What should be in a maths class?

I have recently (well, in the past two years) read two very interesting essays on the teaching of mathematics. At first glance, they seem to be almost diametrically opposed, but I tend to find myself agreeing, overall, with the thrust of both. The essays are Paul Lockhart's A Mathematician's Lament and Conrad Wolfram's TED talk on mathematical education. A couple of quotes from each which provide a brief summary.

From Wolfram's talk (trancript available here):

I want to see a completely renewed, changed math curriculum built from the ground up, based on computers being being there, computers that are now ubiquitous almost. calculating machines are everywhere and will be completely everywhere in a small number of years. Now I'm not even sure if we should brand the subject as math, but what I am sure is it's the mainstream subject of the future.

From Lockhart's Lament:

The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

So, Lockhart thinks we should be teaching mathematics as an art form, and Wolfram thinks we should be introducing more computers into mathematics lessons so that people can concentrate on doing the bits that are actually useful. Which of them is right? Well... both, but mostly Wolfram.

The question we have to ask ourselves before we can even begin to compare the two essays is why do we teach mathematics at all? So far as I can see, the only sensible justification for having mathematics as a subject that everyone in the world should be taught up to a relatively high level is because it's useful. You can't do physics, or engineering, or any sort of science, or do derivatives trading, or even decide which mortgage to get, without knowing quite a lot of mathematics. For this reason, everyone should be taught the basic mathematics that they need to know in order to do these things (or the basics they need to know in order to learn the specific maths they wnat to use).

Note that one corollary of this mode of thinking is that most of the mathematics you learn probably shouldn't be taught in a maths class. It's much easier to learn how to get from acceleration to speed than it is to learn how to differentiate a function. Yes, it is useful to then point out the possible generalisations (getting from acceleration to speed is the same as getting from jerk to acceleration) but I don't see any reason why these topics can't be introduced concretely. I personally have serious trouble doing any calculus that I can't do using physical intuition, and I think it's fair to say that I am an above-average student when it comes to learning maths. Calculus should be taught when you're doing engineering, statistics should be taught when you're analysing the results of experiments, graph theory should be taught when you're trying to solve scheduling problems.

Lockhart, on the other hand, seems to think that we should be learning mathematics because, essentially, mathematics is awesome. I happen to agree with him that mathematics is an exceptionally beautiful art form. I'm happy to sit back and bask in the glory of Cantor's diagonalisation argument, or the ingenuity of Karp's reductions between NP problems, but I'm not sure that I'm willing to contend that everyone should be forced to. Yes, if you want to be a mathematician you have to learn that mathematics is actually an art, but most people who study mathematics don't want to be mathematicians, and most people who study mathematics *shouldn't* want to be mathematicians. For these people, learning about the art of mathematics is little more than an intellectual curiosity, on a par with learning about Titian or Shakespeare.

In other words, Lockhart is right, inasmuch as we want people to study mathematics for it's own sake. Wolfram is right, inasmuch as we want people to study mathematics because it's useful.

Now, I happen to tend strongly towards the idea that the only things we should be teaching in schools are things which are potentially useful, but that obviously isn't the prevailing wisdom - everyone in this country is still forced to do an English Literature GCSE. Lockhart-style mathematics is a perfectly good substitute for art class, or critical theory. Wolfram's mathematics is a necessary prerequisite for doing just about anything else.

The Story of a Proof

I started writing this blogpost about 6 months ago after reading an interesting post by RJ Lipton on why mathematicians prove things. I can't find that post any more (maybe it wasn't by RJ Lipton...?), but it was general speculation on why mathematicians are actually interested in formal proofs of their theorems.

A recent experience of mine in my own research seems particularly relevant here, and I'm going to try and explain what happened (this will, at the very least, satisfy my promise to Michael Brough at about the same time of getting some more serious mathematics into my blog). This is pretty much the first time I've ever tried to explain something technical in a non-technical setting, so let's see how it goes...

Click here to read the rest

Writing Mathematics: Formalism obscures intuition?

I have read quite a large percentage of the books that are currently widely available on the topic of writing mathematics, and several that aren't. I've also been involved in teaching the Mathematical Writing course run at the QMUL. There is one serious issue that I've encountered: the main problem seems to be that most people start with the assumption that you already know everything that you're going to write, and the basic structure of your proof, but don't know the best order to put the words on the paper. In my experience, this is far from true.

