On Stating the Obvious

Several years ago, students I was teaching at the time asked that I try not to connect thoughts with words like “obviously” and “of course”. It wasn’t, they said, that they doubted the logical connection of one idea to the next, but that, from their perspective, these were not “obvious” connections but new ones. These students felt that my using these phrases made it hard to ask questions or seek clarification, because they implied these concepts, despite being new, shouldn’t need further explanation and the students were somehow unsatisfactory themeselves as if they did not understand immediately.

Put in these terms, my students’ case made good sense. In fact, I would even say it made obvious good sense – with the understanding, in using the word here, that in this situation I am actually the one learning, and so better able to tell whether a claim is “obvious” or not. I have attempted – with varying success – to avoid these phrases and others with similar implications when possible in the classroom.

In the school, the teacher has an institutional authority, which is not always used wisely. In other contexts, an attempt to provide an explanation or correction by its very nature is a claim to a similar authority. So, in those contexts as well, the goal in assuming such an authority is not to parade your own superiority, but to make those things you already know become obvious to those you address. I have offered an anecdote here to demonstrate how even normally harmless words can hinder such attempts to communicate. It is always our responsibility to watch our words carefully, but this is especially true when attempting to instruct others.

Reflections on an Old Textbook

Having limited access to the local libraries at present, I have been making some inroads on the set of books that is on my shelves but so far unread.  One of these was Dr. Hutton Webster’s Early European History, which appears (from names in the front) to have been acquired somewhere by my parents and scrounged by myself from some stack of books which they had, eventually, decided to pass on.

This text, as explained by Webster in his preface, is a selection and rearrangement from two previous textbooks, his Ancient History and Medieval and Modern History, chosen to meet then-new requirements put forward during the 1910s by New York’s Regents’ Syllabus and eventually the National Education Association.  It appears that a two-year course in European history was recommend for all or certain high schools, of which this volume met a requirement for the study of “ancient and Oriental civilization, English and Continental history to approximately the end of the seventeenth century, and the period of American exploration”.  The book I have read is the second, or “revised”, edition published in 1924.

After the manner of textbooks, each chapter concludes with several questions for study, which take many forms: factual review, discussion of students’ experiences, reflection on famous (or less famous) sayings or statements about the period covered in the chapter, and what amount to prompts for further research: that is, questions, usually factual or comparative, that could not be answered simply from Webster’s text.  Webster recommends that his text be used in conjunction with readings from original sources (of which he himself had also prepared several collections, though appears not to have reorganized these to match the new recommendations: I believe he cites four or five such volumes throughout this textbook).

Whether such original sources would suffice to answer all of the research questions I am unsure, but from my memories of studying such topics, and my guess as to the extent of these “extended, unified, and interesting extracts” such as would be provided at the high school level, I would guess not – which however raises the question of how much additional research students might have been expected to do.  (The answer, almost certainly, is that this varied extensively from school to school even where this textbook was used: what Webster had in mind, as a college professor writing for high schools, I don’t know how to guess.)

As far as his topic goes, Webster’s story proper moves from early civilizations in Egypt and the Middle East (which appear to be his “Orient”); to Greece and then Rome as unifiers around the Mediterranean; then to the civilizations of the surviving (Byzantine) empire, the Arabic Islamic caliphate and its successor states, and the European states rebuilding from invaded Roman provinces, through years of feudalism and Papal supremacy to Renaissance, Reformation, exploration, and colonization; and finally to some account of France and England through the seventeenth century.

This is recognizeable as the “Western civilization” narrative (at least as it’s generally thought of in America – one suspects European authors might not drop Poland, Russia, and the Austrian Hapsburgs, to say nothing of the smaller central European states, out of the story quite so soon).  It appears to be an arrangement intended by the recommendations mentioned above.  Without access to Webster’s other textbooks, either as constructed for this set of recommendations or in their original form, I don’t know how he would have considered this to fit into history as a whole.

Webster appears to have considered himself primarily an anthropologist, and it’s worth noting some of the peculiarities he displays in his introductory chapter and throughout the book.  He considers history to begin with written records, and for writing to be a prerequisite for considering a society civilized.  He considers “savage”, “barbarian”, and “civilized” to be at least roughly scientific classes, the first indicating a tool-using society without metals, and the first two without writing and likely nomadic.  In this summary I am not fully representating the degree to which Webster acknowledges the lack of clarity in these distinctions.

