# Maths is right or wrong – end of discussion  – or is it?

In 1984 I was a tutor in Operations Research to second year university students. My own experience of being in tutorials at University had been less than inspiring, with tutors who seemed reserved and keen to give us the answers without too much talking. I wanted to do a good job. My induction included a training session for teaching assistants from throughout the university. Margaret was a very experienced educational developer and was very keen for us to get the students discussing. I tried to explain to her that there really wasn’t a lot to discuss in my subject. You either knew how to solve a set of linear equations using Gauss-Jordan elimination or you didn’t. The answer was either correct or incorrect.

I suspect many people have this view of mathematics and its close relations, statistics and operations research.  Our classes have traditionally followed a set pattern. The teacher shows the class how to do something. The class copies down notes and some examples into their books, and then they individually work through exercises in the textbook – generally in silence. The teacher walks around the room and helps students as needed.

Prizes can help motivate students to give answers in unfamiliar settings

So when we talk about discussion in maths classes, this is not something that mathematics and statistics teachers are all familiar with. I recently gave a workshop for about 100 Scholarship students in Statistics in the Waikato. What a wonderful time we had together! The students were from all different schools and needed to be warmed up a little with prizes, but we had some good discussion in groups and as a whole. One of the teachers  commented later on the level of discussion in the session. Though she was an experienced maths teacher she found it difficult to lead discussion in the class. I am sure there are many like her.

## It is important to talk in maths and stats classes

It is difficult for many students to learn in solitary silence. As we talk about a topic we develop our understanding, practice the language of the discipline and experience what it means to be a mathematician or statistician. Explaining ideas to others helps us to make sense of them ourselves. As we listen to other people’s thinking we can see how it relates to what we think, and can clear up misconceptions. Some people just like to talk, (who me?) and find learning more fun in a cooperative or collaborative environment. This recognition of the need for language and interaction underpins the development of “rich tasks” that are being used in mathematics classrooms throughout the world.

I have previously stated that “Maths learning should be communal and loud and exciting, not solitary, quiet and routine.”

## Classroom atmosphere

One thing that was difficult at the Scholarship day was that the students did not know each other, and came from various schools. In a regular classroom the teacher has the opportunity of and responsibility for setting the tone of the class. Students need to feel safe. They need to feel that giving a wrong answer is not going to lead to ridicule. Several sessions at the start of the year may be needed to encourage discussion. Ideally this will become less necessary over time as students become used to interactive, inquiry-based learning in mathematics and statistics through their whole school careers.

Number talks” are a tool to help students improve their understanding of number, and recognise that there are many ways to see things. For example, the class might be shown a picture of dots and asked to explain how many dots they see, and how they worked it out. Several different ways of thinking will be discussed.

Children are encouraged to think up multiple ways of thinking about numbers  and to develop discussion by following prompts, sometimes called “talk moves”. Talk moves include revoicing, where the teacher restates what she thinks the student has said, asking students to restate another students reasoning, asking students to apply their own reasoning to someone else’s reasoning (Do you agree or disagree and why?), prompting for further participation (Would someone like to add on?), and using wait time (teachers should allow students to think for at least 10 seconds before calling on someone to answer. These are explained more fully in The Tools of Classroom talk.

Google Image is awash with classroom posters outlining “Talk moves”. I have been unable to trace back the source of the term or the list, and would be very pleased if someone can tell me the source,  to be able to attribute this structure.

# Good questions

The essence of good discussion is good questions. Question ping pong is not classroom discussion. We have all experienced a teacher working through examples on the board, while asking students the answers to numerical questions. This is a control technique for keeping students attentive, but it can fall to a small group of students who are quick to answer. I remember doing just this in my tutorial on solving matrices, when I didn’t know any better.

Teachers should avoid asking questions that they already know the answers to.

It is not a hard-and-fast rule, but definitely a thing to think about. I like to use True/False quizzes to help uncover misconceptions, and develop use of statistical language. I just about always know the answer to the question, but what I don’t know is how many students know the answer. So  I ask the question not to know the answer, but to know if the students do, and to provoke discussion. Perhaps a more interesting question would be, how many students do you think will say “True” to this statement. It would then be interesting to find out their reasoning, so long as it does not get personal!

## Multiple answers and open-ended questions

Where possible we need to ask questions that can have a number of acceptable answers. A discussion about what to do with outliers will seldom have a definitive answer, unless the answer is that it depends! Asking students to make a pictorial representation of an algebra problem can lead to interesting discussions.

The MathTwitterBlogosphere has many attractive ideas to use in teaching maths.

I rather like “Which one doesn’t belong”, which has echoes of “One of these things is not like the other, one of these things doesn’t belong, can you guess…” from Sesame Street. However, in Sesame Street the answer was usually unambiguous, whereas  with WODB there are lots of ways to have alternative answers. There is a website dedicated to sets of four objects, and the discussion is about which one does not belong. In each case all four can “not belong” for some reason, which I find a bit contrived, but it can lead to discussion about which is the strongest case of not belonging.

