In patriotic duty bound, the Cambridge of Newton adhered to Newton's fluxions, to Newton's geometry, to the very text of Newton's Principia; in my own Tripos in 1881 we were expected to know any lemma in that great work by its number alone, as if it were one of the commandments or the 100th Psalm.

.... Finally, in the earlier section of the Tripos Examinations (officially described as "qualifying for honours", commonly known as "the three days"), there was a rigid rule against the explicit use of a differential coefficient and of an integration-process: we might substitute x+h for x and subtract, dodging onwards to the satisfaction of the examiner; we might use a Newton curve, if we could devise it, to effect a quadrature; but never might we use d/dx or the ∫-sign of integration which were taboo. 

A R Forsyth, Old Tripos Days at Cambridge, The Mathematical Gazette, Vol 19, No 234, July1935 (at JSTOR, unfortunately)

losing streak

The Khan Academy has just got rid of its popular streak metric.  I woke up this morning in a somewhat ratty frame of mind and decided to soothe the savage breast by doing a few exercises on KA (which for a while now has had a feature that suggested exercises to review) - and discovered, with shock and dismay, that KA had completely changed the UI.  It is much more sensible in one respect - the user is prodded into working through a group of exercises on a single subject, rather than reviewing the mixed bag of exercises hauled up by the algorithm.  Unfortunately all this solid good sense is coupled with a new progress display consisting of stacks of leaves, which replaces the former streak bar (and the most recent streak bar was already a step in the wrong direction, replacing the ur-streak bar which told you how long a streak you had racked up). 

Now honestly.  If they were going to lavish this kind of ingenuity on the site, why not give us exercises on Laplace transforms?  Or, to be slightly less esoteric, why not have a bank of exercises on integration?  At present the site simply reinforces the bad mindset of the sort of person who does not use calculus on a daily basis - that is, the attitude that differentiation is the easy one and integration is to be approached with extreme caution, not to say trepidation.  (There are currently NO exercises in integration.)

A while back I read a piece by a British mathematician who commented that the sense for how best to tackle integration came with maturity - one developed one's intuition by doing a wide variety of problems over the years.  I was unfortunately introduced to integration at Smith, at a time when I was profoundly depressed, so I did not then lay down the foundation for this particular sort of mathematical maturity - but the comment filled me with hope.  I felt that if I did problems on a daily basis intuition would come.  And if such problems were readily available online, with instant feedback, I would probably be doing them on a daily basis.  Paltry it may be, but it would be good for me to rack up a streak of a thousand or so.  Well, it is salutary, no doubt, to be made to confront one's sloth: I expect there is a software package with a perfectly serviceable question bank, and sloth has led me down the path of least resistance.

I started doing exercises on the Khan Academy about 8 months ago as a way of taking my mind of crazymaking things I can do nothing about.  I now have 1,755,000 exercise points and loose change; this gives some idea of how crazy I would have been if I had not been working out fiddly little exercises in kinematics and such.  I feel a bit guilty about this, because I could long since have gone past the mathematics I already know by watching videos. The problem is, though, that I hate videos as an instructional tool, and the whole point, after all, was to soothe the savage breast.

The other day, though, I succumbed to the gamification which some see as a flaw in the enterprise. At the time I had every badge it was possible to win without watching a video.  There are many more badges, but these all require watching videos, and I do so LOATHE videos.  Still, I thought I would look at the list of videos and see if there was anything I could bear to watch.  And what should I see but a whole slew of videos on Laplace Transforms!  Something I had never covered in the days when I was studying mathematics!

I should say at once that I had no idea at this point what a Laplace Transform actually was.  The appeal of the topic was simply the name "Laplace Transform."  For reasons that I can't defend, mathematics appeals to what I suppose boils down to a love of kit.  Glamorous names are good, as is some novel sort of notation.  (The Laplace Transform, of course, offers both.)   So I watched 6 or 7 videos, racking up several badges in the process - but the fact is, I really don't like videos.

It was at this point that the policy of giving house room to unread books came into its own.  Back in 1997 I would appear to have bought a book on differential equations under the impression that I would quickly be reading up on differential equations.  (Readers familiar with my publishing career will, I hope, not hold it against me if this optimism was unfounded.)  Now, though,  I pulled the book off my shelf and found a whole chapter on the Laplace Transform!  A chapter which I did find much easier to follow than the video, though without the video it might have gone unread for further countless years. 

