Kevin Houston

In a previous post on the Khan Academy I said in an aside

I think that video’s threat to teaching jobs is greatly exaggerated. When the printing press was invented people probably said “No more need for teachers, you can learn from a book”. Every new technology is predicted to revolutionize teaching and to cause the disappearance of the bulk of teaching jobs. People said it about radio, they said it about film, they said it about TV. They now say it about YouTube videos and laptops.

I was asked for evidence for the “They said it about film” part. Given that when I was school we watched educational films projected on the wall at least someone was eager to use the medium in education so someone must have said it.

However, the quote comes from an interesting source: Thomas Edison.

I believe that the motion picture is destined to revolutionize our educational system and that in a few years it will supplant largely, if not entirely, the use of textbooks.
I should say that on average we get about two percent efficiency out of schoolbooks as they are written today. The education of the future, as I see it, will be conducted through the medium of the motion picture … where it should be possible to obtain one hundred percent efficiency.

Thomas Edison (1922) quoted in Larry Cuban, Teachers and Machines: The Classroom Use of Technology Since 1920.

Whilst researching this quote I came across Quote Investigator which discusses the history of the quote. This led to an interesting article using the quote to comment on the current Let’s-give-all-the-kids-laptops/iPads movement. Of course, I could have easily found a dozen such articles.

And when was television the answer to all educational ills? Well, recently I found a University of Leeds prospectus from 1968 which highlighted the university’s use of television as a teaching method. The hope was that the lecturer would be unnecessary as students could just watch pre-recorded programmes, or at the very least one lecturer could broadcast to many students. In fact, until the major refurbishments following the introduction of student fees, in many teaching rooms here you could see the old sockets used for the coaxial TV cables.

I think the point is that in most situations the best type of teaching comes from real people. Yes, books are good, videos are good, etc, but there is no better to way to learn than to have a teacher there interacting with you. People know this and will pay good money for it whether employing a personal trainer for exercise or attending a educational institution with good access to staff.

Probably, like me, your most inspirational learning experiences came from teachers and not books. Am I right?

Fake quote?
Whilst researching this post I came across a quote spookily similar to the Edison quote above but it seems to be made up. The alleged quote is attriduted to BF Skinner, the famous behaviourist who studied pigeons and rats. He invented a “teaching machine” (which he patented!) that was going to solve the education problem. (Aside: I can remember a children’s educational book about the “World of the Future” that featured the machine. What I remember is that it had a little roll of paper and a small window. A question could be asked, the student would be able to write the answer on the paper, the paper would advance behind the window so the student’s answer could not be changed while the true answer was given. I like the way Skinner had taken into account that students might try to cheat but had not taken into account that we’d communicate with machines via a TV screen.)

Anyhow, according to the source, Skinner viewed the machine as a solution to our education problems (compare with the Edison quote):

I believe that teaching machines are destined to revolutionize our education system and that in a few years they will supplant largely, if not entirely, the use of teachers.

This was referenced with Skinner, 1968, p1. See here.

Well, I’ve checked the works by Skinner in 1968 and can’t find it. (The obvious publication in that year is his major book The Technology of Teaching.) Also, it seems unlikely that Skinner would plagiarize a comment so badly!

What’s funny is that the quote has been repeated many times in other places. Why have those authors not checked their sources. Or am I missing something here?

Over the last few weeks I’ve been thinking about producing more videos. I’ve certainly fallen behind my planned schedule of producing them – too many other tasks get in the way! Anyhow, I have to produce some so that my school can complete a project for HE-STEM. As part of this I’ve had to consider what makes a good video and for me one of the most interesting debates about videos in education has been triggered by the huge success of Salman Khan’s video series. You can see him explain the history and the philosophy behind it in the following TED video:

Looks good doesn’t it? However, his videos have been strongly criticized. This is not surprising, educators were always going to object to an ex-hedge fund manager – backed by Bill Gates and with no educational training – coming in and saying “This is how you do it”. Nonetheless, many of their criticisms have foundation. I’m particularly against the “gamification” of education. Whilst games can be a useful tool in education when you wrap all learning in a game, then students lose sight of the importance of education; they see it just as collecting points for their scorecard.

