Spreadsheets, statistics, mathematics and computational thinking

We need to teach all our students how to design, create, test, debug and use spreadsheets. We need to teach this integrated with mathematics, statistics and computational thinking. Spreadsheets can be a valuable tool in many other subject areas including biology, physics, history and geography, thus facilitating integrated learning experiences.

Spreadsheets are versatile and ubiquitous – and most have errors. A web search on “How many spreadsheets have errors?” gives alarming results. The commonly quoted figure is 88%. These spreadsheets with errors are not just little home spreadsheets for cataloguing your Lego collection or planning your next vacation. These spreadsheets with errors involve millions of dollars, and life-affecting medical and scientific research.

Using spreadsheets to teach statistics

Use a spreadsheet to draw graphs

One of the great contributions computers make to statistical analysis is the ability to display graphs of non-trivial sets of data without onerous drawing by hand. In the early 1980s I had a summer job as a research assistant to a history professor. One of my tasks was to create a series of graphs of the imports and exports for New Zealand over several decades, illustrating the effect of the UK joining the Common Market (now the EU). It required fastidious drawing and considerable time. (And correcting fluid) These same graphs can now be created almost instantaneously, and the requirement has shifted to interpreting these graphs.

Similarly, in the classroom we should not be requiring students of any age to draw statistical graphs by hand. Drawing statistical graphs by hand is a waste of time. Students may enjoy creating the graphs by hand – I understand that – it is rewarding and not cognitively taxing. So is colouring in. The important skill that students need is to be able to read the graph – to find out what it is telling them and what it is not telling them. Their time would be far better spent looking at multiple graphs of different types, and learning how to report and critique them. They also need to be able to decide what graph will best show what they are looking for or communicating. (There will be teachers saying students need to draw graphs by hand to understand them. I’d like to know the evidence for this claim. People have said for years that students need to calculate standard deviation by hand to understand it, and I reject that also.)

At primary school level, the most useful graph is almost always the bar or column chart. These are easily created physically using data cards, or by entering category totals and using a spreadsheet. Here is a video showing just how easy it is.

Use a spreadsheet for statistical calculations

Spreadsheets are also very capable of calculating summary statistics and creating hypothesis tests and confidence intervals. Dedicated statistical packages are better, but spreadsheets are generally good enough. I would also teach pivot-tables as soon as possible, but that is a topic for another day.

Using spreadsheets to teach mathematics

Spreadsheets are so versatile! Spreadsheets help students to understand the concept of a variable. When you write a formula in a cell, you are creating an algebraic formula. Spreadsheets illustrate the need for sensible rounding and numeric display. Use of order of operations and brackets is essential. They can be used for exploring patterns and developing number sense. I have taught algebraic graphing, compared with line fitting using spreadsheets. Spreadsheets can solve algebraic problems. Spreadsheets make clear the concept of mathematics as a model. Combinatorics and Graph Theory are also enabled through spreadsheets. For users using a screenreader, the linear nature of formulas in spreadsheets makes it easier to read.

Using spreadsheets to teach computational thinking

In New Zealand we are rolling out a new curriculum for information technology, including  computational thinking. At primary school level, computational thinking includes “[students] develop and debug simple programs that use inputs, outputs, sequence and iteration.” (Progress outcome 3, which is signposted to be reached at about Year 7) Later the curriculum includes branching.

In most cases the materials include unplugged activities, and coding using programmes such as Scratch or Java script. Robots such as Sphero and Lego make it all rather exciting.

All of these ideas can also be taught using a spreadsheet. Good spreadsheet design has clear inputs and outputs. The operations need to be performed in sequence, and iteration occurs when we have multiple rows in a spreadsheet. Spreadsheets need to be correct, robust and easy to use and modify. These are all important principles in coding. Unfortunately too many people have never had the background in coding and program design and thus their spreadsheets are messy, fragile, oblique and error-prone.

When we teach spreadsheets well to our students we are giving them a gift that will be useful for their life.

Experience teaching spreadsheets

I designed and taught a course in quantitative methods for business, heavily centred on spreadsheets. The students were required to use spreadsheets for mathematical and statistical tasks. Many students have since expressed their gratitude that they are capable of creating and using spreadsheets, a skill that has proved useful in employment.

 

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Statistical software for worried students

Statistical software for worried students: Appearances matter

Let’s be honest. Most students of statistics are taking statistics because they have to. I asked my class of 100 business students who choose to take the quantitative methods course if they did not have to. Two hands went up.

