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.


The problem with videos for teaching maths and stats

The message of many popular mathematics and statistics videos is harming people’s perceptions of the nature of these disciplines.

I acknowledge the potential for conflict of interest in this post –  critically examining the role of video in learning and teaching mathematics and statistics – when StatsLC has a YouTube channel, and also provides videos through teaching and learning systems.

But I do wonder what message it sends when people like Sal Khan of Khan Academy and Mister Woo are applauded for their well-intentioned, and successful attempts to take a procedural view of mathematics to the masses. Video by its very nature tends towards procedures, and encourages the philosophy that there is one way to do something. Both Khan and Woo, and my personal favourite, Rob Tarrou, all show enthusiasm, inclusion and compassion. And I am sure that many people have been helped by these teachers. In New Zealand various classroom teachers ‘flip” their classrooms, and allow others to benefit from their videos on YouTube. One of the strengths, according to Khan, is that individual students can proceed at their own pace. However Jo Boaler states in her book, Mathematical Mindsets, that “Sadly I have yet to encounter a product that gives individualised opportunities and also teaches mathematics well.”

So what is the problem then? Millions of students love Khan, Woo, ProfRobBob and even Dr Nic. Millions of people also love fast food, and that isn’t good as a total diet.

In my work exploring people’s attitudes to mathematics, I find that many, including maths educators, have a procedural view of mathematics, which fails to unlock the amazing potential of our disciplines.

Procedural maths

Many people have the conception that to do mathematics is to work out the correct procedure to use in a specific instance and use it correctly in order to get the correct answer. This leads to a nice red tick. (Check mark) That was my view of maths for a very long time. I remember being most upset in my first year of university when the calculus exam was in a different format from the ones I had practised on. I was indignant and feared a C at best, and possibly even a failing grade. I liked the procedural approach. I felt secure using a procedural approach, and when I became a maths teacher, I was pretty much wedded to it. And the thing is, the procedural approach has worked very well for most of the people who are currently high school maths teachers.

Computation was an important part of mathematics

I recently read the inspiring “Hidden Figures”, about African American women who had pivotal roles in the development of space travel. For many of them, their introduction into life as a mathematician was as a computer. They did mathematical computations, and speed and accuracy were essential. I wonder how much of today’s curriculum is still aiming to produce computers, when we have electronic devices that can do all of that faster and more accurately.

Open-ended, lively maths

In parallel to the mass-maths-educators, we have the likes of Jo Boaler and Youcubed, Dan Meyer and Desmos, Bobbie Hunter and Mathematics Inquiry Communities, Marian Small, Tracy Zager, Fawn Nguyen and pretty much the entire Math-Twitter-Blogosphere spreading the message that mathematics is open-ended, exciting and far from procedural. Students work in groups to construct and communicate their ideas. Wrong answers are valued as evidence of thinking and the willingness to take risks. Productive struggle is valued and lessons are designed to get students outside of their comfort zones, but still within their zone of proximal development. Work is collective, rather than individualised, and ability grouping is strongly discouraged.

I find this approach enormously exciting, and believe that it could change the perception of the world towards mathematics.

The problem of the social contract

Thus I and many teachers are keen to develop a more social constructivist approach to learning mathematics at all levels. However, teachers – especially at high school – run into the problem of the implicit social contract that places the teacher as the owner of the knowledge, who is then required to distribute said knowledge to the students in the class. Students want to get the knowledge, to master the procedure and to find the right answers with as little effort or pain as possible. They are not used to working in groups, and find it threatening to their comfortably boring, procedural vision of maths class.

Some years ago I filled in for a maths teacher for a week at a school for girls from privileged backgrounds. I upset one class of Year 12 students by refusing to use up class time getting them to copy notes from the whiteboard. I figured they had perfectly good textbooks, and were better to spend their time working on examples when I was there to help them learn. Silly me! But I was breaking with what they felt was the correct way for them (and me) to behave in maths class. In fact their indignation at my failure to behave in the way they felt I should, actually did get in the way of their learning.

So who is right?

I guess my working theory is that there is a place for many types of learning and teaching in mathematics. Videos can be helpful to introduce ideas, or to provide another way of explaining things. They can help teachers to expand their own understanding, and develop confidence. Videos can provide well-thought-out images and animations to help students understand and remember concepts. They can do something the teacher cannot.  I like to think that our StatsLC videos fit in this category. Talking head or blackboard videos can act as “the kid next door” tutor, who helps a student piece something together.

Just as candy cereal can be only “part of a healthy breakfast”, videos should never be anything more than part of a learning experience.

We also want to think about what kinds of learning we want students to experience. We need our students to be able to communicate, to be creative, to think critically and problem solve and to work collaboratively. These are known as the 4 Cs of 21st Century learning. We don’t actually need people to be able to follow procedures any more. What we need is for people to be able to ask good questions, build models and answer them. I don’t think a procedural approach is going to do that.

The following table summarises some ideas I have about ways of teaching mathematics and statistics.

Procedural approach Social constructivist approach
Main ideas Maths is about choosing and using procedures correctly Maths is about exploring ideas and finding patterns
Strengths Orderly, structured, safe, cover the material, calm Exciting, fun, annoying
Skills valued Computation, memorisation, speed, accuracy Creativity, collaboration, communication, critical thinking
Teaching methods Demonstration, notes, practice 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. Another learner, source of help, sometimes annoyingly oblique
Attitude to mistakes Mistakes are a sign of failure Mistakes happen when we learn.
Challenges Boredom, regimentation, may not develop resilience. Can be difficult to tell if learning is taking place, difficult if the teacher is not confident.
Who succeeds? People like our current maths teachers Not sure – hopefully everyone!
Use of worksheets and textbooks Important – guide the learning Occasional use to supplement activities
Role of videos Can be central Support materials

If we are to have a world of mathematicians, as is our goal as a social enterprise, then we need to move away from a narrow procedural view of mathematics.

