How to learn statistics (Part 2)

Some more help (preaching?) for students of statistics

Last week I outlined the first five principles to help people to learn and study statistics.

They focussed on how you need to practise in order to be good at statistics and you should not wait until you understand it completely before you start applying. I sometimes call this suspending disbelief. Next I talked about the importance of context in a statistical investigation, which is one of the ways that statistics is different from pure mathematics. And finally I stressed the importance of technology as a tool, not only for doing the analysis, but for exploring ideas and gaining understanding.

Here are the next five principles (plus 2):

6. Terminology is important and at times inconsistent

There are several issues with regard to statistical terminology, and I have written a post with ideas for teachers on how to teach terminology.

One issue with terminology is that some words that are used in the study of statistics have meanings in everyday life that are not the same. A clear example of this is the word, “significant”. In regular usage this can mean important or relevant, yet in statistics, it means that there is evidence that an effect that shows up in the sample also exists in the population.

Another issue is that statistics is a relatively young science and there are inconsistencies in terminology. We just have to live with that. Depending on the discipline in which the statistical analysis is applied or studied, different terms can mean the same thing, or very close to it.

A third language problem is that mixed in with the ambiguity of results, and judgment calls, there are some things that are definitely wrong. Teachers and examiners can be extremely picky. In this case I would suggest memorising the correct or accepted terminology for confidence intervals and hypothesis tests. For example I am very fussy about the explanation for the R-squared value in regression. Too often I hear that it says how much of the dependent variable is explained by the independent variable. There needs to be the word “variation” inserted in there to make it acceptable. I encourage my students to memorise a format for writing up such things. This does not substitute for understanding, but the language required is precise, so having a specific way to write it is fine.

This problem with terminology can be quite frustrating, but I think it helps to have it out in the open. Think of it as learning a new language, which is often the case in new subject. Use glossaries, to make sure you really do know what a term means.

7. Discussion is important

This is linked with the issue of language and vocabulary. One way to really learn something is to talk about it with someone else and even to try and teach it to someone else. Most teachers realise that the reason they know something pretty well, is because they have had to teach it. If your class does not include group work, set up your own study group. Talk about the principles as well as the analysis and context, and try to use the language of statistics. Working on assignments together is usually fine, so long as you write them up individually, or according to the assessment requirements.

8. Written communication skills are important

Mathematics has often been a subject of choice for students who are not fluent in English. They can perform well because there is little writing involved in a traditional mathematics course. Statistics is a different matter, though, as all students should be writing reports. This can be difficult at the start, but as students learn to follow a structure, it can be made more palatable. A statistics report is not a work of creative writing, and it is okay to use the same sentence structure more than once. Neither is a statistics report a narrative of what you did to get to the results. Generous use of headings makes a statistical report easier to read and to write. A long report is not better than a short report, if all the relevant details are there.

9. Statistics has an ethical and moral aspect

This principle is interesting, as many teachers of statistics come from a mathematical background, and so have not had exposure to the ethical aspects of research themselves. That is no excuse for students to park their ethics at the door of the classroom. I will be pushing for more consideration of ethical aspects of research as part of the curriculum in New Zealand. Students should not be doing experiments on human subjects that involve delicate subjects such as abuse, or bullying. They should not involve alcohol or other harmful substances. They should be aware of the potential to do harm, and make sure that any participants have been given full information and given consent. This can be quite a hurdle, but is part of being an ethical human being. It also helps students to be more aware when giving or withholding consent in medical and other studies.

10. The study of statistics can change the way you view the world

Sometimes when we learn something at school, it stays at school and has no impact on our everyday lives. This should not be the case with the study of statistics. As we learn about uncertainty and variation we start to see this in the world around us. When we learn about sampling and non-sampling errors, we become more critical of opinion polls and other research reported in the media. As we discover the power of statistical analysis and experimentation, we start to see the importance of evidence-based practice in medicine, social interventions and the like.

11. Statistics is an inherently interesting and relevant subject.

And it can be so much fun. There is a real excitement in exploring data, and becoming a detective. If you aren’t having fun, you aren’t doing it right!

12. Resources from Statistics Learning Centre will help you learn.

Of course!

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.

Open Letter to Khan Academy about Basic Probability

Khan academy probability videos and exercises aren’t good either

Dear Mr Khan

You have created an amazing resource that thousands of people all over the world get a lot of help from. Well done. Some of your materials are not very good, though, so I am writing this open letter in the hope that it might make some difference. Like many others, I believe that something as popular as Khan Academy will benefit from constructive criticism.

I fear that the reason that so many people like your mathematics videos so much is not because the videos are good, but because their experience in the classroom is so bad, and the curriculum is poorly thought out and encourages mechanistic thinking. This opinion is borne out by comments I have read from parents and other bloggers. The parents love you because you help their children pass tests.  (And these tests are clearly testing the type of material you are helping them to pass!) The bloggers are not so happy, because you perpetuate a type of mathematical instruction that should have disappeared by now. I can’t even imagine what the history teachers say about your content-driven delivery, but I will stick to what I know. (You can read one critique here)

Just over a year ago I wrote a balanced review of some of the Khan Academy videos about statistics. I know that statistics is difficult to explain – in fact one of the hardest subjects to teach. You can read my review here. I’ve also reviewed a selection of videos about confidence intervals, one of which was from Khan Academy. You can read the review here.

Consequently I am aware that blogging about the Khan Academy in anything other than glowing terms is an invitation for vitriol from your followers.

However, I thought it was about time I looked at the exercises that are available on KA, wondering if I should recommend them to high school teachers for their students to use for review. I decided to focus on one section, introduction to probability. I put myself in the place of a person who was struggling to understand probability at school.

Here is the verdict.

First of all the site is very nice. It shows that it has a good sized budget to use on graphics and site mechanics. It is friendly to get into. I was a bit confused that the first section in the Probability and Statistics Section is called “Independent and dependent events”. It was the first section though. The first section of this first section is called Basic Probability, so I felt I was in the right place. But then under the heading, Basic probability, it says, “Can I pick a red frog out of a bag that only contains marbles?” Now I have no trouble with humour per se, and some people find my videos pretty funny. But I am very careful to avoid confusing people with the humour. For an anxious student who is looking for help, that is a bit confusing.

I was excited to see that this section had five videos, and two sets of exercises. I was pleased about that, as I’ve wanted to try out some exercises for some time, particularly after reading the review from Fawn Nguyen on her experience with exercises on Khan Academy. (I suggest you read this – it’s pretty funny.)

