Kevin Houston

I haven’t posted a link to a video so here’s a remedy for that situation. Charles Seife is the author of Zero: The Biography of a Dangerous Idea which really is a book about nothing. His latest book is Proofiness: The Dark Arts of Mathematical Deception. I haven’t read it yet (usually I wait for the paperback version of a book so that my house doesn’t get totally overtaken by books).

Here’s a video of a lecture he gave for Google.

I’ve mentioned elsewhere that the current system of publishing research is flawed. The government, i.e. taxpayers, pays us to do research, we send the resulting papers to journal publishers, we referee the papers and edit the journals for free and then the publishers sell the research back to us for a high price. The result is that the taxpayers pay twice, we work for free and the commercial publishers get rich.

My answer to this was to start charging for my refereeing services. A small change but so far no-one has asked me to referee for something from a commercial publisher so it is really no change! However, a bigger and more effective method of change is coming – an academic boycott against Elsevier, considered by many to be a serious offender in this problem, has been started by Tim Gowers. Read the story over at the Chronicle of Higher Education, home to Prof Hacker.

The Yorkshire Branch of the Mathematical Association recently hosted a talk by David Acheson entitled Proof, Pizza and Guitar. (By the way I’ll be giving a talk on card cheating for the YBMA on Wednesday 8th February at 7.30pm in School of Mathematics. All welcome but a small charge of a pound may be made.)

During David’s talk he gave a proof of Thales’ Theorem. This is a theorem that states the following. For any point in a semi-circle, the angle formed by the lines from that point to the two edge points of the base is right-angled.

Thales' Theorem

This is a good theorem in that, to me at least, it does not seem intuitively obvious (what, it’s always a right angle? Really?) and yet it is easy to convince yourself it’s true by doing some examples.

Thales (c624BC – c547BC) is often considered to be the first scientist because he was the first person (we know of) who looked for non-supernatural reasons for phenomena. Rather than believing lightning or earthquakes were caused by gods he considered more natural explanations. However, his solution to the latter involved the way that the land floated on the sea, i.e., he was totally wrong but here it is the concept of avoiding invoking the gods that counts. In the case of mathematics he is credited with a number of theorems and the main point is that, allegedly, he provided proof.

He is also credited with measuring the pyramids in Egypt. His method is interesting because it does not involve a brute force use of measuring instruments, i.e., get out measuring rods and send people up the pyramids with them. His proof is more elegant than that. He measured the height of a slave and when the sun was such that the length of the shadow of the slave was equal to his height they measured the length of the shadow of the pyramid. From this shadow the height of the pyramid could be found.

Thales' measurement of pyramids

Returning to Thales’ Theorem, its proof is rather simple once one accepts that a triangle has angles adding up to 180 degrees and that an isosceles triangle has two equal angles. (The former statement is probably the most well known theorem in the world. I think it beats Pythagoras’ Theorem because, although most people have heard of that, they usually mis-state it.)

Let’s see what I will call the traditional proof. From the centre point draw a line to the alleged right-angle point to get two triangles. Both are isosceles as, obviously, edges from the centre have length equal to the radius of the circle.

Proof of Thales' Theorem

As in the diagram below we can label the angles with and . Using the fact that angles up to 180 degrees we get that is 180 degrees, i.e., is 90 degrees as required.

Proof of Thales' Theorem

This is the proof I use in my geometry lectures and was the one presented by David Acheson in his talk. He also said that he once overheard someone say that it’s easier than that, you just need to use a rectangle. The idea is that you rotate the triangle 180 degrees about the centre. David wondered if the person was in fact confusing this proof with a proof of the converse of Thales’ Theorem.

Anyhow, this idea led to a lot of discussion after the lecture. Some of us tried to make the rectangle argument work. Rotating the triangle produces a parallelogram (since we rotated by 180 degrees the opposite sides are parallel). The diagonals cross in the centre and have the same length. Such a parallelogram must be a rectangle, hence the angle we are interested in must be 90 degrees.

Of course, we all know and can accept that such a parallelogram must be a rectangle, but the problem is that to prove it rigorously seems to be more involved than the traditional proof above of Thales’ Theorem. In fact I’ve not been able to give a simple proof of the parallelogram result that did not involve something similar to that proof.

This rectangle argument for Thales’ Theorem was naggingly familiar to me and after I had left the talk I recalled that I had seen it in Lockhart’s Lament. This is a document circulated on the web by Keith Devlin a few years ago in which Lockhart laments the state of mathematics education in the USA. It is well worth a read if you have never seen it. However, it makes the claim that the rectangle proof is fantastic.

Unfortunately, Lockhart compares the rectangle proof not with the traditional proof above but with a two-column type geometric proof. This type of proof used to be the favoured method of teaching mathematics. Indeed it does teach rigour effectively but as a downside it squeezes all the fun out of geometry. To compare this type of proof with the rectangle proof is very unfair – from the enjoyment perspective any proof has got to be better than a two-column proof!