I would love to hear your thoughts on this as mathematicians, statisticians, teachers and learners. Do we need to be more careful about the messages our resources such as textbooks and videos give about mathematics and statistics?

The Central Limit Theorem – with Dragons

To quote Willy Wonka, “A little magic now and then is relished by the best of men [and women].” Any frequent reader of this blog will know that I am of a pragmatic nature when it comes to using statistics. For most people the Central Limit Theorem can remain in the realms of magic. I have never taught it, though at times I have waved my hands past it.

Sometimes you don’t need to know.

Students who want that sort of thing can read about it in their textbooks or look it up online. The New Zealand school curriculum does not include it, as I explained in 2012.

But – there are many curricula and introductory statistics courses that include The Central Limit Theorem, so I have chosen to blog about it, in preparation to making a video. In this post I will cover what the Central Limit does. Maybe my approach will give ideas to teachers on how they might teach it.

Sampling distribution of a mean

First let me explain what a sampling distribution is. (And let me add the term to Dr Nic’s long list of statistics terms that cause unnecessary confusion.) A sampling distribution of a mean is the distribution of the means of samples of the same size taken from the same population. The distribution of the means will be different from the distribution of values in the original population.  The Central Limit Theorem tells us useful things about the sampling distribution and its relationship to the distribution of the values in the population.

Example using dragons

We have a population of 720 dragons, and each dragon has a strength value of 1 to 8. The distribution of the strengths goes from 1 to 8 and has a population mean somewhere around 4.5. We take a sample of four dragons from the population. (Dragons are difficult to catch and measure so it will just be 4.)

We find the mean. Then we think about what other values we might have got for samples that size. In real life, that is all we can do. But to understand what is happening, we will take multiple samples using cards, and then a spreadsheet, to explore what happens.

Important aspects of the Central Limit Theorem

Aspect 1: The sampling distribution will be less spread than the population from which it is drawn.

Dragon example

What do you think is the largest value the mean strength of the four dragons will take? Theoretically you could have a sample of four dragons, each with strength of 8, giving us a sample mean of 8. But it isn’t very likely. The chances that all four values are greater than the mean are pretty small.  (It’s about a 6% chance). If there are equal numbers of dragons with each strength value, then the probability of getting all four dragons with strength 8 is 0.0002.

So already we have worked out that the distribution of the sample means is going to be less spread than the distribution of the original population.

Aspect 2: The sampling distribution will be well-modelled by a normal distribution.

Now isn’t that amazing – and really useful! And even more amazing, it doesn’t even matter what the underlying population distribution is, the sampling distribution will still (in most cases) look like a normal distribution.

If you think about it, it does make sense. I like to see practical examples – so here is one!

Dragon example

We worked out that it was really unlikely to get a sample of four dragons with a mean strength of 8. Similarly it is really unlikely to get a sample of four dragons with a mean strength of 1.
Say we assumed that the strength of dragons was uniform – there are equal numbers of dragons with each of the strengths. Then we find out all the possible combinations of strengths from samples of 4 dragons. Bearing in mind there are eight different strengths, that gives us 8 to the power of 4 or 4096 possible combinations. We can use a spreadsheet to enumerate all these equally likely combinations. Then we find the mean strength and we get this distribution.

Or we could take some samples of four dragons and see what happens. We can do this with our cards, or with a handy spreadsheet, and here is what we get.

Four samples of four dragons each

The sample mean values are 4.25, 5.25, 4.75 and 6. Even with really small samples we can see that the values of the means are clustering around some central point.

Here is what the means of 1000 samples of size 4 look like:

And hey presto – it resembles a normal distribution! By that I mean that the distribution is symmetric, with a bulge in the middle and tails in either direction. A normal distribution is useful for modelling just about anything that is the result of a large number of change effects.

The bigger the sample size and the more samples we take, the more the distribution of the means (the sampling distribution) looks like a normal distribution. The Central Limit Theorem gives mathematical explanation for this. I put this in the “magic” category unless you are planning to become a theoretical statistician.

Aspect 3: The spread of the sampling distribution is related to the spread of the population.

If you think about it, this also makes sense. If there is very little variation in the population, then the sample means will all be about the same.  On the other hand, if the population is really spread out, then the sample means will be more spread out too.

Dragon example

Say the strengths of the dragons occur equally from 1 to 5 instead of from 1 to 8. The spread of the means of teams of four dragons are going to go from 1 to 5 also, though most of the values will be near the middle.

Aspect 4: Bigger samples lead to a smaller spread in the sampling distribution.

As we increase the size of the sample, the means become less varied. We reduce the effect of one extreme value. Similarly the chance of getting all high values in our sample or all low values gets smaller and smaller. Consequently the spread of the sample means will decrease. However, the reduction is not linear. By that I mean that the effect achieved by adding one more to the sample decreases, depending on how big the sample is in the first place. Say you have a sample of size n = 4, and you increase it to n = 5, that is a 25% increase in information. If you have a sample n = 100 and increase it to size n=101, that is only a 1% increase in information.

Now here is the coolest thing! The spread of the sampling distribution is the standard deviation of the population, divided by the square root of the sample size. As we do not know the standard deviation of the population (σ), we use the standard deviation of the sample (s) to approximate it. The spread of the sampling distribution is usually called the standard error, or s.e.


Implications of the Central Limit Theorem

The properties listed above underpin most traditional statistical inference. When we find a confidence interval of a mean, we use the standard error in the formula. If we used the sample standard deviation we would be finding the values between which most of the values in the sample lie. By using the standard error, we are finding the values between which most of the sample means lie.

