Making Your Own Sense

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

Tag: teaching

  • Factorising quadratics by focusing on the sum first

    This blog post is about my way of helping students factorise quadratic expressions by inspection, which is the opposite of how most people do it.

    Factorising

    When you multply out two monic linear factors to make a quadratic, the same thing always happens:

    (x+a)(x+b)=x2+ax+bx+ab=x2+(a+b)x+ab\begin{aligned} & (x+a)(x+b) \\ &= x^2+ax+bx+ab \\ &=x^2+(a+b)x+ab \end{aligned}

    You end up with the sum of the two constant terms as the coefficient of xx, and the product of the two constant terms as the final constant term.

    Therefore, if you want to do this process backwards – that is, to factorise a quadratic expression – then you need to think of two numbers that add to give the coefficient of xx and multiply to give the constant term.

    For example, to factorise x2+5x+6x^2+5x+6, you need to think of two numbers that add to give 5 and multiply to give 6.

    There are at least two ways you could go about doing this systematically.

    One way is to think of pairs of numbers that multiply to give 6, and then test them to see if they also add to give 5. So, you’d think of 1×6 and test 1+6=7, which isn’t right. And you’d think of 2×3 and test 2+3=5, which is right. So your factorisation is (x+2)(x+3)(x+2)(x+3).

    Another way is to think of pairs of numbers that add to give 5, and test them to see if they also multiply to give 6. So, you’d think of 1+4 and test 1×4=4, which isn’t right. And you’r think of 2+3, and test 2×3=6, which is right. So your factorisation is (x+2)(x+3)(x+2)(x+3).

    Every maths teacher I’ve ever met tells students to list the product first and check the sum. I think that it’s much better to tell students to do the sum first and check the product.

    Examples

    Let me do several examples to compare the sum first approach with the product first approach.

    Example 1: x2+13x+40x^2+13x+40

    Product first

    I need two numbers that multiply to give 40, which could be 1×40, 2×20, 4×10, 5×8 and I think that’s it. The matching sums are 41, 22, 14, 13. So the numbers I need are 5 and 8 and the factorisation is (x+5)(x+8)(x+5)(x+8).

    Sum first

    I need two numbers that add to give 13. I’ll start at 10+3, and the product is 10×3=30, which is too low. Now I’ll try 9+4, and the product is 9×4=36, which is higher but still too low. Now I’ll try 8+5, and the product is 8×5=40, which is just right. So the numbers I need are 8 and 5, and the factorisation is (x+8)(x+5).(x+8)(x+5).

    Example 2: x2+20x+91x^2+20x+91.

    Product first

    I need two numbers that multiply to give 91. 1×91 obviously, and the matching sum is 1+91=92. So I need something else. What else? 2? Doesn’t go. 3? Doesn’t go. 5? Doesn’t go. 7? Oh yes that does work because 91=70+21, which is 10 and 3 sevens, so 91=7×13. The matching sum is 7+13=20, so that works. The factorisation is (x+7)(x+13)(x+7)(x+13).

    Sum first

    I need two numbers that add to give 20. My first thought is 10+10, and the matching product is 10×10=100, which is too high. Now 11+9=20, and 11×9=99, which is still too high, but lower. Next, 12+8=20 and 12×8=80+16=96, which is still too high, but lower. Next 13+7=20, and 13×7=70+21=91, which is just right. The factorisation is (x+13)(x+7)(x+13)(x+7).

    Example 3: x2+30x+144x^2+30x+144.

    Product first

    I need two numbers that multiply to give 144. What goes into 144? It’s 12×12, so 1, 2, 3, 4, 6, 12 will all work. Have I missed anything? Oh 9, taking a 3 from each 12. Anything over 12 will go with one of the small numbers. Right, so what have we got?
    1×144, but 1+144 is way too big.
    2×72, but 2+72 is too big.
    3×48, but 3+48 is too big.
    4×36, but 4+36 is too big.
    6×24, and 6+24 is just right.
    So the factorisation is (x+6)(x+24)(x+6)(x+24).

