Should we teach maths like music?

"High School Geometry: the instrument of the devil"

Via Scott Aaronson, I read this glorious rant by mathemetician Paul Lockart about maths teaching in schools. 

Now, I don't agree with most of it, and hope to discuss why in a future post, but it is such a pleasure to read a critique that isn't by a laudator temporis acti who took his O-levels in the 1970s and has been sulking about it ever since. This bit, in particular, is splendid:

"“The area of a triangle is equal to one-half its base times its height.”  Students are asked to 

memorize this formula and then “apply” it over and over in the “exercises.”  Gone is the thrill, 

the joy, even the pain and frustration of the creative act.  There is not even a problem anymore.  

The question has been asked and answered at the same time— there is nothing left for the 

student to do. 

"

Much as Stephen Fry is the stupid person's idea of what a clever person looks like, an exam that require the rote memorization of  endless formulae before plugging in a series of random numbers is what some non-physicists  think a physics paper should looks like.

The difference between physics and economics

Graphiness

Straight lines, baby! There was no need to take 3 data points, or it might have spoilt the fit.

Oh well, at least it wasn't bloody Widdecombe.

Great minds think alike

And the pounds took care of themselves

Via Alex at the Fistful of Euros, I learn that John Graham-Cumming, Slayer of Spam, has had similar ideas regarding Benford's law. He has looked at the figures in the forms submitted by the chancellor, the prime minister, and Harriet Harman. The PM and HH expenses follow the distribution, but the chancellor's do not. There are rather more 3's and 4's than you would expect by chance. He has identified the source of these: the chancellor claims exactly £300 for food every month, and reqularly claims £45 for his telephone bill.

He also observes that Hazel Blears has submitted claims for whole number of pounds. Hmm.

Were I to fabricate my expenses, I -or at least, whichever of my sons was working for me full-time whilst also a full-time student  - should be sure to add an appropriate and possibly random number of pennies to each of my more creative claims, if only to add an air of vermislitude to an otherwise bald and unconvincing narrative. [Benford's law probably wouldn't be able to pick up divergences south of the decimal point, as the distribution tends to uniform as the order of the digit increases]

On the other hand, you are allowed to round down in (some of) the boxes on your self-assesment, so it might well be legitimate to do so on an expenses form. 

In any case, I think the best test is reciepted expenses vs. unreciepted expenses. As soon as we see some tabulation at the Guardian project, it'll be time to break out the Benford's.

Benford's law and The Commons

A modest proposal

Have you ever tried to make up numbers? I mean, to write a series of random numbers on a bit of paper? It's a problem faced by fraudsters: how to fabricate a series of plausible data. 

Consider the digits. I had always supposed that, in a given series of numbers, you'd expect the likelihood of a number starting with a 1, a 2, a3,..., or a 9 would be about equal (for a large enough sample of numbers) at ~11%. 

I supposed erroneously. 

Actually, for a surprisingly wide range of data types -from the lengths of rivers in an atlas to amounts of money in an account- numbers beginning with a '1' are much more common, appearing about 30% of the time. Numbers beginning with a '2' occur about 18% of the time, and higher digits with decreasing frequency. This is called Benford's law.

This remarkable result has been put to use in fraud detection, by using it to pick out suspicious sets of accounts for further investigation.

Now, there's a few caveats: it doesn't work for all data - sequentially assigned numbers like bank account numbers, for instance. However, the MPs expenses scandal (via the Guardian data blog) provides a natural test for this, as MPs expenses are all either:

(a) amounts over £250, which had to be supported by a receipt, or

(b) amounts under £250, which didn't. 

Will both sets of figures follow Benford's law? Will neither? Or will most sets of expenses (a) follow the law, while the un-receipted expenses (b) won't? This might raise some interesting questions. 

Does my privilege look big in this?

It's not about you, either, it's about the facts

[UPDATE: PJ has some comments from a different perspective in the comments]

 Laurie Penny, who wrote this fantastic article, has a post up defending Rowenna Davis' string of untruths that I discussed last week. I do not find it persuasive. She writes:

Dear white, straight guys: it’s not about you.

