Review: Everything and More

In 2003, David Foster Wallace wrote an account of how mathematicians have dealt with aspects of infinity throughout history for W. W. Norton & Co.’s series “Great Discoveries”.  At some point I got a copy as a present, and have now read it.  In 300 pages – and I doubt it was supposed to run that long, as Wallace refers to the work repeatedly as a “booklet”, he traces the history of the problem of calculating infinite and infinitesimal quantities, or calculating with them – in short, how math has dealt with the enormously huge and the incredibly tiny – from the Greek philosopher-mathematicians to the 19th century German mathematician-philosophers, with a quick nod towards 20th century math which has (in Wallace’s account) mostly succeeded in demonstrating the 19th century solutions have their own problems and forced mathematicians to figure out how to deal with that.

Wallace’s style is engaging from one sentence to the next, but the overall structure of the book ends up a bit muddled, as he attempts to present at times incredibly complicated concepts in an understandable framework.  For the most part – though as a mathematician (low-grade) I may not be the best person to judge – he succeeds, though I judge the “optional” technical explanations periodically inserted would have been better maintained as integral parts of the book.  The final effect is much like the legendary Japanese garden technique where a great deal of care goes into presenting a final appearance of perfect naturalness: but this by its own rules prevents other qualities of line or symmetry.

Everything and More was a good reminder for me of the puzzles and paradoxes that attract me to some of the more theoretical parts of math, though I would hardly say I understand it all.  At one point, Wallace describes Leibniz – he of the calculus – as a “lawyer/diplomat/courtier/philosopher for whom math was sort of an offshoot hobby”, footnoting this description, “Surely we all hate people like this,” but I suspect Wallace himself of similar gifts, given how easily he, acknowledged mainly as a writer, converses of complicated mathematical topics.  It is also evident from Wallace’s fulsome praise for the man that he had at least one great teacher, a Dr. Goris.

If you have any interest in mathematics – especially its quirks and paradoxes – and are prepared for a reading experience complete with concept-induced headaches, I’d recommend this one.

Opinion on Mathematics: Three Suggested Keys

One of the uses of a blog is as a drafting board available for public perusal.  The following is my current attempt to sum up what I believe are the most important ideas which the student being introduced to mathematics proper – which is to say, in modern civilization, a high school student – must learn.

I. Prerequisites and Goals

For almost all students, for these things to make sense they must already be able to do arithmetic: certainly to have learned addition, subtraction, multiplication, and division; of whole numbers, fractions, and probably decimals; the methods by heart and the tables as well, preferably and probably by the tested method of rote memorization.

In my experience, the methods for whole numbers and decimals are generally well-learned by this age, but the tables are not, and, whether as an additional failure or because of this, lack of understanding of factoring, common denominators, and fraction work in general is quite common.  The fraction methods are reviewed – I sometimes wonder if “taught” would be more applicable – almost as if new in most Algebra 1 texts (9th grade work), so either my standards are unrealistically high or arithmetic standards in general are drastically low.

II. The Three Ideas

A. Equal Change.  One explanation I remember from my own days learning simple algebra (and have occasionally used myself) is the idea that the “equation must balance”.  If you have two equal things, they must change the same way, or how do you know they are still equal?  If two things are unequal, the same operations must be effected, or how do you know if the inequality is still present – or perhaps now reversed?  This is the simplest example of this principle to illustrate; if x + 2 is 7, then by removing 2 from both parts we find that the variable x is here representing 5.  (On which more later.)  But it applies in many, many other places as well.  One of the more difficult ideas for some students to grasp, for instance, is that a formula should do “the same thing” to any number it applies to.  The most common path to enlightenment seems to be doing it again… and again… and again…  “Evaluate x +2 if x = 3.”  Ah, we get 5.  “What if x = 7?”  Er, what do we do with the 7?  “Well, what does the formula tell you to do?”  Um, add two?  It sometimes takes a while for the idea to take.

B. Complete Variability.  The counterpart of understanding the things are always the same, or need to be changed the same way, is realizing how much the things that change – that is, in basic algebra, the variables used to represent unknown quantities – can be changed.  After the initial hurdle of realizing “x” is “any number we want it to be”, this usually is not a significant problem again until students hit trigonometry or calculus, where suddenly they start running into the mathematician’s favorite trick, “Let’s take everything in the denominator of this ratio and call it phi…”  But at that point, it seems to completely stump some students, and most, my guess is, never really do enough work to really understand it.  Personally, I would say I finally “got” the idea around my sophomore year of college.  “We don’t care what it does by itself any more, let’s take this function and shove it in this other function and see what happens.”  In short, the ability to abstract a concept almost completely from whatever reality it might represent.  But while crucial, I wouldn’t say even this is the heart of mathematics.

C. Identifying Relationships.  Most math courses including a bit on ratios when teaching fractions; and a lot more when dealing with probabilities; and “relations” are mentioned as a precursor to explaining “functions”, but the idea that one thing is specifically related to another thing, by an exact kind of relationship, is, I think, often lacking.  Graphing, for instance, can provide a kind of concrete evidence that “this kind of operation ‘looks like’ this”, but it is time-consuming – and often lacking in examples of “this is the kind of thing it represents”.  I’m perfectly happy, myself, playing the number games, but there’s an actuality to a fancy graph.

