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?

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?

Lessons for a budding Social Enterprise from Elevate

Statistics Learning Centre is a social enterprise set up by Dr Nic Petty and Dr Shane Dye after leaving the University of Canterbury. Our aim is to help the world to feel better about mathematics and statistics, by inventing, creating and disseminating resources and ideas to learners and teachers. We believe that facility and confidence with mathematics and statistics is as important as literacy in enabling individuals to participate fully in their world.

We didn’t always have our mission or aim or vision as well articulated, and if asked we tended to give some vague description like – “we make stuff to help people learn maths and statistics.”

StatsLC identifies as a social enterprise because we are driven by a purpose beyond making profit for shareholders, and our purpose is a social good – in this case education. A social enterprise exists in the continuum between a business which operates for profit, and a charity, which is strictly not-for-profit, but measures its effectiveness in different ways. We wish ultimately to be self-sustaining so that we are not at the mercy of grants or contracts with outside providers.

Ākina Elevate

We, the directors, have spent the last eight months, on and off, working on our purpose, customer focus, financials and operations as part of an Elevate course with Ākina. The course is aimed at social enterprises, and we have been participating with between five and eight other social enterprises based in Christchurch, New Zealand.

At our last session Ākina wanted to know what value we have gained from the course, what it does well and what can be improved. Ākina itself is a social enterprise that helps other social enterprises. Social Enterprise is a popular phenomenon, particularly in our area, where recently Ākina hosted the World Forum.


The first unit of four sessions, one morning per week, addressed our impact. We needed to identify what we are trying to achieve, why and how. We talked about vision, mission and purpose. This would help us later to think about who are our customers and who are our beneficiaries.  I still find the delineation between vision, mission and purpose a bit confusing. Our vision has expanded during the course. This is where we are up to now, though it is still a work in progress.

Vision – a world of mathematicians

Purpose: We invent, create and disseminate resources and ideas to enable people to learn and teach mathematics and statistics enjoyably and effectively.

We invent resources to enable people to learn mathematics enjoyably
create and and and and
disseminate ideas teach statistics effectively

As we considered our impact we realised that we are making an impact. We have over 1000 views of this blog daily. There are over 35,000 subscribers on our Youtube channel. Hundreds of children and teachers have been inspired and enthused by our “Rich Maths” events. You can see more about our impact here: Statistics Learning Centre Impact.

We have not been doing well at specifying exactly what impact we aim to have, and measuring it. Originally our impact was with teachers and learners of secondary and university level statistics. However we are now thinking bigger, and wish to create a world of mathematicians.  We truly believe that education is a political act, and knowledge of maths and statistics empowers people, allows greater career choice and enables informed citizenship.


The “customer” or marketing section of the course was the one we felt most in need of, and probably are still most in need of. We learned that we need to ask what problem we are solving and for whom. This has led to serious thought and discussion on our part as we have so many ideas about how we can do good, and for whom. However, the point of social enterprise is that you are not a charity, so need to trade or provide services for money in order to be sustainable. So we need to identify our customers – the people and organisations that will pay money for what we do – either for them or for others.

At the time we were gearing up for a holiday programme, and we used some of the ideas to advertise on Facebook. One outcome of the course is that we have decided we need to employ someone to help with the marketing.


As we already have an accounting package, Xero, and work with an accountant, the need for help here felt less imperative. We have developed different systems in using Xero that will help us analyse our progress. One idea that was valuable was to do with the value of our time. Time and money emphasis did not have to be commensurate in all circumstances. Two sessions on budgets were helpful when thinking about grant applications. We have thought more about cashflow, though a crisis at the end of 2016 had already made us aware of potential problems. We started paying ourselves.

What has become clear throughout the course is that we do not have enough time between the two of us to do all the things we need, as well as maintaining cashflow through contracts. This has helped us to recognise the need to employ someone to cover our areas of weakness, in particular marketing. We also need to develop more passive income streams.


What was extremely valuable in this section was learning about employment contracts and health and safety. We are now formalising our contracts with staff. Being a responsible employer, even for family members, takes a bit of work.

