## Lesson Videos

I upload videos of my classroom mathematics lessons to my YouTube channel. If you’d like to use them to help you learn, search for the relevant topic below within the section that relates to you:

A few other points for you to be aware of:

• These topics are aligned to the NSW Mathematics curriculum. If you live in another Australian state or territory, please use the Mathspace + Wootube portal.
• Some topics below have “related content” – this includes videos that explain connected skills and concepts, but may not be within the scope of the syllabus.
• In some lessons, I teach multiple topics so they don’t fit neatly into a heading below. You can find those videos in Mixed Topics.
• As part of enrichment classes, I’ve taught several concepts that aren’t related directly to the syllabus. You can find those in Mathematical Exploration.

## Stage 4 Mathematics (Years 7 & 8)

Number & Algebra
 Topic This playlist includes… Computation with Integers Ordering & calculating with whole numbers Fractions, Decimals and Percentages Operating & converting between number forms Basic Financial Mathematics Purchasing goods Basic Ratios & Rates Operations and graphical representation Basic Algebraic Techniques Introduction to algebra & pronumerals Basic Indices Operations with positive integer & zero indices Basic Equations Solving simple equations Basic Linear Relationships Number patterns & straight line graphs
Measurement & Geometry
 Topic This playlist includes… Measuring Basic Shapes Length, area & volume Time Units of time & time zones Pythagoras’ Theorem Calculating side lengths Basic Properties of Geometrical Figures Classify shapes; congruent triangles Angle Relationships Angle properties, parallel lines (and related content)
Statistics & Probability
 Topic This playlist includes… Data Collection & Representation Surveys, graphs & charts Basic Single Variable Analysis Measures of location & range Basic Probability Chance of simple & compound events

## Stage 5 Mathematics (Years 9 & 10)

Number & Algebra
 Topic This playlist includes… Further Financial Mathematics Earning, Spending & Investing Further Ratios and Rates Proportion problems & working with graphs Further Algebraic Techniques Algebraic fractions & quadratic expressions Further Indices Operations with negative & fractional indices Further Equations Solving quadratic, cubic & simultaneous equations (and related content) Further Linear Relationships Coordinate geometry techniques (and related content) Non-Linear Relationships Sketching & interpreting curved graphs Polynomials Sketching, factor & remainder theorem Logarithms Define logarithms & use laws Functions & Other Graphs Use function notation & sketch graphs
Measurement & Geometry
 Topic This playlist includes… Measuring Further Shapes Area, surface area & volume Numbers of Any Magnitude Scientific notation & significant figures Trigonometry Angles & sides in triangles Further Properties of Geometrical Figures Similar figures, geometric reasoning (and related content) Circle Geometry Deductive arguments with circle theorems
Statistics & Probability
 Topic This playlist includes… Further Single Variable Analysis Quartiles, box plots, standard deviation Bivariate Analysis Relationships between two variables Further Probability Multi-step experiments (and related content)

## Mathematics Standard 2

Year 11
MS-A1 Formulae and Equations
MS-A2 Linear Relationships
MS-M1 Applications of Measurement (and related content)
MS-M2 Working with Time
MS-F1 Money Matters
MS-S1 Data Analysis (and related content)
MS-S2 Relative Frequency and Probability

Year 12
MS-A4 Types of Relationships
MS-M6 Non-right-angled Trigonometry
MS-M7 Working with Rates and Ratios
MS-F4 Investments and Loans
MS-F5 Annuities
MS-S4 Bivariate Data Analysis (no specific content, but see related content in MA-S2)
MS-S5 The Normal Distribution
MS-N2 Network Concepts
MS-N3 Critical Path Analysis

Year 12
MA-F2 Graphing Techniques (and related content)
MA-T3 Trigonometric Functions and Graphs (and related content)
MA-C2 Differential Calculus (and related content)
MA-C3 Applications of Differentiation (and related content)
MA-C4 Integral Calculus (and related content)
MA-M1 Modelling Financial Situations (and related content)
MA-S2 Descriptive Statistics & Bivariate Data Analysis
MA-S3

