Reflections on maths, learning and maths learning support, by David K Butler
This blog post is about the Solving Problems poster that has been on the MLC wall for more than ten years in one form or another.
I conscripted the game Numbers and Letters seven years ago to help promote the Maths Learning Centre and the Writing Centre at university events like O’Week and Open Day. Ever since then, it has always bothered me how free and easy participation in the Letters game is, while the Numbers game is much less so. This Open Day I had a remarkable idea: instead of stating in the rules that the goal is to achieve the target, and trying to encourage people to take a different approach, what if I just changed the stated goal! I don’t know why I didn’t think of it before, to be honest!
You can read the rest of this blog post, and four other related posts, in PDF form here.
The titles of the five posts are:
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.
Context fatigue is a particular kind of mental exhaustion that happens after having to make sense of multiple different contexts that maths/statistics is embedded in. I feel it regularly, but I feel it most strongly when I have spent a day helping medical students critically analyse the statistics presented in published journal articles.
The problem with maths in context is that the contexts themselves require understanding of their own in order for the maths to make sense. This is nowhere more true than in statistics, where you have to use your understanding of whether you expect the relationship to exist, what direction you expect it to be, and whether you think this is a good or bad thing. The classic one in my head is an old first year statistics assignment where they used linear regression to investigate the relationship between manatee deaths and powerboat registrations in each month in some southern coastal American city. You have to know what a manatee is, what it means to register a powerboat, and why those things might possibly be connected in order for the statistical analysis you’re asked to do to make sense, not least because at least one part of the question will ask you to interpret what it means. When helping students read published articles recently, I’ve had to find out what’s been done to the participants in the study, how things have been measures, what kind of measurements those are, why they’ve been measured that way, and all sorts of little details to decide how to interpret the numbers and graphs that are presented.
Even ordinary everyday word problems are a minefield. Across two recent assignments, some financial maths students had to cope with album sales for AC/DC, flooding of the land a factory is built on including insurance, bull and bear markets, machines in a mining operation, committees with various named positions, road testing electric cars, contraband being smuggled in shipping containers. This is a lot of context that has to be made sense of before you can get a handle on the maths, and there is nothing in the question itself to tell you what any of this context means if you don’t already know. Even if you are already familiar with the context, you actually have to suspend some of your understanding in order to do the maths problem, because it’s much simpler than the actual situation any of the questions are talking about.
All of this interpreting is exhausting stuff! It just tires you out if you have to even a moderate amount all at once. You just feel like you don’t have any more energy to deal with any more today. That feeling there is context fatigue. Yesterday the first year maths students were doing related rates and every question was a new context with little nuances created by the context that had to be dealt with. Those poor students were exhausted after just one problem, letalone three or four.
As teachers, we need to realise that as the people writing the assignment questions, or at least people who have dealt with them before, we are much more aware of the details and nuances of the context than the students are, so we don’t have to work so hard to make sense of them. Not only that but we’re usually simply more experienced in both life and language than most of our students so it’s easier for us. Imagine the context fatigue you would get reading ten research papers in an unfamiliar area in one day (I feel this in real life regularly). That’s the sort of context fatigue your students have just from your assignment questions. Cut them a little slack, and make sure there is adequate time to process the context with appropriate rest time between context-interpretation. Also it wouldn’t be the worst thing to explicitly teach them strategies for making sense of context, such as ignoring the goal, and finding out about what some of the words mean. Strategies can make the work less intimidating, especially in the face of knowing how tiring it is already!
PS: If you’re in charge of tutors in a drop-in support centre, especially one that deals with statistics, please be kind. Context fatigue is real and tends to wear us down some days!
Trig substitution is a fancy kind of substitution used to help find the integral of a particular family of fancy functions. These fancy functions involve things like a2 + x2 or a2 – x2 or x2 – a2 , usually under root signs or inside half-powers, and the purpose of trig substitution is to use the magic of trig identities to make the roots and half-powers go away, thus making the integral easier. One particular thing the students struggle with is choosing which trig substitution to do.
You can read the rest of this blog post in PDF form here.