I'm currently writing up my thesis, and have several new ideas which I've never written down in complete formal detail before. This process is difficult. The ideas in are generally simple enough that I think I could explain them to reasonably bright 15 year old in about 10 minutes given a piece of paper, but sufficiently abstract that to write them down formally has involved pages of writing that even I barely understand. These are proofs that I can (and have) run through in the pub with non-experts on the back of a napkin, and they translate into 10 pages of symbols that even I can barely follow. (I'm working on a post containing an example, but it will require drawing some pictures, so I'll probably get round to it some time over the weekend).

It is all very well saying that one should write an informal verbal summary of what you are going to do before you do it, but when the formalism is so far removed from the key idea, this becomes difficult. Also, when you are writing a maths paper, you have to make damn sure that the "summary" is still technically accurate. There can be no hand-wavey 'look, it just works' and (importantly) no interaction with the audience - you dont' know if they 'get it', so you have to put down everything it might take for them to get it. In my experience, the heavy formalism has often meant that the proofs I write down end up so complicated that I'm not even sure I would 'get' them if I had to pick up the ideas directly from reading the papers. The formalism masks the intuition, and the intuition isn't quite formal enough to be appropriate for the paper.

This is an interesting topic. It's one that several people have no doubt spent a lot of time thinking about, but it's not one that seems to be discussed. Even in a mathematical writing course, the tendency is to focus on technicalities. To be fair, this is usually adequate for undergraduates, and you do need to get the technical stuff right even when writing down simple ideas, but this is something that needs addressing: exposition of mathematics is a difficult skill, and probably one that there's not enough focus on in training academics.

Maybe not everyone has this problem - maybe not everyone has elementary ideas which don't translate nicely into formalisms - maybe their ideas are inherently more technical, or maybe they're just better at finding formalisms than I am - but I do find it interesting that it's happened to me almost every time I've tried to write out a formal version of a proof that I've created myself.

PS - the vast majority of this post was written over a year ago - the posting of it was sparked by having a discussion with Andy in which he mentioned that he is currently having exactly the same problem, and by me finally figuring out how to access the drafts of my old posts...

Philosophy of Education

I just came across this on Rosemary Bailey's website. While I think the ideals are nice, I think in practice I disagree with just about all of it:

My vision of a university was succintly described by David L. King, writing in The Times Higher on 9 April 2004. It is:

• “the belief in a community of scholars and not a confederacy of self-seekers;
• the idea of openness and not ownership;
• the professor as a pursuer of truth and not an entrepreneur;
• the student as an acolyte whose preferences are to be formed, not a consumer whose preferences are to be satisfied.”

I believe that university students should be able to be confident that they are being taught by people who are immersed in the subject in other ways than teaching. I collaborate with a range of scientists on the design of their experiments and the analysis of their data, so I teach Statistics. I still prove theorems in Combinatorics and Algebra, so I also teach those subjects.

I've no idea what a "confederacy of self-seekers" is, so I'm not going to address that point. I don't think anyone believes in a professor as an entrepreneur, one thing that I think people can sensibly expect professors to be, at least in the current system, is teachers. Incidentally, I happen to think that Rosemary is a very good teacher, and one who thinks a lot about doing the best for her students, so I hope she won't be offended by any of this.

The problem I have mostly is with the last point in the bulleted list: "the student as an acolyte whose preferences are to be formed, not a consumer whose preferences are to be satisfied.". Now, that might be a nice model for a university if the point of going to university was to learn about the subject you were studying. The problem is that for the vast majority of people who go to university currently this just isn't the case. They go because in order to get the better paid jobs you have to have a little piece of paper which said that you went to university. They go to university to get this piece of paper and, and I think this is the important part. This system is not the students' fault.

The students in our department are not there because they are particularly interested in mathematics and (and I think this is something that academics sometime struggle with, and I will almost certainly get round to making a separate post about this one day) the vast majority of them are not interested in pursuing a career in academia. The overwhelming majority of students are at university to get a degree because it will help them get a better job. And they are right. In the world we happen to live in people do need degrees to get good jobs, and universities are there to supply these degrees, and to make sure that the degrees are a useful signal of ability for employers.

It might be that the vision of the university described by David King is a good way to train people who want to obtain the sort of detailed knowledge of an academic subject one needs to do research, but it absolutely is not the case that 50% of the population want, or need to obtain that sort of detailed knowledge of any subject. So, while we have a system where something pushing 50% of young people go to university (and, remember, pay the fees that pay the salaries of the people who work at the university).