So I watched the first video about probability and it was like any other KA video I’ve viewed, with primitive graphics and a stumbling repetitive narration. It was correct enough, but did not take into account any of the more recent work on understanding probability. It used coins and dice. Big yawn. It wastes a lot of time. It was ok. I do like that you have the interactive transcript so you can find your way around.

It dawned on me that nowhere do you actually talk about what probability is. You seem to assume that the students already know that. In the very start of the first video it says,

“What I want to do in this video is give you at least a basic overview of probability. Probability, a word that you’ve probably heard a lot of and you are probably just a little bit familiar with it. Hopefully this will get you a little deeper understanding.”

Later in the video there is a section on the idea of large numbers of repetitions, which is one way of understanding probability. But it really is a bit skimpy on why anyone would want to find or estimate a probability, and what the values actually mean. But it was ok.

The first video was about single instances – one toss of a coin or one roll of a die. Then the second video showed you how to answer the questions in the exercises, which involved two dice. This seemed ok, if rather a sudden jump from the first video. Sadly both of these examples perpetuate the common misconception that if there are, say, 6 alternative outcomes, they will necessarily be equally likely.

Exercises

Then we get to some exercises called “Probability Space” , which is not an enormously helpful heading. But my main quest was to have a go at the exercises, so that is what I did. And that was not a good thing. The exercises were not stepped, but started right away with an example involving two dice and the phrase “at least one of”. There was meant to be a graphic to help me, but instead I had the message “scratchpad not available”. I will summarise my concerns about the exercises at the end of my letter. I clicked on a link to a video that wasn’t listed on the left, called Probability Space and got a different kind of video.

This video was better in that it had moving pictures and a script. But I have problems with gambling in videos like this. There are some cultures in which gambling is not acceptable. The other problem I have is with the term  “exact probability”, which was used several times. What do we mean by “exact probability”? How does he know it is exact? I think this sends the wrong message.

Then on to the next videos which were worked examples, entitled “Example: marbles from a bag, Example: Picking a non-blue marble, Example: Picking a yellow marble.” Now I understand that you don’t want to scare students with terminology too early, but I would have thought it helpful to call the second one, “complementary events, picking a non-blue marble”. That way if a student were having problems with complementary events in exercises from school, they could find their way here. But then I’m not sure who your audience is. Are you sure who your audience is?

The first marble video was ok, though the terminology was sloppy.

The second marble video, called “Example: picking a non-blue marble”, is glacially slow. There is a point, I guess in showing students how to draw a bag and marbles, but… Then the next example is of picking numbers at random. Why would we ever want to do this? Then we come to an example of circular targets. This involves some problem-solving regarding areas of circles, and cancelling out fractions including pi. What is this about? We are trying to teach about probablity so why have you brought in some complication involving the area of a circle?

The third marble video attempts to introduce the idea of events, but doesn’t really. By trying not to confuse with technical terms, the explanation is more confusing.

Now onto some more exercises. The Khan model is that you have to get 5 correct in a row in order to complete an exercise. I hope there is some sensible explanation for this, because it sure would drive me crazy to have to do that. (As I heard expressed on Twitter)

What are circular targets doing in with basic probability?

The first example is a circular target one.  I SO could not be bothered working out the area stuff so I used the hints to find the answer so I could move onto a more interesting example. The next example was finding the probability of a rolling a 4 from a fair six sided die. This is trivial, but would have been not a bad example to start with. Next question involve three colours of marbles, and finding the probability of not green. Then another dart-board one. Sigh. Then another dart board one. I’m never going to find out what happens if I get five right in a row if I don’t start doing these properly. Oh now – it gave me circumference. SO can’t be bothered.

And that was the end of Basic probability. I never did find out what happens if I get five correct in a row.

Venn diagrams

The next topic is called “Venn diagrams and adding probabilities “. I couldn’t resist seeing what you would do with a Venn diagram. This one nearly reduced me to tears.

As you know by now, I have an issue with gambling, so it will come as no surprise that I object to the use of playing cards in this example. It makes the assumption that students know about playing cards. You do take one and a half minutes to explain the contents of a standard pack of cards.  Maybe this is part of the curriculum, and if so, fair enough. The examples are standard – the probability of getting a Jack of Hearts etc. But then at 5:30 you start using Venn diagrams. I like Venn diagrams, but they are NOT good for what you are teaching at this level, and you actually did it wrong. I’ve put a comment in the feedback section, but don’t have great hopes that anything will change. Someone else pointed this out in the feedback two years ago, so no – it isn’t going to change.

Khan Venn diagram

This diagram is misleading, as is shown by the confusion expressed in the questions from viewers. There should be a green 3, a red 12, and a yellow 1.

Now Venn diagrams seem like a good approach in this instance, but decades of experience in teaching and communicating complex probabilities has shown that in most instances a two-way table is more helpful. The table for the Jack of Hearts problem would look like this:

Jacks Not Jacks Total
Hearts 1 12 13
Not Hearts 3 36 39
Total 4 48 52

(Any teachers reading this letter – try it! Tables are SO much easier for problem solving than Venn diagrams)

But let’s get down to principles.

The principles of instruction that KA have not followed in the examples:

  • Start easy and work up
  • Be interesting in your examples – who gives a flying fig about two dice or random numbers?
  • Make sure the hardest part of the question is the thing you are testing. This is particularly violated with the questions involving areas of circles.
  • Don’t make me so bored that I can’t face trying to get five in a row and not succeed.

My point

Yes, I do have one. Mr Khan you clearly can’t be stopped, so can you please get some real teachers with pedagogical content knowledge to go over your materials systematically and make them correct. You have some money now, and you owe it to your benefactors to GET IT RIGHT. Being flippant and amateurish is fine for amateurs but you are now a professional, and you need to be providing material that is professionally produced. I don’t care about the production values – keep the stammers and “lellows” in there if you insist. I’m very happy you don’t have background music as I can’t stand it myself. BUT… PLEASE… get some help and make your videos and exercises correct and pedagogically sound.

Dr Nic

PS – anyone else reading this letter, take a look at the following videos for mathematics.

And of course I think my own Statistics Learning Centre videos are pretty darn good as well.

Other posts about concerns about Khan:

Another Open Letter to Sal ( I particularly like the comment by Michael Paul Goldenberg)

Breaking the cycle (A comprehensive summary of the responses to criticism of Khan

Statistics is not beautiful (sniff)

Statistics is not really elegant or even fun in the way that a mathematics puzzle can be. But statistics is necessary, and enormously rewarding. I like to think that we use statistical methods and principles to extract truth from data.

