Reflections on maths, learning and maths learning support, by David K Butler
The “Four Fours” is a very well-known little problem that encourages some creative thinking and use of the order of operations. The purpose of this post is to show you four fourfoursesque puzzles I’ve created which have encouraged some great learning.
You can read the rest of this blog post, and two followup blog posts, in PDF form here.
The titles of the three posts in the series are:
I love wrapping presents. I’d like to say it’s because of the warm glow I have inside from giving a gift to someone else – and that feeling is certainly there to an extent – but I’m sorry to say the main reason is because I like the process of wrapping presents itself.
I like putting the present on the paper and making a judgement of how much paper to cut; I like using the scissors like a knife to cut a clean edge; I like folding the edge of the paper so that it looks nice and clean when you fold it over the present; I particularly like the part where you do the fold-in-the-sides-then-fold-up bit on the sides; and most of all I like the part where it’s all finished and your present is neatly encased in a piece of paper just the right shape with all the bits folded in neatly.
Yes, I know I’m weird.
But I reckon I’m not that weird. My daughter at 10 years old, still likes reciting the alphabet, though she learned to do this 6 years ago. My other daughter at 5 years old, will write her name over and over and over and over, seemingly getting pleasure out of the simple act. A musician will sometimes play a song they know well, for the sheer pleasure it, and almost any person will go up to a piano and play chopsticks. Many people I know like the experience of making scrambled eggs, no matter how many times they have done it before.
It seems that all people derive some pleasure in doing things well that they know how to do well, even though they have done it before. There is something about the repetition that gives you a sense of pleasure. Perhaps your brain likes to have the electrical signals pass down the well-worn paths where it’s not so much effort. Perhaps the experience helps you remember the buzz when you learned it for the first time.
I think perhaps the second reason is pretty accurate because I see myself doing it all the time in my work as well: guessing eigenvalues, calculating integrals, adding fractions and drawing conics. I love them all. I jump at the chance to do them with students in the MLC because I love doing them, no matter how many times I’ve done them before. And every time I do them, I remember with pleasure the first time I figured out how to do them myself.
But whatever the reason, I do get pleasure from doing the integral of ex cosh x or (cos x)2 or 1/(x2 – 1) – integrals I have done a hundred times – and it coming out to the answer it ought to. It’s the same pleasure I get from wrapping a present.
Sometimes you just enjoy doing something you know how to do.
Some years ago, I saw a snippet on the ABC science show Catalyst about insomnia – in particular, the flavour of insomnia where a person has trouble falling asleep at all. They reported on a trial study investigating the effectiveness of a tortuous new treatment for chronic insomnia. (You can find the published research here: Click here to go to insomnia article .)
The usual way to cure insomnia is to retrain your brain and your body to associate the bed with sleep rather than wakefulness. What they recommend is to only go to sleep when you’re really really tired, and if you don’t fall asleep in quarter of an hour, to get up and go to some other room until you feel tired enough to go to sleep again. Eventually, you’ll fall asleep in bed. Then you try again tomorrow night, and the next night, and the next night… Usually it takes a month.
The big problem with it is that people just don’t have the stamina to put themselves through all this for four weeks. Here’s where the radical treatment comes in: you compress the month of practice into 24 hours. The poor participant is put in a windowless room and practises going to sleep, and when they finally do fall asleep, they only get four minutes to sleep before they are woken up to try and fall asleep again. In this way you fit a month’s worth of falling-to-sleep practice in one day. Imagine how desperate you would have to be to sign up for this sort of thing!
Recently, it occurred to me that there are a lot of other skills that take a lot of practice to learn and this practice is usually drawn out over such a long period that people just don’t get through it all. One of these is statistics – in particular, the process of deciding which statistical procedures should be used to analyse your data.
In your standard stats course, the approach to teaching students to make decisions is to get them to do a project. This gives them practice at making decisions a grand total of once. And so students need a whole degree’s worth of projects, and probably years of working as a statistician, to learn how to make decisions. Hence, very few people ever get very good at making them. It’s just like the poor insomniac trying to cure their own insomnia once a night.
But what if you could, like the new insomnia treatment, compress all that practice into a short amount of time? What if you could pick out just the part where you make the decision and get students to make a lot of decisions all at once? Then they might get the necessary experience rather more quickly than the standard approach.
I tried it out last year with the med students. I gave them a quick lecture about how you make the decision of which hypothesis test to use. Then, I gave them 30 research questions and got them to make a decision for each one. They seemed to get the idea of how it worked. So much so that they actually had intelligent questions to ask afterwards!
I’m trying again this year, only this time the Medical School is letting me help design the whole stats teaching program, not just one lecture. Here’s hoping that a little bit of torture for a short time can alleviate months of pain later…
Theis comment was left on the original blog post:
Richard Knowling 27 January 2012:
This is an awesome idea David! I only wish Mike Roberts had still been alive to hear about it!