No, really, listen up. I have been stunned this week by the cybersquall that has erupted over Rowenna Davis’ Guardian article, entitled – although not by her – ‘Stupid White Heterosexual Male’. The article was well written, reasonable, and managed to make points about equality without getting personal, which is unsurprising, as Rowenna Davis is at the tender age of 24 one of the finest and most ethical journalists I’ve ever had the privilege to meet. But the piece got almost as many negative comments as Charlie Brooker’s denouncement of the BNP in the same paper got supportive ones – all because Davis had the temerity to suggest that perhaps white, heterosexual males might not actually need their own anti-discrimination officer at Oxford University of all places (45% private school students, almost entirely white and with a tenacious male bias in finals marks), especially not when Andrew Lowe’s policies included ‘to replace St Anne's college crèche with a finishing school, ban women from the library and save money by getting female students to serve food in halls instead of kitchen staff.’ - [source]

 

You can, and should, read the whole thing here. I was particularly impressed by the claim:

No, really. You might not think that you personally, sitting behind your computer, reading this rant and getting pissy, are part of the problem -but you are. The people who attacked Rowenna Davis’ on-the-money article with such bile and vitriol are part of the problem, even though many of those are the very same hands-up-harries who were the first to condemn the BNP.

Because there is a heartbeat’s space between the blind stupid rage of otherwise sensible people who felt hard done by reading that article and the creeping influence of right-wing policymakers in parliament. There is a heartbeat’s space between the growing tide of otherwise non-idiotic white male resentment in this country and the breathtakingly idiotic racist, homophobic and misogynistic logic with which we have just sent two far-right representatives to the European Parliament. And if you are not prepared to step up, own your privilege and be part of the solution, then, my darlings, you are going to become part of the problem.

Of course, I would have been more impressed to see some actual evidence for the claim - which brings us to the problem with both Davis' article and Penny's defense. 

My objection to Davis article is not "because Davis had the temerity to suggest that perhaps white, heterosexual males might not actually need their own anti-discrimination officer at Oxford University of all places", but because Davis makes a number of statements that are untrue. 

1) Davis claimed that Oxford admitted 5 black students last year. This is untrue. An underestimate by a factor of 9, the correct figure can be found in the document "Undergraduate Admissions Statistics 2008", page 5, table 5, [source]. Happily, Davis has corrected this claim, although the fact she thought it was remotely plausible raises profound, not to say disturbing, questions about her own privilege.  

2) Davis claims that "The only thing harder than spotting the black kid in my college photo was trying to find a woman on my reading list.". This is untrue. You can  readily falsify this for yourself by reading the politics department PPE reading list, as I thought - clearly rather optimistically - that Davis would have done. [In Davis 2007 article (about institutional sexism  at Oxford) in the Times Higher Education supplement she claimed "trying to find a woman on my reading list was analogous to playing "Where's Wally"?" - [source]]

3) Davis claims "Class: Despite over 90% of the country being state educated, just 55% of Oxbridge students come from state schools. New figures suggest that these class divides are getting worse, not better. ". This is untrue. As I wrote in my original comments on Davis article, "it is instructive to compare the results from 2006(2007) ([source], page 3 table 1), when 47.1%(46.8%) came from state schools. Certainly in the short term, things seem to be improving. " . I also wrote "But 98.4% of candidates who are offered pre-qualification places achieve AAA ("Undergraduate admissions statistics 2008"  page 4, table 3). As 28.6% of A grades are awarded to pupils at Independent Schools [source] and  33.8% of Oxford applicants come from Independent schools  ("Undergraduate admissions statistics 2008",page 2, table 1), perhaps the over-representation is not entirely Oxford's fault.".  I hope all numerate readers will appreciate why this is relevant.

These are matters of fact, rather that "privilege". The statements Davis has made do not correspond to reality. They are untrue. It matters not at all how white, male, privileged, disadvantaged or publicly-schooled the person who makes these statements is: they are untrue. 

