The students had been going for a little while on the activity, and I walked over to one group just as they were pulling apart some groupings of cards. I asked them what they were doing and they said “We’re starting again because the one we did didn’t work.”

“What do you mean it didn’t work?” I asked.

”We we’re looking at hat wearing and happiness and we didn’t see anything,” they replied.

I was momentarily shocked as the implication on this began to dawn. These students had made a picture that showed there was no relationship, and decided to take it apart because it didn’t work. That is, in their minds, it only works if there is a relationship!

I said to them I’d love to have them put their picture back, because it’s still good to show there isn’t a relationship. (They didn’t, which made me sad.)

I wonder if they had come to this conclusion just because of their natural thinking, or because their past experience was that if a teacher asks them to look at data then there is always a relationship. Either way it’s a bit of a dangerous thing to set up because we are in a bit of a crisis in medical publishing where only positive results get published.

Perhaps we need to give students more examples of data working effectively to argue a *lack* of relationship.

One of the puzzles is the Seven Sticks puzzle, which I invented:

Seven Sticks

I have seven sticks, all different lengths, all a whole number of centimetres long. I can tell the longest one is less than 30 cm long, because it’s shorter than a piece of paper, but other than that I have no way of measuring them.

Whenever I pick three sticks from the pile, I find that I can’t ever make a triangle with them.

How long is the shortest stick?

I sat down at one of the tables and I could see the students there were working on the Seven Sticks, so I asked them to explain what they had done. (WARNING! I will need to talk about their approach to the problem, so SPOILER ALERT!)

They told me about their reasoning concerning lengths that don’t form triangles, and what that will mean if you put the sticks in a list in order of size. They had used this reasoning to write out a couple of lists of sticks on the tablecloth which showed that a certain length of shortest stick was possible but that a longer shorter stick wasn’t possible. And so they knew how long the shortest stick actually was.

Only they said to me they hadn’t done it right.

I was surprised. “What?” I said. “All of your reasoning was correct and completely logical and you explained it all very clearly. Why don’t you think you’ve done it right?”

Their response was that what they had done wasn’t maths. They pointed across the table to what some other students were doing, which had all sorts of scribbling with calculations and algebra, and said, “See? That’s maths there and we didn’t do any of that.”

I told them that actually what they did was exactly what maths is — reasoning things out using the information you have and being able to be sure of your method and your answer. Just because there’s no symbols, it doesn’t mean it’s not maths.

Only later did I realise the implication of what had happened: these students are coming into a maths degree, which means they have done the highest level of maths at school, and they perceived that something was only maths if it had symbols in it. A plain ordinary logical argument wasn’t maths to them. And a scribbling of symbols was better than a logical argument, even if it didn’t actually produce a solution.

This made me really sad.

I wonder what sort of experiences they had that led them to this belief. I wonder what sort of experiences they missed out on that led them to this belief. I wonder how many other students have the same belief and I will never know. I wonder how I can help all the new students to see more in maths than calculations and symbol manipulation, and allow themselves to be proud of their work when they do it another way.

]]>For years I got at this idea by drawing a little magnifying glass on my function, and a big version nearby to show what you could see through the magnifying glass. But last year I decided to get technology to help me do this dynamically and I made a Desmos graph with a zooming view window that will show you the view on a tiny circle around a point on a curve. (Some of the bells and whistles here were provided with a little help from Andrew Knauft.)

Here’s what the graph looks like, and a link to the graph so you can play with it yourself: https://www.desmos.com/calculator/pa1cudpc07

I’ve used this to help students see that the curve itself really does look straight when you’re very close, and so treating it like it is straight there and saying it has a slope is a reasonable thing to do. Students are usually most impressed and love to play with it. It’s also interesting to put in a function with a kink in it like |x^{2}-1| to show that the kink never goes away no matter how closely you view the function, so having a single slope there isn’t really reasonable.

Every time I’ve had to search my own twitter account to find the tweet where I shared it, and I couldn’t keep doing that, so here it is in a quick blog post for posterity.

]]>The puzzle goes like this:

The Number Dress-Up PartyAll the numbers have come to a dress-up party in full costume. They all know themselves which costume everyone else is wearing, but you don’t know.