Audrey Watters gives a good explanation of the arguments against Khan and various links in her post The Wrath Against Khan which arose as in response to an article in Wired on the Khan Academy.

(Aside: I think that video’s threat to teaching jobs is greatly exaggerated. When the printing press was invented people probably said “No more need for teachers, you can learn from a book”. Every new technology is predicted to revolutionize teaching and to cause the disappearance of the bulk of teaching jobs. People said it about radio, they said it about film, they said it about TV. They now say it about YouTube videos and laptops. (Aside to aside: Currently, the main threat to jobs is funding. Teachers are losing their jobs because governments are cutting budgets. But that’s a different story.))

Another critique, by Frank Noschese, in a post punningly entitled
You Khan’t Ignore How Students Learn contains a great quote:

Khan (along with most of the general public, in my opinion) has this naive notion that teaching is really just explaining. And that the way to be a better teacher is to improve your explanations. Not so! Teaching is really about creating experiences that allow students to construct meaning.

Actually, I doubt Khan believes that teaching is just explaining – although it’s true many of his videos are just explaining – I think the difficulty is that his courses do not do enough to construct the meaning mentioned in the second half of the quote.

I’m a fan of Dan Meyer. He recently blogged about an article on the Khan Academy featured on the TV programme 60 Minutes. Meyer’s post is here but maybe you should watch the report first. (It has a couple of annoying adverts near the beginning but it is worth persevering.)

There is also a follow up video in which the reporter makes a Khan Academy video with Khan. (See it here. One interesting thing to note is how well Sal Khan presents himself. He seems genuinely likeable and self-effacing (see the “My mum wishes I was you” comment). Since the videos I am preparing are about mostly about how to give a presentation it would be a good idea to use him as an example of how important likeability is. I’m guessing that if Kahn was a grumpy man who railed against the world his videos would not be so popular.)

One of the best videos I have seen explaining why Khan’s videos are problematical whilst praising their use in certain situations is by Derek Muller who has a good collection of YouTube videos.

The video relates to science education generally rather than mathematics but I think the point for mathematics is clear too. For example, one can apply the idea of showing common errors when introducing the rigorous definition of limit. This is a subject students think they know when they come to university so don’t pay attention to the correct definition. When asked to define a limit in an exam they write “it’s the thing that gets closer and closer to a number but never actually reaches it”. (Aside: A common complaint I hear from students is that before we taught them limits they understood the concept and afterwards they don’t! Ditto for integrals and Riemann integration.)

Now if you’ll excuse me I’ve got to go and create some online video materials. No, seriously, I have to…

One for lecturers: I’ve been asked to speak at a free workshop entitled Teaching Introductory Mathematical Analysis. This HEA MSOR-funded event will take place at De Morgan House in London (home of the London Mathematical Society) on Wednesday May 2nd. Leaving aside this humble scribe it has a cracking line up:

• Lara Alcock (University of Loughborough), download her book with Adrian Simpson
• Camilla Jordan (The Open University),
• Joe Kyle (University of Birmingham),
• Chris Sangwin (University of Birmingham).

Further details can be found at http://www.mathstore.ac.uk/?q=node/2091.

Last week was another busy week with trips to London and Birmingham. The latter was to give a talk at the annual National Association of Mathematical Advisers. This is a group of people who advise school teachers and politicians regarding the content and implementation of Primary, Secondary, GCSE and A Level Mathematics. (Probably, that’s a gross distortion of what they do but it’ll give you the idea if you’ve never heard of them before.) My talk was on How to Think Like a Mathematician. I’m not sure that it was the sort of talk they were looking for but I gave it a go…

On the Thursday evening there was a conference dinner followed by the comedian Helen Keen who was doing her It is Rocket Science talk. It was certainly an enjoyably funny show but maybe a bit short – though that could be because she speaks so fast. Nonetheless, she is worth watching – according to her website she will speaking at the Cambridge Science Festival this month. Which brings me to the Leeds Festival of Science, and in particular, my talks.

I’ll be speaking 1:30-2:30 on Saturday 10 March and Wednesday March 28th. See the Festival Science brochure and the WP Milne poster for details.

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?