Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Author: davidkeithbutler

  • 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.

  • Where the complex points are: i-arrows

    Once upon a time in 2016, I created the idea of iplanes, which I consider to be one of my biggest maths ideas of all time. It was a way of me visualising where the complex points are on the graph of a real function while still being able to see the original graph. But there was a problem with it: the thing I want, which is to see where the complex points are (or at least look like they are) is several steps away from locating them.

    However, in my original series of blog posts, I actually already created a solution to this problem! I can draw a complex number as an arrow on the real line, which starts at the real part and extends in the length and direction of the imaginary part. Anyway, combining this arrow model of a complex number from an x-coordinate and a y-coordinate produces an arrow in the plane. The point (p+si,q+ti) is an arrow based at the point (p,q) and extending along the journey (s,t) from there. 

    This is the representation I need. I have decided to call them i-arrows.

    You can read the rest of this blog post, and all eight blog posts in the i-arrows series, in PDF form here. 

    The titles of the eight posts in the series are:

    1. Where the complex points are: i-arrows
    2. The complex points on a line using i-arrows
    3. Further updates on the complex points on an unreal line using i-arrows
    4. The complex points on a line in finite geometry using i-arrows
    5. The complex points on a parabola using i-arrows
    6. The complex points on real circles using i-arrows
    7. The complex points on unreal circles using i-arrows
    8. The line joining two complex points using i-arrows

    UPDATE: There was a later blog post in 2024 further investigating the line joining two complex points.

  • 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.

  • My first Maths Teacher Circle

    Last week I participated in my first Maths Teacher Circle . I just want to do a quick blog post here to record for posterity that I did it and it was excellent. I choose to take the practical approach of just relating what happened.

    I had been interested in somehow going to one since I heard about them a while ago, and then the founder of the Aussie Maths Teacher Circles, Michaela Epstein , contacted me through Twitter back in November to ask if I might like to facilitate an activity at an online session in 2021, and of course I said yes. She invited me to a session about mathematical games, and I was so excited to share some of the games I have invented with some interested teachers.

    Of course, the closer it got, the more nervous I got. When I heard there would be 40 or so teachers ranging all through primary to secondary to post-school teachers, I was rather intimidated! But Michaela and Alex  assured me I would be ok and that what I had planned would work. And they also put up with my scatterbrained discussion of random maths stuff whenever I met with them too. So, feeling a little reassured, but still nervouscited (as Pinkie Pie would say), I dove right in feet first last Wednesday morning.

    To start off with, Michaela invited past Maths Teacher Circles participant Samantha  to  set the scene by sharing what she has gotten out of Maths Teacher Circles in the past. This was a nice way to begin by grounding it in a real teacher’s experience. Then Michaela shared the goals of Maths Teacher Circles, which were exploring maths, strengthening classroom practice, and bringing maths enthusiasts together. I was so glad I had come to a place that resonated with all the things I love. It really matched with the goals of One Hundred Factorial, which is probably why Michaela invited me to present in the first place. This was all a really smart way to begin, because it set the tone for the rest of the session. Even when the housekeeping notes about breakout rooms and whiteboards and chat windows came, it was clear that these were there to support the overall vibe.

    Then we had a very quick chat in breakout rooms with a couple of people. We were supposed to talk about Noughts and Crosses too, but we only just made it through the introductions! But honestly I was happy to just have met a couple of friendly faces to help reduce the nervous part of the nervouscieted.

    By this time, so much had happened already, yet it had only been a few minutes. And now it was my turn. Michaela introduced me and I was now responsible for the journey of these 45-ish hopeful people. I put up the rules for Which Number Where, and asked everyone to quietly have a read, then ask any questions they might have. People had some very useful questions in the chat and out loud, and I felt we were ready to try it live. I asked for volunteers and described how to play the game Mastermind-style, with one player being the Secret Keeper and the other players asking questions. After a couple more questions, we were ready to break into groups to play.

    Michaela put people into groups of fourish, and I popped into about half of them to have a chat. I asked people how they were going and played with them for a bit, seeding a different kind of question than the ones they had been asking so far. I found everyone to be gracious and thoughtful and engaged. Such a thrill to meet such wonderful people and play maths with them. These moments when I was in a small group with people were my favourite parts of the session.

    I brought everyone together into the big group to discuss how the game went. I started by asking people if they had a favourite question that was asked. And then people shared any thoughts they had at all about how to use this in a classroom.

