Reflections on maths, learning and maths learning support, by David K Butler
This blog post is about a game I invented in February 2020, the third in a suite of Battleships-style games. (The previous two are Which Number Where and Digit Disguises.)
Once upon a time, I met a His Royal Highness the Duke of Kent.
The story of how that happened was pretty cool from my perspective, but every so often I wonder about it from his perspective. The Duke is the patron of the Royal Institution of Australia, and was in Australia just as they were displaying their installation of the Institute for Figuring’s Hyperbolic Crochet Coral Reef. So they organised a special viewing for him, to which a few key people were invited. One of those people was me, since I was the mathematician involved in the project.
I had a short but pleasant conversation with an old man about geometry and what it was like to be a man crocheting in public. We’re laughing in the photo, so somebody must have said something funny, though I don’t remember what it was. And that was it.
I suppose His Royal Highness meets a lot of people, so I might not make much of a difference to him. But every so often I think about how when he thinks about his time in Adelaide, there I am, if even for a moment (perhaps made just a little more memorable due to the rarity of meeting people standing in a roomful of coral made of yarn). I am part of the life memory of the Queen’s cousin.
Which gets me thinking…
Many of the students who I meet through my work in the MLC, I only meet once. Yet what happens with me will be a part of their memory of their time at university, however small. My short time with them is entangled forever with the events of their lives during their study. Will that moment with me make their memories more or less pleasant? Will it encourage them or discourage them? Will I just confirm their worst fears about themselves, or will it be at all like this student, who one visit to a lab in one class made all the difference?
| From Ashleigh Geiger on Twitter 23 Aug 2020 |
|---|
| For me, at school, the focus on memorising rather than understanding is what fostered a fear of maths. Returning to uni later in life and needing to use maths (biochem and genetics) showed me there was nothing wrong with my abilities, just my perception of my abilities. And the way the maths was taught in my degree showed me how powerful and interesting it can be, and helped me learn to enjoy it again. I never ended up visiting the MLC but having your team wander around in prac classes, peer over my shoulder and tell me I’d nailed it did wonders for my confidence! 😊 So whilst I did have experiences early on that made me fear/hate maths, take heart that they can be turned around by positive and empowering experiences with the right teacher. Love your work! |
| https://x.com/ashleigh_bryar/status/1297305993487896577 |
Every one of these students is important to someone, much more than a Royal who lives on the other side of the world is important to me. And I am there in their memory for better or for worse. This inspires me regularly to be the best I can be with each and every one of the students, especially that very first time I meet them.
Because once upon a time, His Royal Highness the Duke of Kent met me.
One of my favourite puzzles is the Twelve matchsticks puzzle. It goes like this:
| Twelve matchsticks can be laid on the table to produce a variety of shapes. If the length of a matchstick is 1 unit, then the area of each shape can be found in square units. For example, these shapes have areas 6, 9 and 5 square units. |
| Arrange twelve matchsticks into a single closed shape with area exactly 4 square units. |
I will tell you soon why this is one of my favourite puzzles, but first I want to tell you where I first learned this puzzle.
Once upon a time, about ten years ago, I met a lecturer at my university who was designing a course called Puzzle-Based Learning. It was an interesting course, whose premise was that puzzles ought to provide a useful tool for teaching general problem-solving, since they are (ostensibly) context free, so that problem-solving skills are not confounded with simultaneously learning content. Anyway, when he was first telling me about the course, he shared with me the Twelve Matchsticks puzzle, because it was one of his favourite puzzles. The reason it was his favourite appeared to be because the solution used a particular piece of maths trivia.
And the puzzle has become one of my favourite too, but for very different reasons! Indeed, I strongly disagree with my colleague’s reasons for liking the puzzle.
The first thing I disagree with is the use of the word “the” in “the solution”, because this puzzle has many solutions! Yet my colleague presented it to me as if it had just one.
The second thing I disagree with is that using a specific piece of trivia makes it likeable. I do recognise that knowing more stuff makes you a better problem-solver in general for real-world problems, but even so I always feel cheated when solving a puzzle requires one very specific piece of information and you either know it or you don’t. It seems to me it sends a message that you can’t succeed without knowing all the random trivia in the world, when of course it is perfectly possible to come up with acceptable solutions to all sorts of problems with a bit of thinking and investigating.