This week many of the high school maths teachers in New Zealand were exhorted to take part in a Stanford MOOC about teaching mathematics. I am not a high school maths teacher, but I do try to provide worthwhile materials for them, so I thought I would take a look. It is also an opportunity to look at how people with an annual budget of more than 4 figures produce on-line learning materials. So I enrolled and did the first lesson, which is about people’s attitudes to math(s) and their success or trauma that has led to those attitudes. I’m happy to say that none of this was new to me. I am rather unhappy that it would be new to anyone! Surely all maths teachers know by now that how we deal with students’ small successes and failures in mathematics will create future attitudes leading to further success or failure. If they don’t, they need to take this course. And that makes me happy – that there is such a course, on-line and free for all maths teachers. (As a side note, I loved that Jo, the teacher switched between the American “math” and the British/Australian/NZ “maths”).

I’ve only done the first lesson so far, and intend to do some more, but it seems to be much more about mathematics than statistics, and I am not sure how relevant it will be. And that makes me a bit sad again. (It was an emotional journey!)

Mathematics in its pure form is about thinking. It is problem solving and it can be elegant and so much fun. It is a language that transcends nationality. (Though I have always thought the Greeks get a rough deal as we steal all their letters for the scary stuff.) I was recently asked to present an enrichment lesson to a class of “gifted and talented” students. I found it very easy to think of something mathematical to do – we are going to work around our Rogo puzzle, which has some fantastic mathematical learning opportunities. But thinking up something short and engaging and realistic in the statistics realm is much harder. You can’t do real statistics quickly.

On my run this morning I thought a whole lot more about this mathematics/statistics divide. I have written about it before, but more in defense of statistics, and warning the mathematics teachers to stay away or get with the programme. Understanding commonalities and differences can help us teach better. Mathematics is pure and elegant, and borders on art. It is the purest science. There is little beautiful about statistics. Even the graphs are ugly, with their scattered data and annoying outliers messing it all up. The only way we get symmetry is by assuming away all the badly behaved bits. Probability can be a bit more elegant, but with that we are creeping into the mathematical camp.

English Language and English literature

I like to liken. I’m going to liken maths and stats to English language and English literature. I was good at English at school, and loved the spelling and grammar aspects especially. I have in my library a very large book about the English language, (The Cambridge encyclopedia of the English Language, by David Crystal) and one day I hope to read it all. It talks about sounds and letters, words, grammar, syntax, origins, meanings. Even to dip into, it is fascinating. On the other hand I have recently finished reading “The End of Your Life Book Club” by Will Schwalbe, which is a biography of his amazing mother, set around the last two years of her life as she struggles with cancer. Will and his mother are avid readers, and use her time in treatment to talk about books. This book has been an epiphany for me. I had forgotten how books can change your way of thinking, and how important fiction is. At school I struggled with the literature side of English, as I wanted to know what the author meant, and could not see how it was right to take my own meaning from a book, poem or work of literature.  I have since discovered post-modernism and am happy drawing my own meaning.

So what does this all have to do with maths and statistics? Well I liken maths to English language. In order to be good at English you need to be able to read and write in a functional way. You need to know the mechanisms. You need to be able to DO, not just observe. In mathematics, you need to be able to approach a problem in a mathematical way.  Conversely, to be proficient in literature, you do not need to be able to produce literature. You need to be able to read literature with a critical mind, and appreciate the ideas, the words, the structure. You do need to be able to write enough to express your critique, but that is a different matter from writing a novel.  This, to me is like being statistically literate – you can read a statistical report, and ask the right questions. You can make sense of it, and not be at the mercy of poorly executed or mendacious research. You can even write a summary or a critique of a statistical analysis. But you do not need to be able to perform the actual analysis yourself, nor do you need to know the exact mathematical theory underlying it.

Statistical Literacy?

Maybe there is a problem with the term “statistical literacy”. The traditional meaning of literacy includes being able to read and write – to consume and to produce – to take meaning and to create meaning. I’m not convinced that what is called statistical literacy is the same.

Where I’m heading with this, is that statistics is a way to win back the mathematically disenfranchised. If I were teaching statistics to a high school class I would spend some time talking about what statistics involves and how it overlaps with, but is not mathematics. I would explain that even people who have had difficulty in the past with mathematics, can do well at statistics.

The following table outlines the different emphasis of the two disciplines.

Mathematics Statistics
Proficiency with numbers is important Proficiency with numbers is helpful
Abstract ideas are important Concrete applications are important
Context is to be removed so that we can model the underlying ideas Context is crucial to all statistical analysis
You don’t need to write very much. Written expression in English is important

Another idea related to this is that of “magic formulas” or the cookbook approach. I don’t have a problem with cookbooks and knitting patterns. They help me to make things I could not otherwise. However, the more I use recipes and patterns, the more I understand the principles on which they are based. But this is a thought for another day.

The importance of being wrong

We don’t like to think we are wrong

One of the key ideas in statistics is that sometimes we will be wrong. When we report a 95% confidence interval, we will be wrong 5% of the time. Or in other words, about 1 in 20 of 95% confidence intervals will not contain the population parameter we are attempting to estimate. That is how they are defined. The thing is, we always think we are part of the 95% rather than the 5%. Mostly we will be correct, but if we do enough statistical analysis, we will almost definitely be wrong at some point. However, human nature is such that we tend to think it will be someone else. There is also a feeling of blame associated with being wrong. The feeling is that if we have somehow missed the true value with our confidence interval, it must be because we have made a mistake. However, this is not true. In fact we MUST be wrong about 5% of the time, or our interval is too big, and not really a 95% confidence interval.

The term “margin of error” appears with increasing regularity as elections approach and polling companies are keen to make money out of sooth-saying. The common meaning of the margin of error is half the width of a 95% confidence interval. So if we say the margin of error is 3%, then about one time in twenty, the true value of the proportion will actually be more than 3% away from the reported sample value.

What doesn’t help is that we seldom do know if we are correct or not. If we knew the real population value we wouldn’t be estimating it. We can contrive situations where we do know the population but pretend we don’t. If we do this in our teaching, we need to be very careful to point out that this doesn’t normally happen, but does in “classroom world” only. (Thanks to MD for this useful term.) General elections can give us an idea of being right or wrong after the event, but even then the problem of non-sampling error is conflated with sampling error. When opinion polls turn out to miss the mark, we tend to think of the cause as being due to poor sampling, or people changing their minds, or all number of imaginative explanations rather than simple, unavoidable sampling error.