Face it – statistics is necessary but not often embraced.

But actually it is worse than that. For many people statistics is the most dreaded course they are required to take. It can be the barrier to achieving their career goals as a psychologist, marketer or physician. (And it should be required for many other careers, such as journalism, law and sports commentator.)

Choice of software

Consequently, we have worried students in our statistics courses. We want them to succeed, and to do that we need to reduce their worry. One decision that will affect their engagement and success is the choice of computer package. This decision rightly causes consternation to instructors. It is telling that one of the most frequently and consistently accessed posts on this blog is Excel, SPSS, Minitab or R. It has been  viewed 55,000 times in the last five years.

The problem of which package to use is no easier to solve than it was five years ago when I wrote the post. I am helping a tertiary institution to re-develop their on-line course in statistics. This is really fun – applying all the great advice and ideas from ”
Guidelines for Assessment and Instruction in Statistics” or GAISE. They asked for advice on what statistics package to use. And I am torn.

Requirements

Here is what I want from a statistical teaching package:

  • Easy to use
  • Attractive to look at (See “Appearances Matter” below)
  • Helpful output
  • Good instructional materials with videos etc (as this is an online course)
  • Supports good pedagogy

If I’m honest I also want it to have the following characteristics:

  • Guidance for students as to what is sensible
  • Only the tests and options I want them to use in my course – not too many choices
  • An interpretation of the output
  • Data handling capabilities, including missing values
  • A pop up saying “Are you sure you want to make a three dimensional pie-chart?”

Is this too much to ask?

Possibly.

Overlapping objectives

Here is the thing. There are two objectives for introductory statistics courses that partly overlap and partly conflict. We want students to

  • Learn what statistics is all about
  • Learn how to do statistics.

They probably should not conflict, but they require different things from your software. If all we want the students to do is perform the statistical tests, then something like Excel is not a bad choice, as they get to learn Excel as well, which could be handy for c.v. expansion and job-getting. If we are more concerned about learning what statistics is all about, then an exploratory package like Tinkerplots or iNZight could be useful.

Ideally I would like students to learn both what statistics is all about and how to do it. But most of all, I want them to feel happy about doing statistical analysis.

Appearances matter

Eye-appeal is important for overcoming fear. I am confident in mathematics, but a journal article with a page of Greek letters and mathematical symbols, makes me anxious. The Latex font makes me nervous. And an ugly logo puts me off a package. I know it is shallow. But it is a thing, and I suspect I am far from alone. Marketing people know that the choice of colour, word, placement – all sorts of superficial things effect whether a product sells. We need to sell our product, statistics, and to do that, it needs to be attractive. It may well be that the people who design software are less affected by appearance, but they are not the consumers.

Terminal or continuing?

This is important: Most of our students will never do another statistical analysis.

Think about it :

Most of our students will never do another statistical analysis.

Here are the implications: It is important for the students to learn what statistics is about, where it is needed, potential problems and good communication and critique of statistical results. It is not important for students to learn how to program or use a complex package.

Students need to experience statistical analysis, to understand the process. They may also discover the excitement of a new set of data to explore, and the anticipation of an interesting result. These students may decide to study more statistics, at which time they will need to learn to operate a more comprehensive package. They will also be motivated to do so because they have chosen to continue to learn statistics.

Excel

In my previous post I talked about Excel, SPSS, Minitab and R. I used to teach with Excel, and I know many of my past students have been grateful they learned it. But now I know better, and cannot, hand on heart recommend Excel as the main software. Students need to be able to play with the data, to look at various graphs, and get a feel for variation and structure. Excel’s graphing and data-handling capabilities, particularly with regard to missing values, are not helpful. The histograms are disastrous. Excel is useful for teaching students how to do statistics, but not what statistics is all about.

SPSS and Minitab

SPSS was a personal favourite, but it has been a while since I used it. It is fairly expensive, and chances are the students will never use it again. I’m not sure how well it does data exploration. Minitab is another nice little package. Both of these are probably overkill for an introductory statistics course.

R and R Commander

R is a useful and versatile statistical language for higher level statistical analysis and learning but it is not suitable for worried students. It is unattractive.

R Commander is a graphical user interface for R. It is free, and potentially friendlier than R. It comes with a book. I am told it is a helpful introduction to R. R Commander is also unattractive. The book was formatted in Latex. The installation guide looks daunting. That is enough to make me reluctant – and I like statistics!