So, my question is, can anyone prove the “parallelogram with equal diagonals is a rectangle” result without resorting to an argument like my traditional proof of Thales’ Theorem?

In Cal Newport’s Study Hacks blog he recently had a post about a piano student called Jeremy.

Jeremy’s Strategies for Becoming Excellent…

Strategy #1: Avoid Flow. Do What Does Not Come Easy.
“The mistake most weak pianists make is playing, not practicing. If you walk into a music hall at a local university, you’ll hear people ‘playing’ by running through their pieces. This is a huge mistake. Strong pianists drill the most difficult parts of their music, rarely, if ever playing through their pieces in entirety.”

Strategy #2: To Master a Skill, Master Something Harder.
“Strong pianists find clever ways to ‘complicate’ the difficult parts of their music. If we have problem playing something with clarity, we complicate by playing the passage with alternating accent patterns. If we have problems with speed, we confound the rhythms.”

Strategy #3: Systematically Eliminate Weakness.
“Strong pianists know our weaknesses and use them to create strength. I have sharp ears, but I am not as in touch with the physical component of piano playing. So, I practice on a mute keyboard.”

Strategy #4: Create Beauty, Don’t Avoid Ugliness.
“Weak pianists make music a reactive task, not a creative task. They start, and react to their performance, fixing problems as they go along. Strong pianists, on the other hand, have an image of what a perfect performance should be like that includes all of the relevant senses. Before we sit down, we know what the piece needs to feel, sound, and even look like in excruciating detail. In performance, weak pianists try to reactively move away from mistakes, while strong pianists move towards a perfect mental image.”

Of course as we are talking about studying for a public performance of music these strategies don’t translate immediately or perfectly to the study of mathematics. Nonetheless, I’ll give it a go.

1. Avoid Flow. Do What Does Not Come Easy.
The second part is crucial and is where many students make a big mistake. They get good at half of the course and hope that that will get them through the exam. I saw it in contemporaries at university and I see it in some of my students. They do they easy parts, for example focussing on doing the calculations rather than the concepts.

Hence the advice is to do the deeper work. I.e., become competent in not just the superficial calculations but have deeper understanding. For example, don’t just know what the definition is, know why it is the way it is and — hardest of all — know why it is not some other way.

2. To Master a Skill, Master Something Harder.
When I was first studying group theory I went further by studying why equations of degree 5 do not have solutions via radicals – a topic which was not part of the course. I found that doing this not only helped me grasp the basics of the material such as the definition of simple group but also helped with the harder stuff such as Sylow p-groups.

Mathematics is a linear subject in the sense that one concept is built upon another. Despite this we don’t have to — and probably don’t — learn it in a linear way. When learning proofs I would often try to learn the proofs from the end of the course first. One, this told me what was important in the course — if Lemma 2.1 kept coming in the later parts of the course, then it was a good idea to pay attention to it. Two, when it came to study the earlier parts in depth these seemed much easier. (An important point: you have to at least have some idea of what is in the start of the course, you can’t start in the middle, what I am saying is you don’t necessarily need to learn the start in depth to begin with.)

There is another good reason for studying the last half of the course. The harder parts of a course are often there because they are needed as prerequisites for later courses.

3. Systematically Eliminate Weakness.
When studying vector calculus I found that my weak point was calculating accurately the various types of integrals – I could quote the definitions and theorems such as Stokes’ Theorem without problem – but I kept messing up the calculations by forgetting terms, eg, missing the side of a cube. So I got Schaum’s outline series book on vector calculus and did many examples from there until I felt my accuracy had improved.

4. Create Beauty, Don’t Avoid Ugliness.
Hardy famously said there is no room for ugliness in mathematics (I rather less famously said, “Then why do they write so many ugly papers”). But how to make point 4 relevant to mathematics? Well, although it can be seen above that I took action to avoid mistakes (ugliness), my main ambition was to have an overall understanding of the course so that everything fitted together perfectly for me. One method I used to achieve this was to rewrite the lecturer’s notes. I don’t mean word-for-word rewriting but trying to make major improvements. Lecturers rarely provide perfect notes (you try writing over thirty hours of material) and many theorems can be generalized with only a little bit of effort. I sometimes found that when I had a generalized theorem, then I could deduce some of the theorems from the course with ease.

For me, that was where the beauty was! Making it simple.

I was back on the radio reviewing the morning papers. Fortunately, there was a lot of good maths and science stories. Unfortunately, too many to cover. You can hear me (for the next few days at least) at BBC Radio Leeds at approximately 1:25 and 1:55.

The main stories were Alex Bellos: How to Learn to Love Maths in the Guardian, an interview with Patrick Moore in the Daily Mail and an astronomy article in the Daily Telegraph.