So how do we teach this in such a way that it goes beyond school learning and is internalised for future use as efficient citizens?

Teaching suggestions

I have two suggestions. The first is a series of True/False statements that can be used in a number of ways. I have them as part of on-line assessment, so that the students are challenged by them regularly. They could be well used in the classroom as part of a warm-up exercise at the start of a lesson. Students can write their answers down or vote using hands.

Here are some examples of True/False statements (some of which could lead to discussion):

  1. You never know if your confidence interval contains the true population value.
  2. If you make your confidence interval wide enough you can be sure that you contain the true population value.
  3. A confidence interval tells us where we are pretty sure the sample statistic lies.
  4. It is better to have a narrow confidence interval than a wide one, as it gives us more certain information, even though it is more likely to be wrong.
  5. If your study involves twenty confidence intervals, then you know that exactly one of them will be wrong.
  6. If a confidence interval doesn’t contain the true population value, it is because it is one of the 5% that was calculated incorrectly.

You can check your answers at the end of this post.

Experiential exercise

The other teaching suggestion is for an experiential exercise. It requires a little set up time.

Make a set of cards for students with numbers on them that correspond to the point estimate of a proportion, or a score that will lead to that. (Specifications for a set of 35 cards representing the results from a proportion of 0.54 and 25 trials is given below).

Introduce the exercise as follows:
“I have a computer game, and have set the ratio of wins to losses at a certain value. Each of you has played 25 times, and the number of wins you have obtained will be on your card. It is really important that you don’t look at other people’s cards.”

Hand them out to the students. (If you have fewer than 35 in your class, it might be a good idea to make sure you include the cards with 8 and 19 in the set you use – sometimes it is ok to fudge slightly to teach a point.)
“Without getting information from anyone else, write down your best estimate of the true proportion of wins to losses in the game. Do you think you are correct? How close do you think you are to the true value?”

They will need to divide the number of wins by 25, which should not lead to any computational errors! The point is that they really can’t know how close their estimate is to the true value – and what does “correct” mean?

Then work out the margin of error for a sample of size 25, which in this case is estimated at 20%. Get the students to calculate their 95% confidence intervals, and decide if they have the interval that contains the true population value. Get them to commit one way or the other.

Now they can talk to each other about the values they have.

There are several ways you can go from here. You can tell them what the population proportion was from which the numbers were drawn (0.54). They can then see that most of them had confidence intervals that included the true value, and some didn’t. Or you can leave them wondering, which is a better lesson about real life. Or you can do one exercise where you do tell them and one where you don’t.

This is an area where probability and statistics meet. You could make a nice little binomial distribution problem out of being correct in a number of confidence intervals. There are potential problems with independence, so you need to be a bit careful with the wording. For example: Fifteen  students undertake separate statistical analyses on the topics of their choice, and construct 95% confidence intervals. What is the probability that all the confidence intervals are correct, in that they do contain the estimated population parameter? This is well modelled by a binomial distribution with n =15 and p=0.05. P(X=0)=0.46. And another interesting idea – what is the probability that two or more are incorrect? 0.17 is the answer. So there is a 17% chance that more than one of the confidence intervals does not contain the population parameter of interest.

This is an area that needs careful teaching, and I suspect that some teachers have only a sketchy understanding of the idea of confidence intervals and margins of error. It is so important to know that statistical results are meant to be wrong some of the time.

Answers: T,T,F, debatable, F,F.

Data for the 35 cards:

Number on card

8

9

10

11

12

13

14

15

16

17

18

19

Number of cards

1

1

2

3

5

5

6

5

3

2

1

1

Teaching with School League tables

NCEA League tables in the newspaper

My husband ran for cover this morning when he saw high school NCEA (National Certificates of Educational Achievement)  league tables in the Press. However, rather than rave at him yet again, I will grasp the opportunity to expound to a larger audience. Much as I loathe and despise league tables, they are a great opportunity to teach students to explore data rich reports with a critical and educated eye.  There are many lessons to learn from league tables. With good teaching we can help dispell some of the myths the league tables promulgate.

When a report is made short and easy to understand, there is a good chance that much of the ‘truth’ has been lost along with the complexity. The table in front of me lists 55 secondary and area schools from the Canterbury region. These schools include large “ordinary” schools and small specialist schools such as Van Asch Deaf Education Centre and Southern Regional Health School. They include single-sex and co-ed, private, state-funded and integrated. They include area schools which are in small rural communities, which cover ages 5 to 21. The “decile” of each of the schools is the only contextual information given, apart from the name of the school.  (I explain the decile, along with misconceptions at the end of the post.) For each school is given percentages of students passing at the three levels. It is not clear whether the percentages in the newspaper are of participation rate or school roll.

This is highly motivating information for students as it is about them and their school. I had an argument recently with a student from a school which scores highly in NCEA. She was insistent that her friend should change schools from one that has lower scores. What she did not understand was that the friend had some extra learning difficulties, and that the other school was probably more appropriate for her. I tried to teach the concept of added-value, but that wasn’t going in either. However I was impressed with her loyalty to her school and I think these tables would provide an interesting forum for discussion.

Great context discussion

You could start with talking about what the students think will help a school to have high pass rates. This could include a school culture of achievement, good teaching, well-prepared students and good resources. This can also include selection and exclusion of students to suit the desired results, selection of “easy” standards or subjects, and even less rigorous marking of internal assessment. Other factors to explore might be single-sex vs co-ed school, the ethnic and cultural backgrounds of the students, private vs state-funded schools.  All of these are potential explanatory variables. Then you can point out how little of this information is actually taken into account in the table. This is a very common occurrence, with limited space and inclusion of raw data. I suspect at least one school appears less successful because some of the students sit different exams, either Cambridge or International Baccalaureate. These may be the students who would have performed well in NCEA.

Small populations

It would be good to look at the impact of small populations, and populations of very different sizes in the data. Students should think about what impact their behaviour will have on the results of the school, compared with a larger or smaller cohort. The raw data provided by the Ministry of Education does give a warning for small cohorts. For a small school, particularly in a rural area, there may be only a handful of students in year 13, so that one student’s success or failure has a large impact on the outcome. At the other end of the scale, there are schools of over 2000, which will have about 400 students in year 13. This effect is important to understand in all statistical reporting. One bad event in a small hospital, for instance, will have a larger percentage effect than in a large hospital.

