Lockhart’s Lament

A few days ago during #mathscpdchat  someone tweeted a link entitled “Lockhart’s Lament”, and an attached comment saying that Lockhart thought that the current state of maths education was akin to drilling pupils on music scales.

I duly followed the link, and discovered a 25 page essay by Paul Lockhart entitled “A Mathematicians Lament”. Paul Lockhart is a maths teacher in the USA who has previously had a successful career as a research mathematician, and as the title suggests he is not happy about the way maths is taught in the USA.

The essay itself is amazing and provides a lot of food for thought around both the mathematics curriculum and mathematical pedagogy. I would certainly advocate anyone with an interest in maths or teaching spends a bit of time reading it.

While reading it I found that there is a whole lot of similarities between maths teaching here in the UK and over the Atlantic in the US, although I did notice some differences too. I also noticed a real similarity between many of the things Paul was saying and the way I feel about maths.

We clearly share a live and a passion for the subject; share the feeling that school pupils should be allowed to develop and explore the subject in a manner that allows them to build a love for the subject and feel that there are some real errors in our curriculum.

Lockhart talks about the history and development of maths, and how this can enhance ones love of the subject. This is something I agree wholeheartedly with, and that is why I include these things in my lessons. I love to discuss the mathematicians behind the mathematics and I find that pupils do too. He also mentions a feeling that the curriculum is two constrained and doesn’t leave room for much actual mathematics, reasoning or conjecture. I think (in the UK at least) that these things can be included into lessons with the current curriculum if teachers have a broad enough and deep enough knowledge of the subject. I certainly find these opportunities. This week my “low set” year 8 class were told to discuss which game gave more chance of winning, “Rock Paper Scissors ” or “Rock Paper Scissors Lizard Spock“, and was please that four tables independently came up with “Rock Paper Scissors Lizard Spock, because whatever you throw there is less chance of getting a draw, so more chance of winning or losing.” it only took a few minutes too. They then used Sample Spaces to investigate the theoretical probabilities, and played a few games of it!

There are a few issues I intend to explore further and I will no doubt be referring back to “Lockhart’s Lament” again.

Further reading

The essay itself was published with an intro by Keith Devlin here

There is also a series of responses from Keith’s readers and then an answer from Paul here

Matrices

I love matrices, I loved them at school from the very basic matrices that I learned at high school to the more hardcore matrices that were involved in my A Levels I always enjoyed them. Their associative, but non-communicative properties for multiplication fascinated me. As for the things that happen at university with tensors…..

So recently, I have been really enjoying teaching the chapter on 2×2 matrices to my FM AS class. From basic arithmetic with matrices to using the inverses to solve simultaneous equations. I think its fascinating to look at the transformations matrices geometrically, and as think it is certainly one of the topics that shows the links between the different strands of mathematics really well. My class have enjoyed the topic, and have been equally as fascinated as I have by the algebra and geometry involved.

I have uploaded the resources I used to TES here If you do use them , please let me know how they went.

The lessons on the notebook/exported PowerPoint presentation follow the chapter from the Edexcel textbook. I think the order is quite good, although I taught the bit on simultaneous equations prior to reversing transformations.

QTS, Inequality and Political Footballs

This month’s #blogsync, in conjunction with Labour teachers, invites bloggers to write an open letter to Tristram Hunt, the shadow secretary of state for Education. Here is mine:

Dear Mr Hunt,

Welcome to your new role, I think that this invite is a novel and brilliant idea and hope more politicians look to engage with the electorate in a similar manner.

I would like to raise a few points that I feel should be at the forefront of the debate on education and that I hope you will look into, raise in the house if appropriate and even include in your next manifesto if you are inclined.

Qualified teacher status

I think that the current administrations decision to remove the requirement of QTS is terrifying, damaging and dangerous. It removes the professional status of teachers and really does make a mockery of the whole thing. I fully believe that all teachers should have, or be working towards, QTS, or some equivalent and I hope that you do too. Having a qualification guarantees an adequate subject and pedagogical knowledge which enables teachers to ensure that all pupils get the best education possible. Overlooking it at best, I.e. for the top academics, might mean you have extremely brilliant historians standing in front of classes unable to impart any knowledge at all. At worst it might mean unscrupulous heads employing people with neither subject knowledge nor teaching skill to cut costs.