The screenshot displayed on the front page of R Commander

iNZight and iNZight Lite

I have used iNZight a lot. It was developed at the University of Auckland for use in their statistics course and in New Zealand schools. The full version is free and can be installed on PC and Mac computers, though there may be issues with running it on a Mac. The iNZight lite, web-based version is fine. It is free and works on any platform. I really like how easy it is to generate various plots to explore the data. You put in the data, and the graphs appear almost instantly. IiNZIght encourages engagement with the data, rather than doing things to data.

For a face-to-face course I would choose iNZight Lite. For an online course I would be a little concerned about the level of support material available. The newer version of iNZight, and iNZight lite have benefitted from some graphic design input. I like the colours and the new logo.

Genstat

I’ve heard about Genstat for some time, as an alternative to iNZight for New Zealand schools, particularly as it does bootstrapping. So I requested an inspection copy. It has a friendly vibe. I like the dialog box suggesting the graph you might like try. It lacks the immediacy of iNZight lite. It has the multiple window thing going on, which can be tricky to navigate. I was pleased at the number of sample data sets.

NZGrapher

NZGrapher is popular in New Zealand schools. It was created by a high school teacher in his spare time, and is attractive and lean. It is free, funded by donations and advertisements. You enter a data set, and it creates a wide range of graphs. It does not have the traditional tests that you would want in an introductory statistics course, as it is aimed at the NZ school curriculum requirements.

Statcrunch

Statcrunch is a more attractive, polished package, with a wide range of supporting materials. I think this would give confidence to the students. It is specifically designed for teaching and learning and is almost conversational in approach. I have not had the opportunity to try out Statcrunch. It looks inviting, and was created by Webster West, a respected statistics educator. It is now distributed by Pearson.

Jasp

I recently had my attention drawn to this new package. It is free, well-supported and has a clean, attractive interface. It has a vibe similar to SPSS. I like the immediate response as you begin your analysis. Jasp is free, and I was able to download it easily. It is not as graphical as iNZight, but is more traditional in its approach. For a course emphasising doing statistics, I like the look of this.

Data, controls and output from Jasp

Conclusion

So there you have it. I have mentioned only a few packages, but I hope my musings have got you thinking about what to look for in a package. If I were teaching an introductory statistics course, I would use iNZight Lite, Jasp, and possibly Excel. I would use iNZight Lite for data exploration. I might use Jasp for hypothesis tests, confidence intervals and model fitting. And if possible I would teach Pivot Tables in Excel, and use it for any probability calculations.

Your thoughts

This is a very important topic and I would appreciate input. Have I missed an important contender? What do you look for in a statistical package for an introductory statistics course? As a student, how important is it to you for the software to be attractive?

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

Comments

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.

For WHOM?

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.

Introversion/extraversion

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.

Moderation

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.

Reminder!

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 info@statsLC.com 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.

Probability

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. info@statsLC.com or n.petty@statsLC.com.

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.

Claims

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.

Political polls – why they work – or don’t

Political polls – why do they work – or don’t

This is written in the week before the 2017 New Zealand General Election and it is an exciting time. Many New Zealanders are finding political polls fascinating right now. We wait with bated breath for each new announcement – is our team winning this time? If it goes the way we want, we accept the result with gratitude and joy. If not, then we conclude that the polling system was at fault.

Many wonder how on earth asking 1000 people can possibly give a reading of the views of all New Zealanders. This is not a silly question. I have only occasionally been polled, so how can I believe the polls reflect my view? As a statistical communicator, I have given some thought to this. If you are a statistician or a teacher of statistics, how would you explain that inference works?

Here is my take on it.

A bowl of seeds

Imagine you have a bowl of seeds – mustard and rocket. All the seeds are about the same size, and have been mixed up. These seeds are TINY, so several million seeds only fill up a large bowl. We will call this bowl the population. Let’s say for now that the bowl contains exactly half and half mustard and rocket, and you suspect that to be the case, but you do not know for sure.

Say you take out 10 seeds. The most likely result is that you will get 4,5 or 6 mustard seeds. There is a 65% chance, that that is what will happen. If you got any of those results, you would think that the bowl might be about half and half. You would be surprised if they were all mustard seeds. But it is possible that all ten seeds are the same. The probability of getting all mustard seeds or all rocket seeds from a bowl of half and half is about 0.002 or one chance in five hundred.

