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 to guarantee that you actually find these numbers — you shouldn’t have the possibility of having to go through all the numbers to be sure of finding 0, for example.

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.

]]>Here’s how to make fairy bread: take white bread, spread it with margarine, and sprinkle with hundreds and thousands. Now cut into triangles and serve.

Notes:

- It has to be white bread. If you try to make fairy bread with wholemeal bread, or multigrain bread, woe betide you!
- It has to be margarine, not butter. Butter may just be acceptable only if it’s the kind that is spreadable directly from the fridge. It may be that “margarine” means something different in other places in the world, so just in case, what I’m thinking of the butter-like spread made of plant oils that is spreadable directly from the fridge and can spread very thinly.
- Hundreds and thousands are a kind of brightly-coloured sprinkles that are shaped like very tiny balls. If you use chocolate sprinkles, or sprinkles shaped like little sticks, or coloured sugar, then it’s not fairy bread.
- It has to be cut into triangles. Don’t ask me why. Triangles are more magical than rectangles I suppose.

When I went to Twitter Math Camp in the USA in 2017, one of the lunchtimes I made fairy bread for everyone and passed it out. It was heaps of fun seeing people’s reaction to it, which was mostly good, though mixed with various levels of surprise and confusion.

Fairy bread from Australia is delicious! Thank you @DavidKButlerUoA! #TMC17 pic.twitter.com/jwGLkJZARr

— Heather Kohn (@heather_kohn) July 29, 2017

For me, fairy bread is strongly linked to memories of my childhood, and every time I eat it I am surprised again at how good it is. I mean, it’s the stupidest thing: bread and margarine with sprinkles. Yet somehow all the more awesome for that.

And here is where I am supposed to make a point about maths or teaching or maths teaching. But that might ruin the whole thing. Like those horrible people who try to make fairy bread “more healthy” by using wholemeal bread. Honestly people! It’s a party food – just own it!

Actually this reminds me of people who are always trying to get me to make a mathematical moral to my play. Yes there are times when the mathematics people do is deeply meaningful or useful for solving real world problems, and there are other times when it’s just for fun and there is no other purpose to enjoy myself and spend time with good people. Sometimes I need to be left to simply enjoy it, thank you very much.

Oh look, I did make a point. I hope it didn’t ruin the experience too much.

]]>Then, on the morning of the day the students were coming, I had an inspiration and quickly knocked together the Human Galton Board.

The Galton Board is a device that is used to illustrate the binomial distribution, usually with little balls falling through grid of pegs where the balls bounce back and forth from one peg to another, to come to rest in little bins at the bottom. (It also goes by the name “quincunx”, but I was not going to use that word with high school students!) My idea was that you could achieve this with human bodies walking paths, but there was still a problem of not having enough people to create the graph using “bins”. Then I realised that the people could tread the path multiple times and create a graph of where they had arrived by stacking up some sort of block. All I needed was a way for them to get instructions for which way to go, which I knocked together on the train journey to work. Nothing like a last-minute plan!

**The Setup**

I started by drawing a giant Galton board on the ground in chalk, making sure it had 11 circles at its widest end. I carefully drew connecting lines in yellow for left and blue for right (it was easiest to do this diagonally in one colour and then in the other colour). Then I drew big numbers 0 to 10 above the last row. Like this:

I also set up a table beyond that last row, with labels 0 to 10, in order to build our graph. I needed something to use as blocks to make our graph, so I brought down a box of videotapes and placed it at the pointy end of the Galton board.

Finally, I randomly generated sheets with 100 arrows in rows of 10, which had questions to consider printed on the back. These were what the students would be using to trace their paths through the Galton Board. You can download a Word document with the arrows/questions.

**The Activity**

I gave out a sheet of directions to each student, and then demonstrated how to use the directions to walk the Board. The process is this:

- Pick up a block (video cassette).
- Stand on the first circle.
- Choose the next row of your list of directions and look at the first arrow. Follow the path in the direction the arrow describes to get to an circle on the board in the next row forward.
- Repeat with all ten arrows until you reach the end of the board.
- Look at the number where you are standing, then add your block to the pile on the table with the corresponding number.

I discovered quickly I needed to be explicit about only following the lines and that the left and right arrows did not mean moving exactly left and right but always moving forward. Also I needed to be explicit that you had to stand on the first circle before taking your directions. The colour-coding of the directions turned out to be a really good move, which helped them to follow the directions more easily.

Here is a video of three perspectives of the activity in action: from my own perspective as I walk the Board, from the side as the students walk the Board, and from above as the students walk the Board.

I asked the students to do this three times each (or four for the smaller groups), and at the end when they were milling around wondering what to do, I asked them to turn the page over and think about the questions there while they waited. (The list of questions is further down, if you’re interested, and it’s also in the same file as the random directions.)