    Sum first

    I need two numbers that add to 30. How about starting with 10 and 20?
    10+20=30, 10×20=200, too big.
    11+19=30, 11×19=110+99=209, that’s worse. I should be going the other way.
    9+21=30, 9×21=189, still too big, but the right direction.
    8+22=30, 8×22=160+16=176, closer.
    7+23=30, 7×23=140+21=161, closer.
    6+24=30, 6×24=120+24=144, just right.
    So the factorisation is (x+6)(x+24)(x+6)(x+24).

    Reasons for a sum first approach

    The above examples point to the many reasons why I think focusing on the sum first is better than focusing on the product first. I’m going to list them, but they overlap quite a bit, so be prepared for me to repeat myself a lot in the explanations below each reason.

    Reason 1

    Doing the product first requires you to know or figure out what numbers divide into another. Doing the sum first doesn’t require any special knowledge about factors.

    Look at Example 2. We had to figure out that 91 had 7 as a factor at all before we could get to the answer. With the sum, it just fell out along the way.

    You may argue that guessing factors is a really important skill, and I don’t disagree, but honestly students don’t have much practice at that when they start factorising quadratics, and it’s a huge barrier to success. Focusing on the sum first allows them early success without the need for this skill. And you know what, they do a lot of multiplications along the way and might even notice what numbers tend to be multiples of what other numbers.

    Also look at Example 3. The number 144 has a lot of factors, and you kind of need to find all of them to be able to have things to try to see if they come out to the right sum. Most worked examples for students dont even list all the options for factors, but just zero in on the magically right one, picking from an unspoken list in the teacher’s head. With the sums first approach, it doesn’t matter if you missed a factor.

    And look, all the work you’ve done in the past to get good at seeing factors isn’t wasted! If one of the sums is 7+23 so you test 7×23 going for 144, you can actually say to yourself that 144 isn’t a multiple of 7 and just skip that one. I actually think developing this instinct for ways to shortcut the process can be quite an exciting idea to students.

    Reason 2

    With sums first, you can get started right away.

    When you do product first, you have to think of some factors to begin with, and it’s very rare that 1×something is going to work, so there’s this job to do before you can even get started. When you do sums first, it’s not hard to think of a sum that works and you can just get on with it.

    And there’s no wrong place to start either. You can just do a couple and you’ll know then if you’re going in the right direction. (See what happened with the one with 144.) So there’s no need to worry about your first inspiration – you can just get going.

    Reason 3

    With sums first, you feel like you’re getting somewhere.

    When you investigate the sum first, you systematically change them by 1 each time and the product changes along with it, getting closer and closer to the right answer. There is a real feeling of progress, like the work is paying off. And to reinforce the previous Reason, this feeling happens right at the start, rather than having to wait for finding factors first.

    I will concede that you can be systematic with the product first approach too, as you saw in my example with the 144. But to many students, the examples they see seem random, or worse, go straight to the right answer with no trial and improvement. If you do want to do product first, then I recommend being more systematic about it so that students can feel like they’re getting somewhere, rather than waiting for the lightning strike of the right one.

    Note that the feeling of getting somewhere has another advantage: if you’re a long way away from the right result, it makes you feel safer to skip some steps to get there quicker. This way lies developing instincts for when some combinations of numbers are unlikely to work.

    Reason 4

    With sums first, there’s cool things you can help students to notice.

    I personally think the experience of running through several possible sums and testing the products is some excellent fuel for helping students notice cool things, which are totally lost on a products-first approach.

    For example, in the example with 20x20x, the highest possible product happened when the sum was 10+10=20. That is, when it was two of the same number. This is very cool and that way lies completing the square. Also the further apart the numbers were, the further away the next product is from this one. Indeed, the differences were two apart.