Different rules

We hear a lot about comparing apples and oranges. School league tables include a whole fruit basket of different criteria. Schools use different criteria for allowing students into the school, into different courses, and whether they are permitted to sit external standards. Attitudes to students with special educational needs vary greatly. Some schools encourage students to sit levels outside their year level.

Extrapolating from a small picture

What one of the accompanying stories points out is that NCEA is only a part of what schools do. Sometimes the things that are measurable get more attention because it is easier to report in bulk. A further discussion with students could be provoked using statements such as the following, which the students can vote on, and then discuss. You could also discuss what evidence you would need to be able to refute or support them.

  • A school that does well in NCEA level 3 is a good school.
  • Girls’ schools do better than boys’ schools at NCEA because girls are smarter than boys.
  • Country schools don’t do very well because the clever students go to boarding school in the city.
  • Boys are more satisfied with doing just enough to get achieved.

Further extension

If students are really interested you can download the full results from the Ministry of Education website and set up a pivot table on Excel to explore questions.

I can foresee some engaging and even heated discussions ensuing. I’d love to hear how they go.

Short explanation of Decile – see also official website.

The decile rating of the school is an index developed in New Zealand and is a measure of social deprivation. The decile rating is calculated from a combination of five values taken from census data for the meshblocks in which the students reside. A school with a low decile rating of 1 or 2 will have a large percentage of students from homes that are crowded, or whose parents are not in work or have no educational qualifications. A school with a decile rating of 10 will have the fewest students from homes like that. The system was set up to help with targeted funding for educational achievement. It recognises that students from disadvantaged homes will need additional resources in order to give them equal opportunity to learn. However, the term has entered the New Zealand vernacular as a measure of socio-economic status, and often even of worth. A decile 10 school is often seen as a rich school or a “top” school. The reality is that this is not the case.  Another common misconception is that one tenth of the population of school age students is in each of the ten bands. How it really works is that one tenth of schools is in each of the bands. The lower decile schools are generally smaller than other schools, and mostly primary schools. In 2002 there were nearly 40,000 secondary students in decile 10 schools, with fewer than 10,000 in decile 1 schools.

Parts and whole

The whole may be greater than the sum of the parts, but the whole still needs those parts. A reflective teacher will think carefully about when to concentrate on the whole, and when on the parts.

Golf

If you were teaching someone golf, you wouldn’t spend days on a driving range, never going out on a course. Your student would not get the idea of what the game is, or why they need to be able to drive straight and to a desired length. Nor would it be much fun! Similarly if the person only played games of golf it would be difficult for them to develop their game. Practice driving and putting is needed.  A serious student of golf would also read and watch experts at golf.

Music

Learning music is similar. Anyone who is serious about developing as a musician will spend a considerable amount of time developing their technique and their knowledge by practicing scales, chords and drills. But at the same time they need to be playing full pieces of music so that they feel the joy of what they are doing. As they play music, as opposed to drill, they will see how their less-interesting practice has helped them to develop their skills. However, as they practice a whole piece, they may well find a small part that is tripping them up, and focus for a while on that. If they play only the piece as a whole, it is not efficient use of time. A serious student of music will also listen to and watch great musicians, in order to develop their own understanding and knowledge.

Benefits of study of the whole and of the parts

In each of these examples we can see that there are aspects of working with the whole, and aspects of working with the parts. Study of the whole contributes perspective and meaning to study, and helps to tie things together. It helps to see where they have made progress. Study of the parts isolates areas of weakness, develops skills and saves time in practice, thus being more efficient.

It is very important for students to get an idea of the purpose of their study, and where they are going. For this reason I have written earlier about the need to see the end when starting out in a long procedure such as a regression or linear programming model.

It is also important to develop “statistical muscle memory” by repeating small parts of the exercise over and over until it is mastered. Practice helps people to learn what is general and what is specific in the different examples.

Teaching conditional probability

We are currently developing a section on probability as part of our learning materials. A fundamental understanding of probability and uncertainty are essential to a full understanding of inference. When we look at statistical evidence from data, we are holding it up against what we could reasonably expect to happen by chance, which involves a probability model. Probability lies in the more mathematical area of the study of statistics, and has some fun problem-solving aspects to it.

A popular exam question involves conditional probability. We like to use a table approach to this as it avoids many of the complications of terminology. I still remember my initial confusion over the counter-intuitive expression P(A|B) which means the probability that an object from subset B has the property of A. There are several places where students can come unstuck in Bayesian review, and the problems can take a long time. We can liken solving a conditional probability problem to a round of golf, or a long piece of music. So what we do in teaching is that first we take the students step by step through the whole problem. This includes working out what the words are saying, putting the known values into a table, calculating the unknown values in the table, and the using the table to answer the questions involving conditional probability.

Then we work on the individual steps, isolating them so that students can get sufficient practice to find out what is general and what is specific to different examples. As we do this we endeavour to provide variety such that students do not work out some heuristic based on the wording of the question, that actually stops them from understanding. An example of this is that if we use the same template each time, students will work out that the first number stated will go in a certain place in the table, and the second in another place etc. This is a short-term strategy that we need to protect them from in careful generation of questions.

As it turns out students should already have some of the necessary skills. When we review probability at the start of the unit, we get students to calculate probabilities from tables of values, including conditional probabilities. Then when they meet them again as part of the greater whole, there is a familiar ring.

Once the parts are mastered, the students can move on to a set of full questions, using each of the steps they have learned, and putting them back into the whole. Because they are fluent in the steps, it becomes more intuitive to put the whole back together, and when they meet something unusual they are better able to deal with it.

Starting a course in Operations Research/Management Science

It is interesting to contemplate what “the whole” is, with regard to any subject. In operations research we used to begin our first class, like many first classes, talking about what management science/operations research is. It was a pretty passive sort of class, and I felt it didn’t help as first-year university students had little relevant knowledge to pin the ideas on. So we changed to an approach that put them straight into the action and taught several weeks of techniques first. We started with project management and taught critical path. Then we taught identifying fixed and variable costs and break-even analysis. The next week was discounting and analysis of financial projects. Then for a softer example we looked at multi-criteria decision-making, using MCDM. It tied back to the previous week by taking a different approach to a decision regarding a landfill. Then we introduced OR/MS, and the concept of mathematical modelling. By then we could give real examples of how mathematical models could be used to inform real world problems. It was helpful to go from the concrete to the abstract. This was a much more satisfactory approach.

So the point is not that you should always start with the whole and then do the parts and then go back to the whole. The point is that a teacher needs to think carefully about the relationship between the parts and the whole, and teach in a way that is most helpful.