If you pick any two of them and ask them to combine with +, -, × or ÷, they will point out which costume is the correct answer, and they’ll happily do it as often as you want. For example, you might ask for hamburger + bear and they will point to unicorn. (If the answer isn’t at the party, they’ll tell you that too.)

How do you correctly identify the numbers 0, 1, 2 and 3? How do you do it in as few steps as possible?

It is worth clarifying now that when I say “all” numbers I originally meant it to be all the real numbers. (But it is very interesting to think about how the problem is different or even possible if it means all the rational numbers or all the integers or all the natural numbers or some other set of numbers entirely.) It’s also worth pointing out that it is actually possible for you to guarantee that you actually find these numbers — you shouldn’t have the possibility of having to go through all the infinitely many costumes to be sure of finding 0, for example.

It’s also worth clarifying that the rules say you have to ask two *different *costumes to combine with an operation. If you *can *see how using two of the same costume might help you identify actual numbers, then you are thinking along some helpful lines. However, the puzzle is much harder and much more interesting if you have to use two different costumes every time.

I love this puzzle so much! The reason I love it is that it forces you to think about numbers and algebra in a completely different way to any other puzzle or problem I have seen. You really need to think about how numbers are related to each other and how they behave under operations in order to figure out a way to correctly identify these numbers from the way the costumes interact.

Also, in order to tell someone about your solution, or even figure it out in the first place, you need to find a way of talking about or writing about this, which is a lot more difficult than it might seem at first glance. I find it really interesting to see how people attack the problem of describing what they are doing in this problem.

The feel of this puzzle to me is the feel of an abstract pure maths course like abstract algebra or number theory or real analysis, where you are digging deep into how numbers work without reference to specific numbers per se. I would love to go into such a course and use this in the first lecture/tute to get students in the right sort of mindset to attack the rest of the course. I’d also love to do an extension to this as an investigation into how the Euclidean Algorithm for natural number division works. To all those people who haven’t had that sort of training I say you are doing what a pure mathematician does when you think about this problem! You never thought it would be so much fun, did you?

Anyway, there is the Number Dress-Up Party puzzle for posterity. There are several different solutions and they are all lovely. Have fun!

PS: If you feel like seeing how people have attacked this problem, and are ok with spoilers, then check out the replies to this tweet.

This has turned out to be a most interesting puzzle. (Latest version @MathTeachScholl, @joeykelly89, @nomad_penguin.) #MTBoS pic.twitter.com/yFwp7K4gpY

— David Butler (@DavidKButlerUoA) August 11, 2017

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There are many things I could have written about this, but I think I will choose one thing that is about my approach in the MLC to student questions. In the MLC *everyone is worthy to ask both stupid and smart questions*.

My Maths Learning Centre is a place where any student doing coursework at the Uni of Adelaide can visit to talk about their maths learning with a tutor (often me). People come to talk about all aspects of their maths learning in all sorts of places where maths appears, from dividing whole numbers by hand to understanding proofs about continuity of functions between abstract metric spaces. My point here today is that people from both ends of that spectrum and everywhere in between are allowed to ask questions that are about basics and questions that are about deep connections.

Imagine a student who has always been good at maths, who finds things easy and quickly grasps abstract definitions. It is natural for such a student to fold their goodness at maths into their identity, which often means they become extremely embarrassed to show any sign of struggling. They’re supposed to be the smart student and this simple stuff is supposed to be obvious for them. So if they have a question about the basics, they hide it and hope it will come clear eventually.

The thing is, having a question about something simple doesn’t make you stupid, and it doesn’t even make you not smart. Having a question about how to get from line 3 to line 4 is at the very least a sign that you’re paying close enough attention to wonder about that step; having a question about the definition is a sign that you know definitions are important; and having a question about some random bit of algebra or notation you happen to have never seen just shows you want to learn. In my Maths Learning Centre, I try to make it a place where everyone can ask a “stupid” question. Where stupid questions are treated with respect and answered clearly, with encouragement to make sense of what is happening.

Now imagine a student who has always struggled with maths, who just never seems to understand the explanation the teacher is giving the first time, and who struggles to get through the first few of the exercises. It is natural for such a student to fold their badness at maths into their identity, which often means they don’t even try to understand things and just look for some step-by-step instructions they can follow so it will be over with as quickly as possible.

