Making Your Own Sense

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

Category: Reflections

Reflections on learning and teaching and research and life.

  • 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.)

  • More wisdom from the Dodecahedron

    After a long hiatus, I am dusting off my blog and I’ve moved it here to a new home. While I was going through the process of transferring everything here, I re-read the very first post I ever wrote, called Wisdom from the Dodecahedron. And I also found my drawing of the Dodecahedron that the original banner on the very old blog site was based on.

    A net for a regular dodecahedron, with twelve pentagons joined together in two flower-like arrangements. Each pentagon has a face drawn on it, each showing a different emotion, including happy, sad, surprised, suspicious, angry, confused, wistful, and ashamed. Some edges have tabs saying "glue under" and some edges have hands and feet coming off them. There is also a little beret with tabs to help it stand when it is cut out.

    The Dodecahedron is a character from The Phantom Tollbooth by Norton Juster, and he lives in Digitopolis, the city of numbers. Many people prize mathematics for its cool logic and want to hold it up as emotionless as if that somehow makes it better. But here is the first person we meet in Digitopolis and he has on display twelve emotions!

    And that appeals to me a lot, because I find that maths is full of emotion. Frustration, curiosity, surprise, satisfaction, pride, sadness, companionship, wonder, silliness, joy — they’re all there, sometimes in quick succession. Talk to any mathematician about their work and those emotional words are guaranteed to spill out. We’re human, and humans feel emotions, and humans do maths in a human way, which is an emotional way.

    The Dodecahedron reminds me to feel my emotions as I do maths, and to make space for others to do the same.

  • 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. 

  • Space to enter

    This is a photo of the entrance to my Maths Learning Centre. What do you notice?

    Maths Learning Centre entrance. In the foreground on the floor is a big cross made of yellow and blue coloured tape.

    There are many many things to notice in that photo, and if you ever want to ask me about any of them, please do. Today, the thing I want to focus your attention on is the empty space right at the front as you walk in. Every so often someone asks me why I leave that space empty and I don’t put an extra table there, and there are a couple of very good reasons why.

    First, the space isn’t empty: it has the floor graph in it. The floor graph used to be at the back between the % and the 3 on the wall, but one day I realised that I could have a bit of extra space for it if I put it in the entrance. I also hoped it might send the message to people arriving not only that maths is a thing that happens here, but also that we do things a little differently to your regular university maths classroom. One day I will write about the floor graph to tell you all about how we use it, but there is one purpose I want to tell you today: the floor graph helps us to break students out of staring forlornly at their page or screen. The open space on the floor graph gives a sense of physical freedom, which can translate to a sense of mental freedom.

    The openness of the floor graph space was the main reason I moved it to the entrance, actually, because it makes the space easier to enter, for several different kinds of students:

    We regularly get tours of new students or prospective students come past the MLC, and with an open space in the entrance, we can bring those tour groups right into the MLC, rather than standing outside and pointing. The experience is so much realer if you can come right in and stand surrounded by the art and whiteboards. They can remember that we asked them to come all the way in. Without the space in the entrance, we’re just pointing from outside and they miss so much.

    Students who are familiar with the MLC stand on the floor graph for a moment when they arrive and look around to find a good table, or other students they recognise. The empty space allows them to take a moment to make a choice, and to prepare themselves for working in the space.

    Students who are not familiar with the MLC have a place to stand looking lost. (We tutors even call it the “lost soul zone”.) When there was a table in the entrance, people wouldn’t want to be too close to the students already studying there, so newcomers would do their unsure dithering stance outside where we couldn’t see them, and more often than not they would just leave without us ever knowing. The emptiness of the space now means that they can come in without feeling like they’re encroaching on the work of the students and staff already in the room. Just like our regulars, they can prepare themselves for asking for help while actually being in the room. And since they’re in the room, we can see them and go up to them to ask what they are looking for.

    Without an empty space in the entrance, we would not be able to welcome as many new students to the MLC as we do. The emptiness is important to provide space for the complex process of deciding whether and how to engage with us. I am so happy I managed to created the space to enter.

  • 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.

  • Running out of puzzles

    Because people know I run the One Hundred Factorial puzzle sessions, they often ask me if I have a repository of puzzles they can use for their classroom, enrichment program, maths club, or their own enjoyment.

    Sometimes I feel embarrassed because I don’t actually have a big repository of puzzles. Surely since I am a person known for promoting problem-solving and puzzles, I should have such a thing. At the very least I should have a record of the puzzles we did do. But I don’t.

    It turns out my scatterbrained tendency to forget record-keeping is not the main thing that caused this lack of puzzle repository, but only in the last few weeks did I realise what the main cause actually was. It’s that I don’t feel the need for lots of puzzles. A person recently asked for my advice on where to find puzzles and told me the reason was they were worried their maths club would tear through them and so have nothing to do. Only when they gave this reason did I realise I don’t worry about this at One Hundred Factorial. But why?

    Firstly, puzzles are not the main food at One Hundred Factorial. I usually have exactly five activities available: a logic puzzle (eg sudoku), a word/geometry puzzle, an art activity/construction toy, a game, and the Numbers Game. If people get to the end of the puzzles, there is always other stuff to do instead.

    Secondly, and much more importantly, the whole vibe of One Hundred Factorial means that puzzles do not end. I have carefully cultivated a culture encapsulated in the mantra:

    The goal is not the goal.
    The end is not the end.

    What “the goal is not the goal” means is that the stated goal of a puzzle or problem is not the actual goal. The “goal” might be to find the area of a shape, or the probability of some event, or count how many of something there is, or whatever. They are not the goal. The real goals are to learn something, or understand someone’s thinking, or make something beautiful, or find a connection to something else.

    What “the end is not the end” means is that even if you do get to the stated goal of a problem, it doesn’t mean the thinking stops. You can ask if there’s another way, or what the problem would be like if you changed this aspect, or look for a connection to something else, or build something cool out of the answer or process. The truth is there is no end.

    The mantra of “the goal is not the goal, the end is not the end” means that we can get by at One Hundred Factorial with just one puzzle. In fact, we can get by with no new puzzle at all. Maybe someone was at the previous session and we want to continue with the non-end of last week’s puzzle. Or someone saw a random thing during the week that inspired their thinking and turn up ready to include others in their thinking or find out what thinking it might inspire in others. Or someone pulls out a puzzle that’s been done before and wants to find out how other people might think about it.

    As far as I can see it, my approach to cultivating a “goal is not the goal, end is not the end” culture had three aspects:

    1. Constantly ask goal-free, non-end questions like “what are you thinking?”, “is there another way?”, “what would happen if?”, “what can we make?”, “what is this connected to?”.
    2. Notice when other people ask those sorts of questions and run with it. I found that once I became attuned to them, I noticed people asked them a lot more often than I realised.
    3. Provide open-ended things other than just puzzles, like construction toys or art activities. There is nothing like an activity with no goal to foster a more goal free attitude. Even just puzzles with more than one solution foster a more open-ended attitude.

    So that’s how I don’t run out of puzzles: I don’t only use puzzles, and when I do, we go further or in different directions than the puzzle says to.

  • 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.