Most people are there to get a degree because getting a degree is necessary prerequisite to doing whatever it is they actually want to do. Like it or not, the job of the teaching staff in a university is to help them get that degree. If you don't like it, by all means do your best to change it, but while you're accepting money from the university as it exists now, I'm afraid I think you are morally obligated to teach in the way that is appropriate for the majority of students who are there now. More to the point, if you don't like the current system, you're going to have to figure out another way to pay for researchers - at the moment research is cross-subsidised by an awful lot of socially innefficient teaching.

I personally would love to live in a world where universities could all be like Cambridge and Oxford, and where the vast signalling game that is university education as it exists now didn't exist. I admit that this world would probably be a world in which less "blue skies" research got done, and I find it hard to believe that would be a particularly bad thing.

This post is getting long, and I'm not sure my thoughts on the last paragraph of Rosemary's statement are fully-formed, so I won't write much about that for now. I will however finish with one thought that I have on the issue: it would be a remarkable coincidence if the best teachers in the world also happened to be the people who were best at doing original research. I happen to think that I personally am quite a lot better at the former than the latter. I'm not going to give an example of someone whose abilities are skewed in the other direction, but I'm almost certain everyone who has studied or worked at a university already has one in mind.

Security Theatre

Every week this term, on both Tuesdays and Thursdays, I teach a class in the Francis Bancroft building on our university campus. Twice so far this year (so that's 20% of the times I've been into the building), there has been a security guard on the door checking the ID of every person going in.

Now, whatever the merits of having someone check the ID of everyone entering a university building, whatever the point of this exercise is supposed to be, surely it is rendered *entirely* useless if you don't do it every day. If the purpose is to stop people stealing things (there are plenty of old computers which must be worth tens of pounds on the black market) then surely they'll just come in and steal them on the days when you don't do the checks. If the point is... well, I actually have no idea what else the point could be, it's a university campus, not a Government Intelligence building.

There can be literally no point whatsoever in having these security checks unless they're done regularly (in much the same way as there can be no point whatsoever having full-body scanners in airports unless you also perform cavity searches). Security which is this easy to circumvent is not security, it's theatre.

Ten Films in Ten Days

I've recently taking to going to the cinema as a way to pass time during the day. When I tell some people that I like to go to the cinema alone, they look at me as if I have a third arm, but really, going to the cinema is a solitary activity: talking is actively frowned upon! Anyway, I found myself going quite a lot, and in order to amortize the cost, I decided to get myself a Cineworld Unlimited card. This card entitles the bearer to unlimited cinema, as one might expect, for only £13.50 a month. It therefore pays for itself if you go twice a month - I have regularly been going more than twice a week. Bargain.

One catch, you can only get one of these cards by signing up for a twelve-month contract, and I have almost no idea what I'll be doing in three months time, never mind twelve. I therefore made the completely rational and considered decision to get the card anyway, and let future me worry about paying for it. I then figured that: hey, I only need to go about 20 times and it's paid for the entire year. I then figured: hey, why not go 20 times in the first 20 days, then I'll have 345 days worth of free cinema!

Yes, I realise that this doesn't actually make any sense, I do understand the concept of sunk costs, but it sounded like it might be fun, and I'm all for fun. So, in the past 10 days I have been to the cinema 10 times, to see 10 different films. After each one I took brief notes: my reviews come after the fold, along with my reflections on the somewhat ridiculous project.

By the way, for those of you who know me, and know that I'm currently supposed to be writing my thesis, and are wondering what impact this has had on my progress, well I can say for certain that I've not made any less progress in the past 10 days that I did in the previous 10. For those of you who know me a little better, no, that isn't actually a trivial statement.

The Dilemma

I mostly chose this film because it was on at a convenient time (something of a theme through the rest of the week, and my interpretation of "convenient" might have started to get slightly broader). It was... ok. The only bits anyone actually laughed at were the slapstick, and I couldn't quite genuinely believe in some of the really stupid things Vince Vaughan's character did in the name of keeping the plot moving. His relationship with his girlfriend in particular just wasn't believable, and his reasons for lying to her remain utterly unfathomable.