Spoiled by the power of your best tools, you tend to shy away from messy calculations or long, case-by-case arguments unless they are absolutely unavoidable. Mathematicians develop a powerful attachment to elegance and depth, which are in tension with, if not directly opposed to, mechanical calculation. Mathematicians will often spend days thinking of a clean argument that completely avoids numbers and strings of elementary deductions in favor of seeing why what they want to show follows easily from some very deep and general pattern that is already well-understood. Indeed, you tend to choose problems motivated by how likely it is that there will be some "clean" insight in them, as opposed to a detailed but ultimately unenlightening proof by exhaustively enumerating a bunch of possibilities.

What is it like to have understanding of very advanced mathematics, the rest here (ht Tyler Cowen at MR)

useful & cool (dulce et utile)

I was playing around on Khan Academy (as one does).  As one does if one is unable to block out the world and write a book because unable to leave e-mails unchecked for months on end because there is a book to be launched. ('Publish and be damned' takes on a whole new meaning in these degenerate days.)

And!

What to my wondering eyes should appear!

A couple of weeks ago I was playing around on Khan Academy, reminding myself of really basic stuff, trigonometry, bits and pieces, mostly last used a couple of decades ago, needed for less basic stuff.  The answers to the exercises were multiple choice.

Last night I went back to a couple of these exercises.

They had fixed things that weren't quite right.

Instead of multiple choice answers, the player (erm, student) had blanks to fill in.  The player could also click to get a  list of acceptable formats for answers.

So on the one hand you had to work harder -- had to generate the correct answer rather than picking it off a list -- but on the other hand you were less likely to be penalized for not giving the right answer in the right format.

I told my mother about the Khan Academy the other day.

My grandmother, Blanche Spurrier Marsh, was born in 1900; she was a mathematician.  After majoring in math at Randolph Macon she went on to teach, then to be principal of a school. She then married my grandfather, a Southerner who did not want his wife to work.  My mother was born; my grandfather told my grandmother that she could not do two things.  Her job was to look after the child; she could not also work in a school.  What it turned out to mean was that it was fine for my grandmother to go to a school as a substitute, to help out as a favor, but not to have the advantages of a permanent job. (This would imply she needed to do it for the money.)

My mother was a musical prodigy, but she had no aptitude for mathematics.  My grandmother tried to tutor her.  To this day -- my mother is now 78 -- my mother remembers working on problems in long division at the dining room table.  My mother was then 9 -- this would have been 1942.  My grandmother walked up the stairs to the landing, looked down, said: You'll never be anything but a nincompoop!

(My mother has a phobia of computers.  When things go wrong she does not remember that she played the Ballades of Chopin at her senior recital; she remembers that her mother called her a nincompoop in 1942.)

So, ANYWAY, I talk to my mother about the Khan Academy.

Khan says he started tutoring his cousins by phone, made a few videos as a "nice to have" -- and was told they liked the videos better.  Which, he realized, made sense: they didn't have to expose their ignorance, they didn't have to worry about wasting his time, they could go back, replay, shame no longer got in the way of learning.

I think I thought telling my mother about this wonderful resource would lance the wound. 

Or maybe that if my mother went online and did some exercises THIS would lance the wound.

It seems not to work that way.  

My mother did see at once the value of the resource.  She said you would go into a math class where everyone else understood something, and you would pretend to understand, so you fell further and further behind because no one bothered to explain because you had been pretending to understand.

(She never bothered to look at colleges.  One of her teachers asked her about her plans in 12th grade, and she had done nothing, and he was appalled, and pushed her into an application to Rollins, which had an excellent conservatory -- and so she went to college.  Because she was a musical prodigy, and one of her teachers noticed that something had to be done. I think we can agree an educational system ought not to depend on last-minute saves.)

It may be that you have to see the damage a sense of inadequacy can cause over a lifetime to appreciate the value of the Khan Academy.  Khan himself may be too young to understand the full value of what he is offering.  I looked at these exercises, which had been improved in a few WEEKS, and was charmed, disarmed, and for once, among all the madness, hopeful.

don't mention fried ants

(More from T W Körner, The Pleasures of Counting - a passage which makes the end of the book, on which a post currently waits in the drafts folder, all the more inexplicable)

The study of algorithms is part of the constant search for better ways of doing things just like the search for ways of preventing cholera or submarines. But the task does not end there. Having found a better way you must persuade others to adopt it. Learning how to do this can only come from long and painful experience. ... Until the reader has experienced the shock of having her well thought-out and carefully argued proposal thrown out for the most fatuous of reasons, she will not understand why I write this section or why, before any difficult committee meeting, it is important to recite the two phrases: 'You can only resign once' and 'Those who fight too long with dragons become dragons themselves.'