I move to statements who are not untrue, but are simply peculiar. They display a mixture of petty bigotry and misunderstanding of a university.

4a) Davis complains of "competitive tutorials". The whole reason/excuse for the college fee - the 3000 pounds a year extra that is lavished on Oxbridge students - is the tutorials system. Every other university in the country can't afford to provide two-to-one or one-on-one tuition , although some try by cannibalising their research budget. If Davis was so unsuited to the tutorial system - and I agree it is not for everyone, and I am agnostic on the question of whether it is the "best" system - there are 107 universities she could have attended.   It is unfortunate to attend Oxford for 3 years and not realize that "competitive tutorials" are the nominal reason the taxpayer stumps up the additional three grand. Doubly so, to enter a privilege-ridden profession like journalism, where they'll be plenty of  competition. No walk of life is a Caucus-race, and academics would be doing a disservice to their students by allowing them to believe that all must have prizes. Again, this is not unique to Oxford, or even university.

4b)  I thought the statement "arrogant public schoolboys" was bigoted (try replacing the noun in that sentence and you'll see what I mean. Moreover, if the public schoolboy(s) in question were such idiots, thickos who glided into Oxford on the back of the privilege-fairy, they can't have been that hard to out-compete. 

Can they?

I do not believe I have met Davis, but Penny describes her as "one of the finest and most ethical journalists I’ve ever had the privilege to meet". Accordingly, I am certain she will correct the untrue claims she has made on the website of a national newspaper. It is, of course, embarrasing to admit messing up, all the more so when 5 minutes on google combined with numeracy could have avoided the matter, but it is the honest and ethical thing to do. I am delighted to see see has made a step towards this by correcting the ludicrous error in point 1 above. 

Point 4 is more tricky - it is not about facts, but opinions. I am aware that I must be viewing the situation with privilege-tinted spectacles. That said, so must Davis, Penny, and everyone else in the world.

There are two possibilities: either

(a) privilege obscures our view of the world so severely as to prevent us from make true judgements about it; or,

(b) privilege does not obscure our view of the world so severely as to prevent us making true judgements about it.

Suppose (a) is true: my inability to make judgements about the world must also preclude me from accepting the claim that my privilege obscures my view of the world : after all, that's what "being unable to make true judgements about the world" means. This applies not only to me, but Davis, Penny, and everybody else. 

If we are not to lapse into solipsism, therefore, we must accept (b), that we can make statements about the world that are not obscured with privilege.

My initial response to Davis, in the form of an open letter, is here.

[Editted fur spolling]

Fact-Checking Rowenna Davis

An open letter

Dear Ms. Davis,

I write with reference to your recent article on Comment is Free, entitled "Stupid White Heterosexual Male", in which you make a number of statements which are untrue. I am pleased to have the opportunity to correct them, and also to offer some collegial advise as to how you can avoid intellectual humiliation in the future.

Happily, you have corrected your most ridiculous error - a bizarre claim that Oxford admits only 5 black students, out by a factor of 9 [source - document "Undergraduate admissions statistics 2008"  page 5, table 5]. 

(Indeed, your own source continues: "With more than four students applying for every place, competition is intense and the success rate among ethnic minority UK students is nearly 29%, compared with an overall average of 23.7%, but it remains below the hit rate of independent school candidates which is 29.4%."). 

You write:

"The only thing harder than spotting the black kid in my college photo was trying to find a woman on my reading list."

Given you read PPE, you can't have tried very hard. You can read, perhaps for the first time, the Politics department's PPE reading list here for the first year exams. I can see Gutmann, Nancy Rosenblum, Catherine MacKinnon, Anne Stevens, Mary Volcansek, and Sheri Berman. Guinier (of course) is there, as is Fulbrook. The Philosophy reading list is inaccesible outside the university, but are you seriously claiming you were never advised to read Anscombe? 