**The discussion**

Once everyone had gone through the Board the appropriate number of times, I got everyone to come to the table and have a look at the graph we had made. Here is a picture of one from the day.

I asked the students to tell me what they noticed. Most students waved their hands to indicate that it was going up then back down again. A couple of students used words to point out how the numbers in the middle had come up more often.

I asked them why they expected the 5 to happen the most often, and there were a few responses. Some were about how there seem to often be as many left arrows as right arrows, some were about how if it’s a 50-50 chance of getting each, then you’d expect as many rights as lefts and that would put you in the middle.

At this stage I asked them to focus on question number 1 from their list, which was “Can you predict where you will end up based on the list of arrows?”. After several students said you could just walk it out, I refined the question to “Can you predict where you will end up just by looking at the list, without walking it yourself?” Some groups really got into investigating this question, using the existing board to walk and check their answers. Other groups had one person who had figured it out and just told everyone else. I encouraged those other groups to check that the answer was really correct by walking out some to see if it really worked.

Spoiler alert: the answer is that the number you end up at at the end of the Galton Board is the number of blue arrows in the list, regardless of the order they are in.

For most of the groups, I asked them to use their newfound knowledge to take some rows from their sheet and put a block in the right place on the graph without walking the board. This made our graph a lot more bell-shaped than before!

After this, I asked them to consider how many ways there were to get to various numbers. The discussion on how many ways there are to get to 1 was cool, with students saying that you had to pass from one diagonal yellow line to the next and there were 10 blue lines where you could do that. The discussion of how many ways there are to get to 2 could have gone on for quite some time, and if I’d had some more time, I would have gotten the students to break into groups to investigate it, or possibly done it on paper back in the classroom.

After this discussion we went back to our graph and decided another way to explain why 5 was the most common was because there were a lot more paths that get to 5.

And finally I talked to them about the binomial distribution and how it describes how likely it is for something to happen different numbers of times out of a collection, and how Galton had made his Board with marbles in order to demonstrate this distribution, but how much more fun it is to do it with people! I also made the point that a lot of probability situations hide some of the details of what is going on. Our graph only shows the probabilities of getting to specific end points, but completely hides all the individual paths that result in those places. Yet it was thinking about all those individual paths that allowed us to justify the probabilities we got.

**Some thoughts**

I thought this activity went very well considering I only had a couple of hours’ preparation! I was particularly proud of colour-coding the paths and arrows, and of making the graph out of videotape so we could get a good graph by getting students to repeat the process. One thing I liked about it was the physical feeling of going via different paths but still ending up in the same position. If we had had a hundred students each doing it once, I don’t think it would have had the same result.

Next time I might ask the students to chase down the number of paths to get to 2 all the way and to present their arguments. That seemed like a great place to take the investigation. Also I could have focused on the symmetry and considered the number of ways to get to 10, 9 and 8 as well.

A different place to go might be to change the probability of getting a blue arrow and observe what the final graph is like then, even putting the two graphs on the same table to compare. Of course I would need a second set of arrow sheets to do that!

Finally, it ought to be possible to get students to create their own lists of arrows. I could imagine getting the students to roll dice and record left for odd, right for even before you go outside so that the results feel even more random. Then it would be much easier to go back and apply it to a more mundane example. Also you could easily change the distribution and go multiple of three for left, everything else for right.

If I ever do try any of these other ideas, I’ll let you know! If you have any ideas, please put them in the comments so others can be inspired by them!

]]>In a Markov chain, there is a thing that can be in any number of states, and the probability of switching to some other state is dependent only on which state it is in now. So maybe there’s three states the thing can be in, called A, B and C. If it’s in state A then it could switch to state B or C, or it could stay in state A. If it’s in state B, then it could switch to state A or C, or it could stay in state B. But the probabilities of going from A to the other states might be totally different from the probabilities of switching from B to the other states. Described using the word “state”, you might imagine a thing changing a property it has like colour or size, but it can also be used to describe things changing location too. Imagine the states as being literal states in a country, and the thing is a person who moves around between them.

Just like with any other probability situation, you might not be able to predict particularly well what any one individual does, but you might be able to tell what a large collection is doing. There are methods involving matrices and eigenvalues that allow you to make these predictions, but they can be viscerally illustrated with the Human Markov Chain!

**The idea of Human Markov Chains**

The basic idea is to mark out the states as big zones on the floor/ground, and have people walk around between them, using random numbers to tell them which state to go to at each turn. With a large group of people, the number in each state at any given moment settles down to be roughly unchanging from turn to turn, even though there are always people moving. The official term for this is the “steady state”. Note you really do need at least 20 people for this to work properly, preferably at least 30.

**The setup**

I used cloth tape to mark out three lines on the floor coming out from a central point. This created the three states of the Markov Chain where the participants would walk. (I don’t have a photo of this, sorry, but there is a quick view of it in the first seconds of the video below.)

