I went to the movies a month or so ago and saw two movies. I get to go to the movies very rarely, so when I do I like to see movies that are very good. Of course, you can’t know in advance whether they will be very good, and based on last month’s experience the reviews in the paper are no help whatsoever.
Anyway, the movies I saw were “The Lucky One” and “The Five Year Engagement”. In my opinion, the first was good and the second was bad. Let me tell you why…
“The Lucky One” is very good mainly because it is a well-constructed movie. The story it told was simple, but with enough complexity to keep you interested. The characters were well-drawn – both the good and the bad. There were many subtle things to take away from the film, but none of them clouded the overall message, which nevertheless was not rammed down your throat. There was plenty of stuff in it to talk about afterwards. It was a good movie.
“The Five Year Engagement” was just bad. The story it told should have been simple, but it was made unnecessarily complicated by random plotlines. The characters jumped from nice people to complete wackos in the course of single scenes. There were some nice messages to take away about faithfulness and finding the love of your life, but they seemed to appear all of a sudden when someone thought the movie ought to end now. All in all it seemed like a random collection of stuff that someone thought was vaguely connected to the title. And the thing that made it the worst was that it really could have so much better. The idea had such scope for a great film – I really wanted it to be a good film – but it just wasn’t.
My feelings are the same about university courses, especially maths courses. Some have a small number of grand messages that all the ideas fit into neatly; others just seem like random collections of stuff to do vaguely with the title. In some courses each concept is discussed well and the connections between it and the rest of the course are made clear; in others everything seems half-done before you move on to some new and seemingly unrelated topic. Some lecturers give the sense that they have a plan and a story to tell; while with others you feel they just had to do this course today with the materials they had to hand. Finally, some courses leave you with a sense that it was worthwhile being there; while others leave you thinking it was all sort of interesting, but it could have been so much better.
I commend all those lecturers who already know that good course design is like good movie-making – these lecturers have a story in mind to tell, and a plan for telling it well. They know that students need to feel that it was worthwhile turning up.
To the rest of them I say it’s not good enough to have a lot of stuff to say that is good stuff. There has to be a story that draws it all together, and some big ideas you can walk away with. Without these you end up with a course that should be good but just isn’t – you end up with a course like “The Five Year Engagement”.
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!
I went to the Gawler Show with my family the weekend before last, and it was a wonderful day. We had camel and pony rides, patted the animals, looked at all the stalls, bought some toys, got given balloons and generally had a most excellent day.
And as we left, we decided to indulge in some show food. One of the food vans was selling what they claimed to be “The best Dagwood Dogs in the land”. And you know what? They were! If hadn’t already left the show and walked halfway down the street to our car when we had finished, we would have bought another one.
But even as I ate this faboulous Dagwood Dog, I wondered, “Sure it’s good, but why is it so much better than any other one I’ve ever had?” And soon I had quite a list:
It had just the right level of salt. Most Dagwood Dogs are way too salty, but this one was just right.
The batter wasn’t greasy. Instead it was fluffy and light.
The batter had corn and peas mixed into it! I have never seen this before but I’m amazed no-one has ever thought of it.
The flavour was so good I wanted to keep eating it even after the bit with the tomato sauce was gone.
The stick they used had a wider bit at the bottom so you could properly hold onto it.
Later that day, it occurred to me that I very naturally evaluated my Dagwood Dog. It was so easy for me to make the decision of whether it was good or not, and to come up with a list of reasons why it was good.
So why is it so hard to do this when it comes to teaching and learning? When I have a particularly good class, do I stop to think about why it was good so I can achieve it again? When my students fill out a SELT for my seminar, they quickly decide if it was good or bad, but do they give me a list of things that made it good or bad, so I can do better next time?
Yet, it was such a natural thing to do this for my Dagwood Dog. I reckon we could all start using our natural food-evaluation instincts on our teaching and learning, and then perhaps we could claim we have “The Best Teaching in the Land”.
Recently, I’ve heard a lot about the number τ, and I find the whole thing a bit odd.
Here’s how it goes:
The number π is the ratio of a circle’s circumference to its diameter. It’s been known about for thousands of years and is an extremely useful number which appears in all sorts of unusual and unexpected places. It’s not only irrational but also trancendental, which means you can’t write it down exactly using fractions or even square roots. Its decimal expansion begins 3.14159… and a not-too-bad approximation using fractions is 22/7.
People are so enamoured with π that they celebrate π day (14th of March), and π approximation day (22nd of July) – in fact, I will be celebrating π approximation day by writing the digits of π on the street in Adelaide.
But here’s the thing: some people claim that π is not the best number to use as your fundamental circle constant. This is because, if you represent angles as distances around circles (which is what mathematicians do), then π only represents half of the circle. Therefore, these people claim that you should use instead 2π – which they call τ. Vi Heart gives a very impassioned talk on this on YouTube: http://www.youtube.com/watch?v=jG7vhMMXagQ , and others have launched τ day (28th of June) as an alternative to π day.