You write:

"I wonder if those voting for a white, heterosexual male rep have ever faced the reality of the figures. In case they're reading, I'll take the issues in turn. Class: Despite over 90% of the country being state educated, just 55% of Oxbridge students come from state schools. New figures suggest that these class divides are getting worse, not better. "

But 98.4% of candidates who are offered pre-qualification places achieve AAA ("Undergraduate admissions statistics 2008"  page 4, table 3). As 28.6% of A grades are awarded to pupils at Independent Schools [source] and  33.8% of Oxford applicants come from Independent schools  ("Undergraduate admissions statistics 2008",page 2, table 1), perhaps the over-representation is not entirely Oxford's fault.

Regarding the claim "class divides are getting worse, rather than better", it is instructive to compare the results from 2006(2007) ([source], page 3 table 1), when 47.1%(46.8%) came from state schools. Certainly in the short term, things seem to be improving. 

Wisely, you turn from statistics to your own personal experience: "When I was studying there, I felt wedged between overly sexualized bops (college parties) and competitive tutorials with arrogant public school boys." 

I am sorry to hear you have had a negative experience at Oxford. However:

Be honest now: did the sub-editor insert that definition of "bops"? Did you remember that the vast majority of readers won't know what the word means, and won't care? Anyway, the last bop I attended I saw a man dressed as a post-box dancing with a woman dressed as Richard Branson. I can say, without fear of contradiction, I have never felt less "overly sexualized". 

As adults, and all students at Oxford are technically adults, we have control over what we wear, the company we keep, and the bops we attend. At any university, there are plenty of other things to do on a Saturday night.

Further, you complain of "competitive tutorials with arrogant public schoolboys". If you don't like arrogance, prolonged exposure to academics is ill-advised. I am sorry to see you judge a person by the school she attended, but more alarmed that you see "competitive tutorials" as a bad thing. I used to struggle with tutorials. My tutorial partner (as it happens, from a state school) was considerably cleverer that I am, and I always felt like the Red Queen, having to run faster and faster just to stand still. I certainly don't believe the tutorial system is the best possible system for everyone, but it certainly made me into a better physicist. The tutorial system is the reason it cost the taxpayer £3000 more each year for you to study PPE at Oxford rather than something useful elsewhere. You must have known this when you were applying. Had you not wanted this, there are 107 universities in the UK that aren't Oxford and Cambridge. If you found your tutorial partner disagreeable, even for the rather bigoted reason you give, in life and work we often have to put up - even be polite to! - people we dislike. This is not unique to Oxford, or even to university.

So where does this leave us?

There are some great stories to be written about tertiary education in the UK: you could talk about the difficulties of contract research staff, the funding of middle eastern studies departments or how physics is becoming the new classics. You could talk about those 107 universities that aren't Oxford and Cambridge.

It would also be nice if, from time to time, the debate on education could focus the vast majority who did not go to Oxbridge, or even the majority who didn't go to university. You could talk about what's happened to adult education and how to change it. You could talk basic qualifications, and about teaching reading, and about the way political groups manipulate the syllabus in subjects from biology to history.

If it is essential to discuss Oxbridge at such tedious length, why doesn't CiF run a series of articles on Oxford where people wrote that they had quite a nice time, really, didn't do enough work, made some good friends and generally grew up a bit. Just like any other university, except with older buildings  and worse facilities. 

Or, you could write another retread of the "Tales of Terror:  Trapped Among the Poshes!" where the author complains that she was forced to meet people who went to a  school that was funded differently to hers, and spoke with a different accent, and had tutors who expected her to think, and how the college drama society re-enacted Brideshead bloody Revisited 24 hours a day, and in general the sheer awfulness of attending one of the oldest universities in the world, having the taxpayer stump up an additional three grand a year more than your mate who went to London, having contact time and a student-teacher ratio other unis only dream of, and meeting clever, hard-working people from a completely different background to you. The horror, the horror.

The choice is yours.

I remain, yours sincerely,

A Stupid White Heterosexual Male

[Thanks to Ed and Duncan for the tip]

"Is the BNP racist?"

Guess

Matt Wardman spells it out.

[via John B]