# My maths autobiography

#### School maths

I have always loved maths, but the reasons why have changed dramatically over time.

This is my Year 1 work. It reminds me about what I thought it meant to be good at maths: lots of ticks on neat work, especially if it was done quickly.

This attitude was reinforced by my report cards in primary school. A typical one looks like this. Note the focus on speed and accuracy. I loved maths because I was good at it.

Our Year 2 classroom had a corner filled with self-directed puzzle-type problems. If students finished their work early, they could go to the puzzle corner. I recall spending a lot of time there (my report says I was put in an extension group). Looking back, I’m sad that not every student had the same opportunities to engage with these richer, stimulating problems.

Outside of school, I loved doing and making up puzzles. I looked for patterns everywhere. I was always thinking about different ways to count, to organise, and to get things done more quickly. Growing up on a rural property, I had a lot of chores and time to think. For example, I’d think about how many buckets of oranges I could pick in an hour, how long it would take us to fill an orange picking bin, the different ways I could climb the rungs of the ladder, and so on. But I didn’t connect these ideas to maths.

Most of my school maths memories involve doing exercise after exercise from the textbook, but that was fine by me because I could put a self-satisfied tick next to each neatly done problem (after checking the answer in the back of the book!). I remember one high-school maths project to work out the most efficient way to wrap a Kit Kat in foil. It stands out in my memory because it was so different to the rest of maths class.

There were gaps in my knowledge along the way that I tried to cover up. I missed a month of Year 4 due to illness, and a substantial chunk of that time was devoted to fractions. When I got to algebraic fractions in later years, I would furtively use my calculator on simple examples to see if I could work out the right ‘rule’. Now I congratulate myself on having the sense to work it out for myself by generalising from specific examples. In Year 12 I felt embarrassed for using straws and Blu Tack to make visualisations of 3D coordinate geometry; everyone else could do it in their heads. Now I’m proud that I found a tool to help me make sense of the maths.

In Year 12 I hit a big obstacle. All my grades went downhill, including in maths. My maths report card says that I was ‘prone to panic attacks when working against a time constraint’. I don’t remember that, although I do remember crying (which I almost never do) in my maths teacher’s office and thinking that I didn’t know anything. I realise now that much of my maths schooling was about memorisation but not about understanding, and that it had caught up with me by Year 12.

Despite my mostly mediocre grades (I got a D in physics!), I did okay and was offered several university places. My love of the English language drew me to careers such as law, journalism and psychology. But I had also applied for and been offered a place in mathematics. Despite this, I chose to repeat Year 12. I took maths again because I still enjoyed it. The second time around it seemed to make a lot more sense; my scores were 19 and 19.5 for Maths 1 and 2. At the end of Year 13 I was awarded one of the first UniSA Hypatia Scholarships for Mathematically Talented Women. This boosted my confidence and made University study more affordable for a country kid. So, I decided to do mathematics. I also enrolled in a computer science major because I wasn’t sure what kind of job you could get with a maths degree.

#### University days

Most of my undergraduate mathematics experience was the same as high school. I got Distinctions or High Distinctions for all my subjects (except Statistics 3B where I scraped a pass). I did most of my thinking in my head and then committed it to paper. I produced beautiful notes, and would rewrite a page if it had a single mistake on it. On reflection, I had a fairly superficial understanding of mathematics, but knew what to do to get good marks. I got disenchanted in the third year of my four-year degree and briefly considered quitting, but I had never quit something so important so I kept going.

At the end of third year, I had an experience that made me sure I wanted to be a mathematician. I attended a Mathematics-in-Industry Study Group. This is a five-day event that draws together around 100 mathematicians. On the first day, we listen to five or six different companies tell us about a problem they have that needs solving. For example, they might say ‘we want to stop washing machines from walking across the floor when they are unbalanced’ or ‘we want to know the best way to pack apples in cartons’. The mathematicians then decide which problem they want to work on, and smaller groups spend the next three and a half days feverishly trying to find a solution.

