## The missing ingredient in maths education

This article was originally published as a series of posts at the ment2teach blog. That’s why it’s a bit longer than what I usually write – this is actually three posts rolled into one. Enjoy.

Maths is missing a crucial ingredient.

Have you ever prepared a meal without all the right ingredients? I remember the time I was making sushi and forgot to add rice wine vinegar to the rice, and the result wasn’t bad but it tasted pretty flat. Or once, when I was experimenting to try and re-create an old family classic, and the flavour was turning out positively wrong because I was missing the secret and impossible-to-guess component. Trying to cook while missing some of the ingredients you need is a bad idea.

In our maths classrooms, many of us are falling to the same error. It is mostly well-intentioned; many of us don’t know any better because we have (as is typical among teachers) defaulted to the way we were taught, and that isn’t always best practice. But it can be (pardon the pun) a recipe for disaster.

What’s the missing ingredient? Maths is an intensely personal subject – and we have forgotten that. Maths is missing a sense of how personal it is.

A discovery too late
As a primary and high school subject, maths has developed a reputation for being dry and unemotional. It has even entered our vernacular to say someone is “cold and calculating” when referring to a person who lacks empathy and compassion.

This is patently wrong for at least two reasons. Firstly, on the negative side, it doesn’t take long to realise that the mathematical classroom – and its much-maligned cousin, the mathematical exam hall – can be one of the most emotional places in the school, and often in a bad way. Maths anxiety is a documented phenomenon sweeping across many countries in the Western world. (It even has its own Wikipedia page. That’s pretty sad.) But have you ever heard of English anxiety? Or science anxiety? Clearly maths has a unique capacity to make students emotionally worked up, and to ignore this is silly. (In fact, ignoring it is often a big part of what makes it worse, since those suffering from it think they must be the only ones and this only increases their feelings of isolation and nervousness.)

But on the positive side, mathematicians who enter the world of mathematics after school often report back that maths has a thrillingly personal side. They sometimes describe their mathematical endeavours as a collaborative journey of intense mental and emotional energy with a tremendous personal pay-off. The french mathematician Cédric Villani wrote: “Mathematics is about progress and adventure and emotion.” Even for those who struggle to identify with these words, it’s hard to ignore the sentiment that Cédric (and many other mathematicians down the ages) is communicating: maths is undeniably personal.

Why, then, do so many people think otherwise? Well, at least one explanation lies in the way that I’ve laid out these negative and positivde sides – namely, because of time. Maths at school _is_ often taught in a dry and unemotional way. It is often only upon “surviving” high school maths and making it to the university level – where students have far more freedom to pursue the mathematics that interests them, and to do so in an environment that is much further away from the suffocating pressure of endlessly comparing marks and ranks on standardised tests – only then do they often make the discovery that mathematics is something quite different from what they learned in the previous twelve or thirteen years of their life. It is a discovery that most make far too late, not to mention the fact that those who discover it are often the ones who in a sense need it least.

Is it a mountain or a mole hill?
But is this really that important? Let’s just assume this hypothesis is right: that mathematics education is deficient in this particular aspect. What difference does it make whether you’re missing one little ingredient here or there? Sure, the meal as a whole won’t taste as good, but is that such a problem? Answer: yes, it is. The reason why is because ingredients aren’t just about taste. Sometimes, ingredients are about survival.

Take the air you breathe, for instance. You may remember from your high school science class that Earth’s air is a wonderful mixture of ingredients: nitrogen, oxygen, argon and a number of other gases. But did you know that the oxygen level of the air – which is about 20.9% – must stay within a tiny range to remain safe for humans to breathe? If it fluctuates by just a couple of percentage points, the results are hazardous to humans. In this case, getting the ingredients right is a matter of life and death.

This is exactly the case we are in for our mathematics classrooms. And we are past the point of dangerous levels. While there are thousands of examples of effective teaching practice happening across the face of the planet, there are equal numbers – perhaps more – of classrooms where students are simply dying in terms of their desire for and understanding of real mathematics.

