Making Your Own Sense

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

Tag: memory

  • 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.

  • Childhood memories

    Two books I’ve read recently have encouraged me to investigate my memories from childhood. In Tracy Zager’s “Becoming the Math Teacher You Wish You’d Had“, she urged me to think about my maths autobiography to see what influenced my current feelings about maths. In Stuart Brown’s “Play“, he urged me to think about my play history to see what influenced my current feelings and tendencies about play. In the spirit of those two, here are some of my earliest memories about maths and play.

    In primary school, I have very few memories of actually being in a maths class, and all of them are negative. I’ve related two of them already in this blog. One was my memory of doing a maths assignment about one million dollars, where the financial aspect distressed me to tears. Another was my memory of my Year 6 teacher attempting to teach us averages using cricket.

    The only other maths class memory is of a test I did in Year 3. I had been sick with asthma for a couple of weeks and came back to school on the day of a test. I dutifully did the test and actually got almost full marks. The only thing I got wrong was the meaning of the word “net” in the phrase “net weight” as you might see listed on a packet of food. I distinctly remember it being a multiple choice question and ruling out two of the answers as ridiculous, but basically having to guess between the other two. I was angry because how could I possibly know that? Everything else was just logic and so I could figure it out for myself, but you can’t figure out the meaning of a word without more context. Eight-year-old me was an astute little person.

    Across my primary school career, I do remember a strong feeling of pleasure and fascination associated with construction toys. I remember absolutely loving the MAB blocks, in particular the moment when I replaced ten units with a long, and ten longs with a flat and ten flats with a block. Interestingly, my memory is only of the blocks themselves and I can’t pinpoint a year level or a teacher that goes with this. I also remember loving playing with polydrons and attribute tiles, but again the memory is just about the fascination of playing with them, and not about any particular maths class. In fact, thinking carefully about what is around me in these memories, I seem to be in a hall or a library, rather than in a classroom.

    Outside of school, I remember playing a game in each new playground, where I would try to do every part of the play equipment exactly once without crossing my path. Would I have to interpret the slide as both a sliding down and a climbing up in order to do it? Would I end up trapped on the top, or could I finish on the ground where I started?

    At home, we’d build elaborate maze-like cubby houses out of spare mattresses and sheets (we lived in a house where visitors often stayed over). I remember planning these out with my brother with explicit conversations of how we would fit more rooms and pathways into the space of our shared room. I also remember spending hours making designs with a ruler and compass. Or by folding paper several times and cutting out holes then unfolding and sticking on a contrasting colour.

    It seems that for me, geometrical play holds the strongest positive mathematical memories from my primary school years.

    Indeed, my very first memory of primary school is about geometrical play. It’s the moment I walked into my kindergarten classroom for the first time. We walked into a carpeted play area, and the desks and blackboard were some distance away at the other end of the classroom. Here in the play area was a bookcase filled with big thick brown blocks. Some of them were on the floor being made into a car track by some other children. I remember immediately wondering about how the various straight and curved pieces might fit together. I have some vague memories of tying various combinations on other days in kindergarten.

    Earlier than this, one of my only memories of Happy Days Pre-School was getting out the giant foam blocks from the store room under the building and playing with them on the grass.

    It’s funny that so many of my positive mathematical memories are geometrical when now I also have such a love of the structure and behaviour of numbers. Maybe that came later, though my mother says as a very young child I was always “playing number and letter games in my head”. I myself can’t remember doing that, but my mother is a very astute person and I am not about to doubt her observations.

    My earliest memory of any kind is of a cool hard flat greenness. My mother says this is probably a memory of the back verandah at the house we lived in before I was two years old – it had a green-painted concrete floor. I wonder if other people’s earliest memories are about feelings of space and colour. If so, maybe it means we’re all geometers from birth. Or maybe it’s just me.

    What is clear is that it’s hardly surprising that I ended up doing a PhD in finite geometry even though the original undergraduate degree I enrolled in was mathematical physics. I think the fundamental pull towards that geometrical play was calling me all along, considering how strongly I gravitated towards it in primary school despite the rest of maths not being so inspiring.

    If you’re reading this, I don’t know what you might learn from my story. But for myself I realise I am right where I belong.

    This comment was left on the original blog post:

    V Lakshmi 27 September 2017:

    Nice article! Infact, childhood memories have something to learn and plays an important role in future they are like the learning stages check this peace very interesting http://www.publicdebate.in/childhood-happiest-part-life-agree/ 

  • The right order for the fundamental trig identity

    If you google “fundamental trig identity” you will get many many images and handouts which all list the fundamental trig identity as:

    sin2 t + cos2 t = 1

    This is, of course in the wrong order and it should really have cos first then sin, like this:

    (cos t)2 + (sin t)2 = 1

    “But David,” you say, “it’s addition, so it doesn’t really matter what order it’s in does it?” Of course it does! Mathematically it’s the same, but psychologically it’s different. If it really wasn’t different then you would sometimes write cos first and sometimes write sin first, but I can bet you always write it in a particular order. And if you write it with sin first, then you’re making it harder for yourself.

    Let me explain.

    The reason we have the fundamental trig identity is because the angle t there is a piece of the circumference of a unit circle, and cos t and sin t are the coordinates of the points on that unit circle. If I asked you to write down an x-y equation for the unit circle, you would naturally write x2 + y2 = 1 with the x first. But the x-coordinate of a point on the unit circle is cos t, and the y-coordinate is sin t, so of course that means it’s (cos t)2 + (sin t)2 = 1. Writing your trig identity with the cos first makes it easier to make the connection with the equation of the unit circle. If you write it with sin first you’ll have to continually switch it round!

    Also, the order does matter if you’re using hyperbolic trigonometry. Then the formula is (cosh t)2 – (sinh t)2 = 1 and having sin first would be definitely mathematically wrong. For years, I had great trouble remembering which way around this was supposed to go until I realised that the cos and sin were in alphabetical order. From that point forward I always wrote my ordinary trig identity in the same order as the hyperbolic trig identity (in alphabetical order) so that through force of habit I would never get the hyperbolic one wrong.

    So, I recommend you start writing your fundamental trig identity in the right order. It might help you remember and make connections to other things!

    PS: You may be wondering about the way I put brackets in my trig identities. Don’t even get me started.