Sample size

The Central Limit Theorem applies best with large samples. A rule of thumb is that the sample should be 30 or more. For smaller samples we need to use the t distribution rather than the normal distribution in our testing or confidence intervals. If the sample is very small, such as less than 15, then we can still use the t-distribution if the underlying population has a normal shape. If the underlying population is not normal, and the sample is small, then other methods, such as resampling should be used, as the Central Limit Theorem does not hold.


We do not take multiple samples of the same population in real life. This simulation is just that – a pretend example to show how the Central Limit Theorem plays out. When we undergo inferential statistics we have one sample, and from that we use what we know about it to make inferences about the population from which it is drawn.

Teaching suggestion

Data cards are extremely useful tools to help understand sampling and other aspects of inference. I would suggest getting the class to take multiple small samples(n=4), using cards, and finding the means. Plot the means. Then take larger samples (n=9) and similarly plot the means. Compare the shape and spread of the distributions of the means.

The Dragonistics data cards used in this post can be purchased at The StatsLC shop.

Dr Nic, Suzy and Gina talk about feelings about Maths

This hour long conversation gives insights into how three high achieving women feel about mathematics. Nicola, the host, is the author of this blog, and has always had strong affection for mathematics, though this has changed in nature lately. Gina and Suzy are both strongly negative in their feelings about maths. As the discussion progresses, listen for the shift in attitude.

Listen here to the podcast.

And here is a picture of the three of us.

Dr Nic, Gina and Suzy

Dr Nic, Gina and Suzy.

Here are some of the questions we discuss over the hour:

  1. Tell me about your relationship with maths.
  2. How do you think your feelings about maths have affected your life?
  3. If you saw this as an opportunity to talk to people who teach mathematics, what message would you like to give them?
  4. How do you feel about the idea that you could change how you feel about maths?

Videos for teaching and learning statistics

It delights me that several of my statistics videos have been viewed over half a million times each. As well there is a stream of lovely comments (with the odd weird one) from happy viewers, who have found in the videos an answer to their problems.

In this post I will outline the main videos available on the Statistics Learning Centre YouTube Channel. They already belong to 24,000 playlists and lists of recommended resources in textbooks the world over. We are happy for teachers and learners to continue to link to them. Having them all in one place should make it easier for instructors to decide which ones to use in their courses.

Philosophy of the videos

Early on in my video production I wrote a series of blog posts about the videos. One was Effective multimedia teaching videos. The videos use graphics and audio to increase understanding and retention, and are mostly aimed at conceptual understanding rather than procedural understanding.

I also wrote a critique of Khan Academy videos, explaining why I felt they should be improved. Not surprisingly this ruffled a few feathers and remains my most commented on post. I would be thrilled if Khan had lifted his game, but I fear this is not the case. The Khan Academy pie chart video still uses an unacceptable example with too many and ordered categories. (January 2018)

Before setting out to make videos about confidence intervals, I critiqued the existing offerings in this post. At the time the videos were all about how to find a confidence interval, and not what it does. I suspect that may be why my video, Understanding Confidence Intervals, remains popular.

Note to instructors

You are welcome to link to our YouTube channel, and we get a tiny amount of money from people clicking on the ads. Please do NOT download the videos, as it is against YouTube rules, and deprives us of income. Note that we also have a separate pay-to-view channel, with considerably more videos, at higher resolution, with no advertising. Email us at for free trial access to these videos, with a view to providing them for your students on a subscription basis. If you have trouble with reliable internet access, we can also provide the videos as files for your network as part of the licence.

Introducing statistics

Understanding Summary Statistics 5:14 minutes

Why we need summary statistics and what each of them does. It is not about how to calculate the statistics, but what they mean. It uses the shoe example, which also appears in the PPDAC and OSEM videos.

Understanding Graphs 6:06 minutes

I briefly explains the use and interpretation of seven different types of statistical graph. They include the pictogram, bar chart, pie chart, dot plot, stem and leaf, scatterplot and time series.


Analysing and commenting on Graphical output using OSEM 7:13 minutes

This video teaches how to comment on graphs and other statistical output by using the acronym OSEM. It is especially useful for students in NCEA statistics classes in New Zealand, but many people everywhere can find OSEM awesome! We use the example of comparing the number of pairs of shoes men and women students say they own.

Variation and Sampling error 6:30 minutes

Statistical methods are necessary because of the existence of variation. Sampling error is one source of variation, and is often misunderstood. This video explains sampling error, along with natural variation, explainable variation and variation due to bias. There is an accompanying video on non-sampling error.

Sampling methods 4:54 minutes 500,000 views

This video describes five common methods of sampling in data collection – simple random, convenience, systematic, cluster and stratified. Each method has a helpful symbolic representation.

Types of data 6:20 minutes 600,000 views

The kind of graph and analysis we can do with specific data is related to the type of data it is. In this video we explain the different levels of data, with examples. This video is particularly popular at the start of courses.

Important Statistical concepts 5:34 minutes 50,000 views

This video does not receive the views it deserves, as it covers three really important ideas. Maybe I should split it up into three videos. The ideas are the difference between significance and usefulness, evidence and strength of effect, causation and association.

Other videos complementary to these, but not on YouTube are:

  • The statistical enquiry process
  • Understanding the Box Plot
  • Non-sampling error

Videos for teaching hypothesis testing

Understanding Statistical inference 6:46 minutes 40,000 views

The most difficult concept in statistics is that of inference. This video explains what statistical inference is and gives memorable examples. It is based on research around three concepts pivotal to inference – that the sample is likely to be a good representation of the population, that there is an element of uncertainty as to how well the sample represents the population, and that the way the sample is taken matters.

Understanding the p-value 4:43 minutes 500,000 views

This video explains how to use the p-value to draw conclusions from statistical output. It includes the story of Helen, making sure that the choconutties she sells have sufficient peanuts. It introduces the helpful phrase “p is low, null must go”.

Inference and evidence 3:34 minutes

This is a newer video, based on a little example I used in lectures to help students see the link between evidence and inference. Of course it involves chocolate.