So what? At this point, the easy option would be to point the blame at something external, sound angry at someone distant making poor decisions that place us as maths teachers in an impossible position, then dust off my hands and go home. Some easy targets would be the people who have written our current syllabuses and jammed the full of material, making it unfeasible to keep the breakneck pace required to cover everything whilst simultaneously giving ideas sufficient time inside and outside class to resonate emotionally with our students. Alternatively, we could attack the government bodies burdening us with ever-increasing amounts of administrivia that detract from the actual job of classroom teaching, and preventing us from investing emotionally ourselves in the pedagogical process for the benefit of our students. Or, we could pin the blame on parents who, according to the research, are primary contributors to the negative attitudes that many students themselves bring a priori into the maths classroom.

Maths is personal first, procedural second
However, while that would be easier and it might make us feel better for a little while (namely, for the fifteen seconds after you close this post until you forget that you ever read it), blaming someone else isn’t going to help us. Without absolving outside parties of their contribution to this troublesome situation, we still have a responsibility to own our part of the problem and take action to do something about it. And yes, us teachers on the front lines have an enormous opportunity to introduce change in our classrooms and start to make things better. For starters, we need to help our students recover the sense that maths is personal first, and procedural second.

Notice that I said “recover”, not “obtain”. It’s been said that children are born scientists, with an innate desire to observe and experiment, but that our culture as a whole seems to actively discourage this impulse until the point that science becomes a foreign endeavour. I believe that a similar thing can be said about children as mathematicians. All young children seem to display a natural affinity for identifying patterns and solving puzzles. It is not difficult to see the joy in a child’s face when they first begin to master their understanding of numbers as quantity, then as order. The wonder and surprise that appears in a toddler’s eyes when they comprehend the algorithms and strategies needed to re-assemble a wooden puzzle is a beautiful thing to behold. Of course, no child (and few parents) would use any of this language to describe what is happening, but that does not diminish the reality that is there. And my point is that at a certain time, this youthful delight is lost and becomes replaced by a sense of drudgery and loathing when the word maths is uttered. This delight needs to be recovered, and I think that the seed of this emotion remains in every student even if it is dormant.

Furthermore, please don’t see a false dichotomy here. Recognising the importance of mathematics’ personal nature does not mean ignoring its procedural nature. I am not advocating for a diminished view of the skills and techniques learned in the mathematics classroom. Maths is a practical subject, not a theoretical one, and the mastery of practical strategies is as woven into the DNA of mathematics as much as it is into other practical subjects such as music or sport. Indeed, I feel that refusing to see maths as a skill-focused learning area will largely empty it of much of its emotional resonance. (There are few experiences as emotionally gratifying as the satisfaction that comes from skillfuly maneuvring a problem and successfully arriving at a beautiful and insight-giving solution.)

Rescuing the meal
As teachers, we have the privilege and responsibility to make learning personal for our students. It’s easy to lose sight of this in any subject area: History can focus on events and national structures rather than the people shaping them and affecting them; Science can focus on discoveries and models to the exclusion of the people who made them or are using them today; English can focus on techniques and genres in a way that distracts from the characters in them or the readers’ responses to them. But impersonal learning seems particularly endemic in Mathematics. To be called a “numbers person” often implies a form of unsociable eccentricity or a disdain for human relationship – hardly a flattering (or accurate) picture.

So, as I said previously, we must recover the fact that mathematics is personal first and procedural second. What practical things can we do to start moving in this direction? I have three simple suggestions (among others) that we can all start acting on as soon as tomorrow in our classrooms: they are to do with individual conversations, classroom dialogue, and the narrative of our teaching.

Before I explain any of these suggestions, I should point out that if we are trying to produce a personal attitude change in our students, then we must begin with our own personal beliefs and views of mathematics. These suggestions are not simple techniques that can be employed in an emotional vacuum and expected to successfully bring about change. They must come in the context of a teacher whose underlying attitude shows that they themselves are moving in a direction consistent with what these techniques are trying to convey. It’s self-evident that no one can lead their students to a place they themselves have not been.