## Mathematics Extension 1

Year 11
ME-F1 Further Work with Functions (and related content)
ME-F2 Polynomials (and related content)
ME-T1 Inverse Trigonometric Functions (and related content)
ME-T2 Further Trigonometric Identities (and related content)
ME-C1 Rates of Change
ME-A1 Working with Combinatorics

## Mathematics Extension 2

Year 12
MEX-P1 The Nature of Proof
MEX-P2 Further Proof by Mathematical Induction
MEX-V1 Further Work with Vectors
MEX-N1 Introduction to Complex Numbers
MEX-N2 Using Complex Numbers
MEX-C1 Further Integration (and related content)
MEX-M1 Applications of Calculus to Mechanics (and related content)

## I wrote a book!!

Woo’s Wonderful World of Maths – out September 25! International pre-orders open now.

For the last two years, I’ve been writing a book. It’s about all the awesome stuff I’ve encountered that helped me see the beauty of mathematics – and appreciate how it truly is all around us!

I’ve held back from saying anything up until now because I feel like my generation is super-obsessed with posting on social media about doing things… without actually getting out there and doing something. So now that this thing is almost ready to hit shelves thanks to my publisher Pan Macmillan, I thought it was time to get excited about it!

Some of my friends have had the opportunity to read Woo’s Wonderful World of Maths and this is what they’ve said about it:

“Not just a great teacher Woo’s Wonderful World shows Eddie to be a storyteller too. Is there anything the Woo cannot do?” – Adam Spencer, Ambassador for Mathematics, University of Sydney

“Maths is just another “language” – and Eddie Woo makes it so easy and fun.” – Karl Kruszelnicki, Julius Sumner Miller Fellow, The University of Sydney

“A sweeping tour de force of how to engage people with mathematics.” – Matt Parker, author, comedian and maths communicator

“I never thought I’d read a maths book cover to cover, let alone sing its praises. Eddie Woo makes maths fun, accessible and relevant. Now we can all benefit from his extraordinary skill as a teacher.” – Jenny Brockie, journalist and TV host

“Eddie Woo’s gift is in using stories to help us see the way maths breathes life, colour, shape, and rhythm into the world around us. He’s transformed the lives of countless students in his classroom and on Wootube. Now he’s here to change how you see numbers too – whether you think you have a mathematical mind or not!” – Natasha Mitchell, science journalist and radio host

“Eddie Woo is an inspirational maths teacher. Why? Because he can also communicate, connect and write.” – Jane Caro, author

“Learning mathematics is like climbing the stairs of a skyscraper. It’s difficult and can seem utterly pointless. Some educators scream at us from a 10th story window as we look up at them in confusion. Eddie greets you at the foyer and is there beside you while you take each and every step. And once at the observation deck, he’s admiring the beautiful vista with you. Eddie is more than just the maths teacher we all wanted. Eddie is the maths teacher we all need.” – Simon Pampena, Australian Numeracy Ambassador, Numberphile Star

“For a mathematician, Eddie Woo is one helluva storyteller. An excellent and important read from beginning to end.” – Maxine McKew, Honorary Enterprise Professor University of Melbourne

“You probably know acclaimed math teacher Eddie Woo through watching his excellent videos on his WooTube channel. Well, now there is a book, and it’s a winner. A compendium of short essays where Mr. Woo shows how mathematics lies just beneath the surface in practically every aspect of our lives. What makes it sing is that his engaging personality shines through on every page, just as much as it does on video when he is in front of a class.” – Keith Devlin, Stanford university mathematician and author of many popular mathematics books

“Enthusiastic, energetic Eddie Woo explores mathematics in ways that reveal how human and beautiful it is.” – Nalini Joshi, mathematician, University of Sydney

## iPad pro + Apple pencil + 1 mathematics teacher = ?

Articles about pieces of technology generally come in three phases:

• The first phase comes from the press, who’ve either had access to early versions of the device or are there at the launch event. They have stuff pre-written, primed and ready to go so that as soon as the product is officially announced, people will be able to find something as soon as they hit Google.
• The second phase comes from the first people who are actually able to buy the product, generally a few weeks later. These are naturally accompanied by the obligatory unboxing videos (a hilariously fascinating genre of content if ever there was one) and generally give first impressions of each device (they can’t give much else because, you know, they just opened the thing 5 minutes ago).
• The third phase are the “long term” reviews, maybe a month or two later. There are fewer of these because the initial excitement has died down, but people are actually interested in giving their thoughts now that they’ve gotten used to how the device works and can give a more measured explanation of its pros and cons.

And then there are articles like this one, which… well, the iPad Pro was released almost a year ago now. It’s been so long since I first got this device that another iPad Pro has already been released in the intervening time. So why do I think it’s still relevant to write this thing?

Basically, it’s because I’m still getting questions about my iPad Pro wherever I take it. It’s my preferred device for taking notes – so I lug it around with me to any conferences I attend – and pretty much every time I get it out, someone nearby will say, “what is that?” And when I explain that it’s an iPad, I’m almost universally met with surprise. So I guess, despite this tablet’s age, there are still plenty of people out there who don’t know much about it. So hopefully this will give you a bit of a better idea.

This is not a product review. It’s more of a reflection on my specific experience with this pair of devices. But in some ways, it’s the review I couldn’t find. Before I got my hands on an iPad Pro and its surprisingly uncommon accessory, the Apple Pencil, I searched in vain for a review that answered my specific questions. The closest I got was this write-up from AnandTech.

But all of that was to be completely expected, because I’m what you call a very narrow use case. I have very peculiar and idiosyncratic desires and interests in a device like the iPad pro. That’s because I’m a mathematics teacher.

I wouldn’t blame you if you aren’t connecting the dots yet. So let me explain. Though this may not be everyone’s memory of it, the fact is that mathematics is a highly visual subject. From the diagrams of deductive geometry, to the construction and interpretation of graphs, to the notation and equations of calculus – understanding and communicating mathematics is saturated with images and symbols. Not only that, the actual thinking and doing of mathematics usually takes place through the medium of this visual language. Mathematicians think by drawing and writing. Drawing and writing aren’t just representations for communicating mathematical thought; they are mechanisms for constructing mathematical thought. That’s why we call it “working out”.

As a mathematics teacher, I do a lot of this mathematical thinking each day. I do it to remind myself of the processes and common misconceptions in what I teach my students; I do it so my students have solutions to the tests I’ve set them; and I do it to answer questions that my students request help with.

What this means is that I end up using a _lot_ of paper. In the course of a normal school year I will produce hundreds of pages of handwritten notes. People tend to give me a bit of a strange look when they see me scrawling marks on a page, partly because I have a reputation (which is, in some ways, justly deserved) for being really into technology and eschewing traditional ways of doing things. (There’s a false dichotomy operating there – I’ll address that in a minute.) “Why don’t you type those notes? Wouldn’t it be better to have all those notes electronically filed and organised?”

The answer, at least to me, is not straightforward. Yes, it’s great having content in an electronic form. It’s easier to search for myself and to share with others. But there’s a price. Computer keyboards were not designed for entering formulas, equations or mathematical notation. Mice were not built with the construction of mathematical diagrams in mind. That’s not their fault; there are hundreds of other tasks that they excel at. Expressing mathematical thought just isn’t one of them. After more than a decade of practice and thousands of hours using a wide range of software platforms that are intended for this purpose, I still find keyboards and mice a wholly inadequate replacement for a simple pencil and paper. When I enter mathematical script using a keyboard, even though I am quite adept at it, my attention is focused on the typing and not the thinking.