    And I’ve already mentioned students noticing that a certain sum would require 144 to be a multiple of 7 and skipping it, that sort of thing makes the skill of noticing factors feel like a cheat code they’ve discovered, rather than a burden upon them. That sort of noticing is empowering for lots of students.

    More examples

    You’ve probably noticed that all the examples I’ve shown so far have had all positive coefficients, and they’re all monic (the coefficient of x2x^2 is 1). Well it’s time for some examples to deal with that. First I’ll deal with the negative coefficients, then later I’ll deal with non-monic quadratics. Mostly I’ll just do them straight using the sum first approach as if I didn’t know the answers yet, rather than compare them to a product first approach.

    Example 4: x2−13x+40x^2-13x+40

    We have to think of two numbers that add to -13 and multiply to 40.

    Positive numbers won’t add to a negative number, so I need two negative numbers, which will indeed multiply to a postive number.

    (-10)+(-3)=-13, (-10)×(-3)=30, too low.
    (-11)+(-2)=-13, (-11)×(-2)=22, even lower. Need to go the other way.
    (-9)+(-4)=-13, (-9)×(-4)=36, higher.
    (-8)+(-5)=-13, (-8)×(-5)=40, correct!

    So the factorisation is (x−8)(x−5)(x-8)(x-5).

    Example 5: x2+3x−40x^2+3x-40

    We have to think of two numbers that add to 3 and multiply to -40.

    If you think about the product first, there’s twice as many options as there were before, because while 4×10=40, both (-4)×10 and 4×(-10) are -40 and you have to decide which one. If you only think about the product long enough to realise you need one positive and one negative, then you can start your search with sums that add to 3 like this:
    4+(-1)=3, 4×(-1)=-4, not low enough.
    5+(-2)=3, 5×(-2)=-10, lower, so I’m going the right way.
    6+(-3)=3, 6×(-3)=-18, lower.
    7+(-4)=3, 7×(-4)=-28, getting there.
    8+(-5)=3, 8×(-5)=-40, and we’re there.

    I probably could have skipped a couple since there was a long way to go, but it was so pleasant watching it get closer.
    Anyway, the factorisation is (x+8)(x−5)(x+8)(x-5).

    Example 6: x2−11x−26x^2-11x-26

    We have to think of numbers that add to -11 and multiply to give -26.

    I’ll need a positive and a negative number to get a negative product, so let me start with -12+1.

    -12+1=-11, (-12)×1=-12, not low enough
    -13+2=-11, (-13)×2=-26, correct!

    So the factorisation is (x−13)(x+2)(x-13)(x+2).

    What if I had decided to start with something less obvious, like -20+9?

    -20+9=-11, (-20)×9=-180, way too low.
    -21+10=-11, (-21)×10=-210, even lower, so I need to go the other way.
    I’ll skip some since I was so far away.
    -15+4=-11, (-15)×4=-60, getting closer.
    -14+3=-11, (-14)×3=-42, getting closer.
    -13+2=-13, (-13)×2=-26, just right!

    Example 7: x2+11x+26x^2+11x+26

    We have to think of numbers that add to 11 and multiply to give 26.

    Two positive numbers will work, so I’ll start with 10+1.

    10+1=11, 10×1=10, too low.
    9+2=11, 9×2=18, higher so I’m going the right way.
    8+3=11, 8×3=24, closer.
    7+4=11, 7×4=28, too big.

    So there’s definitely a factorisation that will work with roots somewhere between 7 and 8 and between 3 and 4, but there’s not one with integers.

    Example 8: x2+8x+20x^2+8x+20

    We have to think of numbers that add to 8 and multiply to give 20.

    I’ll start with 1+7=8, 1×7=7, too low.
    2+6=8, 2×6=12, too low, but closer, so I’m going the right way.
    3+5=8, 3×5=15, too low.
    4+4=8, 4×4=16, too low.
    But there’s nowhere else to go from here. I’ll never get to 20.
    So this one doesn’t factorise at all.