The blog post references a YouTube video with worked examples, that you can watch here:
Here in Australia, we are at the tail end of a reality cooking competition called “Zumbo’s Just Desserts “. In the show, a group of hopefuls compete in challenges where they produce desserts, hosted by patissier Adriano Zumbo. There are two types of challenges. In the “Sweet Sensations” challenge, they have to create a dessert from scratch that matches a criterion such as “gravity-defying”, “showcasing one colour” or “based on an Arnott’s biscuit”. The two lowest-scoring desserts from the Sweet Sensations challenge have to complete the second challenge, called the “Zumbo Test”. In this test, Zumbo reveals a dessert he has designed and the two contestants try to recreate it. Whoever does the worst job is eliminated.
I find it very interesting that the Zumbo test is the harder of the two tests. In the Sweet Sensations challenge, the contestants can choose to use whatever skills they are already good at, and design their dessert in a way that they can personally achieve. In the Zumbo Test, the contestants have no control over the techniques that are required, and must try to do things they are not familiar with in ways they may not have seen before.
And why am I talking about this? Because my medical students find themselves in similar situations. Our medical students have two projects to do as part of their research curriculum during their third year. One project is a research proposal: they work in a group with a supervisor to plan a hypothetical research project, including ethics, literature review and (this is where I come in) statistics. The other project is a critical appraisal: they work in pairs to analyse a published article, including where it fits in the research, the writing, the importance and (again where I come in) whether the statistics is appropriate.
I have noticed over the years that in terms of statistics, the critical appraisal is harder than the research proposal. A meeting with students about the critical appraisal usually takes twice as long as one for the research proposal, and twice as much preparation for me. Many more students come to me to talk about the critical appraisal, and the ones who do come are more worried about the statistics they find in the critical appraisal than the statistics they need in the research proposal. Why is this?
When watching Zumbo’s Just Desserts, it occurred to me that the reason why is the same as the reason the Zumbo Test is harder than the Sweet Sensations challenge.
When doing your own research you can choose to only investigate questions in such a way to use the statistical methods that you understand. Even if you need a new statistical method, you just need to learn that one. Either way, you have complete control over your own decisions and know the things you are measuring and what they mean. It’s just like in the Sweet Sensations challenge the contestants get to make all the choices and use methods they are familiar with.
On the other hand, when reading someone else’s research, you have no control over the wacky statistical methods they choose to use. Even if they are the appropriate ones (they often are in medicine, actually), the paper almost never describes how the researchers decided to use them – it just says what they used. And they often measure new things in new ways that you don’t deeply understand. It’s just like in the Zumbo Test the contestants have to do things that are new to them in ways that are new to them.
It’s much much harder to understand the statistics in someone else’s research than it is to make your own.
Let’s just hope we don’t eliminate all the students by asking them to do it with less support.
In a recent post (Counting the Story), I talked about how if you look closely at most solutions of combinatorics problems, you’ll see that they actually count the story of constructing the object rather than the object itself.
One exception to this is a problem like this:
“The balloon man has a huge collection of balloons in red, yellow and blue. I’d like to buy 10 for my granddaughter. How many collections of balloons could be made?”
Or this:
“The balloon man has a huge collection of identical balloons. I want to buy 10 and give them to my three grandchildren. How many ways are there to distribute them among the three children, allowing for the possibility that some children might not get any?”
Or this:
“The balloon man has a recurring nightmare about being asked to solve x+y+z=10 for non-negative integers x, y and z. How many solutions are there?”
(The reason I mention the granddaughters, even though I have no granddaughters, is because I wanted to reference the most awesome Rey Casse, who used the first of these three to introduce this type of problem in my Discrete Maths II class back in 1999.)
These three types of problems are usually solved using a method known in the USA as “stars and bars”. Google “stars and bars combinatorics” and you can find out about how it works. This is precisely the method I was taught, though with dots and lines, and I’m not aware of Australians actually giving the method a name.
I am going to present a slightly different approach here. It will come out looking similar to the stars and bars method, but the road to getting there will be a bit different, and is based on telling a story of how to construct all of the possible solutions.