It was transformative as I witnessed, first-hand, mathematics put into action. I also saw how mathematicians creatively and collaboratively approach solving problems. I watched accomplished mathematicians initially not know how to start. I saw them making mistakes. They had intense (but friendly) discussions about whether something was the right approach. It was a defining moment, because it showed me how mathematics is really done, beyond learning mathematics that’s already known, or applying algorithms without a sense of why we would do so.  I saw the true habits of mathematicians in action. I also discovered the important role that communication plays in mathematics, and that I could put my love of the English language to good use.

The transition from doing maths exercises with answers that were ‘perfect’ the first time to the more authentic and messy problem-solving required for mathematical research was not an easy one for me. I found it difficult in my PhD to accept that I was not perfect and that I had to constantly draft and refine both my mathematical ideas and my writing, especially because I had never been taught these skills. But I was helped in being surrounded by more experienced mathematicians who modelled, if not explicitly articulated, that this was how mathematicians really work.

It’s eight years since I was awarded my PhD, and I can now say that I am quite comfortable with this ‘messy’ approach to maths. I like to say that mathematicians are chronically lost and confused, and that is how it is supposed to be. It would be ridiculous for mathematicians to spend their days solving problems that they already know how to solve. So, being uncertain about whether something will work, or uncertain about what to do next, is a natural way for mathematicians to be.

#### Teaching maths

I started teaching mathematics during my PhD. At first I taught exactly as I had been taught, with procedures and algorithms. But I also didn’t want to respond to a student with ‘Because that’s the rule’, so I started trying to really understand why maths concepts worked the way they did. I learned so much more about maths when I started to explain it to others. The way I taught expanded to include visual ways to think about maths, a variety of representations and approaches, and other flexible ways of thinking. It wasn’t natural to me at first (and at times I still solve arithmetical problems in my head by imagining a pen writing the algorithm) but it has immeasurably enriched my own understanding of mathematical concepts.

I also realised that the way I was taught was not the way I wanted to teach, but I wasn’t sure how to change that. I sought ideas from the internet, and eventually stumbled into the early days of the online community that is the MTBoS (the Math Twitter Blogosphere), although I didn’t realise that until much later. I lurked for a long time because I felt like an outsider: I wasn’t a school teacher (what did I know about education?!) and I wasn’t located in North America. Today I couldn’t imagine teaching without the support of my professional community on Twitter which extends all around the world.

Around five years ago I decided that I could help break the cycle of traditional procedural-based teaching by supporting students, particularly preservice teachers, in experiencing maths in the ways that I and other professional mathematicians do. So, I designed a course that gives students these problem-solving experiences alongside learning skills for thinking and working mathematically. I hold these word clouds from Tracy Zager in my head as a reminder and a motivation of what I am trying to accomplish. (You can read more about how I found them here.)

I still love ‘cracking puzzles’ in maths like I did in Year 2, but my love of maths has expanded to include learning how others think about mathematical ideas. In almost every class I see a student think about a problem in a way I’d never imagined, and I love it. Listening to student thinking is why I’ll never tire of teaching, and it helps me to be a better teacher. I can’t wait to learn from you.

# Counting in unexpected ways

It was a delight to recently spend five days working with students and teachers in Alice Springs at the invitation of MTANT, the Mathematics Teachers Association of the Northern Territory. I then spent a week in bed with the flu, which is one reason I’ve recently lost my voice (both physically and online).

The main purpose of the visit was to join the 8th Annual Maths Enrichment Camp at Ti Tree School, in a small remote town 200 km north of Alice Springs. Students travelled from all over Central Australia, some from as far as the Aboriginal community of Hermannsburg, 320 km south (that’s a short distance in the Northern Territory!). The camp runs Friday night to Sunday morning, and is full of fun activities (mathematical and non-mathematical) for kids, and teachers, to engage with. This was my first Maths Camp, and I was thrilled to be invited; thanks @matt_skoss!

