"Practical Math"

Unlike many members of the educational commentariat, Andrew Hacker, writing the the New York Times yesterday, is more concerned with finding a solution to mathematical under-performance than wringing his hands about the problem.  In my book, this automatically makes him about five times more worth listening to than your ordinary educrat.  Dr. Hacker suggests that the formal math standard in American schools is largely not useful either to most workers or to the man as citizen.  He proposes that math courses should instead of the discipline of formal mathematics focus on logical and quantitative reasoning and application.

While he has a point - "useful" mathematical knowledge is far from as common as it should be - I think he misunderstands the real problem.  He begins by citing the number of students failing and doing poorly across the United States in these traditional classes.  He observes that most teachers are dedicated and competent.  He concludes that the problem must therefore be the material.  "Algebra is a stumbling block..."

As a teacher myself, I believe Hacker has conflated two problems.  Many students, I think, find algebra difficult because they have not mastered arithmetic.  I inherit seventh graders, half of whom do not know how many feet in a yard, and the other half are too unsure to volunteer the answer.  Tenth grade students new to me have forgotten or never learned their perfect squares.  Students at all levels are incapable of or unwilling to do unit conversions.

Algebra I would define loosely as the art of manipulating and finding the values of unknown numbers.  Two critical ideas in carrying out algebraic operations (by which I mean mathematical manipulation performed on expressions - "clauses", if you will - with variables), can be taught simply and directly from basic arithmetic techniques.

The first is the concept of the variable itself.  A student familiar with measurements and conversions can be introduced to the idea easily: he is used to answering questions like, "How many meters are in 2400 millimeters?"  Arithmetically, we set the problem up in stages: find the starting amount, and then from the final units set up the conversion factors - which should be memorized by 4th or 5th grade but in any case are easy to look up.  Algebraically, we introduce the idea of equivalence, and mathematical symbols as a language.  The student already knows that, for instance, meters in "math" are m.  Now he learns that "question words" are represented by a symbol: a box, a question mark, or an x.  He learns that "are" and "equals" are (in basic algebra) equivalent.  So we reach the algebraic statement "x m = 2400 mm".  The conversion factor still needs to be reintroduced in its algebraic place, of course, and enough practice done to learn the methods.

The second technique which is an easy extrapolation from basic arithmetic is that of variable operations.  Polynomials, which are (simple version) expressions with multiple variable powers, can be added, subtracted, and multiplied (and therefore exponentiated) almost exactly like ordinary numbers, but with the different powers indicating "place value".  (Division, while also analogous, is slightly different and therefore harder.)  So a student who knows why his arithmetic works the way it does, who has been taught the decimal system, will recognize "place value" and make the connection in most cases without significant difficulty.

In short, I believe the root cause of high school failure is far more likely to be inadequate elementary education than difficulty with the material.

While I think Hacker's reasoning as to cause is flawed, his main concern - is this worth teaching anyway? - is a question worth asking.  I think the answer is still "yes", but with caveats.  I have blogged before about the state of mathematics textbooks in the modern United States: torn between traditional American mathematics and the "unified" approach of much of the rest of the world, having to teach or re-teach concepts not taught successfully before high school, fascinated with fiddly properties at the expense of overview, and so forth.

Algebra or Geometry as disciplines, though, are branches of a formal mathematics which exists - as people realized thousands of years ago - not only for practical purposes but as a training tool for the mind and as an art.  Algebra is valuable for much the same reason that a foreign language is useful, or music, or memory: to cultivate the human spirit in all its facilities.  Certainly most people will not use hyperbolic equations or non-Euclidean axioms in their day-to-day life, any more than most people will use French or German.  Certainly you can enjoy the original Hugo novels - you can also dabble in Newton.

The practical mathematics which Dr. Hacker champions is of course still valuable.  It is found in applied form in physics, chemistry, economics - all things which we say should be normally studied as useful.  But its basic tools can and should be provided before high school.


Identity and the State

President Obama made headlines recently for a speech in which he emphasized the societal foundation of any person's success.  Unfortunately for him, though I believe revealingly, he fell victim to the soundbite era: "If you’ve got a business - you didn’t build that.  Somebody else made that happen."

"And Mr. President, if you become the president, who made that happen?"  This would be a weak question: I believe that Mr. Obama's strongly stated beliefs originate in his own experience, that of accepting and working in the machine.  I suspect his irritation - at "guns and religion", at those challenging ACORN and other interest groups, at the steadfast unwillingness of the American people to give in to his vision - is honest, that it is the bafflement of one who has accepted a view of the world and found it not held as widely as he expected.

