There are many good ways to teach mathematics

There are many good ways to teach mathematics and statistics

Hiding in the bookshelves in the University of Otago Library, I wept as I read the sentence, “There are many good ways to raise children.”  As a mother of a baby with severe disabilities the burden to get it right weighed down on me. This statement told me to put down the burden. I could do things differently from other mothers, and none of us needed to be wrong.

The same is true of teaching maths and stats – “There are many good ways to teach mathematics and statistics.” (Which is not to say that there are not also many bad ways to both parent and teach mathematics – but I like to be positive.)

My previous post about the messages about maths, sent by maths and stats videos, led to some interesting comments – thanks especially to Michael Pye who “couldn’t get the chart out of [his] head”. (Nothing warms a blogger’s heart more!). He was too generous to call my description of the “procedural approach” a “straw-person”, but might have some justification to do so.

His comments (you can see the originals here) have been incorporated in this table, with some of my own ideas. In some cases the “explicit active approach” is a mixture of the two extremes. The table was created to outline the message I felt the videos often give, and the message that is being encouraged in much of the maths education community. In this post we expand it to look at good ways to teach maths.

Procedural approach Explicit but active approach Social constructivist approach
Main ideas Maths is about choosing and using procedures correctly Maths is about understanding ideas and recognising patterns Maths is about exploring ideas and finding patterns
Strengths Orderly, structured, safe, cover the material, calm Orderly, structured, safe, cover the material, calm and satisfying Exciting, fun, annoying
Skills valued Computation, memorisation, speed, accuracy Computation, memorisation, (not speed), accuracy + the ability to evaluate and analyse Creativity, collaboration, communication, critical thinking
Teaching methods Demonstration, notes, practice Demonstration, notes, practice, guided discussion and exploration via modelling. Open-ended tasks, discussion, exploration
Grouping Students work alone or in ability grouping Students discuss as a whole class or in mixed-ability groups
Role of teacher Fount of wisdom, guide, enthusiast, coach. Fount of wisdom, guide, enthusiast, coach. Another learner, source of help, sometimes annoyingly oblique
Attitude to mistakes Mistakes are a sign of failure Mistakes happen when we learn. (high percentage of success) Mistakes happen when we learn.
Challenges Boredom, regimentation, may not develop resilience. Boredom, regimentation, could be taught purely to the test Can be difficult to tell if learning is taking place, difficult if the teacher is not confident
Who (of the learners) succeeds? People like our current maths teachers Not sure – hopefully everyone!
Use of worksheets and textbooks Important – guide the learning Develops mastery and provide assessment for learning. Limits gaps in understanding. Occasional use to supplement activities
Role of videos Can be central Reinforce ideas and provide support out of class. Support materials


We agree that speed is not important, so why are there still timed tests and “mad minutes” .

What is good mathematics teaching?

The previous post was about the messages sent by videos, and the table was used to fit the videos into a context. If we now examine the augmented table, we can address what we think good mathematics teaching looks like.


The biggest question when discussing what works in education is “for whom does it work?”  Just about any method of teaching will be successful for some people, depending on how you measure success. Teachers have the challenge of meeting the needs of around thirty students who are all individuals, with individual needs.


I have recently been considering the scale from introvert – those who draw energy from working alone, and extraversion – those who draw energy from other people. Contrary to our desire to make everything binary, current thinking suggests that there is a continuum from totally introverted to totally extraverted. I was greatly relieved to hear that, as I have never been able to find my place at either end. I am happy to present to people, and will “work a room” if need be, thus appearing extraverted, but need to recover afterwards with time alone – thus introverted. Apparently I can now think of myself as an ambivert.

The procedural approach to teaching and learning mathematics is probably more appealing to those more at the introverted end of the spectrum, who would rather have fingernails extracted than work in a group. (And I suspect this would include a majority of incumbent maths teachers, though I am not sure about primary teachers.) I suspect that children who are more extroverted will gain from group work and community. If we choose either one of these modes of teaching exclusively we are disadvantaging one or other group.

Different cultures

In New Zealand we are finding that children from cultures where a more social approach is used for learning do better when part of learning communities that value their cultural background and group endeavour. In Japan it is expected that all children will master the material, and children are not ability-grouped into lowered expectations. Dominant white western culture is more competitive. One way for schools to encourage large numbers of phone calls from unhappy white middle-class parents is to remove “streaming”, “setting”, or “ability grouping.”

Silence and noise

I recently took part in a Twitter discussion with maths educators, one of whom believed that most maths classes should be undertaken in silence. One of the justifications was that exams will be taken in silence and individually. This may have worked for him, but for some students the pressure not to say anything is stifling. It also removes a great source of learning, their peers. Students who are embarrassed to ask a teacher for help can often get help from others. In fact some teachers require students to ask others before approaching the teacher.


As is often the case, the answer lies in moderation and variety. I would not advocate destroying all worksheets and textbooks, nor mandate frequent silent individual work. Here are some of suggestions for effective teaching of mathematics.

Ideal maths teaching includes:

  • Having variety in your approaches, as well as security
  • Aiming for understanding and success
  • Trying new ideas and having fun
  • Embracing your own positive mathematical identity (and getting help if your mathematical identity is not positive)
  • Allowing children to work at different speeds without embarrassment
  • Having silence sometimes, and noise sometimes
  • Being competent or getting help – a good teaching method done poorly is not a good teaching method

Here are links to other posts related to this:
The Golden Rule doesn’t apply to teaching

Educating the heart with maths and statistics

The nature of mathematics and statistics and what it means to learn and teach them

And thank you again to those who took the time to comment on the previous post. I’m always interested in all viewpoints.