When a government committee was set up in 1978 to consider mathematics teaching in English and Welsh schools, their first step was to set up a survey to find out the opinions and mathematical needs of a representative sample of adults. However, the survey ran into an unexpected difficulty when many of those approached refused to be interviewed.

Both direct and indirect approaches were tried, the word 'mathematics' was replaced by 'arithmetic' or 'everyday use of numbers', but it was clear that the refusal for people's refusal to be interviewed was simply that the subject was mathematics.... Several personal contacts pursued by the enquiry officer were also adamant in their refusals. Evidently there were some painful associations which they feared might be uncovered. This apparently widespread perception amongst adults of mathematics as a daunting subject pervaded a great deal of the sample selection; half of the people approached as being appropriate for inclusion in the sample declined to take part.

Even among those who agreed to take part,

The extent to which the need to undertake even an apparently simple and straightforward piece of mathematics could induce feelings of anxiety, helplessness, fear and even guilt in some of those interviewed was, perhaps, the most striking feature of the study.

There did not appear to be any connection between mathematical competence and occupation group. However,

The feelings of guilt to which we referred earlier appeared to be especially marked among those whose academic qualifications were high and who, in consequence of this, felt they ought to have a confident understanding of mathematics, even though this was not the case.

In view of this, it is clear that mathematicians should not refer to mathematics in advocating a given course of action. By itself this is not a great disadvantage. Darwin's On the Origin of Species is a marvellous example of sustained book-length argument without any recourse to mathematics. Unfortunately, most people are also unwilling to follow sustained argument.

Springer sale

Springer Verlag is having a sale on mathematics books through the end of July, offering "many essential mathematical titles" for half price. So you can snap up Alpay and Vinikov's Operator Theory, Systems Theory and Scattering Theory: Multidimensional Generalizations for a mere 64,90€ instead of the normal price of 129,00€, thereby achieving a saving of 64,10€, or 49.69%. Also many other great and somewhat more affordable items, all here.

radar days

It is generally agreed that the quality of German radar was superior to that of British radar. German radar had been built by engineers, British radar had been lashed together by physicists. In part this represented a deliberate choice of a force march towards a new technology (Watson-Watt's motto of 'Second best tomorrow' rather than 'Best, but next week'), but in part it reflected Britain's industrial weakness and in particular the lack of sufficient suitable engineers.* The strength of the British system lay not in the radar itself but in its integration into the air defence system. (Thus the British refused to believe that the Germans had radar because German fighters took so long to scramble, whilst the desultory German search for a possible British radar was restricted to a system using the single figure metre wavelength which a radar 'ought to use' rather than the primitive 25 metre wavelengths that the first chain actually used.)

* R. V. Jones gives an account of the attempt during the 1930s by the self-made millionaire Lord Nuffield to found an Oxford Colege devoted to engineering. 'According to what I heard at the time this prospect alarmed the strong humanist element in Oxford, headed by the Vice-Chancellor, Lord Lindsay, who sought to palliate the engineering onslaught by persuading Nuffield to broaden his objective. There would be less opposition to the foundation of a new colelge, he said, if Nuffield could disguise his intentions by replacing spcific mention of engeering by some more subtle wording. Engineering was a science, but it made a more direct impact on society, and so it might be fairly described as "social science". Therefore if Lord Nuffield would specify social science as the primary interest, there would be much less opposition to its creation. It was only after the college had been founded and staffed not with engineers but social scientists that Lord Nuffield realised he had been outwitted.

rereading T. W. Körner, The Pleasures of Counting, which has been in storage for some time.

Körner describes the book as 'meant, first of all, for able school children of 14 and over and first year undergraduates who are interested in mathematics and would like to learn something of what it looks like at a higher level.' He adds, 'Listening to a mathematician talking to mathematicians about things that interest mathematicians may well be more enlightening than listening to mathematicians speaking to non-mathematicians about things that they hope may be interesting to non-mathematicians.'

I had taken the book to Yorckschlößchen, where I ordered pommes (pr. pom-mess) and a beer. Sparrows flew down to the table and hopped cautiously at the other side of the plate. I tossed one a chip. It flew off, bearing the chip in its beak. I tossed a chip to another sparrow, which flew off, chip in beak. Soon the Biergarten was full of sparrows flying through the air carrying chips, pursued by other sparrows which had not yet managed to get a chip.

I was reading the book for a piece the LRB may take on information design and James Wood's piece on hysterical realism.