Let’s hear it for the Triangular Distribution!

Telly_Monster

Dr Nic meets Telly monster on the set of Sesame Street

Telly monster is my favourite character on Sesame Street, and a few years ago I was lucky enough to actually meet him. This morning I was delighted to find out from my resident Sesame Street expert that Telly monster is a triangle lover. I too am becoming a triangle lover.

I have learned recently about the triangular distribution. For some reason it is in the New Zealand curriculum and I wondered why, never having used it or seen it in any statistics textbook. I still don’t know the official motivation for including it, but it is a really good idea. The triangular distribution seems to be a useful teaching tool. I say “seems to be” as I haven’t actually used it in a class, but I can see the potential for some very good exercises and learning experiences.

There are three aspects of the triangular distribution that I find appealing.

Multiple models for the same scenario

Using the triangular distribution alongside the normal distribution encourages the idea that the distributions are models of a real-life process. Like many curricula, the previous NZ curriculum included binomial, Poisson and normal distributions. Any questions of what model to use in a specific scenario were determined pretty much by the form of the story. If the story involved continuous data it pretty much had to be “normally distributed”.  This could imbed a false impression that all continuous data was appropriately modelled by the normal distribution.

A nice learning experience is to take some real data such as weights of lemons from a tree, and see how well that is modelled by a normal distribution. This can be enriched by also modelling this as a triangular distribution and seeing how well the two compare. A further extension would be to use a uniform distribution model of the same data. The beauty of this exercise is that it reinforces the idea that a distribution is a model of reality, and that there are different models that may be more or less appropriate for different data and circumstances.

Different characteristics

I find contrasts are helpful for teaching. The triangular distribution provides a nice contrast with the normal distribution in a number of ways.

First is the requirements for specification. The triangular distribution is specified by the maximum, minimum and peak values. When we are making a subjective estimate, such as for completion time for a task, these are three easily pictured amounts – the longest time, the shortest time, and the most likely time for completion. This compares with the normal distribution, for which we need the mean and standard deviation, which would often be drawn from a sample.

The triangular distribution has a finite range, bounded by the maximum and minimum values. In a triangular distribution we can specify, for instance, that the results of a test will lie between 0 and 10. Modelling such a situation with a normal distribution can give results outside the range, as it theoretically goes to infinity in both directions.

The triangular distribution is not symmetric. It can be, but it is not a requirement. Where we have severely skewed data, it may well be that the triangular distribution is a better model than the normal distribution. This helps us when teaching the use of the normal distribution. The contrast is helpful.

Area under the pdf is the probability

The idea that the probability of a continuous distribution is the area under the probability mass function is a difficult one for many students to get their heads around. One way to teach this is to start with a discrete distribution and then cut it up into finer and finer points. But then finding the actual area is problematic. With the normal distribution, the computation is hidden in the calculator, spreadsheet or tables. With the uniform distribution, the computation is trivial and can seem contrived. But areas under the graph in the triangular distribution can be calculated, and the exercise is not trivial. It also very nicely shows how the pdf is not going to give them the probability of a single value.

So there you have it – three important lessons in one tidy little triangle-shaped package.

And to help you use the triangular distribution in your teaching, we have this handout: Notes on Triangle Distributions which you are welcome to use, so long as you leave our branding on it. And if you want your students to have some great practice exercises, they can always join our course!

Conceptualising Probability

The problem with probability is that it doesn’t really exist. Certainly it never exists in the past.

Probability is an invention we use to communicate our thoughts about how likely something is to happen. We have collectively agreed that 1 is a certain event and 0 is impossible. 0.5 means that there is just as much chance of something happening as not. We have some shared perception that 0.9 means that something is much more likely to happen than to not happen. Probability is also useful for when we want to do some calculations about something that isn’t certain. Often it is too hard to incorporate all uncertainty, so we assume certainty and put in some allowance for error.

Sometimes probability is used for things that happen over and over again, and in that case we feel we can check to see if our predication about how likely something is to happen was correct. The problem here is that we actually need things to happen a really big lot of times under the same circumstances in order to assess if we were correct. But when we are talking about the probability of a single event, that either will or won’t happen, we can’t test out if we were right or not afterwards, because by that time it either did or didn’t happen. The probability no longer exists.

Thus to say that there is a “true” probability somewhere in existence is rather contrived. The truth is that it either will happen or it won’t. The only way to know a true probability would be if this one event were to happen over and over and over, in the wonderful fiction of parallel universes. We could then count how many times it would turn out one way rather than another. At which point the universes would diverge!

However, for the interests of teaching about probability, there is the construct that there exists a “true probability” that something will happen.

Why think about probability?

What prompted these musings about probability was exploring the new NZ curriculum and companion documents, the Senior Secondary Guide and nzmaths.co.nz.

In Level 8 (last year of secondary school) of the senior secondary guide it says, “Selects and uses an appropriate distribution to solve a problem, demonstrating understanding of the relationship between true probability (unknown and unique to the situation), model estimates (theoretical probability) and experimental estimates.”

And at NZC level 3 (years 5 and 6 at Primary school!) in the Key ideas in Probability it talks about “Good Model, No Model and Poor Model” This statement is referred to at all levels above level 3 as well.

I decided I needed to make sense of these two conceptual frameworks: true-model-experimental and good-poor-no, and tie it to my previous conceptual framework of classical-frequency-subjective.

Here goes!

Delicious Mandarins

Let’s make this a little more concrete with an example. We need a one-off event. What is the probability that the next mandarin I eat will be delicious? It is currently mandarin season in New Zealand, and there is nothing better than a good mandarin, with the desired combination of sweet and sour, and with plenty of juice and a good texture. But, being a natural product, there is a high level of variability in the quality of mandarins, especially when they may have parted company with the tree some time ago.

There are two possible outcomes for my future event. The mandarin will be delicious or it will not. I will decide when I eat it. Some may say that there is actually a continuum of deliciousness, but for now this is not the case. I have an internal idea of deliciousness and I will know. I think back to my previous experience with mandarins. I think about a quarter are horrible, a half are nice enough and about a quarter are delicious (using the Dr Nic scale of mandarin grading). If the mandarin I eat next belongs to the same population as the ones in my memory, then I can predict that there is a 25% probability that the mandarin will be delicious.

The NZ curriculum talks about “true” probability which implies that any value I give to the probability is only a model. It may be a model based on empirical or experimental evidence. It can be based on theoretical probabilities from vast amounts of evidence, which has given us the normal distribution. The value may be only a number dredged up from my soul, which expresses the inner feeling of how likely it is that the mandarin will be delicious, based on several decades of experience in mandarin consumption.