Another useful session concerned governance, management and operations. As a small enterprise, both of us tend to fill all three roles. At this point we need to get some advice at the governance level – even just having someone to ask us questions and to report to periodically. It can be easy to spend too much time chipping away at the coalface, and losing direction. It can also be seductive to spend all our time discussing visionary ideas for future development, rather than getting on and producing. Like most of life, the answer lies in a balance.

Other thoughts

A common expression in social enterprise is Mission Drift meaning letting the commercial aspects over-ride the social impact focus or mission.

We tend to suffer from something similar, that I call mission lurch. I’m not sure it is the right term, as it is more that we are adapting our mission in order to align it better with activities that will lead to sustainability. Our problem is that we need to be doing some more activities that bring in revenue to sustain our mission.

One big benefit from participating in the programme has been making contact and building relationships with others in similar circumstances. This builds confidence.

Big lessons

For me the big lessons from this course are

  • Articulating our mission
  • Confidence to do something big

A year ago I was quite happy to dabble around in the edges of business/social enterprise. We were not really making enough to keep us going, but had hope that something might change. Over the course of 2017 we have had contracts with Unlocking Curious Minds, to take exciting maths events to primary schools. We have also gained contracts writing materials for other organisations. Our success in these endeavours, along with the help from the Elevate course has helped us to think bigger.

Watch this space!

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.

Educating the heart with maths and statistics

What has love got to do with maths?

This morning at the Twitter chat for teachers, (#bfc630nz) the discussion question was, How and what will you teach your students about life this year? As I lurked I was impressed at the ideas and ideals expressed by a mixed bunch of teachers from throughout New Zealand. I tweeted:  “I wonder how often maths teachers think about educating the heart. Yet maths affects how people feel so much.”

My teaching philosophy is summed up as “head, heart and hands”. I find the philosophy of constructivism appealing, that people create their own understanding and knowledge through experiences and reflection. I believe that learning is a social activity, and I am discovering that mathematics is a social endeavour. But underpinning it all I am convinced that people need to feel safe. That is where the heart comes in. “People do not care how much you know until they know how much you care.” Relationships are vital. I wrote previously about the nature of teaching statistics and mathematics.

Teachers are people

In the culture of NZ Maori, when someone begins to address a group of people, they give a mihi, which is an introductory speech following a given structure. The mihi has the role of placing the person with respect to their mountain, their river, their ancestors. It enables the listeners to know who the person is before they begin to speak about anything else. I am not fluent in te reo, so do not give a mihi in Maori (yet), but I do introduce myself so that listeners know who I am. Learners need to know why I am teaching, and how I feel about the subject and about them. It can feel self-indulgent, thinking surely it is about the subject, not about me. But for many learners the teacher is the subject. Just look at subject choices in high school students and that becomes apparent.

Recently I began studying art at an evening class. I am never a passive learner (and for that reason do feel sympathy for anyone teaching me). Anytime I have the privilege of being a learner, I find myself stepping back and evaluating my responses and thinking of what the teacher has done to evoke these responses. Last week, in the first lesson, the teacher gave no introduction other than her name, and I felt the loss. Art, like maths, is emotionally embedded, and I would have liked to have developed more of a relationship with my teacher, before exposing my vulnerability in my drawing attempts. She did a fine job of reassuring us that all of our attempts were beautiful, but I still would like to know who she is.

Don’t sweeten the broccoli

I suspect that some people believe that maths is a dry, sterile subject, where things are right or wrong. Many worksheets give that impression, with columns of similar problems in black and white, with similarly black and white answers. Some attempt to sweeten the broccoli by adding cartoon characters and using bright colours, but the task remains devoid of adventure and creativity. Now, as a child, I actually liked worksheets, but that is probably because they were easy for me, and I always got them right. I liked the column of little red ticks, and the 100% at the end. They did not challenge me intellectually, but I did not know any better. For many students such worksheets are offputting at best. Worksheets also give a limited view of the nature of mathematics.

I am currently discovering how narrow my perception of mathematics was. We are currently developing mathematical activities for young learners, and I have been reading books about mathematical discoveries. Mathematics is full of creativity and fun and adventure, opinion, multiple approaches, discussion and joy. The mathematics I loved was a poor two-dimensional faded version of the mathematics I am currently discovering.I fear most primary school teachers (and possibly many secondary school maths teachers) have little idea of the full potential of mathematics.

