The CLPD head administrator Cathy told me a story the other day about an experience she had on the train: She was sitting opposite a pair of students, and one was helping the other prepare for a test. The first student was reading out words from a stack of cards and the second was trying to correctly say what they mean. After listening to this for a while, Cathy leaned over and asked what it was they were studying. The students said “pure maths”.
This completely surprised Cathy, because not one word they had said in all that time seemed to be related to maths in any way, and some of them she had never even heard before. Now Cathy has worked in many different areas in her life, many of which were in academic institutions, not to mention her own experience with maths in the past. So it was quite a shock to her that she had never heard these words associated with maths before. “It was like a completely different language,” she said.
My response to this statement was, “Quick Iggle Piggle! Catch Makka Pakka’s Og-Pog before it hits the Ninky Nonk!” And Cathy immediately knew what I was talking about because she, like me, has a young daughter, and therefore watches ABC2 rather a lot.
You, however, may not regognise or attach any meaning to any of the words in that sentence at all. The question is: do you feel like an idiot for not knowing what I’m talking about? Of course not – it’s just that you happen to have never seen the TV show “In the Night Garden”.
So why do so many people admit to feeling stupid for not knowing specialised maths words? If you happen never to have come across that particular area of maths in your life up till now, that doesn’t make you an idiot. It just means you’ve never come across that area of maths in your life up till now.
If you feel stupid when you hear someone using unfamiliar words, just think of a phrase from some other area if life or learning where you’re pretty sure the other person won’t know any of the words. (Such as, “Quick Iggle Piggle! Catch Makka Pakka’s Og-Pog before it hits the Ninky Nonk!”)
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”.
Do you know the story of the Shoemaker and the Elves? Well, I’ve known it since I was very young. It’s a Brothers Grimm, and it goes something like this:
A poor shoemaker is down on his luck and can’t make enough to feed himself and his wife. All he has left is enough leather for one pair of shoes and he works late into the night preparing the leather but falls asleep at the workbench. In the morning the shoes are all made with such fine and perfect workmanship that they are snaffled up quickly by the next person to pass the shop window.
The shoemaker of course buys some more leather and gets it ready and tries the trick again. And again all the leather is sewn into wonderfully well-made shoes. Soon he and his wife are very well off.
Eventually they decide to ask the question of how this is happening, and they hide themselves so they can see who is making the shoes. As it turns out, it’s a team of little elves, who are all completely naked.
The shoemaker and his wife feel sorry for the little elves who have helped them so much and decide to make clothes for them, which they leave out the next night. The elves are so delighted with their clothes that they declare they don’t need to work all night anymore and dance away into the night.
Now those of you who have read Harry Potter may recall a character called Dobby – a house elf, who had to remain in servitude until such time as his master presented him with clothes…
Just a moment! Doesn’t that sound familiar? Of course it does. It’s right out of the Shoemaker and the Elves!
For no reason that I can see, I suddenly came to this realisation this weekend. Jo Rowling rose again in my estimation as being a very clever woman. And I sank just a little in my estimation because I knew this story from when I was very young – why on earth did I not see this connection earlier?
Still, it’s not worth kicking myself over it – this sort of thing happens all the time with learning maths. Students say to me all the time: “I just realised these things were connected! I never knew I didn’t understand how this worked until I suddenly understood how it REALLY worked!”
It’s nice for the feeling to happen to me for a change
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!
The same experience has happened to me several times in the Maths Drop-In Centre recently – with different students from different courses – and it was such a strong pattern I need to talk about it.
The students are doing some algebra involving negative powers on the tops of fractions. Something like this:
\[\frac{1-x^{-2}}{1+x^{-2}}\]
Now they remember this rule (probably from school) which says that a negative power belongs on the bottom of a fraction but as a positive power. And so they do one of these:
\[\frac{1-x^{2}}{1+x^{2}}\]
or
\[\frac{1+x^{2}}{1-x^{2}}\]
Both of these are, of course, TOTALLY WRONG. But the students have a hard time being convinced of this fact.
The problem is, that that rule only works if everything involved in your fraction is multiplication and division. It doesn’t interact with the plus and minus that are trapped there on top and bottom of the fraction. And why doesn’t it interact with the plus and minus? Because the rule is based on the definition of what a negative power means. This is what it means:
\[x^{-2}=\frac{1}{x^2}\]
What this means is that multiplying by a negative power is the same as dividing by the matching positive power. And this gets to the heart of the issue: adding a negative power is not at all anything to do with multiplying it, so the nice “switch to the bottom, make positive” rule just isn’t going to work, because you have to do the addition first.