More examples

Let us look at some more examples:

What is the probability that:

  • I will hear a bird on the way to work?
  • the flight home will be safe?
  • it will be raining when I get to Christchurch?
  • I will get a raisin in my first spoonful of muesli?
  • I will get at least one raisin in half of my spoonfuls of muesli?
  • the shower in my hotel room will be enjoyable?
  • I will get a rare Lego ® minifigure next time I buy one?

All of these events are probabilistic and have varying degrees of certainty and varying degrees of ease of modelling.

Easy to model Hard to model
Unlikely Get a rare Lego ® minifigure Raining in Christchurch
No idea Raisin in half my spoonfuls Enjoyable shower
Likely Raisin in first spoonful Bird, safe flight home

And as I construct this table I realise also that there are varying degrees of importance. Except for the flight home, none of those examples matter. I am hoping that a safe flight home has a probability extremely close to 1. I realise that there is a possibility of an incident. And it is difficult to model. But people have modelled air safety and the universal conclusion is that it is safer than driving. So I will take the probability and fly.

Conceptual Frameworks

How do we explain the different ways that probability has been described? I will now examine the three conceptual frameworks I introduced earlier, starting with the easiest.

Traditional categorisation

This is found in some form in many elementary college statistics text books. The traditional framework has three categories –classical or “a priori”, frequency or historical, and subjective.

Classical or “a priori” – I had thought of this as being “true” probability. To me, if there are three red and three white Lego® blocks in a bag and I take one out without looking, there is a 50% chance that I will get a red one. End of story. How could it be wrong? This definition is the mathematically interesting aspect of probability. It is elegant and has cool formulas and you can make up all sorts of fun examples using it. And it is the basis of gambling.

Frequency or historical – we draw on long term results of similar trials to gain information. For example we look at the rate of germination of a certain kind of seed by experiment, and that becomes a good approximation of the likelihood that any one future seed will germinate. And it also gives us a good estimate of what proportion of seeds in the future will germinate.

Subjective – We guess! We draw on our experience of previous similar events and we take a stab at it. This is not seen as a particularly good way to come up with a probability, but when we are talking about one off events, it is impossible to assess in retrospect how good the subjective probability estimate was. There is considerable research in the field of psychology about the human ability or lack thereof to attribute subjective probabilities to events.

In teaching the three part categorisation of sources of probability I had problems with the probability of rain. Where does that fit in the three categories? It uses previous experimental data to build a model, and current data to put into the model, and then a probability is produced. I decided that there is a fourth category, that I called “modelled”. But really that isn’t correct, as they are all models.

NZ curriculum terminology

So where does this all fit in the New Zealand curriculum pronouncements about probability? There are two conceptual frameworks that are used in the document, each with three categories as follows:

True, modelled, experimental

In this framework we start with the supposition that there exists somewhere in the universe a true probability distribution. We cannot know this. Our expressions of probability are only guesses at what this might be. There are two approaches we can take to estimate this “truth”. These two approaches are not independent of each other, but often intertwined.

One is a model estimate, based on theory, such as that the probability of a single outcome is the number of equally likely ways that it can occur over the number of possible outcomes. This accounts for the probability of a red brick as opposed to a white brick, drawn at random. Another example of a modelled estimate is the use of distributions such as the binomial or normal.

In addition there is the category of experimental estimate, in which we use data to draw conclusions about what it likely to happen. This is equivalent to the frequency or historical category above. Often modelled distributions use data from an experiment also. And experimental probability relies on models as well.  The main idea is that neither the modelled nor the experimental estimate of the “true” probability distribution is the true distribution, but rather a model of some sort.

Good model, poor model, no model

The other conceptual framework stated in the NZ curriculum is that of good model, poor model and no model, which relates to fitness for purpose. When it is important to have a “correct” estimate of a probability such as for building safety, gambling machines, and life insurance, then we would put effort into getting as good a model as possible. Conversely, sometimes little effort is required. Classical models are very good models, often of trivial examples such as dice games and coin tossing. Frequency models aka experimental models may or may not be good models, depending on how many observations are included, and how much the future is similar to the past. For example, a model of sales of slide rules developed before the invention of the pocket calculator will be a poor model for current sales. The ground rules have changed. And a model built on data from five observations of is unlikely to be a good model. A poor model is not fit for purpose and requires development, unless the stakes are so low that we don’t care, or the cost of better fitting is greater than the reward.

I have problems with the concept of “no model”. I presume that is the starting point, from which we develop a model or do not develop a model if it really doesn’t matter. In my examples above I include the probability that I will hear a bird on the way to work. This is not important, but rather an idle musing. I suspect I probably will hear a bird, so long as I walk and listen. But if it rains, I may not. As I am writing this in a hotel in an unfamiliar area I have no experience on which to draw. I think this comes pretty close to “no model”. I will take a guess and say the probability is 0.8. I’m pretty sure that I will hear a bird. Of course, now that I have said this, I will listen carefully, as I would feel vindicated if I hear a bird. But if I do not hear a bird, was my estimate of the probability wrong? No – I could assume that I just happened to be in the 0.2 area of my prediction. But coming back to the “no model” concept – there is now a model. I have allocated the probability of 0.8 to the likelihood of hearing a bird. This is a model. I don’t even know if it is a good model or a poor model. I will not be walking to work this way again, so I cannot even test it out for the future, and besides, my model was only for this one day, not for all days of walking to work.

So there you have it – my totally unscholarly musings on the different categorisations of probability.

What are the implications for teaching?

We need to try not to perpetuate the idea that probability is the truth. But at the same time we do not wish to make students think that probability is without merit. Probability is a very useful, and at times highly precise way of modelling and understanding the vagaries of the universe. The more teachers can use language that implies modelling rather than rules, the better. It is common, but not strictly correct to say, “This process follows a normal distribution”. As Einstein famously and enigmatically said, “God does not play dice”. Neither does God or nature use normal distribution values to determine the outcomes of natural processes. It is better to say, “this process is usefully modelled by the normal distribution.”

We can have learning experiences that help students to appreciate certainty and uncertainty and the modelling of probabilities that are not equi-probable. Thanks to the overuse of dice and coins, it is too common for people to assess things as having equal probabilities. And students need to use experiments.  First they need to appreciate that it can take a large number of observations before we can be happy that it is a “good” model. Secondly they need to use experiments to attempt to model an otherwise unknown probability distribution. What fun can be had in such a class!