Hypothesis tests 7:38 minutes 350,000 views

This entertaining video works step-by-step through a hypothesis test. Helen wishes to know whether giving away free stickers will increase her chocolate sales. This video develops the ideas from “Understanding the p-value”, giving more of the process of hypothesis testing. It is also complemented by the following video, that shows how to perform the analysis using Excel.

Two-means t-test in Excel 3:54 minutes 50,000 views

A step-by-step lesson on how to perform an independent samples t-test for difference of two means using the Data Analysis ToolPak in Excel. This is a companion video to Hypothesis tests, p-value, two means t-test.

Choosing which statistical test to use 9:33 minutes 500,000 views

I am particularly proud of this video, and the way it links the different tests together. It took a lot of work to come up with this. First it outlines a process for thinking about the data, the sample and the thing you are trying to find out. Then it works through seven tests with scenarios based around Helen and the Choconutties. This video is particularly popular near the end of the semester, for tying together the different tests and applications.


Confidence Intervals

Understanding Confidence Intervals 4:02 minutes 500,000 views

This short video gives an explanation of the concept of confidence intervals, with helpful diagrams and examples. The emphasis is on what a confidence interval is and how it is used, rather than how they are calculated or derived.

Calculating the confidence interval for a mean using a formula 5:29 minutes 200,000 views

This video carries on from “Understanding Confidence Intervals” and introduces a formula for calculating a confidence interval for a mean. It uses graphics and animation to help understanding.

There are also videos pertinent to the New Zealand curriculum using bootstrapping and informal methods to find confidence intervals.


Introduction to Probability 2:54 minutes

This video explains what probability is and why we use it. It does NOT use dice, coins or balls in urns. It is the first in a series of six videos introducing basic probability with a conceptual approach. The other five videos can be accessed through subscription.

Understanding Random Variables 5:08 minutes 90,000 views

The idea of a random variable can be surprisingly difficult. In this video we help you learn what a random variable is, and the difference between discrete and continuous random variables. It uses the example of Luke and his ice cream stand.

Understanding the Normal Distribution 7:44 minutes

In this video we explain the characteristics of the normal distribution, and why it is so useful as a model for real-life entities.

There are also two other videos about random variables, discrete and continuous.

Risk and Screening 7:54 minutes

This video explains about risk and screening, and shows how to calculate and express rates of false positives and false negatives. An imaginary disease, “Earpox” is used for the examples.

Other videos

Designing a Questionnaire 5:23 minutes 40,000 views

This was written specifically to support learning in Level 1 NCEA in the NZ school system but is relevant for anyone needing to design a questionnaire. There is a companion video on good and bad questions.

Line-fitting and regression

Scatterplots in Excel 5:17 minutes

The first step in doing a regression in Excel is to fit the line using a Scatter plot. This video shows how to do this, illustrated by the story of Helen and the effect of temperature on her sales of choconutties

Regression in Excel 6:27 minutes

This video explains Regression and how to perform regression in Excel and interpret the output. The story of Helen and her choconutties continues. This follows on from Scatterplots in Excel and Understanding the p-value.

There are three videos introducing bivariate relationships in a more conceptual way.

There are also videos covering experimental design and randomisation, time series analysis and networks. In the pipeline is a video “understanding the Central Limit Theorem.”

Supporting our endeavours

As explained in a previous post, Lessons for a budding Social Enterprise, Statistics Learning Centre is a social enterprise, with our aim to build a world of mathematicians and enable people to make intelligent use of statistics. Though we get some income from YouTube videos, it does not support the development of more videos. If you would like to help us to create further videos contact us to discuss subscriptions, sponsorship, donations and advertising possibilities. or

Mind the gap

Teach the students you have

Our job as teachers at any level is to teach the students we have. I embrace this idea from Dr Kevin Maxwell:

“Our job is to teach the students we have.
Not the ones we would like to have.
Not the ones we used to have.
Those we have right now.
All of them.”

I believe Maxwell’s focus was on the diverse learning needs we have in our classes. I would like to take another angle on this. If students do not have the needed skills to learn what we are teaching, then we need to teach those skills.

In many subjects, content and the skills are largely uncoupled. For example in history, a skill might be to integrate material from two conflicting sources. You can learn this in multiple contexts, and you do not need to know the history of the world up until 1939 in order to study World War II.

In mathematics, there are clear progressions. It is very difficult to learn about trigonometry if you do not have a good working knowledge of the Pythagorean theory. And learning Pythagoras is built on applying formulas, which is built on basic algebra. I admit, that as I write this I can see other approaches, but the point is that later learning in maths is built on earlier knowledge, understanding and skills. Learning in maths is also built on earlier feelings – a post for another day.

The Gaps

There are two gaps we need to mind. The gap between levels of schooling, and the gap between what the preparation the students need, and what they have. I taught at the University of Canterbury for twenty years, and often heard colleagues complain about the level of preparation in our students. I am ashamed to say that it took me several years to realise that if our students do not have the foundation they need to learn what we are teaching, then we need to do something about it. As a result I created a course that started with making sure students knew how to use < and >, and which is bigger out of 0.04 and  0.2. These are necessary in order to make decisions about rejecting a null hypothesis.

Recently at a workshop I asked a group of about forty teachers how many of them have students starting high school who do not have the necessary knowledge of number skills – basic facts and multiplication tables. Every hand went up. There is a gap. I asked them what they are doing about it. Some suggested working in “Communities of Learning” to help primary schools to prepare the students better. This is fine, but what are they doing now! There was some discussion that if we are teaching lower curriculum levels at high school, they may never cover the materials at higher levels.

