Comments on Dr. Paul Lockhart’s Essay “A Mathematician’s Lament”

I was recently sent a link to Dr. Paul Lockhart’s “A Mathematician’s Lament”, an article written in 2002 and more recently (2009) expanded to book form.  (I have not read the book.)  Lockhart – as many have done before him and, doubtless, as many will be doing three hundred years from now – deplores the state of modern mathematical education, criticizing particularly its rigidity, its reliance on rote knowledge, and its disconnection both from the real problems of life and especially from the actual art of mathematics.

I believe I first read this article during Senior Seminar as a senior mathematics student at Hillsdale College, with a professor whose name I have mercifully forgotten as (at least at the time) I found her obtuse and infuriating.  Looking back, I realize that much of the class was focused on the idea of mathematical education – at the time this seemed like a waste of time, though now I am a high school math teacher – and I wonder if perhaps she was trying to practice – and get us to practice – what she preached.  Certainly the only actual mathematics in the class was the senior project assignment, unrelated to the rest of the class.  It came around on the web again at some point a few years ago, and so this time makes the third reading, and I thought I would say something about Lockhart’s piece.

His main premise is that mathematics is an art, and his thesis is essentially that mathematics should be taught as an art.  There is little doubt that Lockhart, in calling mathematics an art, is thinking of the modern idea of “Art” – that is, a unique expression of a unique human mind, without any need for any justification beyond its mere existence.  He says so himself:

“The first thing to understand is that mathematics is an art… Everyone understands that poets, painters, and musicians… are expressing themselves in word, image, and sounds… Why not mathematicians?”

Lockhart follows this theme throughout, advocating experiment and “natural” learning over any kind of orderly or preconceived scheme.  In one sense, he is doing little more than advocating for the Socratic method to be applied to mathematics: teach by question, rather than statement.  So far as that goes, it is good advice, I think – though it leaves me rather puzzled as to how I might apply it, much of the time.  But Lockhart also seems prepared to throw out the notation, the formulas, even the arithmetic tables, if they get in the way of what he sees as “real mathematics”.  I perhaps exaggerate a slight amount, but – despite his periodic qualifications saying he would not advocate this wholesale abandonment of such things – this is the unmistakeable tenor of the piece.  This can be seen clearly at the end, where in his “honest” curriculum he has descriptions like:

“Lower School Math: … Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult.  Multiplication tables are stressed, as are parents, teachers, and the kids themselves.”

(Quite apart from illustrating Lockhart’s feelings, this characterization has a major flaw, being exaggeration or nonsense from beginning to end.  To point out two major problems: The “Hindi symbols” prove a much simpler method for manipulating numbers than previous systems, and have been well-known even in Western countries – which are, especially the United States, Lockhart’s audience – since at least the 16th century: five hundred years.  The multiplication table is, based on my own experience as a teacher, not stressed at all – certainly not stressed enough that I can count on students knowing it by heart.)

This is not to say Lockhart’s complaints are without merit.  When he deals with the high school classes, in language truly intemperate, he hits much nearer the mark.  Much of it is, as he says, flung together without particular rhyme or reason, certainly without any eye towards enlightening the student as to those reasons (although with the benefit of college study of mathematics and a few years teaching I would contend that most of the ideas behind the progression makes sense, even if the application is poor).

There are, though, two questions which Lockhart shows no indication of having asked himself.  The first is inherently important to his thesis, and is this: how is an “art” taught?  His nightmare scenarios of students of art and music working through mindless chores is frightful: but it also calls to mind memories of necessary hours working on scales, arpeggios, and basic theory; or learning to draw straight lines, even curves, and shapes in perspective.  The basic skills of an art are learned, along with simple pieces (and nothing on earth, apparently, can keep an interested student from messing around with his chosen art in spare time); then more complicated skills and harder pieces; and finally, should the student continue his studies, we get the virtuoso, the musician; the famous painter – or, for those of lesser genius, at least a solid competence and satisfaction of a thing well-learned.

Let us then say that mathematics is an art.  While we figure out how to teach its artistry – while we work on learning to ask and answer the numerical and spatial questions, in other words – we cannot simply dismiss the tools of the trade.  The multiplication table, in other words, ought to be learned: it is not simply a convenience or a contrivance, it also happens to be true.