Teaching Confidence Intervals

If you want your students to understand just two things about confidence intervals, what would they be?

What and what order

When making up a teaching plan for anything it is important to think about whom you are teaching, what it is you want them to learn, and what order will best achieve the most important desired outcomes. In my previous life as a university professor I mostly taught confidence intervals to business students, including MBAs. Currently I produce materials to help teach high school students. When teaching business students, I was aware that many of them had poor mathematics skills, and I did not wish that to get in the way of their understanding. High School students may well be more at home with formulas and calculations, but their understanding of the outside world is limited. Consequently the approaches for these two different students may differ.

Begin with the end in mind

I use the “all of the people, some of the time” principle when deciding on the approach to use in teaching a topic. Some of the students will understand most of the material, but most of the students will only really understand some of the material, at least the first time around. Statistics takes several attempts before you approach fluency. Generally the material students learn will be the material they get taught first, before they start to get lost. Therefore it is good to start with the important material. I wrote a post about this, suggesting starting at the very beginning is not always the best way to go. This is counter-intuitive to mathematics teachers who are often very logical and wish to take the students through from the beginning to the end.

At the start I asked this question – if you want your students to understand just two things about confidence intervals, what would they be?

To me the most important things to learn about confidence intervals are what they are and why they are needed. Learning about the formula is a long way down the list, especially in these days of computers.

The traditional approach to teaching confidence intervals

A traditional approach to teaching confidence intervals is to start with the concept of a sampling distribution, followed by calculating the confidence interval of a mean using the Z distribution. Then the t distribution is introduced. Many of the questions involve calculation by formula. Very little time is spent on what a confidence interval is and why we need them. This is the order used in many textbooks. The Khan Academy video that I reviewed in a previous post does just this.

A different approach to teaching confidence intervals

My approach is as follows:
Start with the idea of a sample and a population, and that we are using a sample to try to find out an unknown value from the population. Show our video about understanding a confidence interval. One comment on this video decried the lack of formulas. I’m not sure what formulas would satisfy the viewer, but as I was explaining what a confidence interval is, not how to get it, I had decided that formulas would not help.

The new New Zealand school curriculum follows a process to get to the use of formal confidence intervals. Previously the assessment was such that a student could pass the confidence interval section by putting values into formulas in a calculator. In the new approach, early high school students are given real data to play with, and are encouraged to suggest conclusions they might be able to draw about the population, based on the sample. Then in Year 12 they start to draw informal confidence intervals, based on the sample.
Then in Year 13, we introduce bootstrapping as an intuitively appealing way to calculate confidence intervals. Students use existing data to draw a conclusion about two medians.
In a more traditional course, you could instead use the normal-based formula for the confidence interval of a mean. We now have a video for that as well.

You could then examine the idea of the sampling distribution and the central limit theorem.

The point is that you start with getting an idea of what a confidence interval is, and then you find out how to find one, and then you start to find out the theory underpinning it. You can think of it as successive refinement. Sometimes when we see photos downloading onto a device, they start off blurry, and then gradually become clearer as we gain more information. This is a way to learn a complex idea, such as confidence intervals. We start with the big picture, and not much detail, and then gradually fill out the details of the how and how come of the calculations.

When do we teach the formulas?

Some teachers believe that the students need to know the formulas in order to understand what is going on. This is probably true for some students, but not all. There are many kinds of understanding, and I prefer a conceptual and graphical approaches. If formulas are introduced at the end of the topic, then the students who like formulas are satisfied, and the others are not alienated. Sometimes it is best to leave the vegetables until last! (This is not a comment on the students!)

For more ideas about teaching confidence intervals see other posts:
Good, bad and wrong videos about confidence intervals
Confidence Intervals: informal, traditional, bootstrap
Why teach resampling

The silent dog – null results matter too!

Recently I was discussing the process we use in a statistical enquiry. The ideal is that we start with a problem and follow the statistical enquiry cycle through the steps Problem, Plan, Data collection, Analysis and Conclusion, which then may lead to other enquiries. 
I have previously written a post suggesting that the cyclical nature of the process was overstated.

The context of our discussion was a video I am working on, that acknowledges that often we start, not at the beginning, but in the middle, with a set of data. This may be because in an educational setting it is too expensive and time consuming to require students to collect their own data. Or it may be that as statistical consultants we are brought into an investigation once the data has been collected, and are needed to make some sense out of it. Whatever the reason, it is common to start with the data, and then loop backwards to the Problem and Plan phases, before performing the analysis and writing the conclusions.

Looking for relationships

We, a group of statistical educators, were suggesting what we would do with a data set, which included looking at the level of measurement, the origins of the data, and the possible intentions of the people who collected it. One teacher suggests to her students that they do exploratory scatter plots of all the possible pairings, as well as comparative dotplots and boxplots. The students can then choose a problem that is likely to show a relationship – because they have already seen that there is a relationship in the data.

I have a bit of a problem with this. It is fine to get an overview of the relationships in the data – that is one of the beauties of statistical packages. And I can see that for an assignment, it is more rewarding for students to have a result they can discuss. If they get a null result there is a tendency to think that they have failed. Yet the lack of evidence of a relationship may be more important than evidence of one. The problem is that we value positive results over null results. This is a known problem in academic journals, and many words have been written about the problems of over-occurrence of type 1 errors, or publication bias. Let me illustrate. A drug manufacturer hopes that drug X is effective in treating depression. In reality drug X is no more effective than a placebo. The manufacturer keeps funding different tests by different scientists. If all the experiments use a significance level of 0.05, then about 5% of the experiments will produce a type 1 error and say that there is an effect attributable to drug X. The (false) positive results are able to be published, because academic journals prefer positive results to null-results. Conversely the much larger number of researchers who correctly concluded that there is no relationship, do not get published and the abundance of evidence to the contrary is invisible. To be fair, it is hoped that these researchers will be able to refute the false positive paper.