Now, if you draw out 1000 seeds, it is quite a different story. If all the 1000 seeds drawn out were mustard, you would justifiably conclude that the bowl is not half and half, and may in fact have no rocket seeds. But where do we draw the line? How likely is it to get 40% or less mustard from our 50/50 bowl? Well it is about one chance in 12,000. It is possible, but extremely unlikely – though not as unlikely as winning Lotto. We can see that the sample of 1000 seeds gives us a general idea of what is in the bowl, but we would never think it was an exact representation. If our sample was 51% mustard, we would not sensibly conclude that the seeds in the bowl were not half and half. In fact, there is only a 47% chance that we will get a sample of seeds that is between 49% and 51%.

People are not seeds

Of course we know we are not little seeds, but people. In fact we like to think we are all special snowflakes.  (The scene from “Life of Brian” springs to mind. Brian – “You are all individuals”, crowd – “We are all individuals”, single response – “I’m not!”)

But the truth is that as a group we do act in surprisingly consistent ways. Every year as a university lecturer I tried new things to help my students to learn. And every year the pass rate was disappointingly consistent. I later devised a course that anyone could pass if they put the work in. They could keep resitting the tests until they passed. And the pass rate stayed around the same.

People do tend to act in similar ways. So if one person changes their viewpoint, there is a pretty good chance that others will have also. So long as we are aware of the limitations in precision, samples are good indicators of the populations from which they are drawn.

Here is a link to our video about Inference.

I have described why polls generally work. The media tends to dwell on the times that they fail, so let’s look at why that may be.

Sampling error

Sometimes the poll may just be the one that takes an unlikely sample.  There is a one in a thousand chance that ten seeds from my bowl will all be mustard – and a one in a thousand chance that all will be rocket. It is not very likely, but it can happen. Similarly there is a teeny chance that we will get a result of less than 45% or more than 55% when we take out 1000 seeds. Not likely, but possible. This is called sampling error, and that is what the margin of error is about. Political polls in NZ generally take a sample of 1000 people, which leads to a margin of error of about 3%. What margin of error means is that we can make an interval of 3% either side of the estimate and be pretty sure that it encloses the real value from the population. So if a poll says 45% following for the Mustard Party, then we can be pretty sure that the actual following back in the population is between 42% and 48%. And what does “pretty sure” mean? It means that about one time in twenty we will get it wrong and the actual following, back in the population is outside that range. The problem is we NEVER know if this is the right one or the wrong one.  (Though I personally choose to decide that the polls that I don’t like are the wrong ones. ;))

Non-sampling error and bias

There are other problems – known as non-sampling error. I wrote a short post on it previously.

And this is where the difference between seeds and people becomes important. Some issues are:

Who we ask

When we take a handful of seeds from a well-mixed up bowl, every seed really does have an equal chance of being selected. But getting such a sample from the population of New Zealand is much more difficult. When landlines were in most homes, a phone poll could be a pretty representative sample. However, these days many people have only mobile phones, and which means they are less likely to be called. This would not be a problem if there were no differences politically between landline holders and others. I think most people would see that younger people are less likely to be polled than older, if landlines are used, and younger people quite possibly have different political views. Good polling companies are aware of this and use quota sampling and other methods to try to mitigate this.

What we ask

The wording of the question and the order of questions can affect what people say. You can usually find out what question has been asked in a particular poll, and it should be reported as part of the report.

How people answer

Unlike seeds, people do not always show their true colours. If a person is answering a poll within earshot of another family member, they may give a different answer to what they actually tick on election day. Some people are undecided, and may change their mind in the booth. Undecided voters are difficult to account for in statistics, as an undecided voter swinging between two possible coalition partners will have a different impact from a person who has not opinion or may vacillate wildly.

When the poll is held

In a volatile political environment like the one we are experiencing in New Zealand, people can change their mind from day to day as new leaders emerge, scandals are uncovered, and even in response to reporting of political polls. The results of a poll can be affected by the day and time that the questions were asked.

Can you believe a poll?

On balance, polls are a blunt instrument, that can give a vague idea about who people are likely to  vote for. They do work, within their limitations, but the limitations are fairly substantial. We need to be sceptical of polls, and bear in mind that the margin of error only  deals with sampling error, not all the other sources of error and bias.

And as they say – the only truly correct poll is the one on Election Day.

What mathematicians do

What do mathematicians do?

We ask children what mathematicians do, and the answers include, “they do mathematics”, “they get things right”, and “they answer questions.” Hmm.