    Interlude

    It’s time to stop for a short break. I’m hoping that this set of examples has convinced you that this approach has some merit for helping students understand how quadratic equations work, and indeed making the process a bit more playful.

    I also sneakily wanted to cover some objections people have brought up, such as how you could be sure it doesn’t factorise if there’s infinitely many choices for numbers that add to the x-coefficient.

    I just have one more thing to deal with, which is what do do with a non-monic quadratic. I’m just going to do one example in two ways.

    Two more examples

    Example 9a: 6x2+x−126x^2+x-12

    There is this method called by many “the ac method” which allows you to factorise a non-monic polynomial. I didn’t learn it at school, so I don’t think of it first, but it’s always something people bring up when I talk about factorising quadratics.

    The way it works is you multiply the constant term and the leading coefficient, and then think of two numbers that add to the x-coefficient and multiply to give this new answer. (If your quadratic was ax2+bx+cax^2+bx+c, that means making the sum bb and the product acac, hence the name of “ac method”.) Then you split the x-term into two parts with these numbers as the coefficents and continue from there.

    (For a proof, consider the product of two linear factors:

    (ax+b)(cx+d)=acx2+adx+bcx+bd=(ax)x2+(ad+bc)x+(bd)\begin{aligned} & (ax+b)(cx+d) \\ &= acx^2+adx+bcx+bd \\ &= (ax)x^2+(ad+bc)x+(bd) \end{aligned}

    Notice how the numbers adad and bcbc add to give the x-coefficient and multiply to the same answer as the x²-coefficient times the constant term. There can only be one pair of numbers with a specific sum and product, so if you find these numbers, they will be adad and bcbc and you will be able to do that algebra in reverse. )

    Anyway, this still requires you to find two numbers with a specific sum and a specific product, so you can still do sum first.

    The quadratic is 6x2+x−126x^2+x-12. So I need numbers that multiply to give 6×(-12)=-72 and add to give 1. I’ll need a positive and a negative.

    2+(-1)=1, 2×(-1)=-2, which is way too high.
    3+(-2)=1, 3×(-2)=-6, which is lower, so I’m going the right way, but I have a long way to go. I’ll skip some.
    6+(-5)=1, 6×(-5)=-30, which still has a long way. I’ll skip some more.
    10+(-9)=1, 10×(-9)=-90, which is too far, but quite close.
    9+(-8)=1, 9×(-8)=-72, which is just right

    So I need to split the xx into 9x9x and −8x-8x.

    (You could argue that if I went product first, I might have realised immediately that 8 and 9 would be right, but I can guarantee you that a heap of students would not realise that. This way, they’ll get there in the end.)

    So,

    6x2+x−12=6x2+9x−8x−12=3x(2x+3)−4(2x+3)=(3x−4)(2x+3)\begin{aligned} & 6x^2+x-12 \\ &= 6x^2+9x -8x-12\\ &= 3x(2x+3)-4(2x+3)\\ &= (3x-4)(2x+3) \end{aligned}

    If you wanted this in fully factorised form so that it shows the roots, you’d have to pull out a 3 from one factor and a 2 from the other to get

    (3x−4)(2x+3)=3(x−43)×2(x+32)=6(x−43)(x+32)\begin{aligned} & (3x-4)(2x+3)\\ &= 3\left(x-\tfrac43\right)\times 2\left(x+\tfrac32\right)\\ &=6\left(x-\tfrac43\right)\left(x+\tfrac32\right) \end{aligned}

    (It’s worth noting that for many people, this “splitting the middle term and then factorising twice” thing is the way that you’re supposed to do all quadratic factorisations, including the monic ones, which I can see the appeal of if I’m honest. But I’m not rewriting my entire set of examples now.)

    Example 9b: 6x2+x−126x^2+x-12

    There is a far more prosaic approach than the ac method, which is just to do what I’ve been doing all along but with fractions. Let me show you:

    6x2+x−12=6(x2+16x−2)\begin{aligned} & 6x^2+x-12 \\ &= 6\left(x^2+\tfrac16 x-2\right) \end{aligned}

    Now I’ll factorise the monic quadratric in the brackets there. I need two numbers that add to give 1/6 and multiply to give -2. They’ll have to be a positive and a negative.