Inequality

I stand fully against inequality anywhere, but especially in education. My ideal world would see an end to any inequality. I would bring control of all schools back into the public sector. Removing private education and faith schools entirely is, in my opinion, the only way to create a truly integrated society where everyone has the same opportunities. I think private schools create an elitist culture and increase the gap between rich and poor.

People like James Keir Hardie fought for free and equal education for all, and this is currently under threat. The recent talk of charging the wealthy for state education is dangerous, and risks the reversal of the long fight to make education universal, and not just the property of the upper classes. The wealthiest would surely, if forced to pay, opt for the smaller class sizes and better facilities offered by the private sector, leaving the pupils coming from poorer backgrounds to make the best of a state education being run on a shoestring.

Political football

I think that too often education has become a political football. Governments use it to stamp their authority and this can be very damaging for the pupils. I hope that in the future more safeguards could be put in place to prevent this, and to ensure that the future of the young people of Britain is at the forefront. I would like to see teachers more involved in policy making and perhaps a reduction in the power of the Department of Education. Although I would be terrified if the control were to move entirely out of government control as we would no longer be in a full democracy. I have read about a Scandinavian country (I think it was Denmark) where the secretary of state had two advisers with him constantly, both of who were front line practitioners, and this struck me as an excellent idea.

Those are the three main issues I have at the moment.

Thanks

Stephen

Further Reading

You can read all this month’s #blogsync entries here.

I have written previous about the issues of inequality and education as a political football.

Victoria Coren Mitchell has written this eloquent piece on the state school fees issue.

Chris Hildrew has also written on the inequality issue here.

Tom Sherrington has written this excellent piece on the QTS debate.

Vectors, Despicable Me and Other Stories

This week I have been teaching my year 13s about vectors. It had been a while since they had met vectors, so I asked what they remembered about them and one said: “direction and magnitude, they have direction and magnitude. That’s why vector on despicable me chose his name, because he’s committing crime with direction and magnitude!” See this clip.

I had seen the film before, but a few years ago and I had completely forgotten about this character! It is certainly a clip I will share with students in future! (Along with this classic.)

Videos like these are funny and memorable, and can help to embed the knowledge of vectors into the minds of pupils, thus enabling them to recall the knowledge better in an exam. There is plenty of debate around gimmicks and teaching methods. Whether one should be trying to teach or to entertain, but I don’t see why you can’t do both, as long as the entertainment adds to, rather than detracts from, the learning. And videos like this work better than any mnemonic for embedding knowledge and ideas into the teenage memory banks.

The world of entertainment certainly seems to have strong links to maths, the Simpsons and Futurama have for years included maths, which is cool in itself and I think should give plenty of things to use in lessons. Dr Who has plenty of maths, including happy primes and Fermat’s last theorem. The Big Bang Theory is, quite unsurprisingly, the show which keeps on giving, from Thursdays episode which included the mechanics of cow tipping, through topological problems like which cinema to go to to modelling examples, like when Sheldon extrapolated data on Penny’s serial history while on his first date with Amy.

Perhaps more surpringly is the fact that Disney snuck pi into Mary Poppins and there is clearly a maths enthusiast at Universal Pictures who worked on Despicable Me to include the vector character. Even preschool shows are at it! Peppa pig has reference to the quadratic equation and Ben and Holly’s wise old elf uses the golden ratio! Do let me know if you spot any more.

The mechanics of cow tipping.

The quadratic formula at Daddy Pig’s office, the discriminant is in yellow.

Shuffling and Bringing Mechanics to Life

Last night a colleague and I attended a “Teachers Evening” at Manchester University. We weren’t exactly sure what to expect, but jumped at the opportunity to spend an evening immersed in maths.