Let them see null results

So where does this leave us as teachers of statistics? Awareness is a good start. We need to show null effects and why they are important. For every example we give that ends up rejecting the null hypothesis, we need to have an example that does not. Text books tend to over-include results that reject the null, so that when a student meets a non-significant result he or she is left wondering whether they have made a mistake. In my preparation of learning materials, I endeavour to keep a good spread of results – strongly positive, weakly positive, inconclusive, weakly negative and strongly negative.  This way students are accepting of a null result, and know what to say when they get one.

Another example is in the teaching of time series analysis. We love to show series with strong seasonality. It tells a story. (see my post about time series analysis as storytelling.) Retail sales nearly all peak in December, and various goods have other peaks. Jewellery retail sales in the US has small peaks in February and May, and it is fun working out why. Seasonal patterns seem like magic. However, we need also to allow students to analyse data that does not have a strong seasonal pattern, so that they can learn that they also exist!

My final research project before leaving the world of academia involved an experiment on the students in my class of over 200. It was difficult to get through the human ethics committee, but made it in the end. The students were divided into two groups, and half were followed up by tutors weekly if they were not keeping up with assignments and testing. The other half were left to their own devices, as had previously been the case. The interesting result was that it made no difference to the pass rate of the students. In fact the proportion of passes was almost identical. This was a null result. I had supposed that following up and helping students to keep up would increase their chances of passing the course. But they didn’t. This important result saved us money in terms of tutor input in following years. Though it felt good to be helping our students more, it didn’t actually help them pass, so was not justifiable in straitened financial times.

I wonder if it would have made it into a journal.

By the way, my reference to the silent dog in the title is to the famous Sherlock Holmes story, Silver Blaze, where the fact that the dog did not bark was important as it showed that the person was known to it.

On-line learning and teaching resources

Twenty-first century Junior Woodchuck Guidebook

I grew up reading Donald Duck comics. I love the Junior Woodchucks, and their Junior Woodchuck Guidebook. The Guidebook is a small paperback book, containing information on every conceivable subject, including geography, mythology, history, literature and the Rubaiyat of Omar Khayyam.  In our family, when we want to know something or check some piece of information, we talk about consulting the Junior Woodchuck Guidebook. (Imagine my joy when I discovered that a woodchuck is another name for a groundhog, the star of my favourite movie!) What we are referring to is the internet, the source of all possible information! Thanks to search engines, there is very little we cannot find out on the internet. And very big thanks to Wikipedia, to which I make an annual financial contribution, as should all who use it and can afford to.

You can learn just about anything on the internet. Problem is, how do you know what is good? And how do you help students find good stuff? And how do you use the internet wisely? And how can it help us as learners and teachers of statistics and operations research? These questions will take more than my usual 1000 words, so I will break it up a bit. This post is about the ways the internet can help in teaching and learning. In a later post I will talk about evaluating resources, and in particular multimedia resources.


Both the disciplines in which I am interested, statistics and operations research, apply mathematical and analytic methods to real-world problems. In statistics we are generally trying to find things out, and in operations research we are trying to make them better. Either way, the context is important. The internet enables students to find background knowledge regarding the context of the data or problem they are dealing with. It also enables instructors to write assessments and exercises that have a degree of veracity to them even if the actual raw data proves elusive. How I wish people would publish standard deviations as well as means when reporting results!


Which brings us to the second use for on-line resources. Real problems with real data are much more meaningful for students, and totally possible now that we don’t need to calculate anything by hand. Sadly, it is more difficult than first appears to find good quality raw data to analyse, but there is some available. You can see some sources in a previous post and the helpful comments.


If you are struggling to understand a concept, or to know how to teach or explain it, do a web search. I have found some great explanations, and diagrams especially, that have helped me. Or I have discovered a dearth of good diagrams, which has prompted me to make my own.


Videos can help with background knowledge, with explanations, and with inspiring students with the worth of the discipline. The problem with videos is that it takes a long time to find good ones and weed out the others. One suggestion is to enlist the help of your students. They can each watch two or three videos and decide which are the most helpful. The teacher then watches the most popular ones to check for pedagogical value. It is great when you find a site that you can trust, but even then you can’t guarantee the approach will be compatible with your own.

Social support

I particularly love Twitter, from which I get connection with other teachers and learners, and ideas and links to blogs. I belong to a Facebook group for teachers of statistics in New Zealand, and another Facebook group called “I love Operations Research”. These wax and wane in activity, and can be very helpful at times. Students and teachers can gain a lot from social networking.


There is good open-source software available, and 30-day trial versions for other software. Many schools in New Zealand use the R-based iNZight collection of programmes, which provide purpose-built means for timeseries analysis, bootstrapping and line fitting.

Answers to questions

The other day I lost the volume control off my toolbar. (Windows Vista, I’m embarrassed to admit). So I put in the search box “Lost my volume control” and was directed to a YouTube video that took me step-by-step through the convoluted process of reinstating my volume control! I was so grateful I made a donation. Just about any computer related question can be answered through a search.

Interactive demonstrations

I love these. There are two sites I have found great:

The National Library of Virtual Manipulatives, based in Utah.

NRich – It has some great ideas in the senior statistics area. From the UK.

A problem with some of these is the use of Flash, which does not play on all devices. And again – how do we decide if they are any good or not?