An analogy will help here. I grew up in a bilingual home. My parents moved to Australia from Malaysia more than 40 years ago, so they are fluent in English, but it is definitely not the language they are most comfortable with. Despite their preference to speak in Chinese, they made a very deliberate decision to talk to us primarily in English. But because of their background, they experienced significant difficulties with this. I lost count of the number of times they would pause midsentence, frozen in thought, as they translated on-the-fly from their mother tongue into mine. Sometimes they would simply give up, resorting to finishing their sentence in Chinese and leaving me to try and work out their meaning from the context and their body language. And it is the same for me and mathematics. Analogue input is my mother tongue; I can think in it immediately and without interrupting my flow of thought. So while I recognise the importance of having a lot of my mathematics in electronic form (primarily assessment tasks), I have a hard time imagining doing the lion’s share of my mathematical thinking any other way than with a pencil in my hand.

I guess what I’m saying is that I use different technologies constantly and I love the benefit they bring, but I’ve always been very conscious of the (often hidden) disadvantages they include. Failing to do so always leaves us in an exhausting form of technological idealism, where we are constantly looking at the next big app or device as the “thing that will revolutionize education” – and always leaves us disappointed. (The lesson to learn here is that if someone tells you something has a straightforward solution, they are probably trying to sell you something.) That means I reject the notion that we must completely abandon traditional ways of doing things if we are the kinds of people who embrace technology in the classroom.

I’m convinced that there are healthy ways to combine them for the sake of student learning, which is the real goal – not an attitude one way or the other with regard to technology (which is not the heart of the issue). As a result, I always find conversations that are centred on this technology or that technology to be quite dull. I want to talk about the learners and what they are getting out of the different experiences that various platforms or devices can bring. That’s the real currency I deal in.

Which brings me back to the iPad Pro and Apple Pencil. Almost every single detailed review of the Pencil that I’ve found has been focused on its capacity to enable artistic expression. But my primary interest is in the Pencil’s capacity to enable cognitive expression. And without overstating it, after about ten months of non-stop use, I’m ready to deliver my personal verdict.

I love this thing.

I love it because it supports my learning, my explanations, my organisational structures and my thinking. I’ve found it to be so effective that integrating it into my normal workflow has changed the way I do things as fundamentally as the first time I started using email in the cloud or began storing and manipulating data in spreadsheets rather than word processing documents. There’s been a huge shift. Yet while there’s been a massive amount of change, much of what I’ve done has stayed the same – in fact, in some instances, has become more rooted in ways of the past. It’s a living expression of that tension I was talking to you about before. The iPad Pro and Apple Pencil capture that tension in a unique and powerful way that really resonates with me.

So, I’m a fan. But don’t worry, I recognise that these devices are far from perfect – there are some obvious drawbacks that I also want to point out. First, let’s get down to a bit of context which explains the particular way that I use my iPad Pro.

I already own several devices that bear similarities to the Pro. Most obviously, I have an iPad Air 2 that I use for recording my classroom lessons (which I regularly post onto YouTube). Since this iPad has such a specific purpose, for me it overlaps less with the Pro than you would necessarily think; that’s because the Air sits permanently on top of a tripod in the middle of my classroom during my lessons, so I never connect it to my data projector to show visuals or write lesson notes on it. Its next most common use is to edit and upload those videos. When it isn’t occupied for those purposes, the Air does fill the role of a secondary device quite effectively. Like most people, I use it for light email and internet browsing when I’m away from my desk. Because of its size, it’s very handy just to pick up and go when I have to visit a student or teacher quickly then return to my work station; I can use it easily while standing up or walking, which I think is one of the main benefits of tablets in general.

My primary device is a 13.3 inch Windows laptop. I’ve owned laptops up to 15.6 inch before, but found them too bulky and heavy to carry between my classrooms with all the other gear I lug around on a regular basis (textbooks, exam papers, teacher’s diary, pencil case, my iPad Air and the microphone that goes with it). It’s not just my lack of upper body strength that leads me to say that – during my teaching career I’ve used and broken several carry bags, and the point of failure every time is the straps, indicating that I’m always trying to carry too much stuff. For this reason, I’ve also used 10 and 11 inch netbooks before – but I’ve always found them too much of a compromise to use extensively. The lack of a full size keyboard, the diminutive screen that limits my ability to effectively multi-task (e.g. simultaneously viewing my report spreadsheet while writing my reports in a browser) and the inevitably downsized processor/RAM always prevent me from feeling at home on such a small device.