The irony is, they never finish their exercises, so they never get to be part of that part of a maths class where the early finishers ask the deep and involved questions about theory and beyond-curriculum interesting stuff — the very stuff that can make maths a lot more fun. I know for a fact that students who feel they are bad at maths are intelligent people capable of logical and creative thought, and they deserve to ask their deep questions. So in my Maths Learning Centre, I try to make it a place where everyone can ask a “smart” question. If a student who is struggling asks about infinity or quaternions or what my PhD was about, I will damn well discuss it with them. If they look at the work they’re doing and ask how it is connected to some other bit of maths, we’ll explore that together. That curiosity is a treasure to be prized and I will not squash it by saying we have to get on with the assignment now.

And you know what, it turns out that many a basic question is actually a deep and clever question after all. Recently a student who was struggling asked why it was ok to add two equations together. Not one student in my ten years of working at the MLC has ever asked that question! There must be something really special about the person who asks this question, right? And it’s a really deep question about the nature of equality. I want my Maths Learning Centre to be a place were it is okay for everyone to ask a question that is* simultaneously* stupid and clever.

That’s all I have to say. I believe everyone deserves the chance to ask stupid questions and to ask clever questions and to ask questions that are simultaneously both. They are worthy to have their questions taken seriously and the answers discussed with respect for the humanity and intelligence of the asker. I have to always remind myself to give students the chance to ask these questions when I’m with them, especially students who are struggling to articulate the questions for whatever reason. And maybe if they’re not asking, I’ll sometimes ask the questions for them and we’ll answer them together.

How will you welcome all people in your learning spaces to ask all kinds of questions?

]]>**The inspiration**

In the early 20-teens, we had a show on Australian TV called “Letters and Numbers”. In it, contestants played two games: one where they got a random collection of letters and had to create the longest word they can, and one where they got a random collection of numbers and a random target and had to create a calculation that produced the target. They did this in a 30 second timeframe, counted down by a giant clock on the wall. The Australian show was based on a show which in the UK is called “Countdown” in reference to the clock, which in turn is inspired by a French show called “Des chiffres et des lettres”.

I was thinking of a way to have a combined MLC and Writing Centre activity to engage new students with us so we could talk to them about our services, and of course this TV show popped into my head — it was the perfect combination of low-stakes maths and language. All I needed was a way to do it live in a public place.

**The setup**

In the TV show games, the letters and numbers used are chosen randomly, and I needed a way to do this in a public place. I could possibly have done a computer thing or had a pre-written list, but I really did want them to be random, and I liked the idea of the participants choosing the numbers/letters themselves, so I wanted to choose them from a bag or bucket. The letters use the same distribution as the tiles in a Scrabble set, so I could have just used a Scrabble set for those. But the numbers I would need to make myself somehow. I decided to use pop sticks with the numbers/letters drawn on them, which could be drawn randomly from buckets.

Here are the distributions of letters and numbers I put on the pop sticks:

Vowels | 4A, 6E, 4I, 4O, 2U | (half the vowels in a Scrabbble set) |

Consonants | 2B, 2C, 4D, 2F, 3G, 2H, 1J, 1K, 4L, 2M, 6N, 2P, 1Q, 6R, 4S, 6T, 2V, 2W, 1X, 2Y, 1Z | (all the consonants in a Scabble set) |

Target | Three of each digit from 1 to 9, and two 0’s. | |

Small numbers | Two of each number from 1 to 10 | |

Big numbers | One each of: 25, 40, 50, 60, 75, 100, 120, 125 | (I added some extra on top of what is in the TV show) |

The basic setup is to have a whiteboard with the instructions on it, a space marked out for the letters and numbers puzzles and plenty of space to write solutions.

**Getting started**

For the letters game, randomly choose four vowels from the vowels bucket, and five consonants from the consonant bucket and write them on the board.

For the numbers game, choose three digits from the target bucket, one after the other without replacement. Together these are the three digit target number, which you write on the board. (If the first digit is a 0, then you can treat the rest as a two-digit number if you like, or just rearrange so it doesn’t start with 0.)

Also for the numbers game, choose four small numbers and two big numbers and write them on the board too.