The main reason I love the Twelve Matchsticks puzzle is because it does have many solutions, and because this fact means that different people can have success coming up with their own way of doing it, and feel successful. Not only that, if I have several different people solving it in different ways, I can ask people to explain their different solutions and there is a wonderful opportunity for mathematical reasoning.
Another reason I like it is that people almost always come up with “cheat” solutions where there are two separate shapes or shapes with matchsticks on the inside. I always commend people on their creative thinking, but I also ask if they reckon it’s possible to do it with just one shape, where all the matchsticks are on the edge, which doesn’t cross itself. I get to talk about which sorts of solutions are more or less satisfying.
The final reason is that most of the approaches use a lovely general problem-solving technique, which is to create a wrong answer that only satisfies part of the constraints, and try to modify it until it’s a proper solution. This is a very useful idea that can be helpful in many situations. As opposed to “find a random piece of trivia in the puzzle-writer’s head”.
Which brings me to the second part of the title of this blog post. Let me take a moment to describe two different experiences of being helped to solve this puzzle.
When I was first shown this puzzle, my colleague thought there was only one solution, and his help was to push me towards it. He asked “Is the number 12 familiar to you? What numbers add up to 12? Is there a specific shape that you have seen before with these numbers as its edges?” He was trying to get me closer and closer to the picture that was in his head and was asking me questions that filled in what he saw as the blanks in my head. And there were also some moments where he just told me things because he couldn’t wait long enough for me to get there.
I did get to his solution in the end, and I felt rather flat. I was very polite and listened to his excitement about how it used the piece of trivia, but inside I was thinking it was just another random thing to never come back to again. It wasn’t until much later that I decided to try again, wondering if I could solve the problem without that specific piece of trivia.
So I just played around with different shapes with twelve matchsticks and found they never had the right area. And I wondered how to modify them to have the right area. And I played around with maybe using the wrong number of matchsticks but having the right area, and wondered how to change the number of matchsticks but keep the same area. This experience where I used a general problem-solving strategy of “do it wrong and modify” was much more fun and much more rewarding.
When I help people with the Twelve Matchsticks puzzle, this second one is the approach I take. I ask them to try making shapes with the matchsticks to see what areas they can make. I ask what all their shapes have in common other than having 12 matchsticks. I notice when they’ve made a square with lines inside it and wonder how they came up with that. Can they do something similar but not put the extra ones in the middle? In short, I use what they are already doing to give them something new to think about. I encourage them to do what they are doing but more so, rather than what I would do.
The approach where you have an idea in your head of how it should be done and you try to get the student to fill in the blanks is called funnelling. When you are funnelling, your questions are directing them in the direction you are thinking, and you will get them there whether they understand or not. Often your questions are asking for yes-or-no or one-word answers to a structure in your head which you refuse to reveal to the student in advance, so from their perspective there is no rhyme or reason to the questions you are asking.
The approach where you riff off what the student is thinking and help them notice things that are already there is called focussing. When you are focussing, you are helping the student focus on what’s relevant, and focus on what information they might need to find out, and focus on their own progress, but you are willing to see where it might go.
Interestingly, when you are in a focussing mindset as a teacher, you often don’t mind just telling students relevant information every so often (such as a bit of maths trivia), because there are times it seems perfectly natural to the student that you need such a thing based on what is happening at that moment.
(There are several places you can read more about focussing versus funnelling questions. This blog post from Mark Chubb is a good start .)
From my experience with the Twelve Matchsticks, it’s actually a rather unpleasant experience as a student to be funnelled by a teacher. You don’t know what the teacher is getting at, and often you feel like there is a key piece of information they are withholding from you, and when it comes, he punchline often feels rather flat. Being focussed by a teacher feels different. The things the teacher says are more obviously relevant because they are related to something you yourself did or said, or something that is already right there in front of you. You don’t have to try to imagine what’s in the teacher’s head.