My 13.3 is a very happy compromise between these extremes for me. It’s a Dell XPS 13 that I purchased about two years ago. The screen is a sufficiently high resolution that I can display as much content as I want, and the combination of a high-spec Core i7 processor with loads of RAM and a very handy solid state drive mean that so far it isn’t showing any signs of slowing down. Significantly, the keyboard is a pleasure to use and I can type on it just as fast as I can on any desktop. This is important because it’s the main reason why I decided not to purchase the Smart Keyboard that was designed to pair with the iPad Pro; if I want to do serious typing, I take my laptop. I never intended for the Pro to be a desktop replacement and I wasn’t interested in seeing if it could perform in that role for me.

So then, those are all the ways I _don’t_ use my iPad Pro. What ways _do_ I use it? There are two apps, apart from standard web browsing and email, that dominate my use of the Pro. They are Google Drive (along with its satellite apps, Docs and Sheets) and Notability, which I’ll talk about in turn.

I’ve been into Google Drive for a long time. Back in 2009 I was the intranet coordinator at my school and I drove the school’s adoption of Google Apps for Education. But I would describe myself as having my feet firmly in two camps when it came to using Docs, Sheets and all cloud-based options in general, because while I loved the principle of having everything accessible on all my internet-enabled devices, there were some practical issues that held me back and kept me committed to using desktop software.

The two main issues were formatting and mobile apps. I know it may sound silly, but my time as a semi-professional web, print and graphic designer has really spoiled me in terms of how much control I need over the typography and typesetting in the documents I work with. Things like font face, weight and spacing convey meaning and emphasis. Table layouts and bulleting structure can make the difference between documents that are clear or opaque. Just like a public speaker who has control over their tone, pitch and speed can communicate better than one who does not, software that permits precise control over visual attributes enables us to make documents that are more really digested and understood by readers.

For a long time, Docs and Sheets simply did not give me the amount of freedom I wanted in designing things. But that has improved markedly over the last couple of years, to the point where I can now create a document entirely in Docs and be happy enough with its layout and fonts that I can go directly to print without going anywhere near a desktop program. There are still a few small things in this area where Docs isn’t perfect, the most notable of which is its support for mathematical equations (which is present, but minimal), but I’ve become more and more impressed as years go by and new features are added (for free, I might add!).

The mobile apps are a similar story. I don’t expect an app on a phone or tablet to give me an identical experience to what I’ll get on a desktop in a browser, but got quite a while there was too far a gap between the two. Remembering that the fully-featured versions of Docs and Sheets are kind of like trimmed down versions of their desktop counterparts, and mobile apps are trimmed down versions of their browser counterparts, you can see that if you were on an iPad you were getting a very lightweight experience of spreadsheets and word processing. But recently, the native iOS apps for Docs, Sheets and Drive have gained a core feature set that makes them genuinely useful and sometimes even a pleasure to work with. More and more, I’ve found myself able to do everything I need on my tablet while in my classroom without needing to return to my desk – it’s been liberating.

That brings me to Notability. As I said earlier, I had a very clear use for my iPad Pro in mind from the beginning, so one of the very first things I did during my initial setup of the Pro was to download every note-taking app I could find. Trawling the internet for reviews and soliciting some of my online contacts for suggestions, I settled on the following list of apps to try:

• Notes (i.e. the default iOS app)
• Penultimate
• OneNote
• Notability

There are two apps that I installed but did not include on that list – Explain Everything and Doceri. Both of these are screencasting apps, which I was interested in at a basic level, but as I’ve stated above, was not my main intention for the Pro. Neither app is updated for the Pro’s higher resolution yet, so I haven’t invested any time to really give either of them a proper try.