    Suddenly it seemed my time had run out, so I quickly showed everyone my other two games Digit Disguises and Number Neighbourhoods, and encouraged them to go back to their breakout rooms to keep playing Which Number Where or to try a new game instead. I stayed out in the main room where Michaela made sure I was ready to do a wrap-up when people returned. I very much appreciated being able to think in advance about that part!

    One question Michaela asked was why I chose the game I did. I said I chose Which Number Where because it’s about logic, and not any particular maths topic per se. As someone said earlier, it’s about locations rather than numbers per se, which means it’s really about the yes-and-no questions, and about logical arguments and joining information together, and those are skills you use everywhere in maths, which is why I like it so much. Plus I just love to hear how people think and this game gives me a chance to do that.

    And then it was time for me to participate in someone else’s activity. Toby  and James  shared the Multiple Mysteries game and some problem-solving/proving prompts to go with it. I got to play the game with some lovely other people and join in with the play. It really was a lovely thing to just play around with something that someone else shared that they were excited about. I am very grateful to Toby and James for providing such a great game to play and think about, and to the members of my little breakout room who I had such fun with.

    After this, it turned out that Michaela had read the time wrong and had cut short my activity the first time! So I got to have a few more minutes! I decided to share Digit Disguises properly, and instead of using breakout rooms, to play a game as a whole room with me as the Secret Keeper. Some brave souls shouted out questions and I wrote the questions and responses on a Word document on the screen. After a few questions, I decided that I would stop people and ask them what they can figure out from the information we have so far. This part was just wonderful. People had multiple different ways of gleaning new information about the numbers and their letter disguises from what we already knew, and quite a few of the participants expressed a satisfying amount of delight at these fascinating new possibilities. It was extremely gratifying to have people so excited about something that I am excited about (and egotistically, satisfying that people liked something I had invented).

    At this point, my laptop ran out of battery power and I had to scramble to find the power cord. By the time I came back, things were starting to wrap up, with participants filling out a Padlet with their thoughts. And then it was over. It felt like almost no time at all had passed, which is a good sign that I’ve been deeply engaged.

    After all the other participants left, Michaela, Alex, Toby, James and I had a debrief, which was some lovely discussion about how it went and how cool it was to work mathematically with people rather than just present them with stuff, and just some nice discussion about teaching and learning maths with some lovely people. And after that, I couldn’t help but keep working on  one of the investigations that Toby and James set me off on, because that’s how I roll and is the sign of a good maths problem.

    So that was my first experience of a Maths Teacher Circle. For me, the best part was the chance to think and play together with other teachers. The environment was so safe to just play and talk, and this was very carefully set up by Michaela in the first place, by discussing what was important and how to keep it safe. Being told explicitly that we were allowed to adjust the activities to match the level of the group made us free to play in our own way. And really, everyone was just so gracious and excited and, well, lovely. I am so grateful to have been a part of 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.

  • Quarter the Cross: Connect the Dots

    This blog post is about a new variation on the classic problem, which I call Quarter the Cross: Connect the Dots.

    You can read the rest of this blog post, and four other related posts, in PDF form here 

    The titles of the five posts in the series are:

    • Quarter the Cross (2016)
    • A Day of Maths: Quarter the Cross (2016)
    • David Butler and the Prisoner of Alhazen (2016)
    • Quarter the Cross: Colouring (2020)
    • Quarter the Cross: Connect the Dots (2020)

    Some resources linked from this post:

  • 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

  • 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

  • Questions with a morally wrong answer

    I think asking students questions is an important part of my job of helping students succeed. Good questions can help me see where they are in their journey so I can choose how to guide them to the next step, or can help to make clear the skills they already have that will help them figure things out for themselves. But there is a class of questions that shuts all of this down immediately. Here are some examples:

    • “Did you go to the lecture?”
    • “Have you started yet?”
    • “How many of the exercises have you done?”

    These questions all have answers that are morally Right or Wrong. The answers a student gives make the student out to be a Good Student or a Bad Student. And if a student has the Wrong Answer, they will feel ashamed.

    I know many people who believe it is very important to send students the message that they should go to lectures, start assignments straight away, and do all the exercises. While these are all things students could do to help themselves, they’re not the most important thing to focus on when they are here seeking support from me. They can’t change any of those things right now, so all a question like those does is make them feel ashamed. And, as Turnaround for Children CEO Pam Canto says in this blog post , “shame is toxic to positive outcomes”.

    Shame is the feeling that you are a bad person, that there is something wrong with you. Guilt is a bad feeling about your actions, which is unpleasant, but may make you want to change those actions in the future. Shame is the next level, where you feel you have been exposed as the horrible person you really are. A person who feels shame won’t try to change their actions, they’ll just try to avoid situations that expose them, which will just make the problem worse. I don’t want this to happen to my students, and I certainly don’t want them to think that seeking support from me will expose them to shame, or they will decide not to seek help.