I created posters to tell the people in each state where to go based on what their random number is. They were printed on paper, laminated, and stuck on a pole so they were high enough for everyone to see. You can download a Word document with the signs, including a blank set for making your own.

I created a sheet of random numbers for each person. The idea is that on each turn, you look at the next random number, look at the sign to see where to go, and then move. Each page of the random numbers was different, which the students only took a few minutes to notice and they were very intrigued. This word file has 100 pages of 100 random digits each.

**The activity**

I passed out a sheet of random numbers and a pencil to each student, and got Nicholas to hold the sign where everyone could see it. I explained how the game would work by walking it out myself: everyone would each look at the digit they had, and find that digit on their sign, then go left or right or stay depending on where the digit appeared on the sign. Then we’d do it again with the next digit in our new place, and the next digit, and so on.

Thinking back I reckon I should have gotten a volunteer student to do it for several steps, so that we could see what was happening for one person more clearly. However, this time we just launched into it. I got everyone to start in the corner labelled “A” and then we set off, with me calling out “go” every few seconds, after I had seen everyone arrive in their new locations.

After about ten moves, I asked the students to notice and wonder about how people were arranged and they noticed that everyone was roughly evenly spread across the three corners, even though to start with they were all in one corner. We did a couple more turns so that we could illustrate how this roughly even spread is maintained even though most individual people move on every turn. We also discussed how individual people never settle down to one place and we can’t predict what happens to them, but we can predict what happens to how many there are in each corner.

The coolness of Markov Chains is how we can use the nine probabilities of going from one state to another to create a matrix and use matrix methods to pull out the information of how it will settle down.

Actually, you can investigate it from the other side too. The steady state isn’t just a description of how many people end up in each corner, it’s also a description of how much time any one person spends in each state on average. You could possibly get each person to record which state they end up in and notice that on average they spend roughly equal time in every state.

Then we switched up the signs and put everyone back in corner A and did another game. This time one corner ends up with twice as much as either of the other corners. The point here is that it doesn’t always settle down to be an even spread — you can have other distributions.

Then we switched up the signs one last time and put everyone back in corner A again and did one last game. This time even though everyone started in corner A, everyone ends up in corner C eventually. It was quite funny to see the people arriving in corner C realising that they couldn’t escape, and see the people who hadn’t yet gotten to C all confused about why nobody was moving once they got there only to exclaim “oh” loudly when they arrived because it was obvious when they looked up at the sign. (This sort of situation is called an “absorbing state”.)

Here is a little video of how this last game went:

Today we did Human Markov Chains for an activity with high school students. It went way better than we expected! Here's a quick video of the one with an absorbing state. #MLCUoA pic.twitter.com/4jlPfMe6Ij

— David Butler (@DavidKButlerUoA) May 10, 2018

This last version was pretty cool because we had one person who bounced around for quite some time before arriving at C. If I had had the inclination, I might have started to investigate just how many steps someone might take on average before arriving in the absorbing state, first by getting everyone to do it again and count how many steps it took. This is another thing you can investigate with the matrix methods.

All in all it was a nice little activity to do for half an hour to get a feel for a probability situation quite different to the sorts of things you normally do. I really like the tension between individuals doing wildly random things, but the system being predictable. If I was actually teaching Markov Chains, I’d love to do it with my actual class so that I could give them a bodyscale experience to draw on when imagining these things!

**Some theory**

Just in case you’re wondering, you can set up a nice little matrix to describe what probability there is of moving from any state to any other state. If you have a certain number/percentage in each state you can write that as a column vector, and then how many you expect to be in each state one step later is found by multiplying the matrix and the vector.

You can find the steady state by starting with any vector and applying the matrix over and over and over until the result seems to settle down. This is one of the reasons why you might want to know the powers of a matrix! On the other hand, knowing that the steady state itself is unchanged by the system, you can just solve a matrix-vector equation to go straight to the answer! If A is the matrix and v is the steady state vector, then you must have Av = v, so 0 = v – Av, and hence 0 = (I-A)v. Now we have matrix times vector is equal to 0, and that can be solved by good old row operations!

In fact, I did all of those operations on a general matrix in order to produce a formula for the steady state of a 3×3 matrix, then put it into this Desmos graph so I could fiddle with the matrix to get the results I wanted. I had to do this because I needed my probabilities to be multiples of 1/10 in order to use random digits as my way of telling students how to get around!

**Conclusion**

All in all the activity went much better than I had expected, and I am looking forward to doing it again sometime soon! I want to spend more time discussing what happened and trying to do some predicting and checking our predictions. I just need enough students to make it worthwhile! My hope is that I can convince the Maths 1B lecturer to let me do it with his class(es). We’ll see…

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