Making Your Own Sense

Reflections on maths, learning and the Maths Learning Centre, by David K Butler

Tag: explanations

  • Sticky operations

    This blog post is about a metaphor I use when I think about the order of operations: the idea that the various operations are stickier than the others, holding the numbers around them together more or less strongly.

    You can read the rest of this blog post, and the other posts in the series across the years, in PDF form here. 

    The titles of the five blog posts are:

    • The reorder of operations
    • (Holding it together)
    • The Operation Tower
    • Replacing
    • Sticky operations

  • Zooming in to see the slope

    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 .)

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

    A screenshot of a Desmos graph. There is a red curve which is the graph of a function, and a small circle centred at a point on the function. A dotted line joins this to a bigger circle showing a zoomed in version of what's in the small circle. There is a zoom factor slider.

    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.

  • Remainders remain a puzzle

    My first post of 2018 is a record of some rambling thoughts about remainders. I may or may not come to a final moral here, so consider yourself warned.

    What has prompted these ramblings today was reading this excellent post by Kristin Gray  about her own thoughts on division and remainders. In that post, I saw the following: 

    7÷2 = 3R1

    For some reason, this bothered me. For some reason it’s always bothered me. Today I think I realised what the problem was: In my head “7÷2” is a number, and “=” indicates that two things are equal, but 7÷2 can’t be equal to 3R1 because 3R1 is not a number. It is only today that I realised that 3R1 isn’t a number.

    How do I know 3R1 isn’t a number? Well firstly it’s two numbers. One is a number of groups and the other is a number of objects. I don’t even know how big the groups are – it could be 3 groups of 2 and one left over,  it could be 3 groups of 7 and one left over, or 3 groups of 200 and one left over. I can hear people saying that actually all plain numbers could mean any number of different units, and a 7 could be 7 cm or 7 ducks or 7 groups of 200. But the 1 here is definitely 1 of something, while the 3 is some unknown size of groups of that same something. That seems like a totally different kind of unit issue than with a plain ordinary number.

    Also, if it really is a number, then I should be able to place it on a number line, but where does it go? The 3 I certainly know where it goes, but what about the 1. Where does that live? It lives in a completely different land to where the 3 lives, and I can’t really put it on the number line until I know how big the groups are that the 3 represents.

    It occurs to me that this is a good way of transitioning to a fraction sort of idea. The fact that the 1 is small relative to groups of size 200 and large relative to groups of size 2, and needing to encode this relative size would lead nicely to a need to write this as 7÷2=3+1/2. What an interesting idea.

    My other really big issue is that the “=” sign in this context doesn’t work the way an “=” sign works. If 601÷200 = 3R1 and 7÷2=3R1 then usually the properties of “=” would mean that 601÷200 = 7÷2. But they aren’t equal. I suppose they both produce 3 groups with 1 left over, but that 1 is very different in size relative to the group in each situation! So they’re not really equal are they? Actually, this idea is going back to the same idea I had with the number line, where you need to encode the relative sizes.

    My final problem is that if it really is a number, then surely you’d be able to do operations on it. But I don’t really know how you’d do that. You’d expect that if the 3R1 came from 7÷2, then 2×(3R1) would produce 7, but if it came from 601÷200, then what would 2×(3R1) even mean? I’ve been trying to figure it out, but to no avail.

    It might be possible to do addition and subtraction, if you knew the groups were the same size. In that case 3R1 + 5R3 would be 3 groups and 1 plus 5 groups and 3, so it should be 8 groups and 4. So 3R1+5R3 should be 8R4. It seems you add the two numbers separately, which is actually super interesting. I’m still a bit worried about what would happen if the groups were size 4, say, because then 8R4 is actually the same as just 9. So now it seems like they are a lot more like numbers than I originally thought. This seems like a very interesting thing to investigate.

    As you can see, I’m rather puzzled by remainders and where they stand numerically. I get that the idea of dividing a collection into groups requires us to have a concept of remainder. I just feel weird writing it in this way because these symbolic representations feel like they ought to make numerical and algebraic sense, but here they don’t.

    In Kristin’s post, she floated the idea of the equation being not 7÷2=3R1 but instead 7=3×2+1. This second equation I feel completely comfortable with. It is 100% clear what the numbers are doing and the “=” really is acting as an equality here. I still wonder if there’s a more helpful way of representing the division-producing-a-remainder thing though.