Tangled (3D)

Standard Disney. There isn't really that much more to say. The wicked stepmother (not technically an accurate description, but I can't be bothered to think of a better one) reminded me of Ursula, the hero reminded me of any other Disney Hero, and Rapunzel reminded me of any other Disney Heroine. It was no Aladdin, but I would imagine that everyone's favourite Disney film is just whichever one they happened to see when they were at the age to appreciate it best. On the other hand, it was a lot more fun than Toy Story III.

One thing I did notice is that the only actual good 3D effects came in the adverts at the beginning. In particular, the advert for the 3D Panasonic TV looks awesome - why don't the people who make the films hire the people who do the adverts to do their effects for them?

Black Swan

Surreal ballet psycho-drama. Probably the best surreal ballet surreal ballet psycho-drama I've ever seen, but frankly a little too weird for my taste. Natalie Portman is good, but her character is the sort of part people play when the want to win an award, slightly crazy, which also sums up the rest of the film. Also, while I'm not capable of telling the difference, I can imagine real ballerinas being slightly chagrined at the idea that Portman could possibly be dancing well enough to get the lead part in Swan Lake - did she use a stunt double who could do ballet?

The King's Speech

Colin Firth goes all out to win an Oscar as King George VI, struggling to overcome his speech impediment. All I can say about this film is that you won't be disappointed if you go see it. On the other hand, I also doubt very much if you'll be pleasantly surprised. The best part about it, in my opinion, is the clever title. It's well-acted, well-scripted, and generally well put together drama. If you like this sort of thing, you'll like it. I'm gradually coming to realise that this sort of thing isn't really what I enjoy at the cinema.

Morning Glory

Relatively low budget Rom-Com in which Harrison Ford plays a grumpy old TV reporter who's forced to do breakfast television working for Rachel MacAdams as the bright-young-thing executive producer. Not the sort of thing I would ever admit to going to see normally, as a single heterosexual male, and, quite simply, by far the most fun I've had in the cinema in the entire 10 days! I'm not sure if this was just an exceptionally well-done romantic comedy, or if I just really enjoy romantic comedies, but it was light, funny, actually genuinely touching (I really did care what happened at the end... contrast this with the supposedly "sublime" Brighton Rock). The slapstick was funny, but didn't distract from the plot, and the characters were enormously believable.

Inasmuch as a film like this can be accused of having "themes", one of the big themes was "news vs. entertainment", with Harrison Ford's character constantly griping at having to do lowest-common-denominator stuff when he had won Pullitzers. Well, the feel-good ending (I don't think that counts as a spoiler) tried to convince us that "entertainment" has its place too. The film as a whole couldn't be a better poster-child for that.

The Mechanic

Mindless violence and action scenes courtesy of Jason Statham. One thing that did bother me was the complete lack of any sort of interesting plot. I know that having no sort of interesting plot is pretty much a cliché for this sort of film, but I generally enjoy the sort of ridiculously contrived plot twists that these films have to offer - this one didn't have any that I can even remember.

A Little Bit of Heaven

Another romantic comedy. There is pretty much nothing to this film, and for some reason, which I can't yet quantify (I'm clearly not quite ready to take over from Roger Ebert) I didn't enjoy it nearly as much as Morning Glory. Kate Hudson (who, incidentally, doesn't look at all like a Hollywood Actress to me - kudos) is lively enough as the main character. She is diagnosed with a terminal illness and then, for some utterly unfathomable reason, Whoopi Goldberg appears as God and grants her three wishes. The God bit really does feel as though it's bolted on to what would otherwise be a perfectly serviceable rom-com, which I think might have been part of what bothered me.

Brighton Rock

The posters claim this film is "wonderful" and "spellbinding", if I hadn't been sat in the middle of a row, I would have walked out halfway through. It's gruelling, depressing, and the characters just aren't interesting enough for it to be worth it. First there's Pinkie, who in the space of less than ten minutes (on-screen, but only a couple of days in his world) goes from being afraid to pull a knife on someone to beating the guy to death with his bare hands, for no apparent reason (I've no idea if Graham Green does this transition better in the book). Then there's his pathetic, whining, annoying little girlfriend who, frankly, needs to grow a pair - it's pretty hard to feel sorry for her when you can't possibly begin to empathise with her. There's also the girlfriend's boss who goes from being generally aloof to running around town doing whatever she can for the girl, again, I'm not quite sure why.