Having studied at Manchester University, and knowing that the tower that used to house the mathematics department  has been levelled, I was intrigued to see what the place looked like. The Alan Turing building is brilliant,  and extremely aptly named. The vibe of the place was great and the fact that it was still buzzing with undergraduates and postgraduates at half five was great. Some were even playing backgammon.

The evening itself was set up as two lectures, with an interval. The first lecture was from Professor Oliver Jensen, who’s seat is named after Sir Horace Lamb. His lecture, “bringing mechanics to life”, combined his own specific interest in biofluiddynamics with Lamb’s more general interest in fluid dynamics. It was a great lecture and left me eager to know more, I have a list of topics mentioned which I would like to investigate further. I felt a love of fluid dynamics reawaken,  it had been my favourite topic in the second semester of my second year. It also gave me a good number of lesson ideas for future mechanics classes, and a whole host more answers to the question “what use is maths in the real world?”

The second lecture was entitled “shuffling around, why you should play cards with mathematicians” and was from Dr Charles Walkden. The talk was fantastic. It combined theories on shuffling, modulo arithmetic and mathematical card tricks. It was really interesting and gave me some great lesson ideas, including using standard from to compare the number of ways to order a deck of cards with the number of drops of water in the Pacific. The modulo arithmetic ideas were very timely, as this forms the basis of a “taught round” I’m an up coming year ten competition!

All in all, a great evening,  and I can’t wait for the next!

The great calculator debate

Back in December I posted this blogpost about calculators. It caused quite a stir and prompted many responses. Dave Gale, aka reflective maths, tweeted back with this video, Colin Beveridge, of flying colours maths, responded with this blogpost and there was a much wider debate on twitter with tons of people getting involved on each side. It was brilliant to see. I thought though, that I needed to write a further post to clarify and review what had been said.

In the first instance, I selected a sensationalist title which was intended to catch the eye. I do think, though, that the title may have led people to think my stance was a little more hardline than it actually was. And having seen the views set forward by the alternate position, I think my view has softened further still.

When I wrote the original post I was certainly advocating the banning of calculators in primary classrooms, and I would stick by this now. The opposing case to this was that an inability to subtract two and three digit numbers from 360 was causing a barrier to teaching angles. I would counter this with the statement that subtracting two and three digit numbers from a three digit number is such a basic skill that it needs to be mastered either before moving on to angles, or with angles providing a great opportunity to hone this skill. The other argument was that the government were banning calculators from KS2 tests, but using the same test. On the face of it, this is silly, but I don’t think it is a valid argument for keeping calculators. Rather it is valid argument for altering the tests.

Colin wrote in his post that calculators are not the enemy, but rather it is their misuse. I can see his point here, but I wasn’t advocating we destroy them all, I was advocating that we eliminate their use in primaries and cut it down radically in secondaries. He questioned the necessity of adding 4 or 5 numbers with 5 or more digits together, and this is a point I will concede. My hard line of only using it for trigonometry was perhaps too hard. But I still feel a vast reduction in their use would produce better mathematicians in the long run.

The video Dave sent was of Conrad Wolfram talking about why the future of maths should basically be entirely computational. Conrad feels that we need to stop teaching hand calculation and start teaching only computational mathematics. I feel that this would be an entirely wrong move. Computers can only do as they are told. If we are looking to prove a theorem generally, then we need to be able to hand calculate. Computers can check case after case, but this is not enough for a “proof”, as it is impossible to check an infinite amount of cases. A computer would not have been able to come up with mathematical induction or infinite decent.

A number of people responded along the lines of “What’s the point in learning how to do this when you can use a calculator?” This seems to me to be a ridiculous argument, like saying “Why learn to write when you can use a word processor?” or “Why learn to walk when you can use a mobility scooter?” If we head down that path is won’t be long before we are like the fat oafs in “Wall.E” (see this video) or even completely plugged in, a la “The Matrix”.