On a superficial level, the President's viewpoint is attractive and reasonable.  It contains some idea of a community, working together, holding to a common good; it presents some echo of the polis which, idealized by Plato and Aristotle and Cicero (if in different ways), resounds through the centuries of Western history and still informs the world today.

But Mr. Obama's view has a weak point: a crisis of mis-identification, or at the very least a crucial identification which he has failed to convince the United States' people is either crucial or a true identity - and perhaps more damningly, he has failed to understand that the point is still under debate.

For Mr. Obama, as for much of the academic, legal, and political elite he has grown to belong to,  the people, and specifically the will of the people, is identified with the government instituted by the people.  He does not follow the earlier understanding, where the government was the representative of the people; or even the older idea that the government stands above the people (by breeding or divine will - or both) and is therefore responsible for them.  A representative who represents poorly can be changed.  A father, judge, or a commander, held up as such by the thinnest string of theory or tradition, can be called to account and shamed - or deposed.

Instead in this modern conception, the government cannot be held in check by theories.  A government defined as "the people's will" simply by the fact of its existing as the government has nothing that can be called up against it except natural or divine law, and those having been busily undermined in the name of "reason" and "tolerance" and "relative morality" because everything originates with the people's will - which is, as we already found, simply the government.

I believe President Obama is confused by backlash against his policies.  As the chief executive, he appears is the supreme embodiment of the popular will which is to be enshrined, and he has been busily enacting all the policies he knows are best.  The fact that law and tradition stand in his way is not to be considered an obstacle, in his view: laws change.  In the President's view, I suspect reality changes.  He does not understand, does not know what to do with, a people who believe in lasting truths, in an actual rule of law because it is the law, in not a good or good policy or today's best option but in the Good.

Yet this philosophical confusion is not really the most damning indictment of his speech and his policies.  After all, if he is right - that is to say, if my speculations about Mr. Obama's worldview are correct, and that worldview is itself factual - then he is doing the right thing, and is merely a poor politician, or at least a tactless one.

The worst problem with his attempted solutions is that even on his own terms, it ends up either backwards or tyrannical.  If we take the more positive view, we stop to ask him who these people were that helped.  Certainly, a state of peace is maintained by the government, the servants of the people: the soldiers, police, road crews, and so forth.  But these are for maintenance only.  The next question leads us to the practical problem faced by these bureaucratic-faced dreamers: who takes us beyond subsistence, and who pays the ones who help us to subsist.  The awkward fact that President Obama has yet to acknowledge is that it is the successful men - the Mr. Romneys and Mr. Gateses and, yes, Mr. Obamas of the world - who help the rest of us: by paying outsize shares of taxes, by hiring and training workers, by providing necessities and comforts and then even luxuries.

This puts us in their debt: it is not right to go demanding that "the rich" pay "their fair share" when they already pay, relatively speaking, that and more; when they not only provide those moneys to the government but in the private sectors hire and pay us who are not so wealthy - and to return to the public arena they largely, for good or ill, provide our government from their number.  It may happen, in this day or in a time of war or or stress, that we need not only a normal contribution from these men but an unusual one: but justice demands that, even if we ask it, we ask it - however formally and with whatever force of law behind it - recognizing that we take from them for a common good or need, and not as if demanding a debt owed to us personally.

This is the charitable view: that President Obama, used to seeing so much given out regularly, has forgotten it is given, and sees it only as taken and that by right.

The uncharitable view - which I hesitate to ascribe to him, but you will see argued by others, I have no doubt - is one that reduces to serfdom any private person who is not himself an agent of the government.  More than mere forgetfulness, or blowing on some wind of the age, this would assert that the government has the right and duty to demand whatever is "necessary" - and has forgotten the older theories of reciprocal duties, so that now all is owed to the government, and the government's dole is charity.  And here we find the sinister descendant of what I mentioned above as Mr. Obama's probable political philosophy: we observe that the government has become the State, and the Will of the People, and the actual people reduced to so many resources to be commanded to produce.  No doubt Mr. Obama considers his a benevolent guidance, but he gives no indication that he sees any right of anyone to disagree with that guidance.


Statue Stupidity

Evidence presented during the trial and conviction for child abuse of Jerry Sandusky, former defensive coordinator of the Penn State football team, all-but proved - as had been suspected already from new reports - that there was substantial cover-up of Sandusky's crimes by university officials and staff, probably implicating the late coach Paterno among others.

Some hooligans are suggesting that Paterno's statue be taken down.

Absurd as it is for Paterno to have had a statue on campus while still alive, not to mention still coach, it was and is a measure of what he has meant to the image and idea of the institution.  The image is tarnished, perhaps forever, but taking down the statue is not going to help things.

Should the United States remove Thomas Jefferson's memorial from its capital because the man kept slaves, and perhaps abused them?