All in all, I don't think this film was well done, but I don't think this is it's major flaw. I think it's major flaw is that films like this just aren't fun. I once had a serious argument with a French teacher about a French film called La Vie de Jesus. She claimed the film had an important message to get across, I claimed that that was all very well, but it was so damned depressing that no-one would manage to watch the whole thing to find out what the message was. I'm not sure Brighton Rock has a serious message, but it certainly suffers from the latter problem. Do not watch.

How do you know?

About thirty seconds, ago, I was about to press post. I counted the films in my list, and noticed that I only had 9. I looked back at my notebook in which I've been keeping notes on the films I've seen, and found this entry for "How do you know?" (rom-com with Reese Witherspoon, Paul Rudd and Owen Wilson): "meh".

I think that pretty much sums it up. It was an entirely unmemorable but not at all unpleasant way to pass a couple of hours. Rudd is pretty good as the vaguely believable not-too-perfect male lead, and the whole thing has excellent production values, but, I managed to forget I'd seen it, and couldn't manage any more than "meh" even immediately after coming out - you can certainly afford to wait for the DVD.

The Fighter

My dad is down for two days, which is part of the reason I'm having a hiatus today (the other is that I've been doing "maths-busking" training all afternoon). He has his own Unlimited card, and had been warned in advance that he would be being dragged to see one of the films that I hadn't yet seen. In the end, time constraints meant that we chose to see "The Fighter" at the O2, despite dad's misgivings about watching a boxing film.

I'll leave the first part of the review to him "Well, I'm glad you made me go see that, it was one of the best films I've seen in ages". Later on "I even enjoyed the bits with the fighting". It's basically a character film, following the lives of two brothers. I'm not quite sure what the name is for the genre: filmed almost like a documentary, and based on a true story, but with actors playing the real people: maybe "reconstruction"? The characters were real. You actually care what happens to them. Christian Bale deserves to be nominated for whatever he got nominated for for this film, not only does the character he portrays have exactly the right mix of charisma and pathos for the "boxer turned crack addict" that he's supposed to be, as the end credits roll, we are shown a brief clip of the real Dicky Ward, and Bale has him down to a t.

The characters were very real and the story was inspiring. Definitely my second favourite of the films I've seen so far, but I'm willing to admit that by most people's standards, it would probably be rated as the best.

A few final thoughts

You may have noticed that I claimed earlier I'd be seeing 20 films in 20 days, and wondering why this post isn't entitled "20 films in 20 days: part 1". Well, I've revised my goal downwards - I still aim to have seen twenty by the end of the month, but once a day is going to start to get in the way of me having a life if I try to keep it up.

One thing I've learned from the past ten day is that whatever method I use to predict in advance which films I'll enjoy is currently seriously broken. There is no way I would have seen either The Fighter or Morning Glory if I hadn't been doing this, and they stand out as streets ahead of any of the films I actually did see.

I've also firmed up my belief (which I guess I already had) that I really shouldn't bother reading reviews of films before I go see them - the sort of things I enjoy at the cinema just aren't the sort of things people who are paid by the word to write about films claim to enjoy.

There are several films I like the look of coming out in the near future - I'm sure there will also be some excellent ones that I don't like the look of. Now to figure out how to get to see them.

Little Joys on the Tube

Going the right way, even when the signs try to point you the wrong way (not acceptable at rush hour, but can save you a *lot* of time during the rest of the day).

Having a conversation, and then realising that you're providing amusement for everyone else on the carriage, as otherwise everyone just sits there in complete silence.

Finding exactly the right place to stand on the platform so that you don't have to walk at all when you get to your destination (and pondering where the optimal place to stand is when going to an unfamiliar destination station).

Waiting two minutes for the next train, and getting a seat instead of piling into a crowded carriage.

The sheer inanity of the announcements: "we're being held at a red signal", really? what colour do you use for "go"?

Instrumental Variables

Marginal Revolution linked recently to this paper which claims that overuse of Instrumental Variables (IVs) in econometric studies has led to them being less useful. So far as I can tell, the argument given in the paper is exactly backwards... what am I missing?

For the uninitiated, an IV is a variable which is correlated with the thing we want to study but is unequivocally not caused by it. Let's say we want to study the effect of x on y, then an instrumental variable z is useful if its only effect on y is through its effect on x. To use the language of Pearl's causal graphs, this is equivalent to saying that every path in the causal graph from z to y passes through x.