For that I have two responses. The first relates to the Maxwell quote I started with. “Our job is to teach the students we have.” Our job is to teach the students we have, the things they need to learn. If our students start high school without a good enough grasp of basic facts, then we need to help them to develop them. And we need to work out good ways to do this. I suspect part of the problem is that secondary maths teachers do not have training or knowledge in teacher beginning maths. Do we believe it is not part of our job?

The second response is that there is no point in moving on to later maths if the students’ foundation is weak. Now I say this with some trepidation as I can picture students being held back until they become fluent in their tables. This is not what I mean. One of the participants in the workshop asked me how I would go about setting up a programme to help such students. Obviously this is not a question I could answer on the spot, but here are some ideas and principles.

Ideas and principles for building foundation skills


  • Do not under any circumstances give these students tests with time pressure.
  • Expect the students to be able to learn what is needed more quickly than they would have done when younger.
  • Engage students in deciding what they need to learn and how.
  • Integrate the skills into other activities
  • Make it fun


No time pressure

Read Fluency without fear by Jo Boaler. Read this about Maths trauma. Do not add to the students’ feelings of inadequacy. One possibility if you wish to give a diagnostic test, and want to have some idea of how long they take, tell them they have as long as they need, but after a certain amount of time get them to change to a different coloured pen.

Learn quickly

My experience with teaching adults and teens is that once they realise they can learn, they learn quickly. Believe it. I don’t mean that they can answer questions quickly, but that they will be able to progress more quickly as they have better metacognitive skills, literacy, maturity.

Student agency

This is their learning. Make sure the students know why they need the skills and how they will help them. Talk to students about how they would like to learn them, and let them choose their own reward system if appropriate. Different students will have different areas of weakness, and different ways to improve.

Integrate the skills into other activities

I can’t imagine much worse than an entire maths lesson on basic facts. If we are working on multiplication, this fits well with area calculations. We also need to keep revisiting.

There is a place for well made and used flash cards to improve retrieval. There are multiple posts on using flashcards well. I would recommend them for some students for the last sticky facts, like 6 x 7, 6 x 8, 7 x 8 etc. Those were the ones that got me stumped. However, most knowledge is better gained in context. Create or find rich, open-ended tasks that help develop the skills the students need.

Make it fun

Maths lends itself to games and fun. If you can’t think up a way – find it on line. But if you don’t think it’s fun the students aren’t going to. (Not sure the converse is true, but…)

Mind the Gap

Our aim at Statistics Learning Centre is a world of mathematicians. My dream is for math trauma to be a thing of the past, and for all citizens to embrace mathematical thinking similarly to literacy. As maths and statistics educators we can work towards this. The most important student you have in your maths class is the one who becomes a primary school maths teacher. Make sure she loves maths!

Rich maths with Dragons

Thanks to the Unlocking Curious Minds fund, StatsLC have been enabled to visit thirty rural schools in Canterbury and the West Coast and provide a two-hour maths event to help the children to see themselves as mathematicians. The groups include up to 60 children, ranging from 7 to 12 years old – all mixed in together. You can see a list of the schools we have visited on our Rich Maths webpage. And here is a link to another story about us from Unlocking Curious Minds.

What mathematicians do

What do mathematicians do?

We begin by talking about what mathematicians do, drawing on the approach Tracy Zager uses in “Becoming the Math teacher you wish you had”. (I talk more about this in my post on What Mathematicians do.)

  • Mathematicians like a challenge.
  • Mathematicians notice things and wonder
  • Mathematicians make mistakes and learn
  • Mathematicians work together and alone.
  • Mathematicians have fun.

You can see a video of one of our earlier visits here.

Each child (and teacher) is given a dragon card on a lanyard and we do some “noticing and wondering” about the symbols on the cards. We find that by looking at other people’s dragons as well as our own, we can learn more. As each of the symbols is explained, there follows an excited buzz as children discuss whose dragon is stronger or older, or has more dangerous breath.  We wonder if green dragons are more friendly than red dragons and work together, making a human data table, and using proportional thinking to draw some conclusions.

Dragonistics data cards

A small sample of Dragonistics data cards

Mixed group work

Next, in randomly chosen, mixed level groups of three, the children perform their own statistical investigations. They have randomly assigned roles, as dragon minder (looking after the cards), people minder (making sure everyone is participating) or record minder (making sure something gets written down). They take their roles seriously, and only occasionally does a group fail to work well. The teachers are free to observe or join in, while Shane and I go from group to group observing and providing guidance and feedback. All learners can take part at their own level.

As we visit a variety of schools we can see the children who are more accustomed to open-ended activities. In some schools, and with the older children, they can quickly start their own investigations. Other children may need more prompting to know where to begin. Sometimes they begin by dividing up the 24 cards among the three children, but this is not effective when the aim is to study what they can find from a group of dragons.

Levels of analysis

It is interesting to observe the levels of sophistication in their analysis. Some groups start by writing out the details of each individual card. I find it difficult to refrain from moving them on to something else, but have come to realise that it is an important stage for some children, to really get to understand the multivariate nature of the data before they begin looking at properties of the group. Others write summaries of each of the individual characteristics. And some engage in bivariate or multivariate investigations. In a sequence of lessons, a teacher would have more time to let the learners struggle over what to do next and to explore, but in our short timeframe we are keen for them to find success in discovering something. After about fifteen minutes we get their attention, and get them to make their way around the room and look at what the other groups are finding out. “Mathematicians learn from other mathematicians”, we tell them.


Sometimes groups think they have discovered everything there is to know about their set of dragons, so we have a range of “claims” for them to explore. These include statements such as:

  • Is this true? “There are more green dragons than red dragons.”
  • Is this true? “Changeable dragons are less common than friendly or dangerous dragons.”
  • Is this true? “There are more dragons younger than 200 than older than 200.”
  • Is this true? “Fire breathing dragons are mainly female.”
  • Is this true? “There are no fire breathing, dangerous green dragons.”
  • Is this true? “Strong dragons are more dangerous.”