    And maybe that’s another issue I have with it, that this statement “7÷2=3R1” is about doing an operation and producing a result, as opposed to declaring a relationship, which is what I have come to believe the “=” sign is for. By using something that is not like a normal number and just encodes a description of an answer, are we reinforcing that “=” means “here is the answer”? I don’t know what to do with this question yet, or if it even really matters.

    So there you go. I’ve rambled through a whole lot of thoughts and worries about remainders. I don’t have any conclusions or morals or recommendations. But it’s certainly helped me to try to write it all down. I hope it helped you to read it. I’d love to hear your own thoughts on it.

    This comment was left on the original blog post:

    Deborah Peart 4 January 2018:

    You make a great point! Students have enough confusion around equality and the equal sign. Truthfully it should be expressed
    7/2=(3×2)+1
    Interesting thought!

  • Three hours in the MLC Drop-In Centre

    Last week, I had one of those days in the MLC Drop-In Centre where I was hyper-aware of what I was doing as I was talking with students and by the end I was overwhelmed by the sheer volume of things I had thought about. I decided that today I might attempt to process (or at least list) some of it for posterity.

    A Real Analysis student was worried about a specific part of his proof where he wanted to show that a function was increasing. The function was a kind of step function with more and more frequent steps as it approached x=1. It was blindingly obvious from our sketch of the graph that it was increasing, but every time we tried to come up with a rigorous argument we just ended up saying something equivalent to “it is because it is”. I remember thinking about whether it was just obvious enough to just say it without proof, but shared the student’s desire to make the nice clean argument. I don’t remember now if he ended up with a neat argument or not, but I do remember finishing with the moral that obvious things are often the hardest things to prove.

    Before we even got to this frustrating little bit of the proof, I was reading through what he had done already, and I noticed that he had written “x is < 1” and commented that this wasn’t a grammatical sentence because the “<” contains an “is” already – it’s “is less than”, which makes his sentence “x is is less than”. This led to quite a discussion about the grammar of “<“, which is complicated by the fact that it can be read aloud in multiple ways depending on context. We investigated some other sentences containing < or > he had seen written before to see how it was pronounced there. In my head I was reminding myself to pronounce written maths aloud more often when I’m with students so they can learn the correspondence between speech and writing in maths.

    I was called over to help a Quantitative Methods in Education student with her research assignment because she had mentioned she was struggling with statistics (and I’m the default stats support person). Eventually I helped her figure out that her problem wasn’t statistics at all, but rather that she didn’t understand the wider context of the research, or what the data actually represented, or what the actual goal was. Mainly this came about by me not understanding the context, data and goal and kept asking her questions about those things in order to try to figure out where the statistics fit into what she was trying to do. We talked about how she might go about gaining the understanding she needed in order to come up with the specific-enough questions about variables that statistics would be able to help her to answer. I could see her heart sinking, so I reassured her I would still be here next week to talk about the statistics should she need that support later, but also from our discussion that she would probably be okay with that part when it came.

    I returned to the table I came from to discover even more statistics, this time a statistics theory course in second year of the maths degree. This student was struggling to come up with a linear regression proof which was pitched in the assignment using linear algebra. For this small part of the proof, he had to show that the columns of some matrix were linearly independent. I did what I always do with linear algebra proofs and helped him remember various definitions and facts relating to the vocabulary words in the problem, then we chose a representation of linearly independent from the list we had made. He commented that I go about proofs in a very different way to him, and I said I was merely following the problem-solving advice I had already put up in that big poster on the wall, see? In the end we finished with the moral that coming up with connections between things is a great way to solve problems and to study, even if it means ignoring the goal for a moment.

    I saw a student on the other side of the room that I had talked to months ago and I went to see how she was doing. When I saw her last, she was questioning her decision to study Statistical Practice I and indeed her whole decision to come to university at all, while simultaneously worrying about letting down her son who was the one who had urged her to come to university. Now she revealed she was systematically working her way through the course content so far, making sure she understood everything before classes returned and the last set of new content arrived. She asked if that seemed like a good plan and I said I was so pleased to see her making plans for her study and persevering with her course. I suggested she do some work soon to connect together the ideas in the different topics, since it’s the connections between ideas that will create understanding. I recommended a mind-mapping idea I had seen on Twitter recently (thanks Lisa).

