This week I’ve been running the tutorials for the core first year Health Sciences course. The tutorial is a very light intro into how data is part of communication of health science research, and one of the activities involves the students arranging a set of data cards to investigate relationships between variables. Something happened today that I hadn’t observed before and I need to talk about it.
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.
This week I provided games and puzzles at a welcome lunch for new students in the Mathematical Sciences degree programs. I had big logic puzzles and maths toys and also a list of some of my eight most favourite puzzles on tables with paper tablecloths to write on.
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.
A lot of people introduce the derivative at a point as the slope of the tangent at that point, which to me is quite confusing. It seems to me that the reason we want the derivative is that it is a measure of the slope of our actual function at that point, not the slope of a completely different thing. To me, the thing is that the function itself is pretty much straight if we are close enough to it, so when we’re looking really close, saying it has a slope at this point is a meaningful thing to say.
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 .)
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 |x2-1| 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.
I created the Number Dress-Up Party puzzle way back in 2017 and every so often I stumble across it again when searching Twitter for other stuff. When I stumbled across it today, I decided it was time to write it up in a blog post.
The puzzle goes like this:
The Number Dress-Up Party
All 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 robot + 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 thought about this problem, and are ok with spoilers, then check out the replies to this tweet .
These comments were left on the original blog post:
mau 4 February 2019:
I did not see much of a difference between real and rational numbers: at least, the solution I devised is the same (and no real number can be generated in a finite number of steps, unless we stumbled into a costume specially related to that number: but maybe I am wrong). Integer and natural numbers need a different approach, indeed.
David Butler 11 February 2019:
Yes I didn’t see much of a difference between the real and rational number parties. Or indeed the complex number party either. I think once they’re closed under all four of the operations we’re allowed to use, you don’t get much extra. And yes I agree the natural numbers/integers are very interesting!
This post was going to be part of the Virtual Conference of Mathematical Flavours, which you can see all the keynote speakers and presentations here: https://samjshah.com/mathematical-flavors-convention-center/ . The prompt for all the blog posts that are part of this conference is this: “How does your class move the needle on what your kids think about the doing of math, or what counts as math, or what math feels like, or who can do math?” In the end, it didn’t end up being there, because my computer started dying painfully at the critical time, but I still want to highlight the Virtual Conference anyway because it was a great idea.
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?
Since 2013, the MLC and Writing Centre have been doing a game called Letters and Numbers at Orientation Weeks and Open Days to create interaction with people. I tweeted a photo of one of our sessions during Open Day yesterday and it has attracted a lot of attention, so I thought I might record some details of the game for people to read if they’re interested.
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:
(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 (see “Jack Frost’s Centre”), about how I encourage people to put partial solutions (see “An opening gambit for the Numbers Game”), and how I alleviated the fear caused by the numbers themselves and did something else instead (see “The Arts student’s maths brain”).
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.
UPDATE: Later, I wrote more about the Numbers Game and how to change the goal to make it easier to engage in “Changing the goal of the Numbers Game“.
These comments were left on the original blog post:
Mike Ollerton 13 August 2018
Thank you for writing your description of this game – I absolutely love it.
David Butler 16 August 2018
So glad you like it! I’ve certainly enjoyed doing it for the last five years.
Amy Zimmer 15 August 2018
Thank you so much for writing this up. Can’t wait to try it!
David Butler 16 August 2018
You are most welcome. I hope it goes well for you!
On the 23rd of July 2008, I started my first day as coordinator of the Maths Learning Centre at the University of Adelaide. Today is the 23rd of July 2018 – the ten year anniversary of that first day. (Well, it was the 23rd of July when I started writing this post!)
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.
A year ago, I went to Twitter Math Camp (TMC) and it was a wonderful experience. TMC is a great conference full of all sorts of opportunities for maths teachers to learn from each other in many ways.
The one way I like the best out of all the possibilities is “My Favourite”.
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.
Conclusion
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.
UPDATE: Yes someone did record them and they are in this playlist along with keynotes and My Fabourites from earlier years’ conferences.
Fairy bread, in case you don’t know, is an Australian children’s party food.
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.
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.
This comment was left on the original blog post:
David Roberts 10 July 2018
As I’m sure you know, David, Dutch people love sprinkles of all kinds on bread, and for some reason especially for breakfast. When I was in the Netherlands a few years back, at a supermarket, there were (at least) two whole shelving units for different kinds of sprinkles. I do wonder if fairy bread was introduced via some widely-sold party-food cookbook a few decades back (edit: well, it’s at least 90 years old, according to https://en.wikipedia.org/wiki/Fairy_bread !), where the author/compiler was inspired by this cultural phenomenon.
Last week we were booked in to do Human Markov Chains with several groups of school students, but it turned out there would be a lot fewer of them than we expected, and I didn’t think Human Markov Chains would work very well with under 20 students. I still dearly wanted to do a moving maths activity, and I still wanted it to be about probability, but I wasn’t sure what to do. Then, on the morning of the day the students were coming, I had an inspiration and quickly knocked together the Human Galton Board.
[UPDATE 2023: I’ve been informed that Galton was a horrible racist who invented eugenics. https://chalkdustmagazine.com/features/an-uncomfortable-truth/ Wikipedia says an alternative name is “bean machine”, so this is what I shall be calling it in future. I really like the ring of “human bean machine”.]
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!
These comments were left on the original blog post:
Anonymous 4 August 2023
This looks like a fun, engaging activity for students. Just curious, what is significant about the number of blue arrows? Why does the number of blue arrows determine where you end up?
Thanks for sharing.
David Butler 4 August 2023
With no blue arrows, you have to end up at the leftmost position, which is called “0”.
With one blue arrow, you only move one space rightwards during your journey, so you have to end up on the first diagonal line going to the left, which means you have to end up on the next end position, labelled “1”.
The same thing happens for all the numbers of blue arrows. Each one shifts you to the right one step, and so your final position is also shifted to the right one step.