When teaching my further maths class numerical methods, I often have to field questions as to why we are doing this when if it were needed in the research world a computer would just do it. My answer is always simple, and always the same. “If no-one learns the theory, it will be forgotten, and no-one will be able to programme the computers to do it.”

No, calculators are not the enemy. But if the world becomes too reliant on them then we lose the skills we have built up over the centuries, we lose the ability to construct proofs for general cases, and we lose the beauty and the satisfaction one can get from solving a problem with nothing more than a pencil and paper.

Since the original post, I have realised that this is a wider issue than just calculators. Discussions with colleagues have highlighted that this problem occurs in other subjects when scaffolds are used. Thesauruses can lead to nonsensical sentences in English, for example. Scaffolds can also just mask a problem, pupils can get round something they cant do in lesson (ie subtract 197 from 360), but if it comes up on a non-calculator exam then they will not be able to obtain the correct answer.

Further reading:

From Mark Miller: Removing the cues

and Revision before redrafting (which includes the “greatest” sentence known to man: “a quantity of the most evil inscription is fashioned subsequently to a lexicon”.)

Rugby, and Newtonian Mechanics

Yesterday we went to Headingley Carnegie to watch a game of rugby union. The match was in the British and Irish cup between Leeds Carnegie and Bristol, Leeds won 24-19, if you’re interested in that sort of thing.

About ten minutes into the second half Leeds brought on a prop forward by the name of Sam Lockwood. Sam is affectionately know as “the human wrecking ball” amongst the Carnegie faithful. The nickname has nothing to do with Myley Cyrus, and everything to do with the way he hits the defensive line, which has a similar effect to a wrecking ball.

Anyway, the first time he got the ball he hit the line in his usual fashion and the impact carried him and three defenders a good ten yards. This prompted a conversation about how he would be the last person we would like to run at us. During the conversation we were discussing that it isn’t purely a size thing, Sam is a big bloke but there are others bigger who don’t hit the line anywhere near as hard. We noted that it was down to the combination of size and speed. He moves much faster than most players his size. It was while discussing a size speed trade off, wondering if a player had an optimal size to speed ratio (I think a negative correlation between someone’s mass and their top speed would be fair assumption), that I realised what I was actually discussing was momentum.

Momentum is the product of mass and velocity (algebraically speaking, p = mv). I taught momentum and impulse before Christmas and didn’t think to use rugby at all, which is strange because I watch a lot of it.

I then thought further about the sport, and it occurred to me that there are tons of opportunities to analyse the mechanics of a rugby match. There is the force a player runs at, their momentum, the impulse, the friction exerted be the studs. The clash mentioned above where Sam Lockwood took three defenders ten yards could form the basis for a whole lesson itself! On top of this there is a huge variety of projectile problems one could look at.

The question of forward passes is an interesting one, and the angle the ball leaves the hand, the velocity of the player, the velocity of the pass, if all these vectors were resolved we could get some interesting results.

I am extremely excited about teaching mechanics now, the next mechanics module I teach is M3, and I intend to find some interesting rugby based problems in the content!

Mersenne and his primes

On Thursday my further maths AS class and I arrived at the classroom to discover an interesting slide still displayed on the board from a previous lesson.

My colleague had been teaching a lesson on prime numbers to his year 9 class and the slide in question was about finding new primes, how much money you can earn if you do, why this is and the “Great Internet Mersenne Primes Search” (and its unfortunate acronym).

A discussion ensued about cryptography and the uses of primes. It then moved onto the mathematical monk himself and his work in number theory. In particular that he noticed that numbers of the form (2^p)-1,  where p is a prime, are usually prime. These Mersenne primes have fascinated me for years. How comes so many of them are primes? Why aren’t the all?!

The class were equally fascinated and we had a great discussion. We also managed to link it to a discussion we had had the previous lesson about p vs np, as trying to factorise (2^11)-1 is fairly difficult,  but it is really easy to check if 23 is a factor. The class wondered if they could set a computer to test massive numbers for prime factors. I explained that yes, you could, but it would take so long to check the massive numbers it would be worthless. So if they can find a way to do it quickly they could become very rich.