    2/6+(-1/6)=1/6, 2/6×(-1/6)=1/3×(-1/6)=-1/18, which is not low enough.
    3/6+(-2/6)=1/6, 3/6×(-2/6)=1/2×(-1/3)=-1/6, which is lower but not low enough, and I’ve got quite a long way to go, so I’ll skip some.
    7/6+(-6/6)=1/6, 7/6×(-6/6)=7/6×(-1)=-7/6, so much closer.
    8/6+(-7/6)=1/6, 8/6×(-7/7)=… yeah that won’t work out right.
    9/6+(-8/6)=1/6, 9/6×(-8/6)=3/2×(-4/3)=-2 yay!

    So the factorisation is 6(x+32)(x−43)6\left(x+\frac32\right)\left(x-\frac43\right).

    I have to say I prefer this one to the other one in a lot of ways. But yes a big fly in the ointment is the fraction arithmetic. But honestly this seems to me to be quite low stakes, and it certainly gives a lot of practice! You have to decide how you want to play it.

    Oh, and why did I choose to count in sixths? Well it turns out that in a monic quadratic with rational coefficients, if there are any rational solutions, they’ll be able to be written with the common denominator of the coefficients. (But that’s another story and shall be told at another time.)

    Conclusion

    So, I’ve given a lot of examples to show how the reasoning works when you factorise quadratic expressions by first focusing on the sum and checking the product, rather than the other way around as is more traditional. And I’ve tried to describe why I think it has a lot of advantages. I hope you give it some consideration when you next help students with their factorising.

    (And one little addendum: I think it’s worth considering this for all of your students first, rather than just reserving it for students who struggle with doing products first. Don’t let it become an othering explanation.)

  • Making Maths Foundations

    NOTE: I wrote this article in September 2025 for the University’s Learning & Teaching News, and I thought I would reproduce it here so other interested people could read it.

    David Butler wears a blue t-shirt with many numbers on it, and looks at the screen while holding a pen in his hand. On the table in front of him are mathematical pictures and calculations, as well as brightly-coloured blocks arranged in various places. In the background behind him is the corner of a large artwork made of white pyramids taped together into a bigger pyramid with lots of holes in it.

    Many people feel that their experience with maths up until now prevents them from approaching careers or courses that interest them. For example, a student in an Arts degree might want to become a teacher and must face the government’s numeracy test, or a student in a Music degree might be fascinated by the physics of sound, but intimidated by the calculations involved. The Maths Learning Centre provides an online course called Maths Foundations for just such people. I wanted to share the story of how Maths Foundations was created.

    When I got funding in 2023 from PACE (Professional and Continuing Education) to make a course to support students’ maths skills, I knew exactly what I wanted to do. Lack of confidence with number and algebra is the very thing that blocks a lot of people from pursuing various goals, and I wanted a course that helped students build on their earliest maths experiences to foster that confidence.

    I began to plan what would be in the course. Based on my extensive experience supporting students at the Maths Learning Centre, I brainstormed the number and algebra skills that would be most useful for people studying various sciences, and for people needing to pass numeracy tests for professional registration such as nurses and teachers. I also considered what would be necessary to move on to the MLC’s bridging course MathTrackX, which can be used in place of Year 12 Maths Methods as a prerequisite for various courses such as maths and engineering.

    I took those topics and organised them into four main modules that would allow students to build their number skills and algebra skills at the same time, starting from addition of whole numbers, passing through calculations of areas and volumes and time, and ending with describing lines in the coordinate plane algebraically. The actual process here involved pieces of paper with concepts and skills written on them being physically placed and moved around on the floor and multiple desks. My colleagues and my family were most gracious in accommodating my physical planning process!