Recently in guest workshops I asked about 120 pre-service primary/elementary teachers how many see themselves as mathematicians. Each time, there were about 10% who identified as mathematicians. I then asked them, how many would like the children they teach to think of themselves as mathematicians. It was almost 100% to the affirmative. And then I ask, “Do we have a problem?”

We do have a problem.

I also introduced the idea of maths trauma, that I wrote about in a previous post, and explained that preservice elementary school teachers have been found to have the highest rate of self-identified maths trauma among undergraduate students. The heads were nodding, so I asked who would say they had maths trauma. Nearly one-third of the teachers said they felt traumatised by maths. Some came and talked to me individually after our session, and told me how their fear of maths was restricting the age group they felt they would be comfortable teaching. My message to them, is the very important message I learned from the webinar – “It is not your fault.” They have been taught maths in a way that was not suitable to them, or they many have had one terrible experience that put them off permanently. It is not their fault. And we and they need to do something about this.

Now there is quite a gap between being traumatised by maths, and perceiving oneself as a mathematician. I have, before my mathematical renaissance, been known to say that I was not a mathematician, as I saw myself as an operations researcher or statistician, rather than the abstract-focused (I believed) mathematician. I tend to think concretely, and had perceived that that excluded me from the ranks of true mathematicians. I have also written posts outlining the difference between mathematicians and statisticians (and operations researchers).

But these days, I have become a maths activist. Or maybe a maths whisperer? My mission, for the rest of my life, is to help people, and in particular, teachers, overcome their fear or dislike of mathematics and perceive themselves as mathematicians.

Education is a political act, and knowledge of maths and statistics empowers people, allows greater career choice and enables informed citizenship. (Nic Petty)

I have learned a great deal from the MTBOS or Maths Twitter Blogosphere. I hope one day to attend a Twitter Math Camp (#tmc17), but I fear I am destined always for #tmcjealousycamp where all the wannabetheres lurk. One of the best things was to find out about “Becoming the Math Teacher you wish you’d had” by Tracey Zager. The book is organised in chapters focused on what mathematicians do. We found this inspiring and have spent some more time grouping together our own, Zager’s and others’ ideas of what mathematicians do into the following structure:

What mathematicians do

I put my initial ideas out into the Twitterverse, where they were retweeted and endorsed. I have done some more refining to come up with these.

Mathematicians work in different ways

Mathematicians work together and alone. Too often classroom teaching has focussed on individual endeavour, whereas many people prefer to work in groups, where we can bounce ideas around.

Mathematicians work at different paces. I recently saw a Tweet quoting Jo Boaler who said “There is a common and damaging misconception in mathematics- the idea that strong math students are fast math students.” The person tweeting added, “ It’s not always about speed.” I replied, “Actually, it is never about speed.”

Mathematicians work intuitively and methodically. Sometimes we get a hunch and it turns out to be correct, or useful. Other times we just have to grunt through some ideas and processes to find things out.

Mathematicians estimate and calculate. Sometimes we just need an answer near enough. Often we need to have an idea of the near enough answer so we can check our calculations. Sometimes we need to calculate carefully and with precision.

Mathematicians Strive

This set of ideas would apply to many subjects, but I have found them really useful to encourage bravery in mathematics.

Mathematicians rise to a challenge. When we visit schools with our Rich Maths events, we tell students how mathematicians rise to a challenge. Then when we outline the different activities the can choose from, we tell them that one in particular is very challenging. We have seen many children take great delight in taking on the challenge. You can see it here: Challenging activity

Mathematicians take risks. Too often students are so focussed on getting things correct that it seems too risky to try new things and push boundaries.

Mathematicians persevere. When we see students struggling with a challenging problem, it is really important as teachers to reinforce the characteristic of persevering, and not to “rescue” them. It can be very difficult to hold back, when you are bursting to help them, but short term help is no help. Encouraging them to keep persevering, and recognising what they have already done is far more beneficial.

Mathematicians make mistakes and learn. This is one of the key ideas we emphasise in our visits. It is one of the key principles in the Growth Mindset way of thinking. When we get things right all the time, there is less learning than when we make mistakes.  Sometimes really interesting discoveries come from mistakes. I’d like to add a little aside here that maths teachers ALSO make mistakes and learn. If you have never had a lesson fail miserably, you are not taking enough risks!

I will address the remainder of our characteristics and behaviours of mathematicians in a later post.

Here are the five in summary form:

More detail about what mathematicians do

I would love to hear your opinion – is this what your students think mathematicians do? Is it what you think mathematicians do? Do you make enough mistakes?