**The rules**

The rules of the Letters game that I put on the whiteboard are as follows:

Help to make as many and as long words as we can from these randomly chosen four vowels and five consonants. (Letters may only be used as many times as they are in the list.)

The rules of the Numbers game that I put on the whiteboard are as follows:

Help to make a calculation to produce the target, using some or all of these two big and four small numbers, and any combination of +, -, ×, ÷, and brackets. (Numbers can only be used as many times as they are in the list.)

It’s worth noting that in the original TV show, contestants can choose how many vowels and consonants they want, or how many big or small numbers they want. I dictated a specific number of everything so that we could quickly play the game with anyone who came up, or they could play by themselves if we were momentarily distracted!

**Unwritten rules**

There are some unwritten rules about how we go about doing the Numbers and Letters games that it’s worth making explicit.

For the letters game, small words and proper names are ok. Most word puzzles don’t allow those, but this game is designed specially for public interaction. If someone notices that they can spell Bernita or Spain, who am I to diminish their glory? Plus finding a host of small words like “to”, “of”, “for”, etc is a great start to get some words on the board, and you can often modify them to get them a bit longer words too.

For the numbers game, partial answers are ok. People do stare at the board trying to come up with the answer all in one go, and they need to realise that it’s ok to scribble some working, or have something that is close in order to maybe get closer upon modifying it. In fact, I have written about this before.

It’s also worth pointing out that the numbers game isn’t always solvable. Sometimes when it’s really tricky, what we usually do is try to get as close as we can. On the fly, it’s hard to prove it’s actually impossible, so at some point you need to call it close enough and do a new one.

On the other hand, sometimes the Numbers game is solved in the first few seconds. In that case, we usually try to find many different solutions, encouraging people to try to use different operations or more of the numbers.

**Final notes**

While it would probably work fine to choose the letters and numbers with a computer or app, the buckets have a sort of playful and tactile element that I really like. They also allow us to engage people passing, by going up to them with a bucket and asking them to pull out some popsticks for us. Yesterday every person I asked to choose some sticks for us came over to have a closer look at what we were doing.

I have written about my thoughts to do with the Numbers game three times in the past. I wrote about how the fact it is a game can help people participate when they otherwise wouldn’t, about how I encourage people to put partial solutions, and how I alleviated the fear caused by the numbers themselves and did something else instead.

These games have been part of the MLC and Writing Centre’s identity for five years. We do it at Open Day and also at Orientation every semester and I really look forward to it each time. For new students this sets up a continuity between when students were just visiting university and when they arrive. For existing students, they’ll often seek us out at these times to tell us how they’re doing and engage in something that is a pivotal memory of their early time at university. Also it has become something that all the other service areas expect we do, and they come to join in as well during these events. I’m so glad people of Twitter have become interested, because it really is a fun thing to do.

]]>So much has happened in that time. I have given hundreds of hours of revision seminars, I have written/drawn on tonnes of paper, and used miles of sticky tape and chalk in mathematical artwork, and I have talked individually to over ten thousand students. I can’t possibly distill it all into one blog post, but I can talk about why I believe I am meant to be in this job and still meant to be in this job.

When I went to the interview for the MLC coordinator position, I thought it would be a pretty cool job to have. At the interview, I had the epiphany that it was not just a cool job but it was in fact the perfect job for me, the job I really *needed* to have. Travelling home from the interview, the thought that I might possibly *not* get the job made me cry almost the whole train journey. I remember praying to God that I would find out soon. They called me that very night to say I had won the position!

I still believe that this is the job I was destined to have. In no other job could I have been able to indulge my dual interest in both university pure maths concepts and fundamental maths concepts you meet in primary school. In no other job could I simultaneously help students overcome their crippling fear of mathematics and (sometimes the same students) become research mathematicians. In no other job could I make mathematical art and play an actual legitimate part of my work. Admittedly, I may have *made* some of those things part of my job when they weren’t part of it before, but it was being here in this role at this university that has allowed me to do so.

There are parts of the job that are annoying — interminable meetings, lecturers who take my offer of support as an affront, constant requirements to convince the establishment that what I do is important, semesterly reminders that we just don’t have enough funding to provide the level of support I think is necessary — but overall it is a most wonderful and amazing job.