    The final stage of planning was to use what I had learned from years of interacting with schoolteacher colleagues from early childhood to high school, and choose activities that would help students to learn concepts and practice skills. I was excited to include activities that I know are engaging for people with from a wide range of ages and backgrounds.

    Once this planning was done, the hard yards of making the course began. The team from Learning Enhancement & Innovation (LEI) were champions to help me realise my vision. One particularly long process was making all of the videos I imagined. There were many explanatory videos with physical resources from huge fractal art pieces to play-dough to toy dinosaurs to a bottle of honey. I was able to use the Peer Assisted Study Sessions classroom – which was empty for the summer –  to lay everything out ready to go and to practise, before recording in the Barr Smith Library recording studio. There were also over forty worked example videos, which were each recorded in one take including all the mistakes, to help students understand the messy process of problem-solving.

    On top of the videos, I wrote many pages of text, drew many pictures, created interactive online graphs, designed engaging discussion activities, and coded about seventy randomised practice problem sets. It was a herculean effort, even with the indefatigable support of the LEI team. In April 2024, the course was launched on the AdelaideX platform. Anyone in the world can access the content for free for four weeks, and people can pay for extended access and the opportunity to test themselves with an exam. Now, in September 2025, thousands of students have enrolled, with about 40 so far choosing to complete an exam. There are multiple discussion board posts a day, and MLC lecturer Nicholas Crouch answers them with compassion and verve.

    I am so proud of what I have built, so grateful for the support I had to build it, and so glad students are finding it useful.

  • Making the lie true

    We at my university regularly sell quite a big lie.

    At Open Day and the Ingenuity STEM Showcase and any number of outreach activities, students do puzzles and play with construction toys and walk around with ropes and draw curves on balloons. Whether we say it explicitly or not, there is a message there that says: here at this University, maths is fun. This is a lie.

    Maths at university is not fun. There are hours of video content to watch where the presentation is basically slides or handwritten examples. The classes are presentations, possibly with little quizzes breaking them up, or they consist of doing maths problems similar to the relentless weekly quizzes and assignments. Pictures are rare, making sense by manipulating something with your hands is much much rarer, making sense by moving your body is non-existent. The chances to chase your curiosity are few. The chances to have your own thinking validated and celebrated are fewer. It is very far removed from the experience of university maths the prospective students get when they visit us.

    We are lying to our prospective students. The experience they have of university maths at our events is a lie.

    I do understand that learning does not have to be “fun”, and expecting it to be so all the time is unreasonable and unhealthy. I also understand that ordinary everyday problem-solving and figuring out can feel fun.  I understand as well that play, which is essential to learning deeply, is not the same thing as fun. But there is no denying that the activities we do with prospective students are indeed fun, and that experience is not what it will be like at university.

    Do I want to change the activities we do with prospective students to look as boring as life will be at uni? Of course not. But  there is another way to not lie, and it’s to make your lie true.

    One way I make the lie true is to provide One Hundred Factorial, a weekly games, art and puzzle session where students can experience mathematical play without having to be assessed on it. The sorts of things that happen as a one-off at outreach events happen every week at One Hundred Factorial, and I think it would be a good thing to tell prospective students that this exists. (Writing this blog post is partly to help myself pluck up the courage to suggest to the academics in Maths here that they can do so.)

    Another way is to actually include some of the features in your outreach activities actually in your teaching. I’ve seen the maths academics do an awesome job of running engaging activities and helping students feel like their efforts are meaningful and valued. They’re good at it. What I want to say to them is this: Perhaps you can actually include some whole-body movement or physical models in your university classes, or at very least in your videos. Perhaps you can actually have some free exploration of new ideas without having to immediately write an assignment about it. Perhaps you can keep the idea of celebrating students’ mathematical thought in the very front of your mind more often when they are doing everyday maths problems or answering questions in the lecture. Even just a little more of any of these things might make university maths a little more like the outreach activities you do so well.