    At the next table I came to, some students and one of my staff were having a discussion about function notation, in order to help a student in our bridging course. They were talking about how unfortunate it is that something like f(x) could be interpreted as multiplication in some contexts and function action in another. I agreed that it was unfortunate, but it was just the way maths language has come to work and it was too late to change it now. I related this to how the word “tear” is interpreted differently depending on context, which seemed to help everyone accept the fact, but not necessarily be happy about it! My staff member brought up how it’s made more complicated by the fact that we then write “y=f(x)” or even “y(x)=…”. I flippantly said that this was making a connection between graphs and functions, which is a whole separate discussion. Of course everyone wanted to hear more about this and suddenly I was giving a little seminar on the fly about function notation and graphs and the connection between them. I was keenly aware of everyone watching me, and of the choices I had to make with my words, especially when I made some mistakes along the way. Here’s more-or-less what I said [with some of my thoughts in square brackets]:

    There are many ways to think about functions, but one of them is that a function takes numbers and for each one it produces another number. [I thought about saying it doesn’t have to be numbers, but decided that was just going to distract from the discussion today.] It could be a formula that tells you how to get this other number, or there could just be a big list that says which ones produce which ones. For example, there’s a function which takes 1 and gives you 4, takes 2 and gives you 5, takes 3 and gives you 7, takes 5 and gives you 8. It’s the one that adds 3 to every number. The brackets notation is supposed to highlight that there’s a starting number and the function is a thing that acts on it to produce a result. So we could write something like “addthree(2)=5”. [I’m not sure what motivated me to write this. I’m sure I’d seen someone talk about this on Twitter somewhere, but anyway it seemed right at the time.] Sometimes we want to talk about the function as a thing in its own right so we give it a letter-name like “f” (for function) so that we can talk about it more easily. Or we only have some information about it but don’t know what it really is, so we can give it a letter name until we know its proper name. [I was reminded at this point about a Rudyard Kipling story featuring the origin of armadillos but chose not to mention it. I also thought here that I could have done another example earlier of a function acting like lookupthelist(2), but we’d moved further than that and I didn’t want to go back.]

    And what about the connection to the graph? Well you could draw a nice number line and write down what result each number produces. [I started drawing pictures at this point.] You could visualise this by drawing a line of a length to match the result attached to each number on the line. [I accidentally drew the lines representing the input numbers when I drew my picture, and had to go back and change them later. No-one seemed to mind that much.] You could have a ruler to measure how long each of these is when you needed it. Or you could attach the ruler to the edge of the page and line up the drawn lines with it. And now suddenly you’re locating the end of each drawn line by numbers on two axes. This is coincidentally just like the coordinate grid which is something else we’ve learned about before but not directly connected to functions. In the coordinate grid, the y refers to this second number and the x refers to this first number and you can describe shapes by how the x and y are related to each other. It doesn’t have to be y = formula in x, but that is a useful way to do it if you created your shape from a function.

    I think it was at about this point that I asked the students how they felt about this. My staff member said he quite liked the “addthree(x)” thing, which he’d never seen before. The bridging course student said that really helped her too, as well as realising that there were two different things happening at once when you write y=f(x). I decided to move on at this point.

    The next student was studying Maths 1B and was doing a deceptively simple MapleTA problem which he was stuck on. It listed an open interval – I think his was (-4,5) – and asked him to give a quadratic function with no maximum on this interval, a quadratic function with neither a maximum nor minimum on this interval, and a cubic function with both a maximum and a minimum on this interval. He had answers entered for the first two questions, which MapleTA had marked correct, but he wanted to know why. He revealed a little later he had gotten these answers off a friend and that now he wanted to know how to get them himself. I’m always glad when students do this, because it’s the first step to them really understanding themselves. With the first question, we discussed what it meant to be a maximum, and how that definition played out on an open interval. Once we had done this, I decided that I should take my own advice and make the question more playful, more exploratory. So I asked him what he would need to put in in order to make the answer wrong. He was really intrigued and started thinking through what the function would have to look like in order to have a maximum. After a while, he hit upon the idea that the leading coefficient could be negative, and this is all it would take. This was followed by several attempts to test his theory by putting in bigger and bigger constant terms in order to see if they really didn’t make a difference. They didn’t, and he was most impressed with himself.

    In the second part, we talked about moving the function around until the minimum wasn’t part of the domain. I went to get a plastic sleeve and drew a parabola on it in whiteboard marker, then asked him where he could move it to get what he wanted. At first he rotated it. I congratulated him for thinking creatively but did point out that it’s not the graph of a function like it was before. After that he only moved it up and down. Finally I gently moved it a tiny bit to the left, and he caught the idea that he could move it sideways, which he did until the turning point was outside the domain. The issue then came with how to change the function to do this, and we discussed why his friend’s solution was able to achieve it. Again he went playful and started putting in really big numbers and numbers really close to the boundaries to confirm that yes really he could move it anywhere outside that domain.