This year the Ti Tree Maths Camp attracted around 35 students from Years 4 to 10. Students were divided into three groups and on Saturday rotated through four activities, called ‘Worlds’. Thus, these activities needed to accommodate a broad range of mathematical expertise. To add extra challenge, I rarely work with students in the lower years, so I relied on a couple of trusted friends to help determine whether my planned modifications would be appropriate.

In this post I briefly describe how younger students responded to two of my favourite activities, which I’ve previously written about: The Game of SET and Domino Circles. I doubt that this is going to be revelationary to most teachers, but I am always learning how students make sense of mathematics (younger students, in particular), so I want to record my observations for the future.

### Counting Dominoes

I worked on this problem with a combined group of Years 5 and 6 girls from Bradshaw Primary School and Araluen Christian College in Alice Springs.

Display this image. What do you notice? What do you wonder?

Responses include:

• I notice: that there different numbers of dots on a domino.
• I notice: that there are two groups of dots on each domino.
• I notice: that the dots are different colours.
• I wonder: what is the highest number of dots on a domino? (A fascinating side discussion commenced as we had to resolve whether we meant in total or on one half of the domino. We decided that in a Double 6 set, the highest number is six. What do you notice and wonder now?)
• I wonder: what is the lowest number of dots on a domino? (Zero.)
• I wonder: how many dominoes are there in the set? (My response is usually ‘Good question! I wonder if we can work that out?’ :))
• I wonder: can a domino have more than one instance of the same number of dots? (Yes — I show a ‘double domino’, like 2|2.)
• I wonder: is there exactly one of every combination of numbers of dots? (Yes.)

Usually my next question is to ask students to calculate how many dominoes there will be in a set. Some students start by drawing them all out. For this students I might show an image of a Double 18 set—too many to draw, right? This encourages students to find, and then explain, a formula for the number of dominoes in a Double ‘n’ set.

However, for younger students that I hadn’t met before, I was concerned that this question might be too challenging. Instead, I handed out sets of dominoes and asked students to have a look at them. Then I revealed that each set was missing a domino. Could they work out which domino was missing?

As you might expect, students needed to find a way to organise their dominoes so that they could identify the missing one. Several groups made arrangements like this. It’s easy to spot the missing domino now, right? How else could you have arranged the dominoes to make this discovery?

I gave one large group two sets of dominoes (one paper, one physical) in case they wanted to work in smaller groups. I was delighted to find that they instead used both sets of dominoes in tandem. It looked something like I’ve reproduced in the photo below. Can you spot the missing domino from each set?

There was an interesting moment in the middle of this activity as we discovered that some sets had more than one domino missing, and some sets had duplicates. (Guess who didn’t double-check the domino sets before starting the activity? <blush>) This could have been a disaster, but I took it as a true problem-solving experience for the group. We sorted out our sets eventually!

The rest of the session was largely spent exploring this question:

Is it possible to arrange an entire set of dominoes in a circle so that touching dominoes have adjacent squares with identical numbers?

Once you’ve experimented with a set of dominoes in which the highest number is six, explore whether it is possible for sets of dominoes where the highest number is different.

We finished with a quick ‘Notice and Wonder’ with this short promotional video by Cadbury, in which they set up blocks of chocolate in a suburban street, and knocked them over like dominoes. I wish I could remember all the rich wonderings the students had — they were fabulous!

### Counting SET cards

My chosen activity for my ‘World’ at the Ti Tree Maths Camp was The Game of SET. I’ll briefly recap the game, before talking about how students counted their set cards.

SET is a card game. Three of the cards are shown below. What do you notice?

Students eventually identify a number of attributes of the cards. Sometimes (but not often) they generate more than we need for the game. I acknowledge them and ask if we can focus on four particular attributes: number, shape, colour and shading. We notice that each attribute has three different values that it can take. For example, shape can be ‘oval’, diamond’ or ‘the squiggly thing’.

I confirm that these are all the possibilities of values of attributes. When I work with students in higher grades, my next question is usually as follows.

If a SET deck contains exactly one card of every possible combination of attributes, how many cards are in a deck?