The rules for negative powers are colliding with the rules for addition, and for fractions, with unpredictable results! If only the students had been encouraged more to work from the original definition rather than it being all about remembering a rule. Then maybe the results wouldn’t be quite so unpredictable! If only the students had attempted a few things like this in the past in a situation where someone could notice it and talk to them about it! Then maybe they would have found this glaring gap in their understanding of algebra!
PS: If you’re wondering how to go about simplifying that fraction, then you have to first deal with the negative power using its original definition – which means it will become a positive power on the bottom of its very own little fraction. Like this:
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*.
In various first year maths courses here, the students learn the “bisection method” for finding zeros of continuous functions. (A zero of a function is a number that makes the answer of the function come out to zero – it’s therefore also a point where the graph of the function crosses the x-axis.) It’s based on the Intermediate Value Theorem, which basically says that if the function is below zero at one spot and above zero at another, then is must be equal to zero somewhere in between. Here’s how the process goes:
Find a point where the function is below zero – we’ll call it a – and a point where the curve is above zero – we’ll call it b – and then the actual zero of the function must be somewhere between a and b.
Divide the interval from a to b in half – we’ll call this centre point c.
Now put c into the formula to find out if the function is above or below zero there.
If it’s above zero, then the function goes from below zero to above zero between a and c and so the zero must be between a and c. If it’s below zero, then the function goes from below zero to above zero between c and b, and so the zero must be between c and b.
Now you have a new interval where you know there’s a zero, so divide this interval in half …
And the process continues on and on until you either actually find the zero, or you get as close as you need to be.
It’s a good method, but the problem I have with it is that the numbers you end up working with are much much more precise than you need!
Let me explain what I mean with an example:
Suppose you want to estimate a zero of the function f(x) = x2-2 to two decimal places. Well, we need a point where the function is above zero and a point where it’s below zero. The numbers 1 and 2 ought to do. Now watch:
f(1) is below zero and f(2) is above zero So the zero is between 1 and 2
The midpoint of 1 and 2 is 1.5, and f(1.5) is above zero So the zero is between 1 and 1.5
The midpoint of 1 and 1.5 is 1.25, and f(1.25) is below zero So the zero is between 1.25 and 1.5
The midpoint of 1.25 and 1.5 is 1.375, and f(1.375) is below zero So the zero is between 1.375 and 1.5
The midpoint of 1.375 and 1.5 is 1.4375, and f(1.4375) is above zero So the zero is between 1.375 and 1.4375
The midpoint of 1.375 and 1.4375 is 1.40625, and f(1.40625) is below zero So the zero is between 1.40625 and 1.4375
The midpoint of 1.40625 and 1.4375 is 1.421875, and f(1.421875) is above zero So the zero is between 1.40625 and 1.421875
The midpoint of 1.40625 and 1.421875 is 1.4140625, and f(1.4140625) is below zero So the zero is between 1.4140625 and 1.421875
The midpoint of 1.4140625 and 1.421875 is 1.41796875, and f(1.41796875) is above zero So the zero is between 1.4140625 and 1.41796875
The midpoint of 1.4140625 and 1.41796875 is 1.416015625, and f(1.416015625) is above zero So the zero is between 1.4140265 and 1.416015625
The midpoint of 1.4140265 and 1.416015625 is 1.4150390625, and f(1.4150390625) is above zero So the zero is between 1.4140625 and 1.4150390625
The midpoint of 1.4140625 and 1.4150390625 is 1.41455078125, and f(1.41455078125) is above zero So the zero is between 1.4140625 and 1.41455078125
Any number between these two will round off to two decimal places as 1.41 so our zero will also round off to two decimal places as 1.41.
But look at the numbers we were working with: the final numbers had at least seven decimal place precision, and we only needed two decimal place accuracy!!! I don’t know about you, but it seems a bit silly to me!
So here’s my idea: why don’t we do something in the same spirit, but only use numbers that give us the precision we want? Let’s think: anything between 1.405 and 1.415 will round off to two decimal places as 1.41, so instead of exactly dividing the intervals in half, let’s just use answers with a 5 in the third decimal place. Let’s try again with this focus in mind:
f(1) is below zero and f(2) is above zero So the zero is between 1 and 2
A number between 1 and 2 is 1.5, and f(1.5) is above zero So the zero is between 1 and 1.5
A number between 1 and 1.5 is 1.25, and f(1.25) is below zero So the zero is between 1.25 and 1.5
A number between 1.25 and 1.5 is 1.375, and f(1.375) is below zero So the zero is between 1.375 and 1.5
A number between 1.375 and 1.5 is 1.435, and f(1.435) is above zero So the zero is between 1.375 and 1.435
A number between 1.375 and 1.435 is 1.405, and f(1.405) is below zero So the zero is between 1.405 and 1.435
A number between 1.405 and 1.435 is 1.425, and f(1.425) is above zero So the zero is between 1.405 and 1.425
A number between 1.405 and 1.415 is 1.415, and f(1.415) is above zero So the zero is between 1.405 and 1.415
Any number between these two will round off to two decimal places as 1.41 so our zero is 1.41 to two decimal places.