On-line textbooks

Why would you buy a textbook when you can get one on-line. I routinely directed my second-year statistical methods for business students to “Concepts and Applications of Inferential Statistics”. I’ve found it just the right level. Another source is Stattrek. I particularly like their short explanations of the different probability distributions.

Practice quizzes

There aren’t too many practice quizzes  around for free. Obviously, as a provider of statistical learning materials, I believe quizzes and exercises have merit for practice with immediate and focussed feedback. However, it can be very time-consuming to evaluate practice quizzes, and some just aren’t very good. On the other hand, some may argue that any time students spend learning is better than none.

Live help

There are some places that provide live, or slightly delayed help for students. I got hooked into a very fun site where you earned points by helping students. Sadly I can’t find it now, but as I was looking I found vast numbers of on-line help sites, often associated with public libraries. And there are commercial sites that provide some free help as an intro to their services. In New Zealand there is the StudyIt service, which helps students preparing for assessments in the senior high school years. At StatsLC we provide on-line help as part of our resources, and will be looking to develop this further. From time to time I get questions as a result of my YouTube videos, and enjoy answering them ,unless I am obviously doing someone’s homework! I also discovered “ShowMe” which looks like a great little iPad app, that I can use to help people more.

This has just been a quick guide to how useful the internet can be in teaching and learning. Next week I will address issues of quality and equity.

How to study statistics (Part 1)

To students of statistics

Most of my posts are directed at teachers and how to teach statistics. The blog this week and next is devoted to students. I present principles that will help you to learn statistics. I’m turning them into a poster, which I will make available for you to printing later. I’d love to hear from other teachers as I add to my list of principles.

1. Statistics is learned by doing

One of the best predictors of success in any subject is how much time you spent on it. If you want to learn statistics, you need to put in time. It is good to read the notes and the textbook, and to look up things on the internet and even to watch Youtube videos if they are good ones. But the most important way to learn statistics is by doing. You need to practise at the skills that are needed by a statistician, which include logical thinking, interpretation, judgment and writing. Your teacher should provide you with worthwhile practice activities, and helpful timely feedback. Good textbooks have good practice exercises. On-line materials have many practice exercises.

Given a choice, do the exercises that have answers available. It is very important that you check what you are doing, as it is detrimental to practise something in the wrong way. Or if you are using an on-line resource, make sure you check your answers as you go, so that you gain from the feedback and avoid developing bad habits.

So really the first principle should really be “statistics is learned by doing correctly.

2. Understanding comes with application, not before.

Do not wait until you understand what you are doing before you get started. The understanding comes as you do the work. When we learn to speak, we do not wait until we understand grammatical structure before saying anything. We use what we have to speak and to listen, and as we do so we gain an understanding of how language works.  I have found that students who spent a lot of time working through the process of calculating conditional probabilities for screening tests grew to understand the “why” as well as the “how” of the process. Repeated application of using Excel to fit a line to bivariate data and explaining what it meant, enabled students to understand and internalise what a line means. As I have taught statistics for two decades, my own understanding has continued to grow.

There is a proviso. You need to think about what you are doing, and you need to do worthwhile exercises. For example, mechanically calculating the standard deviation of a set of numbers devoid of context will not help you understand standard deviation. Looking at graphs and trying to guess what the standard deviation is, would be a better exercise. Then applying the value to the context is better still.

Applying statistical principles to a wide variety of contexts helps us to discern what is specific to a problem and what is general for all problems. This brings us to the next principle.

3. Spend time exploring the context.

In a statistical analysis, context is vital, and often very interesting. You need to understand the problem that gave rise to the investigation and collection of the data. The context is what makes each statistical investigation different. Statisticians often work alongside other researchers in areas such as medicine, psychology, biology and geology, who provide the contextual background to the problem. This provides a wonderful opportunity for the statistician to learn about a whole range of different subjects. The interplay between the data and context mean that every investigation is different.

In a classroom setting you will not have the subject expert available, but you do need to understand the story behind the data. These days, finding out is possible with a click of a Google or Wikipedia button. Knowing the background to the data helps you to make more sensible judgments – and it makes it more interesting.

4. Statistics is different from mathematics

In mathematics, particularly pure mathematics, context is stripped away in order to reveal the inner pure truth of numbers and logic.  There are applied areas involving mathematics, which are more like statistics, such as operations research and engineering. At school level, one of the things that characterises the study of maths is right and wrong answers, with a minimum of ambiguity. That is what I loved about mathematics – being able to apply an algorithm and get a correct answer. In statistics, however, things are seldom black-and-white.  In statistics you will need to interpret data from the perspective of the real world, and often the answer is not clear. Some people find the lack of certainty in statistics disturbing. There is considerable room for discussion in statistics. Some aspects of statistics are fuzzy, such as what to do with messy data, or which is the “best” model to fit a time series. There is a greater need for the ability to write in statistics, which makes if more challenging for students for whom English is not their native language.

5. Technology is essential

With computers and calculators, all sorts of activities are available to help learn statistics. Graphs and graphics enable exploration that was not possible when graphs had to be drawn by hand. You can have a multivariate data set and explore all the possible relationships with a few clicks. You should always look at the data in a graphical form before setting out to analyse.

Sometimes I would set optional exercises for students to explore the relationship between data, graphs and summary measures. Very few students did so, but when I led them through the same examples one at a time I could see the lights go on. When you are given opportunities to use computing power to explore and learn – do it!

But wait…there’s more

Here we have the first five principles for students learning statistics. Watch this space next week for some more. And do add some in the comments and I will try to incorporate your ideas as well.