To adapt this question to lower years, I did something similar to what I did with counting dominoes. But instead of removing a card, I asked them to find a way to be sure that they had exactly one card for every possible combination of attributes in their deck.

Their natural ability to group by features that were the same, and to organise in a systematic way, was not unexpected. But I enjoyed seeing the varied ways they went about this.

For example, these two girls made three rows (shown vertically in the photo). Each row corresponds to a colour. Within each row, they grouped first by shape. For example, all the red diamonds, then all the red squiggles, then all the red ovals. Within each shape they grouped by shading. Within each shading they organised by number. One explanation was that, for a particular colour, they knew that there were nine cards for a particular shape. There were three different shapes. So there were 27 cards in one row. There were three rows of different colours. So there were 27 × 3 = 81 cards.

Some students started grouping by colour, but in a different way. In this grid, each row corresponds to one particular shape. Each column corresponds to one particular shading. Each ‘entry’ in the grid contains three cards, grouped by number. There were two other grids like this, each for a particular colour. One explanation was that, for a particular colour, we get a 3 × 3 grid where each entry contains 3 cards. So, each grid has 27 cards. There are three different grids, each corresponding to a different colour. So there are 27 × 3 = 81 cards.

There are four different ways of organising shown in the photo below. In the left bottom half, a student is organising in a way similar to the grid method. Focus on the larger cards in the far right. These girls have nine columns. Each column corresponds to a particular colour and shading combination. For example, the far left column are cards that have purple shapes that are completely filled in. Within each column, they organised the cards in groups of three. The three groups are organised first by shape. Within each group the cards are organised by number. Their explanation is that there are nine columns, each with nine cards. So they have 9 × 9 = 81 cards.

I loved all these ways — and more not described — that students found to count the number of cards they had in their deck. After students had completed their work, we congregated together and went on a tour of the room. Each smaller group explained to the whole group how they had organised their cards and confirmed that there were 81 cards in the deck.

An unexpected advantage of this approach is that students discovered for themselves how to make a SET, because of the natural ways that they grouped cards. In the game, a SET is a group of three cards where, for each of the four attributes, the features are the same across all three cards or different across all three cards. For example, the three cards below are a SET because shape is all-same, fill is all-same, number is all-different, colour is all-different.

Once students understood how to make a SET, we made a new discovery within their work. Consider again the top cards of the 3 × 3 grid shown below. Each row, column and diagonal forms a SET. It’s like a magic square. A magic SET square.

Meanwhile, the students who organised their cards into a 9 × 9 grid decided to keep their columns the same, but rearrange it so that each row corresponded to a particular number and shading combination. I’ve reproduced it below. With a little bit of prompting from me, they discovered that they had a kind of super magic SET square. Can you see what I mean? So cool!

The rest of the session was spent playing the game, and talking over some of our SET-finding and problem-solving strategies. A rough description is here.

What struck me is that these students’ understanding of how to form a SET was much more solid, and developed so much quicker, than many other older students I work with. This is because usually I explain, rather than have them explore. At Maths Camp I was reminded — again — that even in an activity full of moments for discovery, there are still more opportunities to slow down and let students construct knowledge for themselves.

# Too quiet

It’s been quiet around here. A little too quiet.

The first half of this year was a bit crazy. I tried to juggle increased travel and talks with teaching, research and student supervision. It was not always successful. I didn’t quite realise how crazy it felt until I stopped. A week of quietness has helped replenish some rather depleted batteries!

But teaching restarts next week, and along with it, this blog. I won’t post at the rate I did during the #MTBoS30 challenge (which my friend Linda correctly pointed out was rather frenetic), but I hope to manage at least a couple of posts per week to reflect on what happens in class.

The course is Developing Mathematical Thinking. It is a maths elective for pre-service teachers training in primary and middle maths. I’m rather proud of this course, having established it in order to increase students’ problem-solving skills, develop their confidence in tackling unfamiliar problems, promote mathematical communication, and stimulate curiosity. I pair rich mathematical tasks with explicit coaching in mathematical processes and purposeful reflection; I’ve blogged about some of the learning experiences from previous years. It’s as joyful to teach as it sounds. I’ve taught this course twice; the second year we doubled the number of places due to demand. This year, the course was fully enrolled months in advance.

