many people’s instincts about probability are based
on false ideas.
Dice Experiment Theory
• A Geometric Distribution
• 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
X
X = 1 or 2
are the
most likely
scores
Probability
Predicted number in
sample of 60
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0.166666667
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0.138888889
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0.115740741
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0.096450617
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0.080375514
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0.066979595
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0.055816329
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0.046513608
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0.03876134
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0.032301117
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0.026917597
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0.022431331
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0.018692776
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0.015577313
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0.012981094
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0.010817579
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0.009014649
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0.007512207
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A final practical example –
works with year 7 upwards
Poverty
• Distribute squares of CDM bar around class – do it unevenly.
• Obtain class reaction – unfair?
• Calculate mean – arrange it to be 2 if possible.
• 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.
• Students decide how to redistribute squares so median is 2 and
then mode is 2.
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.
• Observe each other teaching – be supportive. Team-teaching can also
help.
• Continue to produce a bank of low and hi-tech activities (cardmatching, 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.
• Plan variety into individual lessons and the organisation of each term.
• Have at least one field trip/visit per year
• Embed practical application/context into each topic – how is it used in
the real world?
• 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.
• Show enthusiasm for the subject!
In Closing …..
• Statistics is a fascinating subject pre and post 16.
• Students really enjoy using it to deal with real-world
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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).
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.