When I started ten years ago, I already knew the pleasure in helping students learn, but since then I have learned the even greater pleasure of letting students help me learn. I have barely scraped the surface of learning first hand about how people think about maths and how they learn maths, and I don’t think I never get to the end of the wonder of it.

Thank you to the other MLC lecturer Nicholas and all my casual tutors for coming along for this ride of teaching at the MLC, for listening to me as I talk through my crazy ideas and plans, and for pushing me to be a better teacher and leader. Thank you to all the other staff of the university that have worked so graciously with me, especially those nearest in the other student development and support roles. Thank you to my new colleagues I have met through Twitter, who make me better as a teacher and a mathematician in so many ways. Most of all thank you to my wonderful wife and daughters for always believing in me, and tolerating my mind ticking over on work things most of the time – I could never do this without your love and encouragement.

It’s been a wonderful ten years at the MLC. I hope the next decade is just as wonderful.

]]>TMC is a truly remarkable conference, as I have described before, both in 2016 when I wasn’t there, and in 2017, when I *was *there. You can click on those links to read what I thought about it back then. The short version is I have never seen or experienced anything quite like it, with such a dense concentration of professional learning and networking, and also making deep connections with other passionate people.

In this last TMC17 post, I want to talk about the thing that I think makes TMC so special: *the attitude. *You see, other conferences might be able to have multiple different ways to share and learn from each other, and they might have opportunities for making real connections with others, but I think they might still fall short without this one special sauce that is the TMC attitude. One way to describe it would be to say the people at TMC come *open and eager to be part of the thinking and the community*.

The first aspect of this TMC attitude is that everyone in every face-to-face session is there ready to learn something. Not just expecting it but *looking out for it *— they are actively seeking for something to learn. More than this, not one person I saw at TMC last year was expecting to have something handed to them by a presenter, as if the presenter was some ordained expert, but everyone was fully expecting to process what they had learned through ongoing discussion with their colleagues, which would happen at breaks and late into the night. Even more remarkable than this, *the presenters themselves *all seemed to expect to learn something in the sessions that they themselves were presenting! (I know I certainly was expecting this.)

You may be thinking that at some other conference of course you are there looking for something to learn, and sure you are. But if I’m honest, sometimes I don’t know if I can go to yet another session that seems to be selling me a product, and sometimes I do go into a PD (Professional Development) expecting someone to just hand me the content without me needing to process it. Some PD experiences are almost designed to be about an expert handing down their product or policy and you don’t hope or expect to think. At TMC you almost always go in with an active attitude.

The second aspect of the TMC attitude is that everyone is there ready to be part of the community. As a presenter, everyone who comes to your session is a colleague you can discuss ideas with later outside the session. As a participant, you know that the presenter is another colleague who has something they are excited about to share and you want to be a support to them. You also feel that everyone around you is on the same journey, hoping to learn something from each other. Outside of sessions there is always someone to talk to and listen to and share both work things and life things with, and you know you can keep up a connection with these people through the medium of Twitter beyond the conference. Even if you don’t feel like engaging with people, you can let it all wash over you and feel part of something bigger.

Other conferences I have been to don’t have the same community feel. Some have been close, but not nearly at the same level. Certainly it’s rare for the presenter of a keynote to be around at the rest of the conference to be part of the community and discuss other people’s ideas, and rare for the audience to act like they are there to support them! It’s certainly true that no compulsory PD I’ve ever been to has had this attitude of community-building! At TMC community-building is always in the background.

I think that the attitude of the people at TMC is what makes TMC so remarkable.

But it doesn’t just have to exist at TMC — I think I can take this attitude to other PDs I go to in order to make them better. I too can go to sessions expecting to be actively learning *something. *and I too can go to sessions actively seeking to make real connections with people, especially in any sessions I present myself. If all of us do this more, then all PDs might be just that bit more like the awesomeness that is TMC.

I wish the TMC attitude for all of you.

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TMC was a wild wild time. At the sessions there were hundreds of amazing maths and teaching ideas to process. At break times during the conference there were things to play with and discuss, with overheard conversations going on all around me about interesting stuff. At the social time outside the conference there were things to play with and discuss, with overheard conversations (and singing) going all around me about interesting stuff. It was exciting and wonderful, and mentally noisy.