Some high school maths teachers struggle with the New Zealand school statistics curriculum. It is embedded in real-life data and investigations. It is not about calculating a mean or standard deviation, or some horrible algebraic manipulation of formulae. Statistics is about observing and wondering, about asking questions, collecting data, using graphs and summary statistics to make meaning out of the data and reflecting the results back to the original question before heading off on another question. Communication and critical thinking are vital. There are moral, ethical and political aspects to statistics.

Teaching mathematics and statistics is an act of social justice

I cannot express strongly enough that the teaching of mathematics and statistics is a political act. It is a question of social justice. In my PhD thesis work, I found that social deprivation correlated with opportunities to learn mathematics. My thoughts are that there are families where people struggle with literacy, but mostly parents from all walks of life can help their children with reading. However, there are many parents who have negative experiences around mathematics, who feel unable to engage their children in mathematical discussions, let alone help them with mathematics homework. And sadly they often entrench mathematical fatalism. “I was no good at maths, so it isn’t surprising that you are no good at maths.”

Our students need to know that we love them. When you have a class of 800 first year university students it is clearly not possible to build a personal relationship with each student in 24 contact hours. However the key to the ninety and nine is the one. If we show love and respect in our dealings with individuals in the class, if we treat each person as valued, if we take the time to listen and answer questions, the other students will see who we are. They will know that they can ask and be treated well, and they will know that we care. When we put time into working out good ways to explain things, when we experiment with different ways of teaching and assessing, when we smile and look happy to be there – all these things help students to know who we are, and that we care.

As teachers of mathematics and statistics we have daunting influence over the futures of our students. We need to make sure we are empowering out students, and having them feel safe is a good start.

STEM, STEM-Ed, STEAM and Statistics

STEM is a popular acronym in educational circles and is used to refer to careers and educational tasks. Though most know that the four letters stand for Science, Technology, Engineering and Mathematics, it can be difficult to pin down what exactly it means. In this post I suggest that there are two related uses of STEM as a description.


The term, STEM, originated in the USA in the late 1990s to describe specific careers and education for these careers. There seems to be no universally agreed-upon definition of STEM. From a careers perspective, the focus is on making sure that there are enough skilled workers in the STEM areas for future development. A common (engineering?) analogy is that of a pipeline. Industrialised nations need people with STEM skills, so need to ensure there are enough people entering and staying in the pipeline in order to fill future demand. There are also identified equity issues, as STEM jobs tend to be higher-paying, and also tend to be dominated by white males. A consequence of the higher demand and pay for people with STEM skills and qualifications is that there is often a shortage of teachers in STEM subjects.

This graphic illustrates the analogy of a STEM pipeline. Are people the drips?

This graphic illustrates the analogy of a STEM pipeline. Are people the drips?


STEM Education

There are multiple ways of viewing STEM Education. One category is specific STEM Projects which I refer to as STEM-Ed, and another is education in STEM subjects, as they currently exist in the school curriculum.


It is believed that one way to encourage children and young people to continue in STEM subjects, is to embed STEM into the curriculum. There has been a move towards specific STEM-based lessons or projects, particularly at middle-school or older primary level. Pinterest is full of attractive STEM-Ed lessons based around engineering and the design process. These include tower and bridge building, making boats to carry certain weights, creating a mechanism that will protect an egg from a fall or launching projectiles a maximum or specified distance. STEM-Ed lessons use a wide range of materials, including Lego bricks, spaghetti, marshmallows, masking tape, newspaper, recycled materials – just about anything you can think of. The makerspace movement ties in with STEM-Ed.

A good STEM-Ed project is described by Anne Jolly in her post Perfect STEM lessons. Anne Jolly suggests that a “perfect” STEM lesson uses an engineering approach as a framework, applies maths and science content through authentic experiences, deals with real world issues, involves hands-on and open-ended exploration with multiple right answers for students working in teams with the teacher in a facilitator role. A STEM project should also engage students in communicating, remove the fear of failure, appeal equally to boys and girls and promote authentic assessment.

It seems that when primary/elementary school teachers talk about STEM, it is usually STEM-Ed they are referring to. Certainly material under the STEM label on Pinterest, a popular source of inspiration for teachers, tends to be STEM-Ed.