Individual conversations
My first suggestion is to change the way we conduct individual conversations in the classroom. We interact with individual students hundreds of times over the course of a single week, and every interaction is a contributor to the overall atmosphere in our class. It is tempting to think that a single conversation is hardly a big deal, and that is true in isolation. But in the same way that a barrage of raindrops can cause a flood in the right quantities, the sheer volume of our student interactions means that every conversation creates an effect on our students and shapes their attitudes toward maths as a subject. Do you encourage students when they make mistakes, or do you berate them? Do you push students to understand why their answer went wrong or are you content to tell them the right answer and just implore them to accept it on your authority? Do you thoughtfully guide students in their struggle with a particular problem or do you just give them the first hint that comes to your mind? Where you sit on the spectrum of each of these questions will be decisive in determining whether your students appreciate and embrace the personal side of mathematics or not.

Whole-class dialogue
Secondly, consider the dialogue you have with your class whenever you stand out the front of your room and converse with everyone at once. How would you describe the kind of verbal and body language you use when you are there? What sense do your words and tone communicate to your students about the subject matter you are engaged in (and that you are attempting to get them involved in too)? Does the way you speak show students that you have engaged in a personal struggle to understand this too, or do you come off as someone who effortlessly understands everything? Is your body framed and moving in a way that demonstrates your emotional engagement with the content you are about to share, or does it betray the fact that you are actually bored by what you are teaching? (And if we are bored with what we are teaching, then why are we ever surprised when students are bored with their learning?) Our success or failure to communicate genuine passion in our teaching will show our students, by example, whether it is worth personally investing in mathematics or not.

Lesson narratives
Thirdly, there is enormous scope for shaping the narrative of our teaching in such a way as to constantly remind students of the personal journeys that lie just beneath the surface of the concepts and skills our students are working to understand. A perfect example of this presented itself to me this term when I was teaching geometry to my Year 8 class. As is reflected by many well-meaning textbooks, Year 8 geometry often begins with a review of the definitions and properties of various shapes (points, lines, polygons etc.) that were introduced in Year 7 and form the foundation of the Year 8 treatment of the topic. But there is nary a whisper of Euclid’s earth-shaking axioms, or Lobachevsky’s astonishing curved world, or Mandelbrot’s wrestle to reconcile the perfection of Euclidean polygons with the jagged self-similarity of the world around us. There is no hint of the fact that geometry is in fact a field in motion – being driven forward by brilliant and determined human beings – rather than a monolithic structure of knowledge delivered down from the heavens since time immemorial. That is why, in my introduction to the topic, I told my class to title this topic, “The Global/Historical Geometry Project”. This name that reflects the fact that they are doing much more than memorising stony propositions. They are, in actuality, dipping their toes into the great rolling ocean of geometric logic and reality – an ocean that humanity has been surfing for centuries.

So, what do you think? Try these strategies in your classroom – and let me know how it goes!

## #aussieED

I’ve written before about how scheduled chats are one of the most powerful “features” of Twitter. I write that with quotation marks because they are more a function of social self-organisation than they are of the Twitter system itself. But nonetheless, the inherently real-time nature of Twitter makes this kind of discussion at home on this platform more than any other.

A few days ago I had the joy of co-hosting the Sunday night #aussieED chat with Brett Salakas (#aussieED founder) and Graham Andre (The Mathematics Shed editor). Brett and his team started this chat last year and it has blossomed into one of the most vibrant communities of educators that I’m aware of, both online and off – they really deserve to be commended for their efforts. The reason I got in on the act this time was that the theme was none other than mathematics (something I was very pleased to see on the agenda of a chat that is intentionally cross-KLA and cross-sector). Since I had a hand in composing the questions beforehand, I also took the liberty of preparing some of my responses ahead of the chat itself (so that I could spend the actual hour interacting with others as much as possible). Here are some of the tweets I sent out (including a handful of images I created specifically for the chat):

In answer to the question: “Maths is either right or wrong.” Agree or disagree? Why or why not?

In answer to the question: Can you be creative teaching maths, if so how?

In answer to the question: Share one tech (app, website etc.) maths tool that you couldn’t be without.

In answer to the question: What is your favourite strategy for engaging your students in mathematical thinking?

In answer to the question: How do you teach maths cross curricular?

There’s much more than that, and especially a lot of fantastic ideas shared by others. Check out the Storify of the chat (part 1, part 2) for more good stuff!