It is worth noting here Webster’s thoughts on race.  He again considers race as it appears in history to be essentially scientific.  Notably he considers the Semitic peoples to be White; and is inclined to see the Pacific and American tribes each as a separate “race” – making five instead of the common three.  However he considers this purely descriptive, likely an artifact of separations in prehistory, and is entirely in favor of what we would call mixed-race relationships: he considers this the obvious thing to have happened in the colonial era, and students are in fact asked to show that mixing of the races is a benefit, if not requirement, for a strong civilization.

On the other hand, Webster does consider that a civilized society is essentially justified in fighting other societies still in a savage or barbaric state, and even subjugating them – although his arguments seem somewhat sophistic.  He appears to assume the barbarian society will always – or as close as no matter – have started the fighting, and is insistent that while conquering the barbarians is all to the good, the conquered people ought to be given equality as soon as practicable: he seems, for instance, to view this as a strength of early Rome, and a failure to completely extend citizenship over later conquests as a great source of weakeness in the later empire.  He dislikes slavery – and while he spends little time on conditions in any colonized area, that may result from the assigned subject matter, or even his editors.   Webster himself seems to have been at least at the fringes of some kind of civil rights activism, at least by the standards of mainstream early 20th century American academia.

At the same time, some of Webster’s judgments are made in ignorance, though whether wilfull or incidental it is often difficult to say, having no really clear knowledge myself of the state of American scholarship at the time.  He does not seem quite aware of the extent of the central and south American native civilizations before European colonization, to say nothing of their North American societies; he considers that only the Chinese and Japanese in Asia had – apparently in his judgment even at the time of his writing – actually reached the point of being “civilized states” which is, by his own criteria, demonstrably false and here I think he really should have known better, though he shows a tendency, as he progresses through the years, to lose track of his essential definition of civilization (writing, with the urban life and establishment of  settled agriculture which he suggests tend to be contemporaneous)  and instead judge societies as only “really” civilized if they possess the most modern technology.

As a textbook, these factual and ethical flaws – together with whatever judgment one may make on the legitimacy of the overall narrative – are its greatest drawback.  Webster’s style is simple, readable, and engaging, and the questions he provides for study, while not entirely consistent in phrasing, number, or seriousness from chapter to chapter, are quite good.

Review: The Silk Roads

Peter Frankopan’s 2015 book The Silk Roads: A New History of the World is best understood as an intellectual exercise in narrative perspective.  As far as I can tell, the book does not depend on original research or even re-interpretation of previous research.  What Frankopan does in this work is to focus entirely on the civilizations of the Middle East, rather than drifting slowly West with progression of technology and global power as is common.

There are obvious successes in this experiment.  Frankopan explores the peripheries of the Roman empire and its western successors in much more detail than the standard western historical narrative allows, from the Persian and other kingdoms that contested Rome’s power to the huge flood of trade between Rome, the Indian states, and other powers – including, at a remove, China.  He describes the rise of Islam and the Islamic states of the Middle Ages in some detail, but including also those which did not turn to Islam – and examining how many of the most successful were those built on an appreciation for knowledge and other fine things, and allowing some tolerance in religion and manners.

The book is also useful, in that it provides an overview of western powers’ influence and interference through the colonial and modern eras.  Bribery and gunpoint were largely the order of the day, together with a general reluctance to treat Asian powers with even the token equality of political niceties.  The most surprising thing to me was the extent to which the British perceived their rival to be Russia – though as the British Empire being seen, in some ways, more important than Britain itself to prestige and prosperity, this is not actually surprising on reflection.

Where the narrative is least convincing is in its treatment of the question of why power moved away from the Middle East – the Ottoman Empire and other regional powers.  Frankopan depends mainly on a technological explanation, to the extent he answers the question at all.  He deals with medieval European powers as largely belligerent insignificant bywaters – not, admittedly, an unfair characterization at many times – and to some extent downplays the expansionism and internal conflicts of the state in the Middle East.  He noticeably avoids the longstanding explanation in the traditional western narrative that European powers in the Renaissance and after benefitted from a renewed interest in learning, and eventually learned the value of tolerance to statesmanship.  It is entirely possible this is done intentionally, another inverting of traditional western focuses: but it is hard to tell.  Usually Frankopan makes it clear when he is making a point, but I would say intention is not signaled in this case.

