# What Mathematics teachers need to know about statistics

28 August 2012
###### Why resampling is better than hypothesis tests and confidence intervals
10 September 2012

My post suggesting that statistics is more vital for efficient citizens than algebra has led to some interesting discussions on Twitter and elsewhere. Currently I am beginning an exciting venture to provide support materials for teachers and students of statistics, starting with New Zealand. These two circumstances have led me to ponder about why maths teachers think that statistics is a subset of mathematics, and what knowledge and attitudes will help them make the transition to teaching statistics as a subject.
An earlier post called for mathematics to leave statistics alone. This post builds on that by providing some ways of thinking that might be helpful to mathematics teachers who have no choice but to teach statistics.

# Statistics is not a subset of mathematics

Let me quote a forum post from a teacher of mathematics in New Zealand:

• “It seems strange to me that Statistics is a small part of Mathematics (which also includes Trigonometry, Algebra, Geometry, Calculus … but for our year 13s and now our year 12s, it’s attained equal parity as one area with all the other branches of maths put together as another area.”

This is very helpful as it lets us see where the writer is coming from. To him, statistics is a subset of mathematics – a small part, and somehow it has managed to push its way to become on equal footing with “mathematics.”
I disagree.
I think we need to take a look at the role of compulsory schooling. It is popular among people who go to university, and even more so among those who never leave (having become academics themselves) to think that the main, if not only role of school is to prepare students for university. If the students somehow have not gained the skills and knowledge that the university lecturers believe are necessary, then the schools have failed – or worse still, the system has failed. Again I disagree.
The vision for the young people of New Zealand is stated in the official curriculum.
“Our vision is for young people:

• who will be creative, energetic, and enterprising
• who will seize the opportunities offered by new knowledge and technologies to secure a sustainable social, cultural, economic, and environmental future for our country
• who will work to create an Aotearoa New Zealand in which Māori and Pākehā recognise each other as full Treaty partners, and in which all cultures are valued for the contributions they bring
• who, in their school years, will continue to develop the values, knowledge, and competencies that will enable them to live full and satisfying lives
• who will be confident, connected, actively involved, and lifelong learners.”

It doesn’t actually mention preparing people for university.
My view is that school is about preparing young people for life, while helping them to enjoy the journey.

# What teachers need to know about statistics

Statistics for life can be summed up as C, D, E, standing for Chance, Data and Evidence.

## Chance

Students need to understand about the variability in their world. Probability is a mathematical way of modelling the inherent uncertainty around us. The mathematical part of probability includes combinatorics and the ability to manipulate tables. You can use Venn diagrams and trees if you like, and tables can be really useful too. The most difficult part for many students is converting the ideas into mathematical terminology and making sense of that.
Bear in mind that perceptions of chance have cultural implications. Some cultures play board games and other games of chance from a young age, and gain an inherent understanding of the nature of uncertainty as provided by dice. However there are other cultures for whom all things are decided by God, and nothing is by chance. There are many philosophical discussions which can be had regarding the nature of uncertainty and variability. The work of Tversky and Kahnemann and others have alerted us to the misconceptions we all have about chance.
An area where the understanding of probabilities and relative risk is vital is that of medical screening. Studies among medical practitioners have shown that many of them cannot correctly estimate the probability of a false positive, or the probability of a true positive, given that the result of a test is positive. This is easily conveyed through contingency tables, which are now part of the NZ curriculum.