Imagine I’m at the balloon man, and I am asking him for a particular combination of colours. One way to do this would be to say how many red ones I want, how many yellow ones I want, and how many blue ones I want, so that the total number is 10. So the number of combinations is the number of ways to choose three numbers (which could be zero) so that they add up to 10. This is precisely the same as solving the balloon man’s nightmare equation of x+y+z=10. Many people teach their students to turn such a problem into this equation and memorise the formula for the number of solutions to such an equation. They may possibly use a variant of the stars and bars to prove that the formula for the equation works.
That’s not satisfying to me. I want to get the answer more directly from the story itself. So how about this scheme for describing how to get the balloons: put the colours in a particular order, say red, yellow, blue. Then progressively either ask the balloon man to get another balloon from this colour, or move over to the next colour.
So if you wanted 3 reds, 2 yellows and 5 blues, you’d say “one of this colour”, “one of this colour”, “one of this colour”, “next colour”, “one of this colour”, “one of this colour”, “next colour”, “one of this colour”, “one of this colour”, “one of this colour”, “one of this colour”, “one of this colour”. If, when you got to a particular colour, you didn’t actually want any of those, you’d just move to the next colour straight away. So if you wanted 6 red, 0 yellow, 4 blue, you’d say “one of this colour”, “one of this colour”, “one of this colour”, “one of this colour”, “one of this colour”, “one of this colour”, “next colour”, “next colour”, “one of this colour”, “one of this colour”, “one of this colour”, “one of this colour”.
You could represent these instructions with pictures. I’ve used a balloon to represent asking for a balloon, and an arrow to represent moving to the next colour. These pictures represent all the stories of choosing a collection of balloons, so now we can count the stories! There are 12 possible symbols and two of them have to be arrows, so the number of possible stories is the number of ways to choose 2 pictures out of 12 to be an arrow. That is, the number of ways is 12C2.
Cool huh?
Now let’s tackle the next problem. Let’s put the grandchildren in a particular order and move along the line. We can give the child we’re up to a balloon, or given them another one, or move on to the next child. Again I can represent this with pictures: a balloon for when we give a balloon, and an arrow for when we move to the next child. And again the number of allocations of balloons to children is the same as the number of ways to choose 2 out of 12 pictures to be an arrow. That is, the number of ways is 12C2.
Nice.
On to the third problem. As I said earlier, many people teach students to reduce other problems to this, and then remember a formula for the number of ways to solve this. I, on the other hand, still tell the same sort of story. This time, I imagine starting with the equation 0+0+0=0, and then moving along the positions. At each stage I can add 1 to the number there currently, or I can move ahead to the next position. As long as I add 1 ten times, it will work. Once more, I can represent this with a picture. I’ve got “+1” for the action of increasing a position by 1, and and arrow for moving to the next position. The picture shows how to get 3+2+5=10 and 6+0+4=10. Except for the change of symbols, the pictures are the same as the other ones, so the number of solutions is still 12C2.
Sweet!
In the traditional stars and bars method, the stars represent objects and the bars represent dividers between them. In my method, the symbols always represent instructions in a story of how the collection/allocation/solution is constructed. And yes the symbols do always match the context of the problem. I find this much easier to remember and apply. Plus it’s cuter!
I was going to have a punchy title for this post, with a big moral to apply to the future, but I’ve decided I’m just going to describe to you what happened yesterday as I tried to learn some Genetics. You see what you can learn from my experience.
Combinatorics is one of my favourite topics in discrete maths – that topic which is all about counting the number of ways there are to choose, arrange, allocate or combine things. I like the idea that I could theoretically find out the answer by writing down all the possibilities systematically and literally counting them, but that I can also come up with a quick calculation that produces the same answer by just applying some creative thought. It’s this creativity in particular that appeals to me, so much so that I don’t call it combinatorics, but “creative counting”.
Of course, not all students share my love of combinatorics. When I look into their books I can see why – they’re full of tables of formulas that split situations into repetition allowed and not allowed, identical objects versus distinguishable objects, and order important versus order unimportant. That makes it seem like it’s all about stimulus-response raw memory, and that is the opposite of creativity!