Included in their reasoning to throw out π and embrace τ is a claim that it’s pedagogically more sound – that it’s confusing for the fundamental constant to only represent half a circle, and that many more formulas are easier to work with and easier to remember with τ rather than π. For example, they cite the trig functions and how they repeat themselves every τ as opposed to every π.
But this is my main bug-bear: of course it’s not easier! The switch in people’s minds from degrees to radians is such a huge jump that whether you use π or τ is really not going to make all that much of a difference! And while many formulas are nicer with τ, others are just uglier (in my mind!).
It just says to me that you can be passionate about something loudly enough and lots of people are likely to agree with you.
But I have one more thing to add: If you were going to work with a new circle constant, I think you should use not 2π, but π/2 – let’s call it η. You see, η represents a right angle, which to me is an extremely fundamental thing in our modern lives. And moreover, it represents the ratio of a semicircle to its diameter. That is, if you want to go from A to B, it’s how many times further you go if you go around a circular path as opposed to in a straight line. That makes a lot more sense to me than either the circumference/diameter, or the circumference/radius. Finally, the trig functions repeat their shape (if not their orientation) every η so the very constant you use would remind you of this simple fact. Yes, if you were going to define a new constant, I reckon η makes heaps more sense than τ.
But of course, I don’t care quite enough about this to make an empassioned speech about it on YouTube, so it’s unlikely anyone will listen. 😉
[NOTE: I do actually respect Vi Hart very much and wholeheartedly support her work in the physical and musical representation of maths, and also her use of YouTube to encourage play in maths rather than rote learning. I just don’t agree with her opinions about π.]
UPDATE 29 August 2011: Ok, so maybe I was wrong about not caring enough to make an empassioned speech on YouTube…
These comments were submitted on the original blog post.
Karl Medlicott 29 April 2014 But why η (ἦτα)?
David Butler 29 April 2014 Hi Karl, I used eta because it seemed like it wasn’t used for much yet, especially not being used for an angle often, and because the capital is H for half, but mostly because it makes a nice pun: “eta pi”. 😉
Karl Medlicott 29 April 2014 … & didn’t the original French metric system use the right angle [the hectograde = 100 grades] as the measure of circular arc? Brilliant! mesures d’arc de méridien hectograde (Hgr) = quart du méridien terrestre décagrade (Dgr) grade (gr) degré centésimal (°) ≈ décamyriamètre décigrade (dgr) ≈ myriamétre centigrade (cgr) minute centésimal (′) ≈ kilomètre milligrade (mgr) ≈ hectomètre décimilligrade (dmgr) seconde centésimal (″) ≈ décamètre centimilligrade (cmgr) ≈ métre
Karl Medlicott 30 April 2014 Cool! …but if we wish to move beyond πι & ταυ we needs must stop talking or referring to either one of them, or defining or naming our own thing in terms of theirs (though puns are always fun). I’d much rather call our “new circle constant” — no, that’s not at all quite right —well, I’d rather call it “q” [actually a SMALL-CAP Q, (pronounced “qu”, as in ancient Latin) which looks rather like a Q, but smaller] which no one else is using for anything at all, & which stands for “quadrant”; & it’s Latin, not Greek!
Karl Medlicott 20 May 2014 … I’ve just this moment read that, in 1958, the eccentric English mathematician Albert Eagle had π/2 as the circle constant — & he used the symbol τ! So, forget “Q”, or “q”; I’ll go with Albert Eagle’s circle constant: τ ≈ 1,570 796 326 794 896 619 231 321 691 639 751 442 098 584 699 687 552 910 487 472 296 153 908 203 143 104 499 314 017 412 671 058 533 991 074 043 256 641 153 32…
Karl Medlicott 21 May 2014 Yes Albert was punning too. τ = 1/2 π.
David Butler 9 July 2014 Thanks for that Karl! Michael Hartl himself has referenced me in the newest version of the Tau Manifesto saying that η simplifies in particular formulas for volume/surface area of spheres in higher dimensions.
Karl Medlicott 20 July 2014 Michael Hartl himself does more than reference you, he agrees with you, save for the “inconvenient factors”, writing “(I liken the difference between τ and η to the difference between the electron charge e and the charge on a down quark qd=e/3: the latter is the true quantum of charge, but using qd in place of e would introduce inconvenient factors of 3 throughout physics and chemistry.)” Thank-you for η! … & thanks for showing me that the original French creators of the metric system knew what they were about when they defined the grade as 1/100 of a right angle — this, with τ, had long disturbed me. … but what would be the canonical definition of η as an equation, mentioning neither π nor τ? Surely not C/r/4? η ≡ … ?
David Butler 23 July 2014 A possibility is to define eta as the ratio of the area of a circle to its inscribed square.
Karl Medlicott 18 August 2014 I’m all with you (& Albert Eagle, & the French mathematicians who devised the original decimal metric system) about right angles — I want η to be the thing! … & then there’s this http://boxingpythagoras.com/2014/06/30/be-smart-use-tau/
David Butler 21 August 2014 *sigh* I don’t really want people to switch to eta. I’m saying that if they were going to make a switch anyway, then I’d prefer eta. What bothers me most is that he thinks he can convince people by telling them they’re stupid — I have never found that a healthy approach to things.