## Whole class and group discussion

Some discussions work well for a whole class, while others are better in small groups or pairs. Matching or ordering paper slips with expressions can lead to great discussion. For example we could have a set of graphs of the same data, and order them according to how effective they are at communicating the aspects of the data. Or there could be statements of possible events and students can place them in order of likelihood. The discussion involved in ordering them helps students to clarify the nature of probability. Desmos has a facility for teachers to set up card matching or grouping exercises, which reduces the work and waste of paper.

Our own Dragonistics data cards are great for discussion. Students can be given a number of dragons (more than two) and decide which one is the best, or which one doesn’t belong, or how to divide the dragons fairly into two or more groups.

It can seem to be wasting time to have discussion. However the evidence from research is that good discussion is an effective way for students to learn mathematics and statistics. I challenge all maths and stats teachers to increase and improve the discussion in their class.

# Proving causation

## Aeroplanes cause hot weather

In Christchurch we have a weather phenomenon known as the “Nor-wester”, which is a warm dry wind, preceding a cold southerly change. When the wind is from this direction, aeroplanes make their approach to the airport over the city. Our university is close to the airport in the direct flightpath, so we are very aware of the planes. A new colleague from South Africa drew the amusing conclusion that the unusual heat of the day was caused by all the planes flying overhead.

Statistics experts and educators spend a lot of time refuting claims of causation. “Correlation does not imply causation” has become a catch cry of people trying to avoid the common trap. This is a great advance in understanding that even journalists (notoriously math-phobic) seem to have caught onto. My own video on important statistical concepts ends with the causation issue. (You can jump to it at 3:51)

So we are aware that it is not easy to prove causation.

In order to prove causation we need a randomised experiment. We need to make random any possible factor that could be associated, and thus cause or contribute to the effect.

There is also the related problem of generalizability. If we do have a randomised experiment, we can prove causation. But unless the sample is also a random representative sample of the population in question, we cannot infer that the results will also transfer to the population in question. This is nicely illustrated in this matrix from The Statistical Sleuth by Fred L. Ramsey and Daniel W Schafer.

The relationship between the type of sample and study and the conclusions that may be drawn.

The top left-hand quadrant is the one in which we can draw causal inferences for the population.

## Causal claims from observational studies

A student posed this question:  Is it possible to prove a causal link based on an observational study alone?

It would be very useful if we could. It is not always possible to use a randomised trial, particularly when people are involved. Before we became more aware of human rights, experiments were performed on unsuspecting human lab rats. A classic example is the Vipeholm experiments where patients at a mental hospital were the unknowing subjects. They were given large quantities of sweets in order to determine whether sugar caused cavities in teeth. This happened into the early 1950s. These days it would not be acceptable to randomly assign people to groups who are made to smoke or drink alcohol or consume large quantities of fat-laden pastries. We have to let people make those lifestyle choices for themselves. And observe. Hence observational studies!

There is a call for “evidence-based practice” in education to follow the philosophy in medicine. But getting educational experiments through ethics committee approval is very challenging, and it is difficult to use rats or fruit-flies to impersonate the higher learning processes of humans. The changing landscape of the human environment makes it even more difficult to perform educational experiments.

To find out the criteria for justifying causal claims in an observational study I turned to one of my favourite statistics text-books, Chance Encounters by Wild and Seber  (page 27). They cite the Surgeon General of the United States. The criteria for the establishment of a cause and effect relationship in an epidemiological study are the following:

1. Strong relationship: For example illness is four times as likely among people exposed to a possible cause as it is for those who are not exposed.
2. Strong research design
3. Temporal relationship: The cause must precede the effect.
4. Dose-response relationship: Higher exposure leads to a higher proportion of people affected.
5. Reversible association: Removal of the cause reduces the incidence of the effect.
6. Consistency: Multiple studies in different locations producing similar effects
7. Biological plausibility: there is a supportable biological mechanism
8. Coherence with known facts.

## Teaching about causation

In high school, and entry-level statistics courses, the focus is often on statistical literacy. This concept of causation is pivotal to correct understanding of what statistics can and cannot claim. It is worth spending some time in the classroom discussing what would constitute reasonable proof and what would not. In particular it is worthwhile to come up with alternative explanations for common fallacies, or even truths in causation. Some examples for discussion might be drink-driving and accidents, smoking and cancer, gender and success in all number of areas, home game advantage in sport, the use of lucky charms, socks and undies. This also ties nicely with probability theory, helping to tie the year’s curriculum together.

# Teaching time series with limited computer access

## How do you teach statistics with limited access to computers?

Last century this wasn’t really an issue, at least not in high schools, as statistics has been a peripheral part of the mathematics curriculum and the mathematics of statistics has been taught as a subset of mathematics.

But this is changing, and it looks as if the change is starting in New Zealand. The NZ school curriculum has leapt ahead of the rest of the world. Statistics is taught at all levels and at the higher levels of high school, statistics is taught as it is actually done in practice – using computers. All analysis is done by a computer package, particularly using iNZight, a purpose-built, free package. The emphasis is on understanding, concepts and critical thinking, rather than the mechanical and slow application of formulas. The rigour has moved from the calculations to the meaning. It is SO exciting!

One big concern for many teachers is access to computers. In many schools there aren’t enough computer suites to schedule the students in for their statistics classes. So how do we deal with this?