Some of the claims are more easily answered than others, and all hint at the idea of sample and population in an intuitive rather than explicit way. Many of them require decisions from the learners, such as what does “mainly” mean, and how you would define a “strong” dragon?

The children love to report back their findings.  Depending on the group and the venue, we also play big running around games where they have to form pairs and groups, such as 2 metres different in height, one of each behaviour, or nothing at all the same. That has proved one of the favourite activities, and encourages communication, mathematical language – and fun! Then we let them choose their own groups and choose from a range of mathematical activities involving the Dragonistics data cards.

The children work on one or more of the activities in groups of their own choice, or on their own. Then in the last fifteen minutes we gather them together to revisit the five things that mathematicians do, and liken it to what they have been doing. We get the children to ask questions, and we leave a set of Dragonistics data cards with the school so they can continue to use them in their learning. It is a blast! We have had children tell us it feels like the first time they have ever enjoyed mathematics. Every school is different, and we have learned from each one.

Solved the puzzle!Three mathematicians showing their work

A wise intervention

The aim is for our event to help children to change the way they feel about maths in a way that empowers them to learn in the future. There has been research done on “wise interventions”, which have impact greater than their initial effect, due to ongoing ripples of influence. We believe that helping students to think about struggle, mistakes and challenge in mathematics in a positive light, and to think of themselves as mathematicians can reframe future events in maths. When they find things difficult, they may see that as being a mathematician, rather than as failing.

Lessons for us

This is a wonderful opportunity for us to repeat a similar activity with multiple groups, and our practice and theory are being informed by this. Here is an interesting example.

At the beginning of the open-choice section, we outline the different activities that the children can choose from. One is called “Activity Sheets”, which has varying degrees of challenge. It seems the more we talk up the level of challenge in one of the activities, the keener the children are to try it. Here is a picture of the activity:

Challenging 9 card

The activity involves placing nine dragons cards in position to make all of the statements true. Originally the packs included just 20 dragons, and by swapping in and out, it is challenging. However, when you have just nine dragons to place, it can be very difficult. Now for the first few visits, when children rushed to show us how they had completed their sheet, we would check it for correctness. However, through reading, thinking and discussion we have changed out behaviour. We wish to put the emphasis on the learning, and on the strategy. Peter Johnston in his book, “Choice words: how our language affects children’s learning” states,

“The language we choose in our interactions with children influences the ways they frame these events and the ways the events influence their developing sense of agency.”

When we simply checked their work, we retained our position as “expert”. Now we ask them how they know it is correct, and what strategies they used. We might ask if they would find it easier to do it a second time, or which parts are the trickiest. By discussing the task, rather than the result, we are encouraging their enjoyment of the process rather than the finished product.

We hope to be able to take these and other activities to many more schools either in person or through other means, and thus spread further the ripples of mathematical and statistical enjoyment.

Dragon Trainer rich mathematical task

I love rich mathematical tasks. Here is one for all levels of schooling. What do you think?

Background to rich tasks

A rich task is an open-ended task that students can engage with at multiple levels. I use the following information from the nrich website when I am talking to teachers about rich tasks.

Some important aspects of rich mathematical tasks

Background to Dragonistics data cards

In this task we use our Dragonistics data cards, which are shown here. For a less colourful exercise you could use 24 pieces of card with numbers 1 to 8 on them.

A small sample of Dragonistics data cards

Each dragon has a strength rating of between 1 and 8, shown by the coloured dragon scales on the right-hand side of the card. The distribution of dragon strengths is not uniform, but is clustered around the middle, and depends to a certain extent on other aspects of the dragon, such as their species, gender and behaviour.

The students will already be familiar with the dragon cards, and each group of students has a set of about 24 dragonistics data cards. As there are a total of 288 dragons, each group will have a different set of dragons. Some may or may not have dragons of each strength rating.

The task

A dragon team trainer says that teams of two dragons chosen at random nearly always have a combined strength of between 7 and 11.
Is this true?
Provide evidence to support your conclusion.

Try it yourself

If you do not have any dragons of your own, make up about 20 pieces of card, with the numbers 1 to 8 on them, so you can explore the problem. Like Tracy Zager, we emphasise the necessity of exploring the maths ourselves before the children.

Possible approaches

What is great about this exercise is that you can explore it experimentally or theoretically. It has a low entry point, as encouraged on Youcubed. This is sometimes called “low floor, high ceiling”.For younger children, it is a good start to take pairs of dragons, add their strengths, and write down the answer. Then they need to work out a recording method, possibly a tally table.  You can have discussions about what it means for the dragons to be chosen randomly. You can also discuss what “nearly always” means.

Recently I used this task with a group of ten-year-olds. After they had made an attempt at solving it, I asked what they thought would be the most common team strength, and one said 9 or 10 because it is in the middle. I should have explored this idea further. What I did do, was start working out how many different combinations were possible. It is not possible to have a team of strength 1, and there is only one way to get a team of strength 2. How many ways to get strength 3? By the time we got to strength 6, they could see a pattern, that the number of combinations is one less than the total strength. So then I jumped to the other end of the distribution, asking “What is the strongest team we could possibly get?” As it happened, they did have two dragons of strength 8 in their set of dragons, so they correctly estimated the answer to be 16. So then I asked how many different ways they could get 16, and using their previous rule, they suggested 15 ways.  Then when I asked them to tell me what they were, they realised that there was only one way. From there we started working down the numbers. Unfortunately this was during a holiday programme, so I didn’t have time to pursue this further. However we will be using this exercise in our rural rich maths events.

Lessons to bring out at different levels

There are three main ways to approach this problem. The first is to experiment by randomly taking pairs of dragons, and recording their total strengths. A simple theoretical model involves thinking about all the possible outcomes and seeing what proportion of the outcomes lies between the chosen values. Then a more refined model would take into account the distributions of strengths for the given dragons.  The learners may well come up with some interesting other ways to go about this.