Boxing Pythagoras 27 September 2014 Hi, Dr. Butler! Thanks for taking the time to read! Honestly, despite my attempt at pithiness with the “Pi is Stupid” and “Be Smart, Use Tau” lines, my goal was not to tell people that they are stupid, but rather to say that we should do our best to prevent the obfuscation of mathematics. Just as I wouldn’t define a circle as ‘two semicircles which share a diameter and endpoints but which have opposite direction,’ I don’t think that the primary constant for describing circles should be defined as the ratio of its Circumference to double (or quadruple) its radius. Re-reading my article, I do now realize that it has an unnecessarily antagonistic tone to it (especially towards the end), and for that I apologize. I tend to get a bit more emotional about my geometry than a person probably ought to get. Thanks, again!
We bought Disney Studio’s newest film “Tangled” on the weekend and I have to say it’s one of my favourite movies ever. It’s certainly Disney’s best movie since “Beauty and the Beast”, and I dearly loved “Beauty and the Beast”. I should warn you now that in order to say what I want to say I’m going to have to reveal a bit of the plot, so let this count as your spoiler alert.
OK. So Rapunzel grows up in her tower thinking that the old lady is her mother and not knowing who she really is. During the film she escapes and goes to the town where there are a lot of sun-shaped motifs. She brings one home to the tower with her on a piece of cloth to remember her experience.
We see her lying on her bed staring at the ceiling, which she has completely filled with painted pictures during her life in the tower. She looks at the sun-shape and notices something remarkable about her painting: the sun-shape from the cloth is there in her paintings, and not just once, but over and over and over, and the repeating pattern sparks a memory of seeing the shape when she was a baby. The music swells as she realises who she really is. In short, the cloth and the paintings spark an epiphany.
But it occured to me that she would never have had this epiphany without two important factors. Firstly, she had to bring the sun-shape home with her on the cloth. Secondly, and more importantly, if the sun-shape had not been in her paintings so many times, she may not have noticed the connection.
And here’s where it relates to learning maths:
We want the ideas we show our students to connect together so that the students realise the true nature of things and the realisation changes them. In short, we want them to have “learing epiphanies”.
I’ve seen it happen for students when learning about subspaces in first-year maths. There are a lot of ideas but they are all highly connected, and sometimes while they are trying to solve a particularly difficult problem they suddenly realise that they’ve been seeing the same pattern over and over and that it all just makes sense.
I want this experience to happen for all my students.
But is it possible to set up these learning epiphanies in advance? It could be argued that epiphanies are highly personal and can’t be engineered. But I think perhaps we can make them more likely by putting certain things in place…
Firstly, the connections between the ideas have to be there all along, just like the sun-shapes in Rapunzel’s paintings. If they weren’t already there, the realisation wouldn’t have been so powerful. We need to make sure that there are patterns in what we do and say from the very beginning.
Secondly, the connections have to be there many times – so many times that once they have been noticed you wonder why you didn’t notice it before. It gives a huge sense of sureness to the realisation that you have, so you don’t just discount it as your imagination. So in our examples and explanations, we need to repeat and repeat the same pattern over and over and over.
Finally, there needs to be an event to start it off, something to help you notice that first connection. Just like Rapunzel’s cloth – she needed the shape to be marked out simply so she could notice it in her own work. So we need to stop and point out the pattern in what we’ve said every so often, and get the students to do activities that hold the patterns up close to each other so they can notice.
I think keeping these things in mind as we choose what examples to show our students, and choose how to present them, and choose what activities to get them to do, may just make it possible to help them have an epiphany like Rapunzel’s.
This comment was left on the original blog post:
“Humane Pain” 26 March 2013 I agree (with both your opinion of *Tangled* and with the learning strategy), but wanted to add an additional benefit besides helping students to grasp the concept: epiphanies also add the element of excitement, that “aha!” or “Eureka!” emotion that is such a rush, such a natural high, that they want to study and learn more and more, in essence, the epiphany becomes a vehicle for motivating the students as well as grasping concepts, and this makes them lifelong learners. It makes maths *fun*.
The other night I was doing a Sudoku, and my two-year-old daughter Charlotte decided she wanted to help, as she always does at any time when I have a pen and paper she could steal.
So she bent over the paper concentrating very hard and a while later she threw her hands in the air and declared in her loudest voice, “I did it!!!”.
And sure enough, she had. Every single box was filled with a letter-like squiggle.
And isn’t this the most basic rule of Sudoku, when you really get down to it? Sure, the symbols in the boxes are supposed to be the digits from 1 to 9 and you’re supposed to have one of each in every column, row and box, but really if there’s not one symbol in every box, you just haven’t finished have you?
I was extremely impressed that she was able to figure out that rule just by watching me for a few minutes!
And, as usual, it made me think of my students. Often they latch on to one idea from their lectures and do their maths problems following that one idea, even though there are quite a few important details that make it actually work. But should I really be upset? They’ve picked out the most important idea and that in itself is impressive. Next time I’ll try my best to let them know how impressive it is before telling them how to be better.