It might seem that the computers are needed every day, but in fact they aren’t. And neither is it necessary to have one computer per student.

## Make them share

I’ve never had a problem when students have had to share computers. I find the people who do share a computer, learn better than those who are trying to work it out on their own. I actively encourage sharing computers in a lab.

I recently had the opportunity to be on the learning end, with computer instruction. The teacher was showing what to do at the front, and we in the class were echoing her steps on our computers. This is not ideal, as it requires everyone to be at the same pace, but as we were adults it was fine. I hadn’t brought my laptop, so I was sharing with another student. I’m pretty sure I learned more, as I got to follow what was happening on both computers, rather than trying to work it out and keep up. I was also able to help my partner, as she would lose track of what was happening when her computer wasn’t doing what it was meant to.

I have found this to be true at all levels, especially when learning a new package. Having two heads at the computer encourages discussion, which is an important element in learning. Students are also more likely to ask questions when they have already discussed a problem with another student. Pairing is so useful that some software companies get programmers to work in pairs, sharing a computer and work desk, because they have discovered that this has benefits.

## Think about what we are trying to teach

I am currently developing resources for a unit in time series analysis, based on the New Zealand curriculum, and using the free software, iNZight. At first glance, you might think that the entire unit would need to be taught in a computer lab. This is definitely not the case. And because of the layout of many computer labs, in fact you are better to stay out of them for most of the unit so that students can work in groups.

I find that it is worthwhile to think about the attitudes, skills and knowledge that we wish our pupils to develop in a unit – in that order of importance! These examples are illustrative rather than exhaustive.

Attitudes – By the end of the topic all students should feel that time series analysis is interesting and relevant (and maybe even fun!).

Time series analysis is pretty straightforward at the beginner level, but can be quite exciting. Once you know the basics, and with a convenient package to speed up the mechanics, you can do some interesting detective work. I would want the students to share some of this excitement, and start to explore on their own.

## Skills

Students should be able to:

• identify elements of a time series, relating them to the real life context.
• write a report on a time series analysis using correct terminology, clear enough for a non-expert reader to understand.
• use iNZight to analyse different time series.

## Knowledge

• Student should be able to explain and apply the following terms correctly: time series, trend, seasonality, stationary, noise, variation

And that is about it really!

So how do we do this, with or without full computer access.

Even with unlimited computer access I would get students to work in pairs for much of the time. I would start away from the computers. First display graphs of time series to the class and get them to write down sentences about them in their pairs. Then share with the class. We should get sentences like, “It mainly goes up, and then it goes down” and “there is a pattern that repeats”. From that the teacher can introduce the ideas of trend, seasonality and noise, modelling the correct use of specialist language.

Then I would talk about the context – or maybe the context should have come first… The time series chosen should be one with an easy to identify context, such as retail sales of recreational goods, or patterns of tourist arrivals. These series are available in New Zealand at Infoshare or in iNZight format via Statslc. Other countries will have similar series available. Again get the students to write down sentences, this time relating them to the context.

Homework could be to find a graph of a time series on-line or in a magazine. Or to make a list of things that might show seasonality.

Next I would get the students onto the computers in pairs. They should have a worksheet like the one here, so that they can work step-by-step through the package at their own pace in pairs. At some time during the class they could swap roles, if one has been instructing and the other operating.

The data set here RetailNZTS4 has four series in it, which show different behaviours. Students should see if they can get all the graphs they need for further analysis.

Four time series compared using iNZight software

The next class is away from the computers again. Here they are writing sentences about the graphs. They should do this alone, and in pairs, and compare in groups. It would be good to have a computer or two available for students to take turns to get any graphs they might find they need. When people are in front of a computer it tends to dominate their thinking and they can produce far too much output with very little thought. Moving away from the computer encourages a more reflective approach.

Then start on another data set. I would use the one about accommodation, AccRegNZTS13 comparing the seasonal patterns of occupancy in different regions of New Zealand. If there are enough computers, the students can spend one day creating the graphs and exploring, then the next day writing it up. Maybe different groups could take different regions, and find out why the pattern is the way it is for that region, then report back to the class.

Then the teacher may like to give some of the mathematical background to how a computer package would go about producing the output.RetailNZTS4

The learning is in the writing and the talking.

The point I’m trying to make is that you actually need to move away from the computers quite often. If you are REALLY stuck for computers you could even print off (and laminate?) the outputs from the different time series, so that the students can study and discuss them. Number or name them for easy reference, and have question sheets to go with them.The computer is only the tool, and with a bit of creativity, we can still teach the important attitudes, skills and knowledge with limited computer access.

I am aware as I am writing this that it is some time since I taught a class of high school students. I would be thrilled to hear comments from the “chalk-face” as to how realistic you think this is! And of course other suggestions will be welcome for teaching a computer-rich subject in a computer-poor environment.

Having said that, one of my experiences as a trainee teacher was having to teach my first lesson to a class at Rotorua Lakes High School during a powercut – which meant no computers and no OHP. We did desk-checking (how you can use pen and paper to look for bugs in code) and it went surprisingly well.