## Increasing Student Engagement

Today I’m giving a presentation on Increasing Student Engagement. Here are some links I refer to during my session:

## Index Noughts & Crosses

I’m endlessly searching for new ways to present and engage with old ideas. Some of the most wonderful experiences come from seeing or working with familiar things in a new way. So I was delighted when I discovered the game Nuxo on the iOS App Store recently.

It’s a simple game and if you’ve got an iOS device you can go ahead and play it yourself. But as I played it I realised there was so much scope for using this in the classroom. I was teaching index laws recently so I modified Nuxo into a game I call Index Noughts & Crosses. You can see how it works below:

## Guest blog at ment2teach

It has taken me a while to think of myself as a teacher who has something to offer to other teachers. In many ways I am still very young and inexperienced, and there’s no shame in admitting that – it would be a greater mistake to pretend otherwise! Yet at the same time, I’ve come to a point in my career where I’ve made enough mistakes and experienced a broad enough range of situations that I actually do have some things of value to pass on to pre-service and early-career teachers. That’s why I’m excited about ment2teach, a new initiative launched by Jennifer Michalski.

Among other things, ment2teach is a space and a platform to facilitate mentoring relationships between teachers. I know how important mentoring is – I was incredibly fortunate to have spent my first six years teaching in a faculty where basically all the staff took it upon themselves to mentor me in one form or another – and I believe it’s one of the best ways to learn. Life’s too short to learn from your own mistakes – if you want to get anywhere, you need to piggyback on the experience of others.

Apart from offering myself as one of the mentors there, I’m also contributing to the ment2teach’s blog. In fact, my post there – “The missing ingredient” – is the first one published on the site. It’s the first in a series, and over the next two weeks I’ll be posting the rest. Happy reading!

## Riding the wave: why I started a new Youtube channel

I really admire intentional people. I mean the kinds of people who think about what they want to achieve, set goals to get them on the right path, and then go out to strategically accomplish their objectives. Those people inspire me: they show me what’s possible when you set your mind on something and then pursue it with all your strength and discipline.

But I’m not one of them. At least, I definitely don’t feel like one. Many of the things I would consider my greatest achievements have arisen from things that took place accidentally or out of my control. I feel like a surfer who just happened to be at the right spot at the right time to ride a wave into the shore.

For instance, my Youtube channel – which is what I’m most well known for at the moment – was never intended for wide public consumption. I started shooting and uploading videos for just one boy I taught two years ago. He was very unwell and missed huge amounts of school due to hospital visits, so I put things up on the channel so he could keep up with us when he was away. This is a little embarrassing now, but I initially made the videos public simply because I didn’t want to deal with the headache of adjusting privacy settings on every single thing I uploaded!

Since the videos were freely available, others started asking if they were allowed to watch too. It started with the other students in my class – no one else knew I was doing this, and they only knew because they saw me setting up my recording equipment each lesson. From there, it spread to students in other classes, mainly Extension 2 students who had glossed over the 2-Unit topics in class and wanted an actual explanation of things like Series and Sequences. And before I knew it, students from other schools – and even other countries! – were watching along.

All of this blew me away. As I said, none of it had been intentional. I never set out to have a popular Youtube channel; it just kind of happened by itself. But at the same time, it wasn’t all that surprising. Mathematics is, at its essence, the same everywhere – and in fact for most of school across the world, it’s compulsory – so it wasn’t that unusual that I found a wider audience of students. What surprised me most, though, was when the people who commented and emailed me weren’t students. They were teachers.

Why would teachers be watching my videos? The videos are aimed at students, not educators. But then I realised why it made sense. In mathematics – not to mention a hundred other areas of human endeavour – one of the key ways to learn is by example. Yes, you can explain the concepts or principles as much as you like – but what does it look like when you _use_ those principles? What do those concepts look like in action? This is why observations form such a helpful part of initial (and continued!) teacher education: there’s nothing quite like seeing someone else actually do something to help you wrap your head around how you might do it.

And that’s what my videos were helping to provide. Just one example of how to go about explaining things. Not the only way, nor the best way – just one way. And apparently there’s a need out there for this kind of thing. Then I thought to myself – are there things I would say to or upload for teachers if I had a channel just for them? And I realised the answer was yes.