Statistics or Calculus? Do both!

This post is prompted by two 17 year old boys, Cam and Thomas, who are about to enter year 13, the final year of High school in New Zealand. They are both academically capable, with highly educated parents. And both boys are struggling with a dilemma – should they  take Calculus or Statistics at school this year. I suspect their maths teachers are pushing for calculus, whereas their parents appreciate the value of statistics.

Let’s take a look at the alternatives and see if we can help. (This makes no pretense of being a balanced view – that’s what comments are for!) Note that this is based on the New Zealand curriculum, which has a recently introduced strong emphasis on statistics. The assessment structure for this includes a full statistics subject in the final year for the first time in 2013. New Zealand is in the somewhat lonely position of leading the world in this area; statistical societies in other countries are watching. (And for you in the Northern Hemisphere who may be feeling confused, it is currently our summer holidays, and school starts back in early February.)

Take Calculus

Calculus is “proper” mathematics. It is elegant, and neat, and you get right answers. You don’t have to write sentences. Ever! Most of the problems are nice and theoretical, so you don’t have to deal with “word problems”. The teachers like Calculus, and fight over who gets to teach it. They feel confident in what they are doing. They have taught it for years and don’t need to do anything new. There are oceans of on-line videos, games and resources to help students. Khan academy videos are useful. But you don’t need to have access to the computer room to do calculus. Parents are more likely to know calculus (though well forgotten) than statistics. Calculus is needed for important subjects such as engineering, physics and… Hmm can’t think what else! Oh yes – more calculus. It is a good mental discipline that helps with problem-solving skills. It can be pretty fun if well taught. Besides people tell me that statistics is the easy option for people who can’t do calculus.

Take Statistics

Statistics relates to life. It is messy and often the answers aren’t clear, so interpretation and thinking are important. You will need to write reports and express yourself on paper. This will help you develop your critical thinking skills and communication skills. You have to understand contextual material such as biology, economics or sport.  Innovative teachers are excited about the changes in the curriculum, and are embracing the new material as an opportunity to learn and develop themselves as well as you.  As New Zealand is leading the world by introducing resampling, randomisation, bootstrapping and time series analysis at high school level, the on-line resources are few, but those extant (and in our pipeline) are focussed for your use.  Parents are not familiar with statistics, but will find what you are doing interesting.  You get to do most of your calculations on the computer, just as real statisticians do.  You will never find yourself asking “Why do we need to learn this?” because it is obvious how it is a part of your life. You will be better able to discern truth from lies on the internet. You will find yourself looking at the world differently.

Statistics is needed for many subjects: psychology, biology, engineering, management, marketing, medicine, sociology, education, geography, geology, law and journalism. It also widens the possibilities in the study of arts subjects such as History and English.

So which should Cam and Thomas take?

Here is our advice – all students who possibly can, should take statistics. Those who are planning to be engineers, physicists, maths teachers or statisticians (yes!) should take calculus as well. Simple really!

What about my own sons – the jazz pianist and the movie maker – what would I have advised them at this point? Statistics all the way. Neither one had use for calculus, nor the aptitude, but both would have benefited from statistics.

I’ll let you know what Cam and Thomas decided.

Organising the toolbox in statistics and operations research

Don’t bury students in tools     

In our statistics courses and textbooks there is a tendency to hand our students tool after tool, wanting to teach them all they need to know. However students can feel buried under these tools and unable to decide which to use for which task. This is also true in beginning Operations Research or Management Science courses. To the instructors, it is obvious whether to use the test for paired or independent samples or whether to use multicriteria decision making or a decision tree.  But it is just another source of confusion for the student, who wants to be told what to do.

Tools for statistics and operations research

A common approach to teaching hypothesis testing in business statistics courses, if textbooks are anything to go by, is to teach several different forms of hypothesis testing, starting with the test for a mean, and test for a proportion then difference of two means, independent and paired, then difference of two proportions. Then we have tests for regression and correlation, and chi-squared test for independence. These are the seven basic statistical tests that people are likely to use or see. I would probably add ANOVA, if there is enough time. Even listed, this seems a bit confusing.

An introductory operations research course might include any number of topics including linear programming, integer programming, inventory control, queueing, simulation, decision analysis, critical path, the assignment problem, dynamic programming, systems analysis, financial modelling, inventory control…And I would hope some overall teaching about models and the OR process.

Issues with the pile of tools

Of course we need to teach the essential tools of our discipline, but there are two issues arising from this approach.

The obvious one is that students are left bewildered as to which test they should use when. Because of the way textbooks and courses are organised, students don’t usually have to decide which tool to use in a given situation. If the preceding chapter is about linear programming, then the questions will be about linear programming.

The second issue is that unless students are helped, they fail to see the connections between the techniques and are left with a fragmented view of the discipline. It is not just a question of which tool to use for which task, it is about seeing the linkages and the similarities. We want to help them have integrated knowledge.

Providing activities to help with organisation

In both my introductory courses I attempted to address this, with varying degrees of success.

In our management science course we end the year with a case of a situation with multiple needs, and the students were to identify which technique would be useful in each instance. Then the final exam has a similar question, with specific questions about over-arching concepts such as deterministic and stochastic inputs, and the purpose of the model – to optimise or inform. This is also an opportunity to address issues of ethics and worldview.

In the final section of the business statistics course we have a large bank of questions for students to work through, to give them practice in deciding which test to use. I was careful to make sure that there was more than one question related to each scenario, so that students would not learn unhelpful shortcuts, such as, if the question is about weight loss, the answer must be paired difference of two means. I also analysed the mistakes given in multichoice answers, to see where confusion was arising, sometimes due to poor wording. From this I refined the questions.