For example, suppose we want to measure the effect of a change in the price of apple on the demand for apples. It may seem difficult to do this directly, as an increase in demand is likely to lead to an increase in supply, and so price is affected by demand. A potential instrumental variable is the weather. If the the weather is favourable for growing apples then more apples will end up being grown, and this is presumably independent of demand - the increase in apples will have a predictable increase in the supply (and therefore the price) and we can measure the effect on demand.

Now, the authors of this paper claim that:

A Tragedy of the Commons has led to overuse of instrumental variables anda depletion of the actual stock of valid instruments for all econometricians. Each time an instrumental variable is shown to work in one study, that result automatically generates a latent variable problem in every other study that has used or will use the same instrumental variables, or another correlated with it, in a similar context. We see no solution to this. Useful instrumental variables are, we fear, going the way of Atlantic Cod.

As I said, I think this is exactly backwards. It is not the fact that new papers are produced which use these instrumental variables in a new context which introduces the latency problem: it reveals a latency problem which already existed. The previous studies were already invalid. The new studies just reveal the fact.

E.g. Imagine that people buy more apples when they have high levels of vitamin D in their blood (because apples are a substitute for fish oil). Then you have to correct for the effect of the Sun on vitamin D levels when you're using the sun as a proxy for apple demand. The problem here though, is that the weather conditions already weren't a good IV for demand in apples. The fact that a new study appears to demonstrate this is not a "Tragedy of the Commons" in any meaningful sense - the study has not been made any worse, we've just found out that it was bad.

On the other hand, Alex Tabarrok, who is more of an economist than I am, appears to be taking the paper vaguely seriously, and not to have noticed this. As I said before: am I missing something?

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Super Saver Delivery - classic price discrimination?

I've just ordered a book from Amazon. I selected the Super Saver Delivery option, and Amazon informed me that I should expect this book before February 9th. Now, I'm not exactly a stranger to ordering books from Amazon. I spent in the region of £500 last year, and I'm very nearly 100% confident that I will receive this book before the end of the week (that's February 5 for those who are keeping track). Why does Amazon do this? The two delivery options "Super Saver Delivery" and "Standard Delivery" are, as far as I can tell, completely indistinguishable in terms of service. The only difference being that one pays something in the region of £2 per order for the latter, and nothing at all for the former.

I can think of several reasons why they would exaggerate times: it's a pleasant surprise when your book arrives before it is supposed to (but then why not claim that all deliveries will take "up to 2 weeks"; it also covers them in case delivery takes longer than they might expect (but, again, why not give an even more exaggerated time). I can also think of reasons why they might want to introduce a "second-class" delivery service, in order to justify charging for their "first class" service (ie, the normal delivery rates that one would expect them to charge anyway). Similar to the way that second class customers on trains are deliberately inconvenienced in order to make first class more attractive. However I don't think this is what's going on here.

I think Amazon is indulging in some good old-fashioned price discrimination. As I said earlier, I'm a pretty regular customer; I order something on the order of £50 worth of books a month. As such, I know how long it takes for my books to be delivered, and I'm not even vaguely tempted to press the "standard delivery" option when it comes to the payment page. I'm exactly the sort of person who is most price-sensitive when it comes to deliveries, and I'm in a position to know that I can get the same service more cheaply.

Contrast Dear Aunt Doris, who orders a book off Amazon once a year for her nephew's Christmas present. She needs to get her delivery when she wants it, and she has no idea that "standard delivery" is the same as "super saver delivery". I mean, why would it be? That would just be crazy, surely? So Dear Aunt Doris pays the extra £2 for her one book a year, because she's not the sort of person who gets the chance to notice that she's paying for exactly the same service she could have gotten for free.

This is almost textbook price discrimination. You charge a lower price for the same service to those customers who are most likely to go look elsewhere, and most likely to be in a position to know exactly how much the service is worth. It's fun noticing phenomena you've read about in books in real life. I vaguely wonder if this theory of Amazon delivery pricing is testable (well, it obviously is if you work for Amazon...) but either way, I'm fairly confident it explains what's going on.

What I'm reading now

Once upon a time, when the Sparknotes message boards still existed* they had one called "what we're reading now" where people were supposed to write ongoing reviews of the books they were currently reading. What always surprised me what the number of people who were reading one book at a time! So, here's a list of the books that I currently haven't finished, but do intend to finish (it's probably incomplete, as it only includes ones that are by the side of my bed right now).