Should Nelson's statue come down from Trafalgar square because he was an adulterer?

Should you have to put on your resume that you've "borrowed" supplies from your office, exaggerated your hours, gossiped about your boss, encouraged and laughed at stories that weren't true, yelled at your children because you had a bad day?

Paterno's legacy before this scandal was that of a great coach and an upright man; if it turns out he was more human than we realized, then Diogenes is right once again.  While we cannot trivialize the evils committed, it would be willfully ignorant and vindictive to pretend not to see the case from Paterno's side; what should you do when a good friend and subordinate is charged with something this abominable?  If you prefer to believe Paterno's motives were more venal, it is still not difficult to understand the desire to sweep under the rug something which would threaten his prestige - and maybe even his job: coaches have been fired for less cause than crimes committed by a subordinate.

If Penn State takes down the statue, what they say is that they refuse to acknowledge the good he did that earned him his place; that they want to disassociate themselves from something like the last sixty years of their sporting history.  If the crime was Paterno's, then such a heinous act might deserve the erasure.  But when the misdeed is not even his, and in fact evidence suggests that what allegations he knew he dealt with as required legally - even if he failed morally by not ensuring justice was timely - then we are looking at overreaction.


No, But Thanks

About a year ago, a passed a high school student I had never seen before on the sidewalk while meandering around my neighborhood of Del Ray.  He stopped and asked me how I was doing, which I thought was a little odd, before he continued, "Didn't you used to teach science at T.C. Williams?"  To which I had to answer that I had not.

Last week, I was picking up students from the airport for our second session, and one particular plane I was waiting for also contained a contingent of young Marines, in what the sci-fi books call "civvies" but which the Corps no doubt has an idiosyncratic term for, clearly looking for a handler, I mean superior.  Of the dozen or so, nearly every single one took a couple steps towards me before correctly identifying me as Not A Marine.  (I realized later this might have had something to do with the khakis and blue t-shirt I was wearing -  equally it might have had something to do with my parents teaching me to stand up straight.)

At contra on Friday, an older woman I was talking me remarked that she could tell I was "obviously a dancer" from the way I moved, which I found amusing considering the students I teach and the fact that all my family I have almost certainly done the least dancing.  (Once again, perhaps all those admonitions to stand up straight and not scuff my feet had some effect?)

I seem to be a person who looks like people.  Perhaps I should have been an actor - or at least a stunt double?


Notes on Giselle

On Thursday evening I went to see the Paris Opera Ballet's performance of Giselle.  Unfortunately, the vast majority of everything I know about ballet I have learned some time in the last ten months by osmosis, so this is more a ramble than an actual review.  I have now reached the point where I can usually spot the big mistakes, of which there was one: the male lead stumbled badly during a variation in the second act.  The third male dancer with a solo (if you have a solo, but your character does not have a name, where does that put you in the hierarchy? I have no idea) also was somewhat unsteady during his pas de deux.

I still find it mildly confusing that ballets are often, if not usually, referred to as [choreographer]'s [ballet].  I realize that the dance is the main thing, but I am much more familiar in most cases with the music.  For instance, the Mariinsky will be in town in October with what I would have naively called Prokofiev's Cinderella, but the playbill calls it Ratmansky's Cinderella.

At any rate, Giselle's music was composed by one Adolphe Adam, a French composer who I happen to know now was almost exactly a contemporary of Berlioz, though musically less ambitious (and perhaps therefore less famous).  For the most part the score is good but not superb - fairly unremarkable, in fact.  However, at one point a climax of the dance of spirits in the second act is marked in the choreography by a repeated figure of what I will call a traveling arabesque.  Physically it is incredibly impressive; and the music here abandons its quiet eeriness for a dramatic build.   Unfortunately, Adam's score is here, if not actually in a major key (and I think it might be), still somewhat martial rather than ominous, making the effect somewhat comical, which breaks the mood rather thoroughly.  It was, true, not helped by the stage of the Kennedy's opera house, which registers every footstep, and thus had the effect of emphasizing the beat and therefore the marche of the music.  (I imagine that is a common problem though - I can not think there are too many stages anywhere in the world specifically designed to deaden the sound of pointe shoes.)

On the whole, though, the show was excellent.  The company struck me as a very technically focused one (where the Bolshoi, which did Coppelia a while ago, seemed more dramatic - though the ballets themselves are also very different).  The set was superb - I am still accustomed to college and community theater sets, so I find these fancy professional ones especially striking by comparison.  I particularly noticed the acting: while ballet acting is by necessity somewhat stylized, the "style" here was not overwhelming the story.  I imagine that there has to be some tendency similar to the actor who keeps trying to "do Shakespeare" instead of focusing on his part, but it was not evident here.