Examples of the questions for test choice in hypothesis testing

Management thinks there is a difference in productivity between the two days of the week in a certain work area. The production output of a random sample of 15 factory workers is recorded on both a Tuesday and a Friday of the same week. For each worker, the number of completed garments is counted on both days.

A restaurant manager is thinking of doing a special “girls’ night out” promotion. She suspects that groups of women together are more likely to stay for dessert than mixed adult or family groups. For the next two weeks she gets the staff to write down for each table whether they stay for dessert, and what type of group they are. She asks you to see if her suspicion is correct.

A human resources department has data on 200 past employees, including how long, in months, they stayed at the company, and the mark out of 100 they got in their recruitment assessment. They ask you to work out whether you can predict how long a person will stay, based on their test mark.

A researcher wanted to investigate whether a new diet was effective in helping with weight loss. She got 40 volunteers and got 20 to use the diet and the other 20 to eat normally. After 6 weeks the weights (in kg) before and after were recorded for each volunteer, and the difference calculated. She then looked at how the weight losses differed between the two groups.

Comment on the questions

You might notice that all the examples are in a business context. This is because this is a course for business students, and they need to know that what they are learning is relevant to their future. Questions about dolphins and pine trees are not suitable for these students. (Unless we are making money out of them!)

The master diagram

The students to work through these multiple choice questions on-line, and we offered help and coached them through questions with which they had difficulty. By taking my turn with the teaching assistants in the computer labs, I was able to understand better how the students perceived the tests, and ways to help them with this. The result is a diagram, or set of diagrams which shows the relationships between the tests, and a procedure to help them make the decision. I am a great believer in diagrams, but they need to be well thought out. Many textbooks have branching diagrams, showing a decision process for which test to use. I felt there was a more holistic way to approach it, and thought long and hard, and tried out my diagrams on students before I came up with our different approach. You can see the diagrams here by clicking on the link to the pdf which you can download: Choosing the test diagrams

The three questions which help the students to identify the most appropriate test are:

  1. What level of measurement is the data – Nominal or interval/ratio?
  2. How many samples do we have?
  3. What is the purpose of our analysis?

I made an on-line lesson which takes the students through the steps over and over, and created the diagrams to help them. Time and again the students said how much it helped them to fit it all together. Eventually I made the following video, which is on YouTube. I suspect it must be coming up to summary time in courses in the US, as this video has recently attracted a lot of views, and positive comments.

The video is also part of our app, AtMyPace: Statistics along with two sets of questions to help students to learn about the different types of tests and how to tell them apart. You can access the same resources on-line through AtMyPace:Statistics

It is important to see the subject as a whole, and not a jumbled mass of techniques and ideas, and this has really helped my students and many others through the video and app.

Best wishes for the holiday season

It is Christmas time and here in Christchurch the sun is shining and barbecues and beaches are calling. I am taking a break from the blog for the great New Zealand shut-down and will be back in the New Year.

Thank you for all the followers and especially your comments, Likes and ReTweets.

The Sound of Music meets Linear Programming

“Let’s start at the very beginning – a very good place to start. When you read you begin with A, B,C!” When you do statistics you begin with…probability? the mean? graphs?

Begin at the end

But really, is the beginning a very good place to start? Sometimes, we need to begin at the end. And sometimes we need to go back before the beginning. Always we need to think about where to begin, because it is seldom obvious, and copying what other teachers and textbooks have done is often a bad idea.

Linear programming

Take Linear Programming, the flagship technique of Operations Research. Most text books start with a simple two variable example, one that can be drawn on a Cartesian plane. They begin by defining the decision variables and the objective function. Next they formulate the constraints and explain the non-negativity conditions. Then finally they get around to solving the problem – often through a graphical approach, and applying it to the trivial real-life imaginary example they started with.

Here is a better approach, with Linear programming as the example:

First ensure all the class members have the prerequisite mathematical skills for what you propose to teach. If they are not good at drawing equations on a plane, you will need to teach them again, or use a different approach such as using Excel Solver. If students are not sure which way around > and < signs go, you will need to go over it. If English is their second language you will need to make sure you explain words like constraint, objective and optimum. This won’t hurt the native English speakers either.

Second think about your destination. When children learn to read, they generally know what the outcome is going to be. They will be able to look at words on a page and make sense of them. When you learn to drive, you know the outcome – you will be able to get safely from one place to another behind the wheel of a car. When we learn to bake cakes, we like to have pictures of the finished product so that we can see where we are headed. Yet somehow we try to teach as if it is a voyage of discovery with no vision of the end. Now discovery is good, if it pertains to how we get to or understand a process, but students need to know what they are learning. It also helps to have a purpose. Reading, driving and baking are all purposeful, with a clear outcome. The same should be true of linear programming (or confidence intervals or decision trees or fitted lines or just about anything else we are learning.)

You give the students an illustration of the completed LP model of the problem, preferably complex enough to be realistic. You show them how it can be useful, and give them a chance to explore the model. This is SO much easier now that we have Excel and Solver to look after the solving. Let students find out all about one model and then another and another, before you begin to show how to formulate. When people know what they are trying to produce, the reasoning behind the steps is more obvious.

Linear Regression

The same approach can be applied to teaching Linear Regression analysis. First we need to make sure that students understand what a fitted line on a graph is. Get them to interpret several fitted graphs, and use them to make predictions and write statements about the nature of the relationships modelled. Then show how to make the fitted graphs once they know why they need to.