The warm responses to this course make me very happy. But I’m particularly excited now by the opportunity to incorporate new ideas and to refine existing ones, based on the rich discussions at conferences, workshops and on Twitter, the thoughtful #MTBoS blogposts, and reflecting on past experiences through this blog. I’ve spent the last two days planning and reorganising. The past year has been one of growth, and I’m looking forward to infusing my teaching with my evolving perspectives on how best to learn and to teach mathematics.

I hope you’ll join my reflective journey through the course, starting next week.

# Wrapping up the #MTBoS30 challenge

It just occurred to me that if you don’t follow me on Twitter (shame on you!), you might wonder why the steady stream of blog posts has slowed.

Since joining the MathTwitterBlogosphere, I’d been wondering what I could add to a community that I was currently only taking from — their thoughts, activities and enthusiasm.  A few people suggested starting a blog, but I wasn’t sure that I had anything to say that hadn’t already been said by people far more eloquent than me. In the afterglow of NCTM 2016 and with an encouraging word from Tracy Zager, I started thinking seriously about blogging. Then in May, I came across #MTBoS30 — the 30 day blogging challenge started by Anne Schwartz — and so I dived right in.

After successfully finishing #MTBoS30 I wanted a new #MTBoS challenge. John Rowe and I (well, mostly John!) are busy putting something together. We hope you’ll get involved; more very soon.

#### Five things I learned from #MTBoS30

How do I know what I think until I see what I say?
— EM Forster

1. Sometimes I am so incredibly stupid. I know that writing clarifies thoughts; the best way for me to progress with mathematical research is to write down, organise and explain what I know so far to help identify gaps in my understanding. So why wouldn’t this be true for other areas of my professional life? I tell my students to write, write, write. I should take my own advice more often.
2. It is easier for me to commit to do something every day than to try and remember to do it frequently. Since I’ve stopped posting every day, I’ve written many great posts in my head — but no-one can read them there (right, Elham :))?
3. Given that this blog helps me clarify my own thoughts, I shouldn’t be concerned about how many people are reading (which is why I am definitely not obsessively monitoring the statistics …. right). But, since I can tell that people are looking (if not reading), it’s nice to know whether I’m just shouting into the wind or saying something that lingers. So now, when someone else’s blog post or idea resonates with me, I try to comment on their blog or by tweeting.
4. A blog is a great repository of explained ideas — for myself and for others. For example, it was easy to share resources from a recent teacher PD session by writing a post about it. It is much more coherent than a link to a Dropbox folder (although that’s useful too). Plus, other people benefit — including me next time I run the session.
5. (I wrote the section header, walked to the photocopier, then came back and forgot what #5 in the list was going to be …)

# Better student talks

Having a mathematical idea is only one part of the equation (excuse the pun!); we also need to be able to communicate it. A good example is Fermat writing next to his conjecture that he (purportedly) had a proof that the margin was too small to contain. If Fermat really did have a proof, then he could have saved mathematicians 350 years of frustration by going to the effort of communicating it with us!

There are many different dimensions to how we want our students to be able to communicate. Is it in written or oral form? It is to be polished, or in intentionally draft form for revision? Does it present the resolution of mathematical work, or is it work in progress? Who is the intended audience — peers, novices, experts, themselves?

The way we write or talk, and the skills we need for the process, depends on which of these dimensions are at play. I think it is unreasonable to expect that students can produce polished writing or give engaging talks without explicit coaching on how to do this — particularly in a mathematical context.

In this post I want to focus on what I’ve done to improve students’ skills in presenting completed work. These ideas build on and adapt (in places, minimally) the great structures that my colleague A/Prof Lesley Ward has put in place in the Mathematical Communication course in our undergraduate mathematics degree.