Frankopan writes clearly but is not a great stylist, and his idiosyncracies occasionally distract from his story.  He is, in dealing with the most recent events, perhaps too optimistic: even another five years perspective casts a pall on his views both of the Arab Spring and Chinese political intentions.  However on the whole I both enjoyed the book and found it very informative.  I would say the author succeeded in his goal of presenting a summary history in a new perspective – though next best, I suppose, to a similar work from an author native to the region – and would recommend it as a way to engage in the exercise yourself.

Sum 15

And now for something completely different.

I wanted an activity for my spare class periods at the end of this week, so I wrote up rules for a card game.  I wanted to use familiar card game concepts, but also put a slight mathematical challenge in, to make it a little more than just a time-waster.  It also had to be something that I could scale to different class sizes.

Cribbage immediately came to mind, an in fact my first run was essentially just the play from cribbage expanded for more players.  This was challenging, especially as I tried to run it with the whole class in one game using multiple decks.

The second run I split the class into smaller groups, and also tried awarding points to multiple players for each play, which one of my three groups got the hang of but proved unrealistic.

The third run I kept the class split up, and reworked the ruleset to essentially what’s below, which looks more like a rummy variant.  15s were changed to 14s, and the limit of 31 changed to a reset on multiples of 15.  I hadn’t worked out a clean method of scoring for this class.

The fourth class I ran again split up, and the last class I ran a game with the whole class and multiple decks.  The fourth and fifth runs came out relatively smoothly, though I wouldn’t call the reception enthusiastic.

“Sum 15” is a working title and referred in the end to the reset rule for multiples of 15.  I haven’t come up with anything better yet.

PLAY
Deal 6 cards to every player
Place the remaining deck in the middle and turn a starter card face up
Each player going around left of the dealer must play a card in turn until all cards dealt have been played
Cards played remain face-up in the middle unless a set is made

SETS
A player who plays a card to make a set picks up those cards and places them face down in front of them.
A set is made if:

  • The last two cards played sum to 14; for sums, Ace is 1, Jack 11, Queen 12, King 13
  • The last two cards make a pair
  • The last three cards are in order, for instance 2-3-4 or Q-J-10

SUM FIFTEEN
If a player can play a card so that the total value of all cards in the middle is a multiple of fifteen (15, 30, 45, 60, etc.), then that player picks up ALL the cards in the middle.
Then place a new starter from the remaining deck face up and continue in turn beginning with the next player.

WINNING THE GAME
When all players have played their six cards, the player with the most cards picked up wins

There’s a limit on how many players you can play with off a single deck.  I think a maximum of six players per deck is the correct rule of thumb, but I ran a game with 14 players with only two decks.  How much of this was my students missing point plays (and so needing fewer restarts) I’m not sure.

Review: Whistling Vivaldi

Whistling Vivaldi, by Dr. Claude Steele, currently provost of Columbia University, is mainly a summary of studies performed to investigate “stereotype threat”, a term coined to refer to decreased performance as a result of perceived negative expectations.

Steele opens by discussing what he calls “identity contingencies” – the fact that some things in life that we have to deal with will depend on who we are or who we are seen as being.  Stereotype threats are presented as instances of this, and the majority of the book is dedicated to examples of various experiments done to demonstrate that they actually exist – and perhaps most disturbingly, can be easily created artificially but intentionally simply by imposing divisions on a group and attaching expectations.

The remainder is spent discussing ways to address the problem.  The method Steele mentions more often focuses on creating positive expectations or otherwise offsetting the negative ones, by using vocabulary meant to be less threatening, by specifically addressing a negative stereotype fear with reassurances, or other techniques to create positive expectations among a population that would typically be stereotyped with negative ones.  He also briefly mentions addressing these problems by making sure that students learn to work in the ways that do work already for groups with high performance.

The circumstances under which the book was recommended to me – to say nothing of the title – suggested to me that Steele’s work would be reliant on anecdotes of mainly emotional value, an impression which proved quite misleading.  In fact I actually enjoyed the book quite a bit and would recommend it.  I found it disjointed in places: the “disjoints” come when he mentions various experiments or discoveries related to his main topic, and then reverts to the main point.  In a way the book is far too short – another way of looking at these rough connections would be to emphasize one of the book’s chief values, that Steele sticks to his point and doesn’t try to do too much.

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?