In last week’s post I talked about histograms. Students should learn to interpret histograms and other graphs before they are required to make their own. Having to read off pie charts should help immunise them against their use.

I was in a computer lab with some students from another first year statistics course, and noticed that the first thing they were taught was how to calculate the mean and standard deviation, including the finite population correction. Was this really the most interesting way to get them introduced to the joys of data analysis and interpretation? Why start with the mean, one of the most difficult concepts in statistics?

Work backwards from the end

There is an interesting technique used for teaching skills to children with special needs. When you teach a blind child to tie shoelaces, you start at the end. You do all but the last part, and let them finish it off. This gives a sense of success and purpose. Then gradually you add the steps backwards, so that they start earlier on in the process. This also means that the part of the skill that is getting the most repetition is the new part, not the part already mastered. The same is true of memorisation. Memorise the last line first, then the last two lines etc. I suspect the same approach may well apply to more abstract skills. Maybe we should teach how to read and critique a statistical analysis, then how to write one, then finally how to do the analysis.

The spiral approach is popular, in which topics are revisited each year and built on.  I would like to incorporate principles of mastery learning along with that. Mastery learning is based on the premise that you must master a skill before moving on to the next one. This is difficult to implement in a classroom, with mixed level of ability, but is more easily enacted with the help of a Learning Management System.

New math had odd beginnings

I was born in the early 1960s and was in the first cohort of children to learn “new math(s)”, devised in the US as a reaction to the humiliation of seeing the Russians put Sputnik into space before them. Even in New Zealand we were not immune to the influence of the Cold War on education!  I loved our bright new textbooks,  which started with Set Theory – even at age 6. Every year the first page of the text book had diagrams of herds of sheep, prides of lions and other sundry collections.  I loved the Venn diagrams and the intersections – even cardinal numbers, but to this day I’m not sure how that connected with mathematics, and learning to add and subtract. And to this day I ask, “What were they thinking?” It appears that set theory is the foundation of all mathematics, and thus these mathematicians decided to start there, baffling teachers and parents alike, who were alienated by these words and symbols.

I have no doubt that the intention was to improve learning, but it seems ill-advised now. I wonder how our attempts will be viewed with the benefits of 40 years of hindsight. These days constructivism is a popular, though not universal, theory and approach to learning. The idea is that we create knowledge through adding new ideas and experiences onto our current knowledge. Sometimes that involves undoing erroneous or primitive knowledge.

Sometimes a good approach is historical – to imitate in the learner (in an accelerated form) the learning process through which mankind has progressed, preferably missing out the stupid bits. (Roman numerals are fun for some children, but pretty pointless once you realise the power of zero). It is certainly worth contemplating as an alternative approach.

This post has touched on ideas regarding the sequencing of a learning/teaching approach. There are many considerations and serious thought needs to go into where we start. Sometimes we need to start at the end.

Question questions

Ooooh – new data!

There is nothing like a new set of data, just sitting there in the computer, all ready for me to clean and graph and analyse and extract its secrets. I know I should be methodical in my approach, but sometimes I feel like a kid at Christmas, metaphorically ripping open the presents as I jump from graph to procedure, and back to graph again. I then have to go back and do it properly, documenting my approach and recording results, but that’s okay too. That can reveal a second lot of wonders as I sift and ponder.

This is what we should be enabling our students to do. Students need to catch the excitement of making a REAL graph of REAL data and finding out what it REALLY tells them. I have already blogged about the importance of real data in teaching, so those of you who have recently started following you might like to take a look. I also gave some suggestions on how to get real data.

I once dabbled in qualitative research. My PhD thesis used mixed methodology, which entailed recording interviews, transcribing and coding. It seemed like a fun idea at the  time of my research proposal. Sifting through the interviews for gems of insight, getting to figure out common themes and finding linkages and generalities, seems appealing. And it was effective – I came up with a new idea for measuring educational effectiveness through opportunity to learn. But given the choice I won’t be doing it again. I truly admire qualitative researchers, as it takes so much more work than good old quantitative research. Much of it is just slog, reading and coding the interviews. It is really important and totally valid as far as I can see. It’s just that it’s a little – dare I say it – boring.

But I digress. This post is meant to be about questions. The questions you ask in class, the questions in the textbooks, the questions in on-line exercises and the questions in the tests and exams at the end of the unit of work.

In another previous post I lamented how “Statistics Textbooks suck out all the fun.” I cited the work by George Cobb, reviewing textbooks in 1987.

“Judge a book by its exercises and you cannot go far wrong,”  said George Cobb.

It’s still true. The questions are what matter.

I have developed a course for learners who lack confidence in mathematics. There are on-line lecture videos and notes with audio, there are links to other materials, but where the real learning takes place is in the questions. Statistics is not a spectator sport – you have to get in and do it. Things can look easy when you see someone else do them like Olympic diving and producing Pivot-Charts in Excel, playing the piano and developing a linear programming model. But these skills require practice to become proficient. However there is no point in practising the wrong thing, or practising doing the right thing wrongly. Both these can happen when questions and feedback are not well designed.

Recently I have been immersed in questions. I am developing on-line materials for a textbook, and my own on-line materials for supporting high school students and teachers who are struggling with New Zealand’s innovative and world-leading statistics curriculum. From the textbook I have had to select problems to work through in demonstrations. For my own course, I am devising my own questions. As I do this I have become intimately involved with the NCEA questions, (National Certificate of Educational Achievement) as this is how the students will ultimately be tested. This combination has caused me to think a lot about how questions can help or hinder learning.