The five apps on my list are, by contrast, note-taking apps through and through. I used each one extensively for about a week before moving onto the next one, so that I could gain a deep sense of each app’s strengths and shortcomings. So here’s my app-by-app overview:

Notability came out as the clear front-runner. Probably the two most distinctive and frequently useful features were (1) its ability to modify text/diagrams after they are written/drawn and (2) its seamless integration with Google Drive, which makes the content I create in Notability instantly more shareable and useful. (From the PDF I can print in high quality since it exports my writing as vectors rather than bitmaps, or I can just email it to someone if that is more convenient. I’ve done both of these several times at work over the last three months and have been very pleased with the output.)

I do want to say something about my overall writing experience across all apps – in other words, what I found to be true because of the way Apple has designed the Pro and the Pencil. Essentially, it feels like magic. After about 2 months of daily use, I reached the point where it felt equally natural to write on the Pro as I did on paper. I no longer notice things like the smoothness of the glass and the Pencil itself, which I remember bothering me at the beginning. In fact, I even find it easier to write with the Pencil over long periods of time because I only have to apply minimal pressure to make clear marks (when I write on paper, out of habit I press quite hard with my pen).

These have been said many times before, so I won’t dwell on them: the lack of discernible lag and the highly reliable palm rejection are the main things that make the Pencil feel so compelling in normal use. My handwriting on the Pro looks exactly my handwriting on paper, and that’s because Apple has successfully engineered the Pencil and Pro so that you don’t need to adjust the way you write to use them effectively.

And this is what makes the whole experience so effective for me as a mathematics teacher. Essentially the Pro has taken all the handwriting I would normally be doing – and as I’ve established, that’s a lot – and supercharged it by integrating it into an electronic workflow. I have several “subjects” set up in Notability, each of which syncs automatically to a separate folder in Google Drive. The synchronisation happens seamlessly in the background whenever I close a document – which means I don’t have the live version backed up like I do in Google Docs or Microsoft OneNote, but if I want to manually trigger the app to update a file on Drive then all I have to do is close and re-open it. This means that whenever I need a document, no matter how long ago I’ve made it, I can get to it from any of my internet-connected devices. This has been useful on dozens of occasions already – taking minutes on a meeting and immediately emailing them out to staff, pulling up an example worked solution to a question posed by a student about a topic we’d looked at several weeks prior, and displaying the solutions to a past exam paper (along with my live annotations) on a data projector for the whole class to see, among many other examples of how this has been useful to me.

One feature that I particularly want to highlight is the ability to select and rearrange or cut and paste marks on the page. I want to point it out because it’s a feature I didn’t even know I wanted – but once I understood it and got used to it, found myself using it constantly (literally, hundreds of times a day) and really miss being able to do it when I return to writing on paper. This takes one of the very best features of typing and imports it into the sphere of handwriting. Anyone who writes frequently – whether it’s emails, essays, reports, articles or fiction – basically takes it for granted that we write (and think) in stops and starts. It’s rare to find someone who can think of the perfect words and grammar to convey their meaning in the exact order and at the exact speed to type them down. Most of us need to write, delete, rephrase, and edit several times before we get something we’re satisfied with. Personally, I tend to write/type out my ideas and then read my sentences back to myself as if I were speaking them to get a sense as to whether a sentence is clear or not. I like to ensure that my paragraphs have a measure of rhythm to them, and this usually requires copious editing.

Proofreading sentences (and equations!) is equally easy on a hard copy as it is on a digital copy. But the actual act of reworking something so that it’s clearer or more powerful is immensely easier to do on a computer. The ease with which you can rearrange and replace phrases (or mathematical symbols) is one of the killer features on a digital word processor, something that we often take for granted when all we do is type all day. And being able to do that with the handwritten script I produce through the Pencil is truly like having the best of both worlds.

Here, at the end of this article, let me mention something which is usually the first thing you find in a review about the iPad Pro: its size. Putting together this write-up has taken me such a long time that Apple has already released its “next” version of the iPad Pro – the 9.7 inch version (i.e. in the same form factor as the iPad Air). However, given my time to settling into using the original Pro, I think 12.9 inches is the best size for this device.