So the last reason the Twelve Matchsticks is one of my favourite puzzles is that it reminds me to use focussing questions rather than funnelling questions with my own students, and I think my students and I have a better experience of problem-solving because of it.
Quarter the Cross is one of my favourite activities of all time, whether in maths or just life. I learned about it way back in 2015 and have been mildly or very obsessed with it ever since. This blog post is about one particular version of the Quarter the Cross problem you might like: the colouring version!
You can read the rest of this blog post, and four other related posts, in PDF form here
The titles of the five posts in the series are:
Some resources linked from this post:
I’ve just started teaching an online course, and one module is a very very introductory statistics module. There are a couple of moments when we ask the students to describe how they interpret some hypothesis tests and p-values, and a couple of the students have written very lengthy responses describing all the factors that weren’t controlled in the experiments outlined in the problem, and why that means that the confidence intervals/p-values are meaningless. When all we wanted was “we are 95% confident that the mean outcome in this situation is between here and here”.
It’s happened to me a lot before. Many students in various disciplines are extremely good at coming up with worries about experimental design or validity of measurement processes, and so they never get to the part where they deal with the statistics itself. They seem to treat every problem like the classic “rooster on the barn roof” problem, essentially declaring that “roosters don’t lay p-values” and choosing not to answer the question at all.
Don’t get me wrong, I really do want the students to be good worriers: they should be able to think about experimental design and validity and bias and all those things that impact on whether the statistics answers the question you think it does. But what they can’t do is use it to avoid talking about statistics at all! There are quite a few students who seem to be using those worries to discount all statistical calculations, and to sidestep the need to understand the calculation processes involved. “Your question is stupid, and I refuse to learn until it’s less stupid,” they implicitly say.
The weirdest part is that the assignment or discussion questions don’t usually discuss enough details for the students to actually conclude there is a problem. They say “the groupss were not kept in identical conditions”, but nowhere does it say they weren’t. I realise that in a published article if it doesn’t say they were then you might worry, but this is just an assignment question whose goal is to try to make sense of what a p-value means! Why not give the fictitious researchers the benefit of the doubt? And also, take some time to learn what a p-value means!
I do realise it’s a bit of a paradox. In one part of the intro stats course, we spend time getting them to think about bias and representativeness and control, and in another part, we get grumpy when they think about that at the expense of the detail we want them to focus on today. It must be a confusing message for quite a few students. But on the other hand, even when reading a real paper, you still do need to suspend all of that stuff temporarily to assess what claim the writer is at least trying to make. It’s a good skill to be able to do this, even if you plan to tear down that claim afterwards!
I am thinking one way to deal with this is to start asking questions the other way around. Instead of asking only for “what ways could this be wrong?”, ask “how would you set this up to be right?” And when I ask about interpreting a p-value, maybe I need to say “What things should the researcher have considered when they collected this data? Good. Now, suppose they did consider all those things, how would you interpret this p-value?” Then maybe I could honour their worries, but also get them to consider the things I need them to learn.
Last year I invented a game called Digit Disguises and it has become a regular feature at One Hundred Factorial and other events. But before Digit Disguises came along, there was another game with a similar style of interaction that we played regularly, and this blog post is about that game. The game is called “Which Number Where?”
In 2016 I created the iplane idea, which allows you to locate the complex points on a real graph. Ever since I had this idea, I have wondered on and off about the complex points on a circle. It’s time to write about what I’ve found.
The titles of the seven posts in the series are:
It’s been four years since I came up with the idea of iplanes as a way to organise the complex points on a graph, and in the intervening time I have thought about them on and off. For some reason right now I am thinking about them a lot, and I thought I would write down some of what I am thinking.
The titles of the seven posts in the series are:
This blog post is about a piece of the MLC learning environment which is very special to me: the date blocks. It’s a set of nine blocks that can be arranged each day to spell out the day of the week, the day number, and the month. I love changing them when I set up the MLC in the morning, so much so that since the face-to-face MLC closed due to COVID-19, I brought them home and have been changing them each morning here in the dining room. The story of how this object came into the MLC is the reason it is so special to me.