I would love to convince students of the creative side to combinatorics, and relieve them from the burden of memory, but I also need to help them learn to solve the problems they are asked to solve. Somehow I am able to do all this in myself. If I can figure out how I do it, I might be able to pass it on to students.
Only recently have I come to realise what it is I do to be creative, avoid memorisation and still succeed in solving problems: I tell a story. Whenever I see a counting problem, I construct a story of how the things we are counting are constructed, which proceeds in stages in a time order. Then I count the story rather than the objects themselves!
If you think about it, this is what is often going on in the explanations for the common formulas. Take the number of permutations of n objects. You imagine constructing this permutation by choosing an object to go first, then an object to go second out of the remaining objects, and so on. There are n choices for the first object, which leaves n-1 choices for the second, and so on. Since the number of choices at the next stage is the same regardless of the choice at the previous stage, you can multiply them all together to get a total of n×(n-1)×…×2×1(also known as n!).
Did you notice? The thing we counted along the way was the number of choices at each stage of the story! We counted the story, not the permutations. Look closely at worked examples for combinatorics problems and you’ll see the same thing happening almost all the time. What they describe is a story, and then they count the story.
Working with a student recently, I pointed out that the key to success with creative counting is coming up with this story, and suddenly everything came together for him. He saw the common thread that connected everything, and was able to come up with his own solutions. He even came up with alternative stories for the same problem, and managed to explain to himself why to different-looking situations had the same calculation by constructing a similar story for both. I know one student doesn’t prove that it will work for all students, but it does show it’s possible!
One final thing that help stories foster creativity is the fact that multiple stories will produce the correct answer. This allows you to celebrate each students’ choice, making it more personal, and therefore more creative. Take the following example:
Suppose you have a televised singing competition with 30 contestants, and 12 must be chosen to go to the live shows. These people will be announced one by one on the show. If Johnny, who has the most compelling backstory, has to be chosen and has to be announced within the last three in order to increase suspense, how many possible announcements are there?
Let’s see. We need a story for how the list is constructed.
We could choose which position Johnny goes in, and then put everyone else in. That’s three places for Johnny, and then 29 choices for the first other position, 28 for the next, and so on. Every choice in the story goes with every other choice after it, so we get 3×29×28×…×13.
Another way would be to choose two people to be with Johnny in that final three, then arrange them. Then choose the 9 people to be the rest of the list and arrange them. So we’d get
9C2×3!×27C9×9!.
We could also just make the list as we go, couldn’t we? Put down someone’s name first (who could be anyone except Johnny) and then another name (again anyone except Johnny or the first person) and so on, until we get to the last three. This is 29×28×27×26×25×24×23×22×21 so far.
Now you can have anyone left including Johnny, and then again and again. So that’s 21×20×19 for the last three, giving 29×28×27×26×25×24×23×22×21×21×20×19 in total. But just a second, that counting arrangement isn’t guaranteed to put Johnny in the last three! Maybe we can fix that. Why don’t we count the end-of-stories that don’t end up with Johnny in the last three: 20×19×18 and take them off?
So we get 29×28×27×26×25×24×23×22×21×(21×20×19-20×19×18).
Finally, what if we try this one: choose the 11 other finalists, then out of those 11, chose the first person, the second person, all the way up to the ninth person. Then choose which position Johnny was in. Then chose which of the two orders the final two people were in. That would be 29C11×(11×10×9×8×7×6×5×4×3)×3×2.
That’s four ways to tell the story, and so four ways to count the callout list, not to mention slight variations on these ways. How’s that for creativity? Not just that, but you have probably done quite a bit of maths checking that they indeed do all produce the same answer.
Of course, you do need to know a few general principles, such as what dictates when multiplication or addition (or even division or subtraction) are useful to help you count. It also doesn’t hurt to know how to figure out the number of ways to choose a collection r things out of n things (nCr), and the number of ways to arrange n things (n!), so you can use these to make more complex stories. Once you have these elements, you can count a whole lot of things by telling the story of how you make them, and you don’t need any new formulas. That is, you can have the freedom to be creative.