So, in light of the opportunity that’s standing right there – to offer practical and thought-provoking content to hundreds of pre-service, early career and even experienced mathematics teachers out there, for free – I’m starting Wootube²: videos for anyone and everyone interested in mathematics education. I’d be lying if I said I have a huge amount of time to invest in the channel – this is most definitely a side project – but I already have tons of ideas for material to post that I hope will be helpful in developing teachers and cultivating constructive discussion around teaching and learning mathematics. If you are a maths teacher and you have ideas or requests for things you’d like me to do on the channel, please contact me and let me know. Otherwise, subscribe and stay tuned!

## What is this pizza worth?

First, answer the questions on this survey.

Second, check out the results here.

## A response to: the transcience of sharing

Last week, Simon Job – the creator of MathsLinks and its attendant sites – wrote a post called The transcience of sharing.

Simon is a sharer par excellence, not to mention a generally thoughtful and down-to-earth guy. So when he talks (types), I listen (read). Essentially, in his post he is posing this very valid question:

Why is sharing happening on social media (where it is transient) rather than on platforms that are clearly built for it and superior to it in almost every way (e.g. MathsLinks)?

This is a question I’ve thought about too – and it’s bugged me. Over the last few months, these have been the thoughts percolating around my head.

1. (a) It’s where the community is active, which motivates the poster. In the right space, at the right time, it will gain a responsive audience and that response is a very powerful motivator.
(b). It’s where people visit, every day and for no particular reason, which is how the viewer sees it in the first place. People come to dedicated sites like MathsLinks when they (i) are after something, (ii) have the presence of mind to look for what someone else has made/found first, and (iii) have the time to commit to browsing for a little while. That happens far less often than people pulling up their social media feed of choice (which seems to happen reflexively once people get to a bus stop or train station these days).
2. Precisely because it does not aim to preserve, only the trendy and really engaging things bubble up to the top (either through Facebook’s black magic sorting algorithm or Twitter’s more organic system of retweets).
3. I alluded to this above, but MathsLinks (and other similar repositories like TES Australia and Scootle) has become its own worst enemy by being so good. There are hundreds of objects there – which is awesome, but also means that a new user doesn’t even know what’s there or where to begin. There’s awesome stuff there but (coming back to the time issue that has already been identified) someone needs to commit to searching thoughtfully through it to find what will be useful to them in the present moment. This is an issue with faculty resource files just like it is for MathsLinks.

So what can be done to improve the situation? I have a handful of thoughts, corresponding to the points above.

1. Clearly, MathsLinks is awesome as it is. We just need to connect it with the community more effectively. I feel like this is a market problem – it’s a great product, in a quiet spot. Stick it in the middle of George Street and it’ll go nuts because people will be exposed to it more frequently and the conversation about how good it actually is will spread from there. How practically to do that in our context is another question entirely, though.
2. Maybe there needs to be a dedicated team (and by team, I mean more than just Simon) of people dedicated to capturing those cool posts when they come up on social media and then preserving them. We don’t want to discourage the spontaneous sharing and ensuing discussion; we want to leverage it and keep it somewhere that it can be found for future reference.
3. Perhaps we need to do something like a “weekly featured resource”? I have considered doing something like that in my department with “my best lesson this week” as a regular feature of faculty meetings. It would just help people become aware of the riches that are hidden away there, rather than letting them gather digital dust in the cellar of the internet.

Just some food for thought.

## Working out irregular areas

What strategies can we use to find out the area of the top playground? What information will we need?

## My not-so-new workspace

12 months ago I moved to a new school and had the fun task of setting up my desk from scratch (again). It was very bare back then – this is how it looks now:

As you can see, the whole thing is rather more filled out than before. My shelves are now looking a little healthier (or they may need to lose a bit of weight, depending on your preference) and my screens seem to have grown and multiplied. (Seriously, trying to do any kind of timetabling on a single monitor? Practically an invitation to madness.)

I’m feeling a lot more settled and directed heading into 2015 – I guess that’s to be expected since this is my second year in the role. Nonetheless, there is plenty of change afoot – a couple of new staff in my faculty, and I’m very excited about the energy they’re going to bring to the table. One thing’s for sure: there are interesting times ahead!