But, oh mathematical ones, do not despair – the rules are still the same, it’s just the vigour with which we state them that has changed.

Comment away!

Post Script

In case anyone is interested, here are the outcomes which now have a probability of 1, as they have already occurred.

  • I will hear a bird on the way to work? Almost the minute I walked out the door!
  • the flight home will be safe? Inasmuch as I am in one piece, it was safe.
  • it will be raining when I get to Christchurch? No it wasn’t
  • I will get a raisin in my first spoonful of muesli? I did
  • I will get at least one raisin in half of my spoonfuls of muesli? I couldn’t be bothered counting.
  • the shower in my hotel room will be enjoyable? It was okay.
  • I will get a rare Lego minifigure next time I buy one? Still in the future!

The Knife-edge of Competence

I do my own video-editing using a very versatile and complex program called Adobe Premiere Pro. I have had no formal training, and get help by ringing my son, who taught me all I know and can usually rescue me with patient instructions over the phone. At times, especially in the early stages I have felt myself wobbling along the knife-edge of competence. All I needed was for something new to go wrong, or or click a button inadvertently and I would fall off the knife-edge and the whole project would disappear into a mass of binary. This was not without good reason. Premiere Pro wasn’t always stable on our computer, and at one point it took us several weeks to get our hard-drive replaced. (Apple “Time machine” saved me from despair). And sometimes I would forget to save regularly and a morning’s work was lost. (Even time-machine can’t help with that level of incompetence.)

But despite my severe limitations I have managed to edit over twenty videos that now receive due attention (and at times adulation!) on YouTube. It isn’t an easy feeling, to be teetering on the brink of disaster, real or imagined. But there was no alternative, and there is a sense of pride at having made it through with only a few scars and not too much inappropriate language.

There are some things at which I feel totally competent. I can speak to a crowd of any number of people and feel happy that they will be entertained, edified and perhaps even educated. I can analyse data using basic statistical methods. I can teach a person about inference. Performing these tasks is a joy, because I know I have the prerequisite skills and knowledge to cope with whatever happens. But on the way to getting to this point, I had to walk the knife-edge of competence.

Many teachers of statistics know too well this knife-edge. In New Zealand at present there are a large number of teachers of Year 13 statistics who are teaching about bootstrapping, when their own understanding of it is sketchy. They are teaching how to write statistical reports, when they have never written one themselves. They are assessing statements about statistics that they are not actually sure about. This is a knife-edge. They feel that any minute a student will ask them a question about the content that they cannot answer. These are not beginning teachers, but teachers with years and decades of experience in teaching mathematics and mathematical statistics. But the innovations of the curriculum have put them in an uncomfortable position. Inconsistent, tardy and even incorrect information from the qualification agency is not helping, but that is a story for another day.

In another arena there are professors and lecturers of statistics (in the antipodes we do not throw around the title “professor” with the abandon of our North American cousins) who are extremely competent at statistical mathematics and analysis but who struggle to teach in a satisfactory way. Their knife-edge concerns teaching, appropriate explanation and the generation of effective learning activities and assessments in the absence of any educational training. They fear that someone will realise one day that they don’t really know how to devise learning objectives, and provide fair assessments. I am hoping that this blog is going some way to helping these people to ask for help! Unfortunately the frequent response is avoidance behaviour, which is alarmingly supported by a system that rewards research publications rather than effective educational endeavours.

So what do you do when you are walking the knife-edge of competence?

You do the best you can.

And sometimes you fake it.

I am led to believe there is a gender-divide on this. Some people are better at hiding their incompetence than others, and just about all the people I know like that are men. I had a classmate in my honours year who was at a similar level of competence to me, but he applied for jobs I wouldn’t have contemplated. The fear of being shown up as a fake, or not knowing EXACTLY what to do at any point stopped me from venturing. He horrified me further a few years later when he set up his own company. Nearly three decades, two children and a PhD later I am not so fastidious or “nice” in the Jane Austen meaning of the word. If I think I can probably learn how to do something in time to make a reasonable fist of it and not cause actual harm, I’m likely to have a go. Hence taking my redundancy and running!

When I first lectured in statistics for management,  I did not know much beyond what I was teaching. I lived in fear that someone would ask me a question that I couldn’t answer and I would be revealed as the fake I was. Well you know, it never happened! I even taught students who were statistics majors, who did know more than I, and post-graduate students in psychology and heads of mathematics departments, and my fears were never realised. In fact the stats students told me that they finally understood the central limit theorem, thanks to my nifty little exercise using dotplots on minitab. (Which was how I had finally understood the central limit theorem – or at least the guts of it.)

I’m guessing that this is probably true for most of the mathematics teachers who are worrying. Despite their fear, they have not been challenged or called out.

The teachers’ other unease is the feeling that they are not giving the best service to their students, and the students will suffer, miss out on scholarships, decide not to get a higher education and live their lives on the street.  I may be exaggerating a little here, but certainly few of us like to give a service that is less than what we are accustomed to. We feel bad when we do something that feels substandard.

There are two things I learned in my twenty years of lecturing that may help here:

We don’t know how students perceive what we do. Every now and again I would come out of a lecture with sweat trickling down my spine because something had gone wrong. It might be that in the middle of an explanation I had had second thoughts about it, changed tack, then realised I was right in the first-place and ended up confusing myself. Or perhaps part way through a worked example it was pointed out to me that there was a numerical error in line three. To me these were bad, bad things to happen. They undermined my sense of competence. But you know, the students seldom even noticed. What felt like the worst lecture of my life, was in fact still just fine.

The other thing I learned is that we flatter ourselves when we think how much difference our knowledge may make.  Now don’t get me wrong here – teachers make an enormous difference. People who become teachers do so because we want to help people. We want to make a difference in students’ lives. We often have a sense of calling. There may be some teachers who do it because they don’t know what else to do with their degree, but I like to think that most of us teachers teach because to not teach is unthinkable. I despise, to the point of spitting as I talk, the expression “Those who can, do, and those who can’t, teach.” One day when the mood takes me I will write a whole post about the noble art of teaching and the fallacy of that dismissive statement. My next statement is so important I will give it a paragraph of its own.

A teacher who teaches from love, who truly cares about what happens to their students, even if they are struggling on the knife-edge of competence will not ruin their students’ lives through temporary incompetence in an aspect of the curriculum.

There are many ways that a teacher can have devastating effects on their students, but being, for a short time, on the knife-edge of competence, is not one of them.

Take heart, keep calm and carry on!