    Once upon a time, I realised that I was causing a student shame, and I decided that I would give myself a new principle.

    Never ask a question that has a morally wrong answer.

    This is one of the rules I use to evaluate if my question is useful and choose a better alternative.

    For example, I could ask “Did you go to the lecture?”, but there is definitely an answer to this question that is morally wrong and having to give that answer will cause shame. Do I really want to know if they went to the lecture? How will that help? Maybe what I really want to know is what the lecturer has to say about the topic, since that might be useful. In that case, I could ask “What did the lecturer have to say about this?” The student doesn’t have to reveal their attendance status to answer this question, thus avoiding the shame. Even better would be to avoid the awkward moment where they have to reveal they don’t know, and say, “It would be useful to know what the lecturer says about this. Can you tell me what they said, or tell me where we might go looking for that?”

    For my second shame-inducing question of “Have you started yet?”, the first simple fix is to remove the “yet”. That implies they should have started already. The second fix is to think about why I want to know this? Maybe I want to know what they’ve done already so we can build on it. In that case I could just ask “What have you done so far?”, since that’s directly asking for the information I want. But there is still an implication that they should have done something, so causing shame if they have to reveal they’ve done nothing. So instead I could ask “What are you thinking about this problem?” or maybe “How do you feel about this problem?”. These let me get into their head and heart and I can help them move on from there. I might be able to ask them about what they’ve done so far later, or it might not even be important because they’ll tell me what they need to help themselves.

    This second example highlights another principle, which is to ask open ended questions, preferably about student thoughts and feelings. This makes it much easier to ask questions without morally wrong answers, because there are no specific predetermined answers in particular! (Asking open-ended questions is actually one of the factors in SQWIGLES, the guide for action I give to myself and my staff at the MLC.)

    So, I urge you, think about whether the questions you ask have a morally wrong answer, and if so, try a more open-ended question that is less likely to cause the shame that is so toxic to success.

    These comments were left on the original blog post:

    Todd Feitelson 12 September 2020:

    Thanks for this, David. I appreciate as always the attention you give to the details of your interactions with your students. It can be excruciating to have to consider every word you speak, but it’s really important. It’s a big source of stress and anxiety for me. But, it’s critical.

    I do wonder about the difference between a direct question (“Were you at the lecture?”) and a less-direct question (“Can you tell me what the lecturer said?”) As you stated, you want to avoid that awkward moment when the student might have to admit not being there, but it feels way more awkward (almost like a trap) to ask the indirect question. For me, setting up the situation and relationship where the direct question can be seen as just a way to gather the information without judgement is the trick. My goal can be to get that all out on the table and move forward, with the student seeing me (I hope) as an ally.

    I’m dealing with younger students, but there is also some value in helping students see that going to the lecture is a valuable tool in learning. It’s water under the bridge once they’re asking for help, but they are asking for help, so it seems important to help them learn for the next time. In a shame-free way — that’s the challenge.

    (Somewhat irrelevantly, it reminds me of what my English teacher wife wants to say when kids come and ask her how to improve their vocabulary scores on standardized tests — “Have read a lot since you were ten.” Not so helpful, so she remains positive and forward-looking, and avoids the shaming.)

    Thanks for writing!

    David Butler 12 September 2020:

    Very helpful thoughts Todd. A decent relationship where you can ask questions to keep them accountable is definitely something that is desirable. For me, this is usually the first time I’ve met them so I don’t have that luxury. I will say that when I have used “It would be good to know what the lecturer said…” and we’ve looked it up, then they often say later that going to the lecture or at least consulting the notes is the thing they learned today.

  • Changing the goal of the Numbers game

    I conscripted the game Numbers and Letters seven years ago to help promote the Maths Learning Centre and the Writing Centre at university events like O’Week and Open Day. Ever since then, it has always bothered me how free and easy participation in the Letters game is, while the Numbers game is much less so. This Open Day I had a remarkable idea: instead of stating in the rules that the goal is to achieve the target, and trying to encourage people to take a different approach, what if I just changed the stated goal! I don’t know why I didn’t think of it before, to be honest!

    You can read the rest of this blog post, and four other related posts, in PDF form here. 

    The titles of the five posts are:

    • Numbers and Letters (2018)
    • An opening gambit for the Numbers Game (2017)
    • Changing the goal of the Numbers Game (2020)
    • Jack Frost’s centre (2015)
    • The Arts student’s maths brain (2017)