The second question Lockhart forgets to ask is which students are being trained.  He is, I think, primarily motivated by the problems, not at the elementary level, but in the high school curriculum.  He assumes, I think, that students will actually learn their arithmetic naturally, when evidence suggests to me that the biggest problem is that they do not.   Lockhart’s method might be very well if he were teaching only future mathematicians – I think it would still be lacking, to concentrate on the questions of math, and let the notation and system and facts slide, but it might work – but of the students in any given math class, few will go on to make mathematics a career.  To return to the artistic comparisons for a minute, consider whether a mathematician and a portrait painter need the same lessons in drawing.  Surely not.

Lockhart also is not at all clear how his methods would actually be implemented.  It is one thing to say that a child faced with an intriguing problem will put time and effort into it and find an answer, and that this is better than memorizing some formula.  It is another to try to figure out how to move from a single example to a full instruction in a subject.  He ignores, too, the fact that some children are simply not interested in these kinds of questions at anything more than a trivial level, if that.  For example, note that not everyone enjoys mathematical puzzles like sudoku – or even the related but more linguistic crossword.  Perhaps Lockhart believes that if a student is not interested, this is an inherent fault and inherently lies with the teacher; I do not.

A final problem is what he means by “mathematics”.  The word is used to cover a vast variety of related topics, from elementary arithmetic, to the variable formulas and their uses which we call algebra, through series and the calculus, or set theory and groups – and so on, up to the areas where mathematical exploration is ongoing.  The Greeks, perhaps, thought of most “mathematics” in almost purely geometrical form – the form, incidentally, where Lockhart’s methods might have their clearest application – although they also explored various number relationships, especially ratio problems.  The modern student of mathematics does not have quite so simple a task before him.

It is safe to say, with the vast majority of humanity throughout known history, that “mathematics” is a subject a free citizen ought to be familiar with.  It is more difficult to pin down exactly how much the average man “must” know.  Algebra?  Even the end of a high school algebra textbook approaches techniques rarely used outside engineering or architectural applications, as far as I know.  Economic or statistical methods should have obvious applications from day to day – but all but the simplest (useful enough, true) require other background.  In terms of practical problem-solving, geometry and trigonometry both consist about half of the incredibly obvious and half the relatively esoteric – and even so, probably most common problems can be solved with the simple application of a few easily learned formulas.

To my mind, education in mathematics should primarily be intended to provide a student with the tools he needs to solve problems and the inclination to do the work himself.  A good mathematics course should therefore provide the necessary facts – tables, formulas, patterns, or what have you – together with sufficient and sufficiently interesting problems to illustrate where those tools should best be used.  Lockhart’s method, at least as he talks about it, might be compared to asking a teenager to fasten two boards together, without informing him that nails exist.  A clever one might find a nail and infer its use; or come up with another solution entirely to solve the problem (a spare shoelace he happened to have in his pocket, say); but for the kid to abandon the problem as a bad job seems just as likely.  A teacher must introduce both problems and tools: the question of whether to introduce tools and then illustrate the projects requiring their use, or to first demonstrate the problem and then introduce the learner to the necessary tools, is a vexed one.  A correct answer likely depends on all of the variables of the tool, the problem, the teacher, and the student, and it would be a very rare teacher who gets the combination correct every time.

I am not sure to what extent this actually disagrees with Lockhart.  I do not like where his methods would seem to end up, and I do not think he has given much thought to the question of early mathematical – that is to say, arithmetic – education.  But I am not sure how much he would disagree with the statement I made, either.  Perhaps these things are further clarified in his book?

Fiction, War, and the Question

It would be pretentious of me to suggest that I read fiction in pursuit of some grand academic pursuit of truth or beauty (or whatever).  The fact is, I enjoy a good adventure story, and reading fiction – especially the science fiction or fantasy stories I prefer – satisfies that desire.  In fact, I probably over-indulge – an easy failing, as I read quickly enough that a large number of pages can take not that much of my time away from other things.

It is true, though, that the mode of fiction – the story, known to be unreal – allows authors, sometimes by accident, sometimes deliberately, to pose questions in a form which might not otherwise be possible.  Much of science fiction is also what is called “speculative fiction”: attempts to answer – or sometimes just ask – the question, “What if?” by providing an imagined context in which the answer becomes obvious (given the premises of the story) or the question becomes pressing (given the conditions set out in the narrative).  Other times, there is no particular point or question to a story in that speculative sense, but characters are free to offer remarks which have bearing not only on their imagined situations but (intentionally or not – and sometimes it is hard to tell) on real life difficulties.