    The experience prospective students have in your outreach activities doesn’t have to be a lie. You can make the lie true.

  • Why mathematical induction is hard

    Students find mathematical induction hard, and there is a complex interplay of reasons why. Some years ago I wrote an answer on the Maths Education Stack Exchange describing these and it’s still something I come back to regularly. I’ve decided to post it here too.

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

  • Other(ing) Explanations

    Most people who teach mathematics are aware that it’s useful to have alternative explanations for concepts, and useful to have different ways to approach problems.

    Given enough time, you are guaranteed to come across students for whom the standard explanation isn’t working today (as long as you give students a chance to tell you about their understanding).

    Having worked with thousands of students one-on-one, I have tried quite a few alternative explanations and methods for many things. Sometimes they’re whole different approaches; sometimes they’re just little tweaks. Sometimes they are just a different order of the sentences you might otherwise say; sometimes they use physical manipulatives like the floor graph or play dough. Many teachers, like me, have such a bank of alternatives.

    The problem is… Well, you can see it already in the way I’ve talked about these explanations: I have called them “alternative”, as opposed to “standard”. They are different, unusual, other. And the students know they are. A student who always has to have the other explanation can come to feel that they themselves are other.

    A prime example of this is when the “dumb class” use physical toys to learn, whereas the “smart class” only uses symbols. (I use “dumb class” and “smart class” because that’s what the kids call them. Don’t fool yourself into believing that they don’t.) If you set up this sort of dichotomy, then any child who ever has to use the physical tool to help them understand knows they are stupid.

    Another example is when mathematicians do not provide pictures when showing how to work out problems, and only provide them when someone doesn’t understand the text version. Students come to think that pictures are only for the “dumb kids” who aren’t capable of understanding the text alone, and they try to avoid drawing them, even if they could solve a problem ten times faster with one.

    Obviously if the first explanation you try doesn’t help a student, then you do need to try another one – I never want people to stop providing alternatives!

    But perhaps the explanation you use as the standard one doesn’t have to be the standard. Perhaps the other one you usually save for second might work as the first explanation for all the people the standard one works for, and also a few more. Each new explanation needs a bit of consideration to decide if maybe it can supplant the one you usually use first. At the very least, when you hear or think of an alternative explanation, don’t say, “I will keep that in mind for my struggling kids.”

    Even better, perhaps we should more often just provide more than one explanation to begin with, rather than just one. No explanation can possibly work for all possible students, and even the “smart kids” will benefit from having more than one way to think about something. So maybe we can avoid othering people by simply giving more options from the outset. For example, to stop students feeling like they’re a “dumb kid” when you draw pictures, you can just draw pictures for everyone a lot more of the time.

    So please, do seek out and try other explanations, but make sure you are careful for them not to become othering explanations.

    These comments were left on the original blog post:

    Kris Lindeblad 10 September 2022:

    Thank you, thank you. Not only do I heartily agree, but you have given me new vocabulary to talk about it. “othering” will stay with me. ~k

    David Butler 11 September 2022:

    I’m glad.

  • Arbitrary mnemonics

    A mnemonic is a mental trick to help you remember things.

    People use them all the time for all sorts of things, like the traditional colours of the rainbow (ROY G BIV), the order of the letters in the English alphabet (a song to the tune of Twinkle Twinkle Little Star), the order of operations (BODMAS or PEMDAS), which months have 31 days (“30 days hath September…” or your knuckles), and which kind of camel has one or two humps (Dromedary starts with D which has one hump; Bactrian starts with B which has two humps).

    The purpose of a mnemonic is to connect something that is hard to remember to something that is easier to remember. If you can remember the mnemonic and the connection, then you can remember the thing. They are especially useful for things that are arbitrary, where there is no obvious or no particular reason why they are the way they are (such as the number of days in each month).

    However, there are a lot of things that most people don’t need mnemonics to remember, and it seems to me they tend to be the things that make sense to them — things that are already connected to other things in an obvious or natural way. Indeed, the very connectedness of things to each other is what causes the sensation of understanding. You feel you understand things when they are highly connected to other things, and you often don’t have to try to remember things that you understand.