A Week in December - Sebastian Faulks

One of those "small world" type novels, small snatches of the lives of a variety of people living in London in late 2007. Fairly well-observed, and it's probably going somewhere. I can't imagine feeling as much for the characters in this as I did for the characters in Birdsong, but you can't expect anyone to write Birdsong twice. One serious gripe: the kids have accounts on YourPlace where they get "jabbed" by their friends who can post messages on their "doormats"; there is a virtual world called Parallax where the inhabitants refer to the real word as TL and trade in Vajos; there's a major bank about to collapse called "Allied National" and the lead singer of Girls from Behind is one of the leads on a reality show called It's Madness. It's slightly jarring living in this parallel world... even Al Qaeda has been given a different name. This make suspension of disbelief slightly harder, and I've pretty much no idea why you would choose to do it.

The News Where You Are - Catherine O'Flynn

Another novel set in the modern world (for some reason that's the only sort of novel I've been able to get into lately). To be honest, I've only read the first dozen or so pages. Has a fairly light touch, and I think it'll be an easy-going distraction. I doubt very much it will be any more than that, but I'm willing to be surprised.

Bridge The Silver Way - David Silver

A collection of bridge stories. My favourite books of bridge stories are the Chthonic books by Daniel Kleinmann. Unfortunately, there are no more of them for me to read, so I'm trying some others. These are readable, but not spectacularly good. I wouldn't recommend them.

The Time Paradox - Philip Zimbardo

This has been in the pile for a while. It's a book about time by the guy who did the Stanford Prison Experiment, and who has since done a lot of work on the psychology of time. It seems interesting, and seems to agree with . I want to know the contents of this book, but for some reason am struggling to actually read it. Will get there eventually.

The 4-Hour Body - Tim Ferris

Having read the 4-hour work week, agreed with most of it and subsequently ignored the vast majority of its advice, I couldn't wait to get my hands on this one. Fully of various reasonably well-researched tips and tricks about diet, exercise, sleep, etc. A lot of the things in this actually seem more actionable without going to too much effort than those in his previous book. I'll no doubt be playing with a few of them over the next few months. If any of them seem interesting/worthwhile/dangerous I'll probably blog about them at some point.

God is Not Great - Christopher Hitchens

Got this for Christmas, have been dipping in and out of it recently. Hitchens intellect is scary, he's a proper old-school Oxford academic (I just looked up to check that he actually did go to Oxford - I've no idea if I'd ever known that for certain before, it just feels as though he did). It's fun to read just for the sheer amount of stuff the man knows - his impressive takedown of the idea that Kosher food laws are in some way hygiene-related is the last chapter I read. Basically, I'm just basking in erudition. I'm sure I'll learn a few things too.

Priceless - William Poundstone

Yet another popular book about behavioural psychology. Extremely well-written, refers to several experiments and theories that I've never heard of, and gives the best intuitive explanation of Prospect Theory I've ever seen. I have actually finished this book, but I keep dipping into it because it was, simply, the best book I've read on one of my favourite subjects. Well-written, well-explained, well-researched, the chapters are a nice length to dip in and out of. I'm trying to think of something I didn't like about this book, but I'm struggling. I sort of which I'd decided to give the books in this post star ratings, so I could give this 6 out of 5.

Causality - Judea Pearl

After I wrote my post on Simpson's Paradox on Less Wrong, I decided I really should learn more about the theory of causality that I casually reference at the end. Partly because I kinda feel I should, but also because it just seems awesome. It's actual mathematics though, not exactly bedtime reading, although it hasn't been too tough so far.

Expert Political Judgement - Philip Tetlock

Tetlock's study is pretty famous. He basically concludes that "foxes" do better at predicting political events than "hedgehogs". To quote wikipedia: foxes "draw on a wide variety of experiences and believe the world cannot be boiled down to a single idea" hedgehogs "view the world through the lens of a single defining idea". I'm not sure how much more there is to get out of reading the whole book than what I just wrote in the paragraph above, but I've started so I'll finish. I've also promised Jess a summary - does that count? Or do I need to do it more detail?

So, there you go. There are no doubt others lying around that aren't in this pile next to my laptop. The pile is usually about this big. I can't understand how some people can read one book at once - what do you do when you're not in the mood for that one? (I sometimes even struggle to decide on one book to take on the Tube...). If I remember, this may become an occasional series.