A screenshot of some STEM tasks found on Pinterest

A screenshot of some STEM tasks found on Pinterest

Education in STEM subjects

In order to encourage and enable students to continue on to STEM careers, they must study the individual subjects that make up STEM. At school level, maths, physics and chemistry are often the areas where students make decisions that limit their later opportunities in STEM areas. (Where they leak out of the pipeline?) This is where teachers of STEM subjects have a part to play. Tying their subjects to authentic, real world contexts and teaching using STEM-Ed projects can help engagement.

However, there is also a need to learn the mathematics that does not appear in a “good STEM lesson”. Current mathematics education thinking aims to enable children to become mathematicians, not just engineers. To quote Tracey Zager’s excellent book, “Becoming the math teacher you wish you’d had“, mathematicians take risks, make mistakes, are precise, rise to a challenge, ask questions, connect ideas, use intuition, reason, prove, and work together and alone. Mathematics curriculum overlaps well with STEM-Ed in the areas of measurement, geometry and statistics. Number skills are practised in context. However, to provide enough exposure to other areas of mathematics, specific STEM-Ed lessons would need to be carefully designed. I suspect that there are areas of the curriculum that are more effectively learned through other methods than STEM-Ed.


There is a push to add Arts to STEM, making it STEAM. The relevance of this addition depends on viewpoint. It does not seem relevant to include Arts when talking about high-shortage career paths. But at the same time, STEM jobs also require other skills, not the least being communication skills. There is a strong link between fine art and technology, through design. The inclusion of A in STEM also depends on the definition of Art. The term “Arts” can include painting, music, dance, literature, film, design and the even the humanities. Including these into STEM (as a career or subject description) seems a trifle incongruous and begs the question whether there is anything that is not included in STEAM. Physical education and foreign languages?

However, when we look at STEM-Ed, there is a rationale for the inclusion of art. Good design does have an artistic component, as is all too clear when we look at some communist-era architecture and much amateur web-design. And written and oral communication are well-developed in many STEM-Ed projects.


Statistics clearly has a place in both STEM and STEM-Ed. There is a demand for statisticians, and people who can use statistics in what they do. The study of the discipline of statistics gives important insights into the nature of variability in our world. STEM-Ed projects could involve collecting and analysing data in a non-trivial way, though I have not seen evidence of this. The barrier to this is the statistical understanding of the teachers creating the STEM-Ed tasks, and points to an area where statistics educators need to be involved. Another barrier can be the time taken to collect an adequate sample, clean the data and analyse it. This is why specific tasks need to be designed for this.

Concerns about STEM and STEM-Ed

We do need to think about the focus on STEM and wonder about the philosophical underpinnings. Are we educating our students to provide workers for the industrial machine? Is this the right thing to be doing? I found a very interesting book: Philosophy of STEM Education: A Critical Investigation by Nataly Z. Chesky and Mark R. Wolfmeyer . They ask these important questions.

STEM-Ed also needs to be approached carefully. Dayle Anderson, a lecturer in science education emphasised at a Primary Maths Symposium that teachers need to keep their eye on the learning. When a project is engaging it can be seductive to think that the learning is taken care of. There are so many demands on time in a school, that STEM-Ed lessons need to be well-designed for specific learning.

STEM, STEM-Ed and 21st Century Skills

I am quite taken with the 4 Cs of 21st Century Skills, which have been defined as Critical thinking and problem solving, Communicating, Collaborating and Creativity and innovation. These correspond well to the five Key Competencies in the New Zealand curriculum – thinking, using language, symbols, and texts, managing self, relating to others and participating and contributing. These skills are needed by people who work in STEM jobs. They need to be able to communicate and work with others.

These 21st Century skills can be developed in STEM-Ed lessons, as students are required to work together, solve problems, think, innovate and communicate their results.

Closing thoughts

Overall I am excited about STEM and STEM-Ed. A knowledge of how the world around us works is empowering to all, whether or not they join the STEM pipeline. Making mathematics, statistics and other related subjects more relevant and desirable is always going to be a good thing. Statistics educators need to be involved make sure that statistics is a vital part of STEM-Ed.

Any suggestions on how this is best achieved? And if you are interested in STEM-Ed, please Like and Follow our Facebook page to keep up with the discussion and find out about our contribution.