## Data

When people talk about “statistics”, more often they are talking about data and information than the discipline of statistical analysis. Just about everyone is interested in some area of statistics. Note the obsession of the media for reporting the road toll and comparing with previous years or holiday periods. Sports statistics occupy many people’s thoughts, and can fill in the (long) gaps between the action in a cricket commentary. Weather statistics are vital to farmers, planners, environmentalists. Hospitals are now required to report various statistics. The web is full of statistics. It is difficult to think of an area of life which does not use statistics.The second thing we want to know about a new-born baby is its weight.
Just because data contains numbers does not make it mathematics. There are arithmetic skills, such as adding and dividing, which can be practised using data. But that’s about it when it comes to mathematics and data. These days we have computer packages which can calculate all sorts of summary values, and create graphs for better or worse, so the need for mathematical or numeracy skills is much diminished. What is needed is the ability to communicate ideas using numbers and diagrams; by communication I mean production and interpretation of reports and diagrams.
The area of data also includes the collection of data. This is taught at all levels of the NZ curriculum. Students are taught to think about measurement, both physical and through questionnaires. Eventually students learn to design experiments to explore new ideas. Some might see this as science or biology, social studies or psychology, technology and business. There are even applications in music where students explore people’s music preferences. Data occurs in all subjects, and really the skills of data analysis should be taught in context. But until the current generation of students become the teachers, we may need to rely on the teachers of statistics to provide support. There are wonderful opportunities for collaboration between disciplines, if our compartmentalised school system would allow them.

## Evidence

Much data is population data and conclusions can easily be drawn from it. However we also use samples to draw conclusions about populations. Inferential statistics has been developed using theoretical probability distributions to help us use samples to draw conclusions about populations. Unfortunately the most popular form of inference, hypothesis testing, is counter-intuitive at best. Many teachers do not truly understand the application of inferential statistics – and why should they – they may never have performed a real statistical analysis. It is only through repeated application of techniques to multiple contexts that most people can start to feel comfortable and get some understanding of what is happening. The beauty is that today the technology makes it possible for students to perform multiple analyses so that they can learn the specific from the general.
The New Zealand school system has taken the courageous* step to introduce the use of resampling, also known as bootstrapping or randomisation, for the generation of confidence intervals. This is contentious and is causing teachers concern. I will dedicate a whole post to the ideas of resampling and why they may be preferable to more traditional approaches. I empathise with the teachers who are feeling out of their depth, and hope that our materials, along with the excellent ones provided by “Census at School” can be of help.
I have no doubt that educators all over the world are watching to see how this goes before attempting similar moves in their own countries. Yet again New Zealand gets to lead the world. Watch this space!

### *In the popular British television show, “Yes Minister”, the public servant, Sir Humphrey, would use the term “courageous” to describe a proposal which was probably right, but also likely to lose votes.

##### Dr Nic

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2. Richard Catterall says:

While this is a very good summary of the interesting things to be studied in probability and statistics, nothing about it suggests that they are not branches of mathematics. A quick perusal of Ian Stewart’s excellent book: 17 Equations That Changed The World will show that there is a chapter on the equation for the so-called Normal Distribution curve and the areas of statistics and probability; but there is so much more to mathematics that affects our everyday lives, and ALL of it is important for “preparing young people for life, while helping them to enjoy the journey” as the NZ curriculum intends. The enthusiasm of a statistician for her branch of mathematics is laudable; but please do not let that enthusiasm spill over into a belief that that branch is any more important than all the other branches that we mathematics teachers know and love.

3. […] Pure mathematicians tend not to like statistics much as it isn’t exact. It’s a little bit sullied by its contact with the real world. However Operations Research goes a step further into the messy world of reality and evaluates the cost of each type of error. Decisions are often converted into dollar terms within decision analysis. Like it or not, the dollar is the usual measure of worth, even for a human life, though sometimes a measure called “utility” is employed. […]

4. Hi,
I understand both sides of the argument,an Richard is wrong. It actualy matters a lot that people can understand evidence, and engage in reasoning about data, because pretty much all citizens are required to do so. It doesn’t matter in anythng like the same way that they grasp calculus, or Euclidean geometry. People die regularly becuase members of my own profession don’t understand basic evidence. Basic arithemtic, fractions, pecentages and decmals come first, but for me, statistics is next in importance for modern lives.

• Dr Nic says:

Hi Anthony
I totally agree, but mathematicians just love the mathematics. I’m working on an app to help teach people about probability. Our stats one has been well received. My blog on screening touches on how medical people don’t understand probability.