Students want to pass

Students want to pass

Like it or not, the main motivating force for most students is to pass the test. For some students passing means getting an A or close to 100%, while for others a scrape through is a cause for celebration. But students want to pass. And generally they want to do this with the minimum amount of work. The student attitude is at odds with teachers, who are wanting to increase the amount of work students do, so that they really understand, and catch the excitement, rather than do the minimum to get by. Questions are the way to get them. The questions the students work on should lead them to both goals simultaneously. The students need to feel that their time and effort is moving them toward their goal of passing. The teachers need to engage this effort and seduce the students into seeing how worthwhile and useful, interesting and exciting the subject is.

Good questions

Good questions will do this. A good question will have context, real data and meaning. Statisticians don’t care about x. Mathematicians do. Asking students to interpret the Excel output of a regression of Y on X is a mathematical question and has no place in a statistics textbook or course. Asking how sales are affected by temperature, or grades are affected by time spent doing homework – these are meaningful examples.

A good question needs to test the thing you are trying to test. If you want to know if the student understands the implications of variability, getting them to calculate the standard deviation by hand is not going to do it. If you want a student to know how to use their calculator to find binomial probabilities, then that is what you should ask. But if you want them to be able to identify times when the binomial distribution is a good model of reality, then the question needs to be relevant to that.

There need to be enough questions. By working through multiple examples students come to understand what is specific to each context, and what is general to all examples. This sounds like “drill”, and I am a firm believer in consistent effort on worthwhile questions.

There has to be good feedback. Students need to be able to find out if they are correct, or “on the right track” as so many of my students ask me. Problem is, if you give them the answers, sometimes they just read them and we are back to the “statistics or operations research as a spectator sport” effect. And sometimes students don’t realise the nuances in what they have written, thinking it looks like the model answer, when really they have missed something vital. Often the teacher has to look after this, which requires a lot of time, though we are exploring ways of using on-line quizzes and exercises to enable more targetted feedback, more promptly.

Whatever the approach, we need to make sure that the questions we ask students to work on are leading them to discover the joy of statistics and operations research as well as passing the course.

Embrace Change

I love graduations. At the University of Canterbury the academic staff act as marshals,

Dr Nic in PhD regalia - I love to dress up!

helping the graduands to be in the right place at the right time in the right order wearing the right clothes and doing the right things. I have acted as a marshal for some years and love helping people to have a good experience. I love graduations because of the accomplishment they represent, and the efforts the student, the parents and the staff have made for these young people to complete their qualifications. This graduation was pretty special, as it was the class that had to cope with repeated earthquakes, snowfalls and other disruptions in the last half of their degrees. They are the students who had to adapt to being taught in tents or on-line, and who, at the beginning of each exam, were warned what to do in case of an aftershock and told to keep their wallet, phone and keys with them at all times. They are the students who rallied together to support each other and the community and shovel silt.

I also love the ceremony at graduations and dressing up in fancy clothes. I love the music and singing Gaudeamus. I cry during the National Anthem. And I love the speeches, full of hope and encouragement and advice. This graduation Emeritus Professor John Burroughs spoke and made two points. The first was to know yourself and what you can do and what you can’t. He wanted to be an All Black, but couldn’t. But he became a prominent figure in law in New Zealand. Sadly he said he couldn’t do mathematics, which I wish was an admission seen as similar to saying one couldn’t read. Why is it that people think it is okay to be bad at math? Or even something to be proud of? moving on…

The second point Prof Burroughs made is pertinent to the teaching of Statistics and Operations Research. He recalled the advent of the ballpoint pen when he was at school. Until then he had been at the mercy of dip or fountain pens. Then when the ballpoint pen arrived it revolutionized writing. His teachers weren’t impressed and often insisted that students stick with fountain pens so as not to ruin their penmanship. It was an example of technology and improvement and change and people’s reaction. When he was a lecturer in law at the University of Canterbury he eschewed computers and was probably called a Luddite, though not to his face. In his later career he has had to embrace the new technology, including Facebook, twitter, Google and the like (the Like?). And he has enjoyed it. He wishes he had ridden the wave at the time.

There will always be change. Prof Burroughes’s advice to the graduates was to try to anticipate and enjoy change. Change equals opportunity.

And now I get to the point. The widespread use of powerful computers has changed Statistics and Operations Research. What will not change is change. There will continue to be advances in the accessibility of our disciplines to the masses. And we need to embrace this. When I learned Statistics and Operations Research in the early 1980s there was little computing power available. We used Eton tables, and solved two-variable LPs on cartesian planes. We performed matrix operations and stochastic simulation by hand calculation. We learned Revised Simplex by hand. We used the Poisson approximation of the binomial distribution as that avoided tables going too high. When we used MiniSPSS we were allowed ten runs in which to produce a linear regression, and the emphasis was on the production rather than the interpretation of output.

That was then, and this is now, and I think too many teachers of Statistics and Operations Research have not moved on. There is certainly evidence of this in the textbooks. Recently a colleague and I reviewed all first year Operations Research textbooks, examining their treatment of Linear Programming. One of the textbooks was a later edition of one I had used in 1981. The later edition used the same example to teach LP. Much of what was in these textbooks did not recognize the powerful opportunity the spreadsheet provides to explore and understand models.

I have also been reviewing statistics textbooks, though there are too many to be exhaustive. Statistics textbooks too often are stuck in the days of the fountain pen, rather than embracing the great possibilities that are there with the power of the computer.

I challenge all teachers of Operations Research and Statistics to examine what they do and ask if it is the same way that they were taught. If the answer is yes, then some more thinking is called for. We have such amazing opportunities to teach so much better, to use real data, to make a real difference, that to be stuck in the old methods, using tables and formulas is close to a crime.