    So, a mnemonic helps you remember arbitrary things, and un-arbitrary things often don’t need much assistance to remember because they make sense.

    What happens if you advocate that learners use a mnemonic for something that is understandable? I think that it sends a signal to learners that the thing is arbitrary – because they know implicitly that arbitrary things are what mnemonics are for – and since it’s arbitrary, they shouldn’t attempt to understand it. So they don’t try. They just try to remember.

    For example, to remember which of sin(.), cos(.) and tan(.) are positive for angles in which quadrants, many people use the mnemonic All Stops To Central (or something similar), to remember it’s all of them in Q1, only sin(.) in Q2, only tan(.) in Q3 and only cos(.) in Q4. But I have met so many learners who have not the slightest clue why this is the truth, and don’t even expect there to be a reason. The fact that it’s a mnemonic signals to them there is nothing to understand. On the other hand, when you remind them that sin(.) is the y-coordinate of the matching point on the unit circle, and the y-coordinate is positive in the top half of the circle, you can see the light go on and the sigh of relief that they don’t have to try to remember any more.

    So my advice is just to be careful with mnemonics. I would recommend not introducing them too early. Help your learners try to make sense of things as much as they can, and when there are a few spots left that are arbitrary and they have trouble remembering them, then you can introduce a mnemonic to help remember. Otherwise, you may signal to them that what they are learning is arbitrary and they shouldn’t attempt to understand it.

  • The Solving Problems Poster

    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.

  • Replacing

    I have had many people say to me over the years, “But algebra is easy: just tell them to do the same thing to both sides!” This is wrong in several ways, not least of which is the word “easy”. The particular way it’s wrong that I want to talk about today is the idea that doing the same thing to both sides is somehow the only move in algebra, because it’s not even the most important or the most common move.

    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

  • Twelve matchsticks: focus or funnel

    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.
    Three shapes made with twelve matchsticks each. The third is a rectangle five high and one wide. The second is a square three wide and three high. The first is an irregular shape like a two-by-two square with two little squares attached near one corner.
    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.

  • Twitter and how not to treat my students

    I have learned a lot from Twitter about how to treat my students, and most of it has been through being treated in ways I do not like. Recently I have been searching my own tweets to find things I’ve said before, and as I’ve dipped into old conversations, several unpleasant feelings have resurfaced when I read the way I’ve been treated. I don’t want to make my students feel that way, so I want to avoid doing those things to my students.

    So, here are some ways I don’t want to treat my students, based on ways I have been treated on Twitter. To my shame, I have done most of these to others on Twitter too, and I am trying hard not to. I know most of the people who have done this to me will be mortified to know they have, so I am not going to call anyone out here. I just want to share what I have learned.

    So, here is a list of things I don’t want to do to the students, because I don’t like it when they happen to me:

    • Offer solutions when they haven’t asked for any.
    • Interrupt their problem-solving process.
    • Ignore their feelings when they express them.
    • Tell them their feelings are wrong.
    • Respond with a story about me rather than seek more from them.
    • Completely ignore the main point of what they said and respond to just one word or phrase.
    • Respond to them telling me something they like by giving recommendations for new things.
    • Respond with a fire hose of even more technical terminology.
    • Tell them they are wrong to be confused.
    • Tell them the thing they like is wrong because I like something else.
    • Discount their success by pushing to the extension straight away.
    • Respond to everything with sarcasm.
    • Focus only on the bit I think is wrong.

    My original plan was to elaborate on each of these, but I have kept coming back to this post for months and feeling overwhelmed with that task, so I think it’s time to just push send. And maybe it’s a good thing each time I read this to have to imagine what each of these things looks like. I’m hoping it’s useful to you to have to think about what these might mean too. But of course if you want me to explain a particular one of them more, do ask and I will do my best.