Körner went on:

The methods Tizard had used to discover how radar could be used came to be called 'oeprational research'. Those who used the phrase found it hard to define what exactly it meant and to explain what was new about it. Certainly it involved the application of science not merely to the invention of weapons, but to the choice of tactics in their use. However, it also required the kind of collaboration between scientista dn military typified by the radar 'Sunday Soviets' in which senior scientists, Staff Officers, junior research workers and serving officers 'straight from the heat of battle' met informally and where anyone could say anything to anyone. Whatever 'operational research' meant precisely, it was something that could be copied, and the idea of operational research spread through the British and then the American armed services. Nothing comparable occurred in Germany....

In January 1943, the Germans shot down a British bomber carrying a new radar. Examination showed it to operate at an incredible ten centimetre wavelength. Göring commented bleakly, 'I expected them to be advanced, but frankly I never thought they would get so far ahead. I did hope we could at least be in the same race.' The German military had not asked for such a radar and, since the proper role for German science was to supply what the military asked for, it had not produced such a radar.

Adam Smith overboard in Economist balloon debate

In an age where you need to be numerate to do almost anything else (from building bridges to conquering disease), governments anxiously compare their performance in mathematics with that of competitor nations. This month a new cry of alarm came from America, where a National Mathematics Advisory Panel, established by George Bush in 2006, reported that “without substantial and sustained changes” the country was doomed to “relinquish its leadership” in the world of numbers as the century wears on.

America has long masked its difficulty in educating enough mathematicians by importing lots of ready-made talent, especially from East Asia and the former Soviet Union. But the problems are real enough. As the panel noted, the share of American students doing degrees in maths or related areas fell from 32% in 1994-95 to 27% in 2003-04. And the share of maths-related doctorates at American universities that went to American citizens or residents fell over the past four decades from 80% of the total to less than 60%. The panel concluded that America's problems become apparent when students start to study algebra—for most, their first encounter with genuinely abstract thinking.

The Economist reports on scarcity in native mathematicians in the US.

The language used in this piece is somewhat odd. Try this on for size: 'America has long masked its difficulty in producing olive oil by importing it from Italy and Greece.' 'America has long masked its difficulty in producing Brie by importing it from France.' 'Britain has long masked its difficulty in producing wine by importing it from France, Germany, Italy, Spain, Portugal, Australia, New Zealand, Argentina, Chile and America.'

From an economic point of view, gearing up the American educational system to produce more capable mathematicians might be far more expensive than simply offering fast-track visas to mathematicians of proven ability who have had the bad luck to be born in countries with nasty political systems (I say 'nasty'; OK, 'nastiER'), underfunded universities, poor infrastructure, a scarcity of malls, fast-food outlets, first-class orchestras - WHATEVER it is that makes one place look less appealing than another to a mathematician of proven ability.

From a nationalistic point of view, um, isn't there something a bit fishy about this soul-searching? 'We're not producing enough second-generation, third-generation, fourth-generation, fifth-generation, n+1th-generation American mathematicians!!!!! We've had to fall back on first-generation Americans!!!!!!! The country is doomed!!!!!!!' If a State of Emergency has been declared because we are not producing enough Cherokee, Navajo, and other certifiably aboriginal American mathematicians, it has passed without comment by the Economist. Fact is, it's a country of immigrants. Today's Chinese math whiz, freed from the one-child-family rule, is tomorrow's parent of a gaggle of Chinese-Americans, some of whom may be math whizzes, all of whom will be American.

The real objection to America's poor showing in mathematical education is not a matter of economics, it's a matter of human rights. For reasons that are never entirely clear to me, religious beliefs, however loopy, are generally treated with respect even by those who don't share them. Mathematicians, however, are drawn to something that has no church and no tax-exemptions: a world outside this world of accidents, waiting to be discovered, a world of beauty, elegance and wit. Because of the strongly utilitarian bent of our educational system, because of the assumption that EVERYONE, regardless of aptitude or inclination, must achieve a certain level of competence, those drawn to mathematics are often forced to study it in a social group consisting primarily of contemporaries who regard it with unqualified loathing. We don't require Jews to study the Talmud in a class of bored, resentful Christians; we don't require Muslims to study the Qur'an in a class of bored, resentful Jews; we don't require Christians to study the Gospel in a class where they are outnumbered by Jews and Muslims 10 to 1. We do throw young mathematicans to the, ahem, unenlightened, with the result that too many end up unqualified to engage with those who should have been their peers.

A few years ago I gatecrashed a class on partial integration given by a Chinese lecturer at Columbia; the thing that stays with me is the wit he brought to the business of converting the seemingly unintegrable to something more tractable. If every American schoolchild could be taught by someone with his gifts we could count ourselves lucky.