#### Brainstorming characteristics of good and bad talks

Students have heard many talks, likely more bad than good! I start by asking them to brainstorm in their groups the features of talks they’ve heard. They write them on the whiteboard under the following headings:

• Delivery: volume, rate, articulation — how the message is transmitted.
• Language: choice of words, grammar, using highly effective phrases — how the message is conveyed.
• Organisation: sequences and relationships amongst the ideas — how the content is structured.
• Content: what is said and how it is adapted to the listener and the situation.
• Visual aids: slides, physical props — how the aids reinforce the major points and stimulate the audience.
• Other: any ideas that don’t fit elsewhere.

There is now a collective understanding (or at least appreciation) of what to aim for and what to avoid. After this, I hand out the final presentation rubric which will be used to score the final project talks. This handout is minimally adapted from one used at Harvey Mudd College.

(Large whiteboard images here and here.)

#### Structured skill-building

Each week we start one class with a mini-talk (the other class starts with a Visual Pattern; thanks Fawn!). The aim is to slowly build skills, and to gradually improve. Talks increase in duration, and in other requirements.

• Mini-talk 1: Getting started! A week in advance of the first talk, students choose their talk partners (usually a friend). I also hand out a mathematical fact (Collatz Conjecture or Koch Snowflake) with a few ideas for what to talk about, but they need to research some more themselves. They deliver their three-minute talks at the desks. After the talk, the listener thinks of one thing the other person did well and tells them. All elements are designed to be the least threatening option that I can think of.
• Mini-talk 2: Build confidence; make improvements. Students choose different talk partners a week in advance. They re-present their fact; the aim is to revisit what went well last time and what could be improved.
• Mini-talk 3: Standing up; using whiteboards. I select student pairs from each student’s usual work group. Students give three-minute talks about their project topics at the whiteboard.
• Mini-talk 4: Present a new topic; listener steps up. I select student pairs from among the class. I give out new mathematical facts (Prime Numbers or Fibonacci Sequence). The only other change from Talk 3 is that the listener must ask at least one question.
• Mini-talk 5: Preparing visual aids. I select student pairs from among the class. Students need to prepare something beforehand to write, draw or show. They present the ‘other fact’ from Mini-talk 4.
• Mini-talk 6: Putting it all together. Self-selected groups of three. Students give four-minute talks about any mathematical topic of their choosing (not their project!). Must use visual aids to engage their audience. Students self-evaluate afterwards according to the presentation rubric. Students also practice time-keeping and ‘not panicking’ when shown the ’30 seconds to go’ yellow card.

I haven’t talked here about guiding students in selecting which parts of the mathematics to draw out and which details don’t need to be told; how to tell the mathematical story is an important skill to develop that I’m not going to cover in this post.

You can get a PDF of my guidelines here. What I’ve set out can of course be adapted and expanded; the key is continual skill development and exposure to giving talks. By the time students get to the final 10-minute presentation they are fairly comfortable with giving talks about mathematical ideas to their peers. The final project talks are also far more enjoyable to listen to because of it!

What tips do you have for developing presentation skills in students? I’d love to hear about your strategies.

# Tracy Zager’s word clouds

Over the past year I have held this image in my head as a reminder and a motivation. It comes from Tracy Zager‘s 2015 NCTM ShadowCon talk ‘Breaking the Cycle‘, which is mandatory viewing. (If you have limited time, stop reading this post and go watch Tracy’s talk instead.)

At the same time that Tracy was giving her talk in Boston in April 2015, I was doing some last-minute preparation for my own talk (that same day!) for maths teachers in Adelaide, ~17,000 km away. But I was procrastinating by looking at Twitter. Fawn Nguyen, the live-tweeter for Tracy’s talk, tweeted out an image of these word clouds that stopped me in my tracks. Tracy had articulated so well what I felt but hadn’t been able to put into words. I grabbed a copy of the image, worked it into my presentation, and was talking about it that afternoon. I had no idea at that stage who Tracy was; I hadn’t yet heard her say a single word, but her message was resonating with me loud and clear through the flurry of ShadowCon tweets.