I am going to, in this post, offer two examples from recent reading which reflect on the place of women in war.  The United States armed forces have, for several years, been moving towards integrating women in combat with men: attitudes towards this vary tremendously, both on ideological grounds (most obviously, conservative Christians vs. radical egalitarians) and on judgments of practicality (relative average strength differences: significant or not? – sexual tension: unavoidable or not? a cause for how much discipline? a cause to abandon the integration? – and so forth).  A significant portion of “modern” science fiction (which I am taking to mean written since the 1980s, or in other words, most of what I read) tends to assume the egalitarian viewpoint will triumph eventually – and the attitude is older than that, if not as widespread or settled.  The Lensman series – one of the first major science fiction series, published in the 1950s – includes women as “equal” protagonists – “equal” being in quotation marks because the work seems incurably dated, even by today’s standards.  Heinlein’s Starship Troopers (1959) makes a sort of concession by calling women better pilots, but the grunts are men and the issue is mostly ignored.  Frank Herbert’s Dune (1965) is an interesting case, in which the importance of men and women is presented as equal, but traditional-type social structures are assumed to have endured, for the most part.  This is of course a very incomplete picture, but will serve to demonstrate some of the history in the genre.

More recent writers – Orson Scott Card in Ender’s Game  and his other works (now, ironically, under fire by egalitarians for some of his other Mormon beliefs), David Weber (of Honor Harrington fame, though it also shows in the rest of his work), Elizabeth Moon (Girdish, Familias Regnant, and Vatta storyworlds, among others), and others – pretty much have accepted that integrated militaries will be the order of the future, by analogy to the success of sexual integration in normal business.  (While complaints do still persist on this basis in real life – not to mention differences of opinion – the current hit show Mad Men should have reminded the public how much has changed even in fifty years.  Also, interestingly, the named authors and most of the others I have read assume – whether unthinkingly or for narrative purposes – that the argument will continue as well.)

All of this is prelude to the material I wish to quote.  From Lois McMaster Bujold’s The Vor Game, I offer a brief excerpt.  Miles, our hero, has messed up, and is brought in for a dressing down by his commanding officer:

Metzov was still expanding, self-stimulated. ‘In a real combat situation, a soldier’s equipment is vital.  It can be the difference between victory and defeat.  A man who loses his equipment loses his effectiveness as a soldier.  A man disarmed in a technological war might as well be a woman, useless!  And you disarmed yourself!’

Miles wondered sourly if the general would then agree that a woman armed in a technological war might as well be a man… no, probably not.  Not a Barrayaran of his generation.

And the story moves on, thought unanswered, dismissed, ignored – and hanging there, making the thoughtful reader wonder.

Bujold, at least in the Vorkosigan series just quoted, is clearly and deliberately telling an adventure story, not attempting commentary.  On the other hand, John Birmingham’s time-travel series which begins with the novel Weapons of Choice is equally clearly a serious attempt to comment on various social issues, within the framework of an action novel.  The premise: a 2020s American-led, combined arms task force is, by a freak accident, transported (and scattered) into World War II.  Chaos ensues.  As the various commanders try to explain 21st-century social expectations to FDR and company, Birmingham has Einstein (yes, that Einstein) say:

‘You seem to come from a militarized society, Admiral; the ease with which your men and women in uniform mix togehter.  The way you don’t appear to heed the race or creed of your comrades.  Some might see that as enlightened, and I suppose it is.  But you could also see it as the defensive response of a society that has been fighting for so long it has shed itself of all trappings save those needed to wage war.  You can see the same thing happening here and now, to a lesser extent.’

Now, Birmingham’s task force is coming from a future war which is implied to be a sort of second outbreak of mass terrorism, and his Admiral answers that these things were happening even in peace – but again, the thoughtful reader starts wondering: from World War II – which brought us Rosie the Riveter – to the Korean War, Vietnam War, defense-conscious ’80s, Gulf War I, Balkan interventions, Gulf War II – how long has the United States been honestly at peace for the last seventy years?  How did that affect us – even if not in the ways this fictional Einstein suggests?

I am not going to attempt to answer the questions raised, by either selection, at this time; the point was to demonstrate the unique place of fiction in discussion of important questions.