We lost around twenty minutes of matrices time, but we have plenty of time to make it up. I think all pupils left with a deeper and broader mathematical knowledge and a healthy thirst to know more- which is at least as important.

Newton, Raphson and Numerical Methods

I have recently taught Numerical Methods to my FP1 class. The topic itself is not one of my favourites, but I think there are definitely places that fun can be had when teaching it.

I have uploaded the notebook presentation and exported PowerPoint here.

Interval bisection is fairly dry, and there are always many questions on “Why do we have to do this, can’t a computer do it?” my usual answer here is that computers can do it, but someone needs to tell them how, so if they re going to be able to program computers to do this they need to understand it.

Linear interpolation is more fun, and it is fairly easy with a bright class and effective prompting to get them to work out how to do it themselves, which they always find satisfying.

The part of the topic I like the best is Newton-Raphson method. It is the bit which requires the most maths, rather than just an ability to substitute. I also find that it is the bit that generates most discussion. I pose questions such as “Why is it sometimes divergent?”, which encourages deeper thinking.

I also like to discuss Newton and Raphson, my students are well aware of Newton by the time the topic comes round. They have always, of course, heard the apple anecdote (be it true or not) and know about gravity. If I have taught them any calculus topics previously then they know about his great work in that field, his battle with Leibnitz and his fractious relationship with Hooke. But they don’t know a thing about Joseph Raphson.

Why would they? When I first taught the topic I spend ages trawling the internet to see if I could find a picture to include, but there was nothing. About all that is known of the poor fellow is the date he was admitted to the royal society, his signature, that he attended Jesus College and coined the word pantheism. What makes it worse is that the so-called “Newton-Raphson” method isn’t a method they came up with at all, but rather it’s entirely Raphson’s work! The reason for the name is that 50 years or so after Raphson published his method, Newton published his own more complex hence inferior version of the method. So his most notable achievement is a victim of Stigler’s Law.

A Book Review: The Simpsons and their mathematical secrets

Well, that was quick. I can hear you say. And you would be right. When I wrote the review of “Fermat’s Last Theorem” I didn’t realise how close to the end of this one I was!

I was extremely excited about this book. The last book I read by the author was my favourite book ever, it contained lots of maths and it was about The Simpsons, a show which I have long been a fan of. It didn’t disappoint.

I’ve spent the last 4 years struggling to find time to read, but I managed to get this book read in under a week, which itself shows how much I enjoyed it. My partner even commented last week that it must be good as I never seem to put it down!

The book itself is superb. Simon again takes the reader on a journey through the history of maths, he covers a couple of things mentioned in Fermat’s last theorem, but generally the maths topics covered are very different.

For decades, the writers of The Simpsons have been sneaking maths into their show, and Simon uses these jokes and references to take us on a journey through maths. He also explores why there are so many mathematicians on the writing team, and how the series has evolved.

While reading the book, I felt that we were merely scratching the surface of geekiness that is The Simpsons (and its sister show Futurama) and that Simon could in fact write an entire library of books around the same topic. The book is written  so that the maths gets increasingly more complex, and there are even 5 “exams” for the reader to sit. The clever part is that even though the episodes mentioned aren’t in chronological order, there is a real feeling of chronology while reading it, and anecdotes about the writers etc are blended seamlessly in with the maths. I have mentioned that there could be a library of books on the topic, and I would love there to be, but I cannot see this happening because the closing chapter brings the book to such a perfectly satisfying conclusion.

The maths sections of the book are well written, and are set out in a way that should be accessible for those with a decent high school mathematical background. Simon uses the appendices to set out more complicated maths, which is great, and appendix 5 especially took me some time to get my head around!

I really enjoyed this book, and would recommend it to anyone with a passing interest in maths, or The Simpsons.

You can see Simon on numberphile discussing the maths involved in the book, and his earlier title “Fermat’s Last Theorem” here

You can buy his books here

You can follow Simon on twitter