In July 2015, I was honoured to give the Hanna Neumann keynote at the Australian Association of Mathematics Teachers (AAMT) biennial conference (the Australian equivalent of NCTM but with far fewer people). In my talk ‘More than mathematics: developing effective problem solvers’, I set out a case for incorporating into our classrooms the creative, active and collaborative ways in which professional mathematicians work, with examples from my own experience. In the middle of my talk, I said something like the following:

This brings me back to the cat in the dark room. Andrew Wiles, the British mathematician famed for resolving Fermat’s Last Theorem, describes mathematical research like exploring a completely dark enormous mansion. You stumble around bumping into the furniture but gradually you learn where the furniture is. After a while — perhaps six months or so — you find the lightswitch, you turn it on, and it’s all illuminated. Then you move into the next room and spend another six months in the dark.

Mathematicians are chronically lost and confused, and that is how it is supposed to be. It would be ridiculous to think of mathematicians spending their days solving problems that they already know how to solve. Instead, we spend a lot of time uncertain about whether something will work, or uncertain about what to do next.

Mathematicians grow to feel quite comfortable with this kind of uncertainty, but I suspect that most of our students do not. So, let’s shift to thinking about our students. Put yourself in the mind of your typical student. What words would they use to describe maths?

Cue, from Tracy’s talk, the word cloud about school maths (on the left of the above image), the word cloud from mathematicians (on the right), and the slide that defines my questions: ‘How do we, in our classrooms, shift from here to here? To help our students experience mathematics as a curiosity-driven, joyful, beautiful, endeavour?’ I spent perhaps five minutes in the middle of a 50-minute talk on this slide, but to me, it is one of the cornerstones.

For some reason my talk struck a chord. I’ve since been delighted by invitations to share the message — and Tracy’s slide — with hundreds of teachers at conferences around Australia. And I was amazed earlier this year to have a senior mathematics professor stop me in the corridor to say something about ‘that word cloud’. Turns out that one photo of my AAMT talk was shared at the Australian Council of Heads of Mathematical Sciences — and it was me standing in front of Tracy’s word clouds. I am beyond ecstatic that, even for the briefest moment, this question was in the minds of the leaders of Australia’s university mathematics departments and the Australian mathematics community.

These word clouds remind me what my purpose is. It is to orient my own students towards the creative, active and collaborative ways in which professional mathematics work, and to help them experience mathematics as a curiosity-driven, joyful, beautiful, endeavour. And, it is to help others position their own thinking and teaching towards this goal.

These word clouds prompted a ‘fourth-grade teacher at heart’ maths coach from Maine and a university mathematician from Australia to become friends and collaborators. Being able to meet Tracy at NCTM 2016 is one of the reasons that I finally decided to make the trip, and it opened up an abundance of other friendships, cemented mainly at #MTBoSGameNight. (That’s Tracy’s work again; she is the master of weaving together and strengthening the threads of this community.)

It doesn’t feel right to finish this post without mentioning Tracy’s new book, due out in December 2016. I’ve been fortunate enough to have a sneak peak at a couple of chapters, and it is good. Put it on your Christmas wish list. (The Australian distributor of Stenhouse Publishers is Hawker Brownlow Education.) If you can’t wait for December, you can read her blog now.

Friendships like this are why I advocate Twitter and the #MTBoS to every teacher I meet. We have so much to learn from one another. All it takes sometimes is one tweet to get it started.

# My motivating images

This morning I read Sara Van Der Werf‘s post, ‘The Story of 2 Words & One Simple Tweak to Get All Students Talking‘. Her post is great, and you should definitely check it out, but what prompted me to write this post was the photo I’ve included below. Sara has it pinned up near her desk. If I walked in and saw it, I might think it is there to prompt any number of curriculum-related objectives. So I was surprised and delighted to find that it’s an artefact of an interaction with a student. Sara keeps it “to remind me to dig deeper than my first gut reaction to a student’s words or actions.”