Then again, to go back to what is taught, you never really get away from numerators and denominators as you go on in math, and yet they’re mostly taught as “well this is how you deal with fractions”, and when they show up again in Final Boss form in (say) a rational expression, the student can sometimes be baffled.

This part I’m explaining really badly.  In short, easy to understand words, you have to realize that, “It’s the relationship, stupid!” but you also need to understand what that relationship means… before you go on to abstracting it all for fun.

III. A Caveat

I teach at a school which is not “academically focused”.  I have bright, serious students for the most part, but few of them have or have had academic ambitions.  My guess is that, mathematically, I get not “the best of the best” but “the best of the rest”.  I’m also an inexperienced teacher, so it’s difficult for me to say how many of the problems that drive me nuts are really my own fault.  And finally, this is all fairly vague – as I said above, a draft – so if it’s a little bit incoherent in places please forgive me.  That said:

IV. Suggested Improvements

The biggest, “Wait, what?” moment occurs regularly each year when we hit the part of the book that briefly reviews unit conversions, and it turns out several students have no idea what they’re doing.  If I wrote an algebra textbook, it would start with the things.  No, you don’t get variables yet.  We’re going to convert kilometers to millimeters, and calculate volumes, and change it to feet, and convert that to inches, and then figure out how much it costs at $3 per cubic decimeter.  And we’re going to do that for a month and a half.  Then, and only then, when you’re thoroughly acclimated to the idea that we change take this number and change it in to whatever other number we want by throwing unchangeable fractions at it; when you’ve carried and canceled and squared and divided not just a few odd numbers to the tens’ place but bunches and bunches of units that end up in the final answer not at all; when you are able, out of sheer desperation if nothing else, to convert miles per gallon to knots per milliliter (okay, maybe that’s a bit far) just because I say so; then, then I give you your first variable.  And it won’t be a number.  It will be a really simple problem: 36 in = 3 x.

The difficulty, of course, is that I’m not sure how to go on from there keeping the same emphases I’ve just talked about.  But observe that this at least would start students on what I’d consider the right track: they deal with equality (36 inches is 1 yard;100 centimeters is 1 meter; 5280 feet is 1760 yards) until it’s not a stranger but an assumption.  The numerator and the denominator must be equivalent.  Rates?  They both have to change.  They deal with ratio and fractions and the exact relationships until they start assuming everything is related.  And yet, all those fine units and equivalences don’t matter, because they can be whatever we want them to be, and we can change them until they are.

I don’t claim it’s a perfect scheme.  As I said, I don’t know how to implement it – I don’t have a plan, just a place to start, and a place that could probably use improvement.  I’m also probably not going to do it, until I figure out (in my infinite wisdom) how to use a textbook to teach a subject instead of teaching a textbook on using a subject, or find time to actually write my own (instead of writing a blog).

But if it’s helpful, here’s my idea.  And if it’s not helpful, at least it’s on the internet in all it’s fleeting perpetual glory, for you to laugh at and me to shake my head over in twenty years.

More on Math

As I have been meaning to do for several years now, I finally looked up a copy of Euclid’s Elements and got to reading.  A complete text – and the one I have been reading – can be found here, in Greek and in a translation a former professor would refer to as a “Bad English Version” done by a physics professor at the University of Texas, Dr. Richard Fitzpatrick.

Compared to the modern text I use at school, I was immediately struck by one thing in particular: Euclid’s proofs rely on a lot of actual drawing, marking off, and otherwise constructing bits and adding pieces to the diagram he starts with.  In contrast, the proofs in my current textbook rely mainly on learning a bunch of principles, and then applying them to other diagrams.

The problem with this second approach, I have found, is that it frustrates most students, who are “proving” things which are perfectly obvious – hey, look, the computer-drawn graphic of a rectangle with some lines on in results in some congruent lines!  Euclid’s approach says, “Fine, smart-aleck, you draw those equal lines.  No, you don’t get a ruler, that’s not how the game works, and anyway how did that ruler get so precise in the first place?”

Off-hand, I can’t remember how my own geometry instruction in high school compared, though vague memories suggest it was somewhere between the two.  I am the sort of person who is perfectly content to play with any sort of logical system for any good or bad reason (up to and including general time-wasting and neglect of chores), so I’m not sure I would have attached any importance to the difference if I did not have to teach the subject now.

Of course, the one thing worse than completely abstracted proofs has to be the ridiculously contrived “real life” problems.  My own conclusion so far is that for most practical applications – not that I have to deal with any often, so I may be way off the mark here – a few essential facts and a basic understanding of trigonometry will get you much further much faster than messing around trying to understand the “application” of the proof of some triangle theorem to a particular scenario.  This is not to write off geometry – the “essential facts” have to come from somewhere.

I am not sure what to make of this – or how exactly to apply it a month and a half into the school year, anyway.  To summarize my current thoughts on the subject: geometry should be a hands-on course, essentially a puzzle game, and stick to obvious applications if any – maybe looking at Kepler’s laws, or something like that.

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?