Extension questions

A teacher can encourage further thinking with questions such as:

Would this answer be the same for every group of dragons? Is it possible to find a set of dragons so that the only team strengths are between 5 and 11? What would happen if you had teams of three dragons. Does it make a difference if you select one team at a time, and shuffle, or divide into lots of teams and record, before shuffling? How many different team possibilities are possible? What if you only had green dragons – would this make a difference?

Show them the maths

It is important to point out the mathematical skills they are exercising as they tackle rich tasks. This task improves number skills, encourages persistence and risk-taking, develops communication skills as they are required to justify their conclusion. At higher levels it is helping to develop understanding of probability distributions, and you could also introduce or reinforce the idea of a random variable – in this case the team strength.

It would also be interesting to look at the spread for single dragons, two dragon teams and three dragon teams. With enough repetitions (and at this point a spreadsheet could be handy) the central limit theorem will start to be apparent. As you can see, there is great potential to expand this.


We need to look at ways this is also applicable in daily life, and not just for dragon trainers. The same sort of problem would occur if you had people buying different numbers of items, or different weights of suitcases. You might like to think of the combined strengths as similar to total scores in sports events. At higher levels you might discuss the concept of independence.

So rich – so many possibilities! Thoughts?

Mathematics and statistics lessons about elections

Study elections in mathematics because it is important

Too often mathematics is seen as pure and apolitical.  Maths teachers may keep away from concepts that seem messy and without right and wrong answers. However, teachers of mathematics and statistics have much to offer to increase democratic power in the upcoming NZ general elections (and all future elections really). The bizarre outcomes for elections around the world recently (2016/2017 Brexit, Trump) are evidence that we need a compassionate, rational, informed populace, who is engaged in the political process, to choose who will lead our country. Knowledge is power, and when people do not understand the political process, they are less likely to vote. We need to make sure that students understand how voting, the electoral system, and political polls work. Some of our students in Year 13 will be voting this election, and students’ parents can be influenced to vote.

There are some lessons provided on the Electoral Commission site.   Sadly all the teaching resources are positioned in the social studies learning area – with none in statistics and mathematics. Similarly in the Senior Secondary guides, all the results from a search on elections were in the social studies subject area.

Elections are mathematically and statistically interesting and relevant

In New Zealand, our MMP system throws up some very interesting mathematical processes for higher level explorations. Political polls will be constantly in the news, and provide up-to-date material for discussions about polls, sample sizes, sampling methods, sampling error etc.


It would be great to hear from anyone who uses these ideas. If you have developed them further, so much the better. Do share with us in the comments.

Suggestions for lessons

These suggestions for lessons are listed more or less in increasing levels of complexity. However I have been amazed at what Year 1 children can do. It seems to me that they are more willing to tackle difficult tasks than many older children. These lessons also embrace other curriculum areas such as technology, English and social studies.

Physical resources

Make a ballot box, make a voting paper. Talk about randomising the names on the paper. How big does the box need to be? How many ballot boxes are being made for the upcoming election? How much cardboard is needed?

Follow the polls

Make a time series graph of poll results. Each time there is a new result, plot it on the graph over the date, and note the sample size. At higher levels you might like to put confidence intervals on either side of the plotted value. A rule of thumb is 1/square root of the sample size. For example if the sample size is 700, the margin of error is 3.7%. So if the poll reported a party gaining 34% of the vote, the confidence interval would be from 33.3% to 37.7%.

You can get a good summary of political polls on Wikipedia.

From NZ maths  – On the Campaign Trail (CL 4)

Figure it Out, Number sense  Book 2 Level 4 – has an exercise about finding fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.

From NZ maths – Whose News (CL 4)

This is a guide to running an analysis on the level of representation of different geopgraphic areas in the news. The same lesson could be used for representation of different parties or different issues.

Graphical representations

The newspapers and online will be full of graphs and other graphical representations. Keep a collection and evaluate them for clarity and attractiveness.

How many people will be employed on election day?

This inquiry uses a mixture of internet search, mathematical modelling, estimation and calculation.

  • How many electorates are there?
  • How many polling booths per electorate?
  • How many people per booth?
  • How long are they employed for?

Fairness of booth provision

  • Is the location of polling booths fair?
  • What is the furthest distance a person might need to travel to a voting booth?
  • What do people in other countries do?

The mathematics of MMP

This link provides a thorough explanation of the system. A project could be for students to work out what it is saying and make a powerpoint presentation or short video explaining it more simply.

Overhang and scenario modelling

Overhang occurs when a party gets more electoral MPS elected than their proportion allows. Here is a fact sheet about overhang and findings of the electoral review. Students could create scenarios to evaluate the effect of overhang and find out what is the biggest overhang possible.

Small party provisions

How might the previous two election results have been different if there were not the 5% and coat-tailing rules?


Different ways of assigning areas to electorates get different results. The Wikipedia article on Gerrymandering has some great examples and diagrams on how it all happens, and the history behind the name.

Statistical analysis of age and other demographics

Statistics should be analysed in response to a problem, rather than just for the sake of it.
Suggested Scenario: A new political party is planning to appeal to young voters, under 30 years of age. They wish to find out which five electorates are the best to target. You may also wish to include turn-out statistics in your analysis.

Resource: Enrolment statistics by electorate – some graphs supplied, percentages for each electorate.

Statistical analysis of turn out

In the interests of better democracy, we wish to have a better voter turnout. Find out the five electorates with the best voter turnout and the worst, and come up with some ideas about why they are the best and the worst. Test out your theory/model by trying to predict the next five best and worst. Use what you find out to suggest how might we improve voter turnout.

Resource: Turn out statistics – by electorate or download the entire file

Happy teaching, and fingers crossed for September.

