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SACE Stage 1 Mathematics STATISTICS

Session 4

The syllabus, do and don’ts, projects and resources.

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The Syllabus

Group Exercise: Look closely at the syllabus and see what we have and have not covered in the workshops.

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Investigations for teaching and learning

A central theme of these workshops has been the use of investigations and activities as a vehicle for the teaching and learning of statistics.

This raises the question of how one obtains appropriate investigations and activities.

There are many resources available, especially on the internet, and a list some potentially useful sites will be provided

BUT…

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Investigations for teaching and learning

Suitable problems and data can be difficult to find.

You need to be clear at the outset about why you are trying to develop a particular investigation.

Some points to consider follow.

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What is the purpose?

A data investigation is useful only if it achieves a particular purpose.

For example, the road safety data serves as a vehicle to: Illustrate the use of histograms Illustrate the use of descriptive statistics Illustrate the cycle of investigation by

successive refinement of the problem being solved

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What is the purpose?

The Titanic investigation serves as a vehicle to: Illustrate the use of tabulation and graphing

for categorical data Illustrate the use of boxplots. Illustrate the cycle of investigation by

successive refinement of the problem. Together with the road safety data, provide

a contrast between the different variable types

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What is the purpose?

When developing an investigation for a teaching and learning role you should have in mind a specific area of the syllabus that is to be explored.

When you find a good problem/dataset, it is natural and appropriate to use it for whatever you can.

But it must serve a specific purpose in relation to the syllabus.

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Engaging the students

A data analytic investigation will work best when the students become actively involved in the process.

They must see that there is a real problem to solve and be enthusiastic about its solution

This means that the context must be understandable and of at least some interest

More importantly, the problem must be genuine.

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Engaging the students

Be wary of investigations where there is high enthusiasm for the subject in general or for some peripheral aspects, but the statistical value of the investigation is next to nil.

For example, the idea of an investigation of “some data from the AFL” may appeal to a large body of students BUT: There are almost no good problems that we

can collect data for and solve in this area.

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Engaging the students

A second example is a poorly designed “Estimate the proportion of red smarties” type of sampling activity.

It is clearly great fun! But it will be almost useless for teaching

sampling if: There is no worthwhile problem to solve. We can’t say whether sampling was random. We can’t compare estimates to the correct answer. The fun aspect is a distraction rather than

motivator.

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Role of statistics

The heart of a data analytic investigation must be a problem(s) that requires the use of certain statistical techniques.

Investigations where the statistics are an afterthought or peripheral are unlikely to be effective.

One must discourage students from emotional distractions and ensure they stick to the statistics.

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Correctness

The statistical methods must be used correctly.

In the road safety example we concentrated on a problem that in itself is not enough to answer the bigger question.

But it was a problem where the methods could be used convincingly and correctly.

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Correctness

Sometimes potentially interesting problems are unsuitable because the available methods cannot be applied correctly.

For example, environmental data are interesting and worthwhile but almost always require advanced methods.

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A Check List for Investigations

Is it real, engaging and worthwhile?Can I get real data associated with the problem?

Does it give access to the knowledge I want the students to learn in a manner that is statistically correct?

How do the Road Accident problem, Titanic problem and the Straw activity shape up?

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Don’t force the issue

Not everything is best developed through data analytic investigations.

For example, sampling is developed far more effectively through the straw activity rather than “solving a real-life problem”

If an investigation or activity looks too contrived, it is probably not worthwhile.

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Don’t force the issue

There is still the need for traditional delivery and practice of some of the detailed technical points

But these will be more meaningful and better motivated in the context of previous investigations and activities.

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Don’t force the issue

For example, in discussing the histogram we need to deal with technical issues relating to bin-width. That is: How should we choose the number of bins? Can the choice of bins influence conclusions?

An appropriate time to consider this is after an investigation in which the histogram played a key role in forming conclusions.

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Student projects

The guidelines for developing investigations for use in the classroom apply to student project work as well.

If possible, the student should choose their own problem and in the course of doing so ensure that …

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Student projects

It is a real, engaging and worthwhile problem.

They can get real data associated with the problem.

They can apply the knowledge they have learned to solve the problem in a correct manner.

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Real, engaging and worthwhile

Students should: Consider their interests. Consider their family/school network. Read widely. Consider ethical issues.

Students should not: Take on a huge task. (KISS principle) Acquire data and try to make a problem

out of it.

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Data collection

Often, secondary data is of little use. The statistical analysis has already been done.

When collecting their own data students should: Use random sampling, if a sampling is

involved. Use randomisation in any experiments. Collect sufficient data. Use family connections or friends as

appropriate.

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Data collection

When collecting their own data students should not: Badger people in public (in person,by

phone or letter). Collect petrol in glass bottles. Measure speeds of cars with radars.

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Applying knowledge

Students, with guidance from the teacher, should ensure they are using their knowledge correctly.

They should not “throw every possible thing they know at the problem”.

If they do they are probably doing something wrong.

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Applying knowledge

They should be made to feel confident that a simple and relatively short analysis is fine so long as it is correct and heads toward a solution to the problem.

They should be happy that establishing ‘no difference’ is as good as establishing a difference.

They should be aware that they are probably not going to prove any result at year 11 level, but will be able to form a conjecture.

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Some examples

Does gypsum or sand work better to improve drainage in soil?

Does castration or testicle insertion result in the heavier sheep at sale age?

Which form of advertising works best for Pizza Haven - flyers or magazine advertising?

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Some examples

Does Mozart’s music improve performance in cognitive tasks?

Are stock-broker chosen portfolios better than randomly chosen folios?

Do Duracell batteries last up to 10 times longer?

Are the reaction times of Year 2 students different from Year 8 students?

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Some examples

Are the TER scores of students who attended a school for R-12 different from those who entered at Year 8?

See past Quantitative Methods examination for some more ideas.

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Resources

There are MANY. But like all resources you may need to use a number to get what you want.

Some suggested resources are …

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Resources

Two on-line texts. The second site has particularly good applets:

http://www.anu.edu.au/nceph/surfstat/surfstat-home/surfstat.html

http://davidmlane.com/hyperstat/index.html

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Resources

A site which uses the problem solving approach to teach statistics. It has modules which can be downloaded, although not easily.

http://www.stats.gla.ac.uk/steps/glossary/index.html

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Resources

This site is a good interactive site for normal calculations:

http://psych.colorado.edu/~mcclella/java/zcalc.html

A text Moore, D.S.(1993) Introduction to the Practice of

Statistics, W.H.Freeman, San Francisco.

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Resources

The site of the Statistics Teacher Network Newsletter .

http://www.bio.ri.ccf.org/docs/ASA/stn.html

It is a joint production of NCTM and the American Statistical Association and contains articles on resources, class projects and graphing calculators. Numerous other journals are available.

See the syllabus for further references, surf the net and ask your colleagues.