    Now we moved on to the third part and we discussed how cubics look and tried to figure out how to make the two turning points higher or lower than the end points. He wanted to use derivatives, so I let him do that, and it was quite a long journey, though he learned a lot about derivatives and specifying function formulas along the way.

    At the end, he asked if he had done all of this the correct way. I replied that it was definitely a correct way. He took this to mean there was a better way, and I said there was possibly a faster way. This meant a further long discussion about specifying zeros of functions and talking about whether we needed them to be zeros or if the function could be higher up or lower down. When we had finished this he asked me to explain the steps of this method again so he could write them down. I said that he didn’t need to remember the method because he’s never going to see this question again. The point is not to remember methods of solving all questions, but to learn something, and look how much stuff you learned today!

    Even though it was very close to my home time, I decided to talk to one more student. This one was doing a course specifically designed for the “advanced” maths degree students. He had been set a problem in Bayesian statistical theory. This is not familiar to me, so he spent a lot of time explaining what was going on to me. There were a whole lot of things in his working that I thought could be solved by simply referring to facts he had learned in previous courses, and I asked if he had done those courses. He revealed that they were in fact prerequisites for being allowed to do this course, at which point I said it was okay to use results from prerequisite courses. The biggest problem at the end was that he had assumed he had to integrate the normal distribution density function by hand, because the lecturer had gone to the trouble of giving its formula in the assignment. I assured him that it was in fact impossible to find areas under that distribution by hand, and he was allowed to use a table or technology. Sometimes you just have to tell a student what’s impossible so they can do it another way.

    So that was my day. I had three hours in the MLC and talked through so many ideas about learning and studying and maths. I had to make so many decisions about my words and actions and teaching tools. And I was left with a lot to think about. I hope you’ve found it interesting too.

  • All dogs have tails

    In maths, or at least university maths, there are a lot of statements that go like this: “If …., then …” or “Every …, has ….” or “Every …, is …”. For example, “Every rectangle has opposite sides parallel”, “If two numbers are even, then their sum is even”, “Every subspace contains the zero vector”, “If a matrix has all distinct eigenvalues, then it is diagonalisable”. Many students when faced with statements like these automatically and unconsciously assume that it works both ways, especially when the subject matter is new to them. This post is about a way of helping students see the problem.

    You can read the rest of this blog post in PDF form here. 

  • Disjointed independence

    There are two terminologies in probability which many students are confused about: “independent” and “disjoint”. The other day I was working with a student listening to their thinking on this and I suddenly realised why.

    You can read the rest of this blog post in PDF form here. 

    The above post has a special diagram to explain independence, which you can download in various formats:

  • The trig functions are about multiplication

    When I was taught trigonometry for the first time, I learned it as ratios of sides of right-angled triangles.

    You can read the rest of this blog post in PDF form here. 

  • A story instead of stars and bars

    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.

    Three collections of balloons, one red, one yellow and one blue. Below are two rows of symbols. The first row of symbols has three balloons, an arrow, two balloons, an arrow, and five balloons. At the end of the row is a little collection of three red, two yellow and five blue balloons. The second row of symbols has six balloons, then an arrow, then another arrow, then four balloons. At the end of the row is a little collection of six red and four blue balloons.

    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.

    A collection of purple balloons with three stick figures next to it representing children. Below are two rows of symbols. The first row of symbols has three balloons, an arrow, two balloons, an arrow, and five balloons. At the end of the row is a picture of the three children with the first holding three balloons, the second holding two and the third holding five balloons. The second row of symbols has six balloons, then an arrow, then another arrow, then four balloons. At the end of the row is a picture of the three children with the first holding six balloons, the second holding none, and the third holding four balloons.

    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.

    An equation x + y + z = 10. Underneath is the equation 0 + 0 + 0 = 0 with the zeros in faded purple. Below are two rows of symbols. The first row of symbols has three boxes with +1, an arrow, two boxes with +1, an arrow, and five boxes with +1. At the end of the row is the equation 3 + 2 + 5 = 10. The second row of symbols has six boxes with +1, then an arrow, then another arrow, then four boxes with +1. At the end of the row is the equation 6 + 0 + 4 = 10.

    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!

  • Making sense of the effective population size formula

    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.

    You can read the rest of this blog post in PDF form here. 

  • Counting the story

    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.