Our three morning sessions were an island of calm and quiet in this tumultuous sea.

At our first session, Megan and I got everyone started on how the iterative crochet works, and from that point forward we explored where these simple ideas could take us. Somehow it produced six hours of wonderful calm.

We sat and focused on our own fluffy explorations, or quietly played with the existing corals, or clamly discussed the events of the conference so far or other thoughts that were flitting through our heads. Most of us had the conference Twitter feed on to look at, but it was almost as if it was outside us and we were observing from afar. In that room we were somehow created a protected space where we could recharge our energy to engage in the wild ride that was the rest of TMC.

I think there were two features of the activity we were doing that helped to create this magical space. The first is the slowness of crochet exploration. In other mathematical domains, exploration can be a chaotic whirl of sound and colour, with frenzied scribbling and conversation. But the exploration in crochet is slow and can take hours to get to the stage where the structure is visible in the thing you are making. The second is the smallness of the actions you perform to do the exploration. The mechanism you use to produce these corals uses small stitches that hold your attention in a tiny bubble between the tips of your fingers, almost causing the rest of the world to fade into the background. Joey described the strange feeling of looking up and finding how big the world is, after you’ve spent the last half an hour looking at the tip of a crochet hook.

I didn’t mean for our workshop to have this haven-like property but I am so glad it did. It really made an amazing positive difference my experience of TMC, and I don’t think I would have had nearly as wonderful a time without it. By the third day I was aching to get there to be part of our little crochet cuddle. If I ever go to TMC again I would want to find or make a similar space to have those quiet times. In the year since, I think I have forgotten how important that quiet time of focus is, and I think in the next several months I will try to recapture it again.

Thank you to the wonderful crochet crew for the quietness.

Hey #MathYarns crew from #TMC17. I found this photo of us while looking for something else on my phone. It was such a great time at those morning sessions. pic.twitter.com/cTwQNRrjqO

— David Butler (@DavidKButlerUoA) February 1, 2018

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My Favourite is a short presentation of 5-15 mins about one of your favourite maths teaching things. Anyone attending TMC can put in a proposal to present one, and they are presented during the big whole-group sessions at the start/end of each day one after the other. Glenn Waddell recorded all of the My Favourites from 2017, and you can view them in a YouTube playlist here.

I have a few reasons why My Favourite is my favourite.

**Everyone is worthy: **I love the fact that anyone at TMC can do a My Favourite and everyone is welcome to. Anyone can become a conference presenter by simply having something to say and asking to say it, and you get to do it in front of the whole conference. At TMC17 I said one of the remarkable wonderful things about TMC is that everyone is worthy to present. My Favourites is one of the ways the TMC organisers really make that real.

**Hearing things I didn’t know I needed: **In the rest of the TMC conference you go to the sessions because you wanted to hear something in particular. The description in the program said something that made you want to hear more about that particular thing. With My Favourites you don’t know what you will hear, but it may just be the very thing you needed to hear today. You get to see the things that others love, and often the simple things that make a difference to their daily teaching lives. I would never know about some of those things without My Favourites and I am so grateful for it.

**Excitement in the everyday: **I love seeing the excitement of teachers who love something about their job. It reminds me that as hard as teaching is, you can find things worth being excited about within it, and that these things are many and varied. Seeing a variety of different people excited about a variety of different things gets me excited too. I want to go back to my teaching space and look for the things I might share at a My Favourite if I had an opportunity to do so.

**Bravery: **Finally, I love seeing the bravery of people sharing something about themselves with less than a day’s notice. It’s amazing to see this sort of courage especially in people who might be intimidated by the people they admire being in the audience. Yet this is the power of TMC. We know that those in the audience will listen and encourage. As Edmund Harris said, “everyone is worthy to present, but wants to be in the audience.” The encouraging environment helps the My Favourites presenters to be brave, which is a testament to what community can achieve.

I won’t be at TMC this year, and to am dearly hoping someone records the My Favourites again so I can see them! If you are going to TMC please sign up to do one, so that we can all hear you excitedly and bravely share things we didn’t know we needed to hear.

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