Graphs – beauty and truth

Graphs – beauty and truth (with apologies to Keats)

A good graph is elegant

I really like graphs. I like the way graphs turn numbers into pictures. A good graph is elegant. It uses a few well-placed lines to communicate what would take a paragraph of text. And like a good piece of literature or art, a good graph continues to give, beyond the first reading. I love looking at my YouTube and WordPress graphs. These graphs tell me stories. The WordPress analytics tell me that when I put up a new post, I get more hits, but that everyday more than 1000 people read one of my posts. The YouTube analytics tell me stories about when people want to know about different aspects of statistics. It is currently the end of the North American school year, and the demand is for my video on Choosing which statistical test to use. Earlier in the year, the video about levels of measurement is the most popular. And not many people view videos about statistics on the 25th of December. I’m happy to report that the YouTube and WordPress graphs are good graphs.

Spreadsheets have made it possible for anyone and everyone to create graphs. I like that graphs are easier to make. Drawing graphs by hand is a laborious task and fraught with error. But sometimes my heart aches when I see a graph used badly. I suspect that this is when a graphic artist has taken control, and the search for beauty has over-ridden the need for truth.

Three graphs spurred me to write this post.

Graph One: Bad-tasting Donut on house occupation

The first was on a website to find out about property values. I must have clicked onto something to find out about the property values in my area, and was taken to the qv website. And this is the graph that disturbed me.

Graphs named after food are seldom a good idea

Sure it is pretty – uses pretty colours and shading, and you can find out what it is saying by looking at the key – with the numbers beside it. But a pie or donut chart should not be used for data which has inherent order. The result here is that the segments are not in order. Or rather they are ordered from most frequent to least frequent, which is not intuitive. Ordinal data is best represented in a bar or column chart. To be honest, most data is best represented in a bar or column chart. My significant other suggested that bar charts aren’t as attractive as pie charts. Circles are prettier than rectangles. Circles are curvy and seem friendlier than straight lines and rectangles. So prettiness has triumphed over truth.

Graph Two: Misleading pictogram (a tautology?)

It may be a little strong to call bad communication lack of truth. Let’s look at another example. In a way it is cheating to cite a pictogram in a post like this. Pictograms are the lowest form of graph and are so often incorrect, that finding a bad one is easier than finding a good one. In the graph below of fatalities it is difficult to work out what one little person represents.

What does one little person represent?

A quick glance, ignoring the numbers, suggests that the road toll in 2014 is just over half what it was in 2012. However, the truth, calculated from the numbers, is that the relative size is 80%. 2012 has 12 people icons, representing 280 fatalities. One icon is removed for 2013, representing a drop of 9 fatalities. 2011 has one icon fewer again, representing a drop of 2 fatalities. There is so much wrong in the reporting of road fatalities, that I will stop here. Perhaps another day…

Graph Three: Mysterious display on Household income

And here is the other graph that perplexed me for some time. It came in the Saturday morning magazine from our newspaper, as part of an article about inequality in New Zealand. Anyone who reads my blog will be aware that my politics place me well left of centre, and I find inequality one of the great ills of the modern day. So I was keen to see what this graph would tell me. And the answer is…

See how long it takes for you to find where you appear on the graph. (Pretending you live in NZ)

I have no idea. Now, I have expertise in the promulgation of statistics, and this graph stumped me for some time. Take a good look now, before I carry on.

I did work out in the end, what was going on in the graph, but it took far longer than it should. This article is aimed at an educated but not particularly statistically literate audience, and I suspect there will be very few readers who spent long enough working out what was going on here. This graph is probably numerically correct. I had a quick flick back to the source of the data (who, by the way, are not to be blamed for the graph, as the data was presented in a table) and the graph seems to be an accurate depiction of the data. However, the graph is so confusing as to be worse than useless. Please post critiques in the comments. This graph commits several crimes. It is difficult to understand. It poses a question and then fails to help the reader find the answer. And it does not provide insights that an educated reader could not get from a table. In fact, I believe it has obscured the data.

Graphs are the main way that statistical analysts communicate with the outside world. Graphs like these ones do us no favours, even if they are not our fault. We need to do better, and make sure that all students learn about graphs.

Teaching suggestion – a graph a day

Here is a suggestion for teachers at all levels. Have a “graph a day” display – maybe for a month? Students can contribute graphs from the news media. Each day discuss what the graph is saying, and critique the way the graph is communicating. I have a helpful structure for reading graphs in my post: There’s more to reading graphs than meets the eye; 

Here is a summary of what I’ve said and what else I could say on the topic.

Thoughts about Statistical Graphs

  • The choice of graph depends on the purpose
  • The text should state the purpose of the graph
  • There is not a graph for everything you wish to communicate
  • Sometimes a table communicates better than a graph
  • Graphs are part of the analysis as well as part of the reporting. But some graphs are better to stay hidden.
  • If it takes more than a few seconds to work out what a graph is communicating it should either be dumped or have an explanation in the text
  • Truth (or communication) is more important than beauty
  • There is beauty in simplicity
  • Be aware than many people are colour-blind, or cannot easily differentiate between different shades.

Feedback from previous post on which graph to use

Late last year I posted four graphs of the same data and asked for people’s opinions. You can link back to the post here and see the responses: Which Graph to Use.

The interesting thing is not which graph was selected as the most popular, but rather that each graph had a considerable number of votes. My response is that it depends.  It depends on the question you are answering or the message you are sending. But yes – I agree with the crowd that Graph A is the one that best communicates the various pieces of information. I think it would be improved by ordering the categories differently. It is not very pretty, but it communicates.

I recently posted a new video on YouTube about graphs. It is a quick once-over of important types of graphs, and can help to clarify what they are about. There are examples of good graphs in there.

I have written about graphs previously and you can find them here on the Collected Works page.

I’m interested in your thoughts. And I’d love to see some beautiful and truthful graphs in the comments.