See? Isn’t that nicer?
I like it anyway.
This comment was left on the original blog post:
David Roberts 10 May 2011: This is a really good approach. It also demonstrates the flexibility one has in choosing terms in a sequence that approximate a real number. The midpoint algorithm is really just a demonstration of the sandwich theorem, and in the absence of any other information about picking a point in an interval to define the next term in the sequence, the midpoint is as good as any (think of it as having an uninformative prior, for example). But if you have some idea about the sort of numbers you want to pick (preference towards 3dp numbers ending in a 5 in this case), then this gives a much faster approach to the final answer.
As I was leaving the other day, a student said that she would come to see us the next day to ask some questions about her assignment. She said she had tried to do as much of it herself as she could, and had only done 70% of it.
The “only” made me start – she had done most of it herself but that wasn’t good enough because she still had to ask for help. And somehow in the hurry of the moment, this came out of my mouth: I said, “And how much did you do on your own last time?”
It was her turn to start – “Oh!” she said, “A lot less I suppose.” And then I had to keep walking or risk missing my train. But as I walked the incident ran around my mind: it’s amazing how many people can discount the evidence of their own problem-solving ability, simply because they still need help.
I’ve seen it before, but I’ve never seen a way to fight against it. I’ve always tried to tell them that they can do it, and point out the bits they did do, but it always seemed to wash over them without leaving an impact. I’ve been focussing on a single moment of problem-solving.
What I’ve learned from my thirty second conversation is that perhaps I should help people focus on more than just today. Instead, maybe I can help them look at their journey so far and focus on the improvement.
But my student has given me even more: she’s given us something concrete to focus on: the amount they have done on their own. This is so much more tangible than “problem-solving ability” because it’s plain numerical data. The student can compare this time to last time and feel success as long as they’ve done that little bit more on their own.
Phew! I submitted our article for the MERGA conference last week and now I feel like I’ve come out of hibernation: I’m standing blinking in the sunlight wondering what happened to everything I was doing before I started work on the article. (One of those things was this blog, which is why I’ve been quieter than usual lately.)
One thing that caused me to descend deeper into research-hibernation was when I stopped to check the word count after getting halfway through what I wanted to say, only to discover that I was already 1500 words over the limit. I had to sacrifice a lot of what I had planned to say, and was left feeling like my research had a lot of loose ends flapping about everywhere.
This is not the feeling I get from maths research. I’ve published very short articles in maths journals before, but felt no qualms about them at all because they were all tied up. I don’t mean I had finished everything there could be to do. No, I mean that there was a proper result – something about which you could really say, “This is it. This is true. This is why it’s true. And that’s all I need to say.” It’s neat.
Education research is not neat. I’m always left with the feeling that you haven’t said anything. It seems more like, “This is sort of it. This is what might possibly be considered reasonable. This is why I think I might be in some way justified by holding the belief that this might possibly be reasonable. A lot more could be said but I have to stop now.” See? Not neat.
My experience researching maths has left me with the feeling that things ought to be neat, and I stress myself out trying to tie up the loose ends in my education research. What I’m learning to realise that loose ends are the way things are in education research and saying why you think something is possibly reasonable is actually enough.
We’ve been running Drop-In Centre tutor training recently, and as part of the training we discussed the statistics on how students use the Centre. The focus of this post is the following graph:
The graph describes how many students visited the Centre various numbers of times across 2010.
There are many things you could notice about this graph, but one thing you might notice is that almost half of the students who use the Centre only visit once.
We all discussed the possible reasons for this, ranging from those students who only come to be reassured that they’ll be fine with their maths this year, to those who came for one specific assignment in a course like Geology, to those who didn’t like the way we helped them and never came back.
But one trainee Nick made a very important point which is the one I wanted to share with you:
Although there are many reasons why it may be true, the shocking truth is that for most students, we only get one chance to help them. Therefore, the way we treat the students and the words we use with them are important every single time because that time we talk to them may be the only time.
Suddenly our responsibility seems so much greater…