That got me thinking about what hangs above my desk. So, of interest to possibly no-one but me, here is the annotated version of my current office pinboard. (An unmarked image is here.) Yes, there are lots of penguins!

1. I’m a fervent follower of David Allen’s ‘Getting Things Done’ principles. This flowchart is a reminder of how to deal with the never-ending stream of stuff that comes ‘in’.
2. A visual reminder (artist now unknown) of the SUCCESs Model from the Chip and Dan Heath’s ‘Made to Stick’ — if we want an idea to be ‘sticky’, there are six principles worth trying to incorporate: Simple, Unexpected, Concrete, Credible, Emotion, Story. These are front-of-mind when I prepare any talk (from keynote talks to weekly classes). When I think about the inspiring ShadowCon and Ignite talks from NCTM, they have most or all of these elements.
3. Most (all?) work starts at the left of this diagram — messy and uncertain — and ends at the right with clarify and focus. A reminder that it’s natural to start in the messy phase.
4. A ‘parchment’ delivered by three of my students a few years ago, advising me (as requested) of their group members for a project. It was originally sealed with wax and a ribbon. The fact that they went to the effort to follow through on an in-joke we had reminds me about forming strong, close relationships with each student.
5. 3D bridge designs by high-school students from workshops that we run. Students use coordinate geometry to plan their bridge and prepare the data file. We turn it into G-code and print the bridges. They then compete as to which bridge can hold the most weight. Reminds me about how creative students can be if we give them a chance.
6. Writing advice from Hugh Kearns and Maria Gardiner from Thinkwell. Includes ‘nail your feet to the floor’.
7. A regal invitation to dinner last year with The Duke of Kent, the Governor of South Australia, and fourteen other ‘young emerging scientists’. Probably the only time I will need to research how to greet a member of the Royal Family.
8. LaTeX commands for making citations. I can never remember these.
9. This PhD Comic about writing. It’s exactly my approach to book writing. Don’t tell my editor.
10. Emergency coffee.
11. A BART ticket reminding me of NCTM 2016 in fabulous San Francisco.
12. A thoughtful gift from a student thanking me for supporting her through a challenging semester.
13. An enlarged copy of the geocaching stickers I made for our Japan Jaunt in 2014.
14. Apparently this is how my partner imagines me lecturing, or otherwise addressing a crowd. Huh.
15. The beautiful Prime Climb Hundreds Chart to remind me of the wondrous things that we can Notice and Wonder about in mathematics. (It doesn’t have to be ‘real world’ to be interesting!)
16. The Knoster Model for managing complex change. When I feel overwhelmed, frustrated, confused, anxious or otherwise unhappy with how a project is going, I review this chart. I don’t use it to be prescriptive, but it helps me reflect on what dimension might be missing from a successful project.
17. More advice from Hugh Kearns and Maria Gardiner, this time for overcoming the inertia of starting a task. It is so faded that you can’t read it, but it says two things. The first is ‘Park on a hill’, which means to leave off at an easy point to pick back up from. The second is ‘Action -> Motivation -> Action (-> Reward)’. It is common to think that you should wait for motivation before starting, but oftentimes it is action that leads to motivation. Spend 5-15 minutes on a task you don’t want to do; if you aren’t motivated, you are free to stop. If you are motivated, keep going!
18. Fractal stamps from when I visited Macau in 2006.
19. Nametags from most (but not all!) of the conferences I’ve been to since January 2015. Can you spot NCTM 2016?
20. A beautiful Hiroshige woodblock calendar. I’ve been to Japan four times in two years. I find woodblock museums to be incredibly peaceful places.
21. Right above my monitor is where I put the little pieces of paper with big advice (and with room codes, phone numbers, grammar reminders and the like). I have a problem with saying yes and with being helpful. I don’t follow my own advice nearly enough.

Finally, just to round it out, here is the pinboard outside of my office. It is a rotating collection of funny and pointed cartoons. The intention is to encourage students and colleagues to stop and chat — but only if my door is open.

What does your pinboard look like? What are your motivating images? I’d love to know!