Let me take a step back before I justify that opinion. Okay, I get it: everyone is gobsmacked by the Pro’s size when they first see it. It’s always the first thing people comment on when they see me using it. That’s unsurprising for two reasons: (1) the Pro’s size is the only fact about it you can notice within 1 second of seeing it, and (2) everyone is mentally comparing the Pro to a 9.7 inch tablet, because that’s what they’re used to. The first point is obviously unavoidable, but I think the second point is actually an honest mistake. I don’t think the Pro is trying to be a better version of the Air; I think it’s trying to be something else altogether (hence the Pro moniker). If you’ve read this far into this post turn you probably won’t be surprised when I say I think the Pro is just the right size at 12.9 inches because of what I am comparing it to: an A4 piece of paper. The writable surface of each is basically identical, and I don’t want either of them to be any smaller than they are right now.

There are other benefits, too. I’ve tried doing Split Screen on a 9.7 inch screen and it feels just like it did on those 10 inch netbooks I mentioned earlier: cramped and not designed for this purpose. But I’ve done legitimate multi-tasking on the Pro numerous times and found it quite a comfortable experience (most frequently with Notability on the left and iBooks, email or Safari on the right).

So, here ends my rambling set of thoughts on the iPad Pro and Apple Pencil. I felt compelled to write about them because they are that rare instance of technology that has genuinely managed to make me adjust my workflow because I get access to significant new benefits this way. There are lots of things I haven’t said but I’ve already written far too much here – so I hope it’s helpful to some of you and if you’ve got further questions, feel free to send them my way!

## Working Mathematically

Learning mathematics is about understanding and mastering content and skills. The content includes topics such as algebra, trigonometry and probability. In the NSW syllabus, the skills are called “Working Mathematically”…

That’s how my primer on Working Mathematically begins. I’ve been wanting to put together something like this for a long time, but the impetus to actually write it all down in a systematic way came when I was training some of our year 11 students to be peer tutors for year 7 students at our school. It occurred to me that while they were probably familiar with some of the ideas, most of them had never been formally introduced to the language of Working Mathematically – and language is powerful. It helps us see, appreciate and work toward things more effectively.

## Reference Sheet – helpful or not?

So, last night BOSTES published the long-awaited “Reference Sheet” that HSC 2016 students will receive in the final exams for Mathematics, Extension 1 and Extension 2.

The response in the teaching community that I’ve seen has been mixed. There has mostly been very positive feedback, but I’ve spoken with many who are (wisely) a little more skeptical. Is it a good idea? Is it “dumbing down” the course?
As one of the teachers who gave feedback early on in the development of this sheet, I’ve been thinking about this for a while. Is it appropriate? Is it even helpful? Like the standard integrals sheet it replaces (which, as an interesting piece of trivia, is actually a page out of the old book of log tables), I think that most strong students will not really use this sheet and it will not change the way they think about or learn mathematics. It was certainly never my experience as a student that I relied on the standard integrals sheet, because I had used those results so often that I inadvertently memorised them.

But this sheet is not designed for students who would have been in that camp (and most maths teachers, it should be noted, would be in this category – that’s part of why we’re maths teachers). This sheet is designed to help out the student who struggles a little more and can’t access a number of questions in the final exam because he can’t quite remember how the cosine rule ends. Or which sign belongs where in difference of cubes. This sheet is going to help them and give them a tiny bit of assistance – just like we all would use in the real world if we were trying to do something and couldn’t remember the formula. We’d look it up on Google! This is an attempt to make assessment less contrived in that way. Of course it isn’t perfect – every solution is a compromise with strengths and weaknesses. But I personally welcome the change.

## 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:

## Matrices Quiz – worked solutions

Hey class – as promised, you can view the worked solutions to your Matrices Quiz. Complete your corrected solutions and then start working on the AT5!

## 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: