Some Thoughts on Mathematics

It is a well-rehearsed complaint that the United States is "falling behind" the rest of the world in technical studies - meaning mathematics and science.  It is paradoxical, then, that the general United States curriculum features specific study of advanced mathematical topics - Algebra and Geometry.  Sometimes requirements have even advanced up to basic calculus or at least its underlying principles (indignified with the moniker "pre-calculus").  Most other countries teach Mathematics, perhaps glorified with a grade number, throughout middle and high school.

Speculating on the explanation for this oddity, I would suspect something like the following happened in American education: Years ago when "high school" would have been considered fairly advanced education, algebra and trigonometry and other advanced math would reasonably have been studied, possibly as electives; essential math would have been learned earlier.  Mathematics instruction likely suffered from the overall decline in American education over the (later part of the?) 20th century.  The language problems, I suppose, are documented better because writers felt them more critically.  Either the courses were never dropped in name, or people noticing the problems codified the "Algebra" and "Geometry" to be studied; however the rigor of the courses was slackened and the material diluted with the addition of other elements which were no longer being learned earlier - or which were part of newly expanding fields and felt to be "necessary".

As evidence, I have mainly the (I am tempted to say absurd) amount of review contained from year to year in the average modern US mathematics textbook series.  Admittedly mathematics is a subject I have some affinity for, but in my estimation the material generally spread over the three years from pre-algebra to second-year algebra either could be condensed into at most two years; or should be taught earlier; or is (at least from a conceptual standpoint) superfluous to the subject at hand.  An "Algebra II" class - I am speaking here from experience - by the end of the first month of classes can still be reviewing material theoretically learned up to three years before in "Pre-Algebra" (possibly even the first semester of that course) - linear equations.  True, the more advanced course has more detail and harder problems; I am not convinced that excuses the state of affairs.

I do not think - at this point in my career, at least - that the problem is the (nominal) focus of American courses.  The integrated approach adopted most famously by most Asian schools does have its advantages, mainly in maintaining a unity in the discipline; the focused study of a particular branch of a subject has off-setting advantages, mainly in ability to explain details rather than teaching by rote.  In disadvantages, the "subject-based" approach of American textbooks does tend to create artificial distinctions of one thing from another that ought to be known as related; however, an integrated approach obscures the focus achieved by distinguishing between the parts of a field of study.  Beyond that I am not qualified to comment - except to mention that more American textbooks seem to be actually using an essentially integrated approach, while maintaining their supposed subject matter in name only.

A more pressing problem is the study habits expected of students.  When a conscientious American parent wants the student to master a subject, and is willing to sacrifice grades if necessary, all is well and good.  When a lazy parent (to say nothing of the student or teachers) with "high expectations" wants the student to have an "A" but could care less about the subject, there is a problem - and the Asian parent determined that his child will have an A and prepared to expect that and make him work for it creates an advantage.  (I dislike relying on overplayed stereotypes, and can say from personal experience that - which should be obvious to any observer of human nature - not every Asian parent is in this regard an "Asian parent".  Some of them are quite prepared to look the other way on instance of, for instance, cheating; or even to berate teachers who attempt to discipline that behavior.  On a general comparison, however, the stereotype does hold true for fairly clear cultural reasons.  The comparison is therefore useful.  Also, I know next to nothing of European education, which is the other reasonable comparison point.)

So much for the problem.  What about a solution?  One obvious comment is that the best method of improvement must be increased expectations, even demands, on the part of parents and schools.  The effect of this simple change can be most clearly seen in modern America within the home- and classical-schooling community, where concerned parents created a demand - and have often been part of creating a supply - of improved, or at least diverse, curricula for language study both in English and the Classics.  Contrast this with mathematics education, especially in the public schools, where the number of available curricula in print has been steadily decreasing.  I believe there is a total of two courses remaining put out by large publishers, and one has much greater presence as best as I can ascertain.  My knee-jerk reaction is to blame Federal government meddling with standards, together with a human inclination to follow the Next Greatest Thing, but I could be wrong - the point is that with no serious competition and a largely captive market, innovation and diversity seems to be quickly becoming a matter of who has prettier pictures.

My own suggestions are limited by unfamiliarity with lower elementary curricula.  A makeshift solution I would propose as an upper school teacher would be to spend seventh (and maybe eighth?) grade focusing on practical math, starting wherever necessary and working up to whatever difficulty level is needed for scraping by (and hopefully more than scraping by) in life but without particular emphasis on unifying principles of "subjects"; the unifying principle of mathematics generally is (in my opinion - this would require an entire other essay) description and problem-solving.  With that foundation in place, you have a fourteen-year-old student who is capable of doing things like working out a simple budget (say, not overspending his allowance), working out basic problems in compound interest, calculating the price of a room's carpet, or not drawing to an inside straight (for reasons other than every book ever written with a card game in it saying so, not that that is a bad reason).

A high school can then teach those subjects which will be either necessary to a future career (as an engineer, architect, or the like) as electives, and require those useful to the formation of a thoughtful citizen ("Let no one ignorant of geometry...").  The question "how much math?" is a fascinating one I am not prepared to answer in the general case.  I am prepared to say that it would be easier to teach algebra thoroughly if a text did not interrupt - and perhaps need to interrupt - the fairly logical progression through the various variable functions with silliness about translations which should have been learned earlier, and bits of a trigonometry (for instance) which could either be learned later or cheerfully ignored.  Every calculus textbook I ever remember seeing not only confined itself politely to a strictly logical presentation of the calculus, but made assumptions of what the student knows.  If we imagine a calculus textbook written on the same principles as the average "Algebra 2", it would start by presenting methods to solve equations in one variable and would reteach the quadratic formula; in between types of derivatives there would be a discussion on graphing complex numbers.  An actual calculus text teaches a subject; I am not sure what the algebra book is trying to do.


Review: John Carter

My roommate has a friend who has a fast car and (apparently) too much free time, so the two of them and myself and my other roommate and another friend who was visiting went off last weekend to see John Carter at a movie theater farther from where I live than was strictly necessary.  In case you have not heard yet, John Carter is a terrible movie (this is objectively true) based on the first of Edgar Rice Burroughs' badly dated books (this is my opinion), A Princess of Mars, about John Carter (surprise?) and his adventures mostly on Mars.

There were things I liked about the movie anyway.  The visual character and setting design was mostly well done - and shamelessly true to the spirit of the book, for whatever that is worth.  The alien aliens, not to be confused with the aliens-who-are-actually-human-(sort of) are also done reasonably well.  In fact, for about the first half hour, I admit I was thinking, "Huh, this might actually be better than the original book."

That feeling did not last, and the many reasons why can be summed up by saying that the filmmakers have no idea how to tell a story - specifically, no one told them that trying to tell two types of story in the same movie takes serious skill and is incredibly hard.  On the one hand, the story is one more variation on Rescue the Maiden Fair; on the other hand, the villains are being egged on - even controlled - by the Ancient Conspiracy.  It is clear that someone involved knew this was not going to work, because while the first plot is resolved the second plot (er, ha) is neither explained in terms of character (any apparent motivation on the part of the behind-the-scenes schemers is nonexistent) or resolved in terms of story.  And yet they are still there - maybe it was somebody's pet brainstorm.  As far as I can tell, the second angle exists only as a sequel hook, but a word to the wise: sequel hooks are bad ideas if the movie is terrible.  (On the other hand, if they had removed the characters in the Big Bad Conspiracy from the movie, but left the Superweapon they give the bad guys, the movie would have been about 500% better and it would have a sequel hook.  Did no one suggest this?)

The end result is a chaotic mess: the moral of this review is not to see John Carter, if it is even still playing in a theater near you (it was a flop and will almost certainly end up losing money, so I suspect it may be out of many theaters, especially with that other movie everybody is talking about...)


For the Love of the Game

The soccer season is officially upon us, beginning for me yesterday when I received my initiation as a highschool referee in a JV & varsity double header.  As a refereeing effort, the varsity game (in which I was probably fortunate to only be on the line) might charitably be called a disaster - but at least no unjust goals were awarded or denied, and the clearly better team won.  At some point I would like to play again, and probably coach at some point, but for now it's just good to be near, to be part of the beautiful game - even when some particular matches are fairly ugly.


The Hard Heart of Conservatism

"...If there were a king over us all again and he sought the counsel of a mage as in the days of old, and I was that mage, I would say to him: My lord, do nothing because it is righteous or praiseworthy or noble to do so; do nothing because it seems good to do so; do only that which you must do and which you cannot do in any other way."
- the advice of Ged the Archmage to the future king Arren in Ursula K. LeGuin's The Farthest Shore
The great rhetorical Achilles' heel of conservative reasoning is that almost by definition it subscribes to the aphorism of some supposed sage, "If it is not necessary to change, it is necessary not to change."  By trivial example this can be shown an untenable philosophy if subscribed to as a Kantian imperative.  If I have by habit drunk a glass of red wine each night with dinner, there is still no harm in drinking a glass of white, or shocking all traditionalists whatsoever by sampling the hard cider, or even abstaining from alcohol altogether.  The conservative therefore who defends the current status quo, whatever it may be, finds himself faced with trying to explain away particular supposed injustices for which a progressive (no matter when or in what form the progress is advocated) purports to have a solution; the conservative who argues against measures already implemented in the name of progress finds himself arguing for a hypothetical - a tricky proposition at best - or else dismissed as nostalgic, no matter how accurate his analysis of the problems brought on by the newfangled ideas.

But despite the rhetorical difficulties, the wonder of the human condition is that life goes on: sometimes better, sometimes worse; sometimes with more variation and sometimes with less; sometimes with tragedy and sometimes with wonder.  All the brilliant policies of all the brilliant men brought together in all the superlative conferences and committees in the world (quite often, I must confess I find them, summing to far less than the total of the parts) have not managed to account for the mysterious ways in which humanity moves.  To take but one example, economists of great renown are reduced either to noting the obvious facts of life - that I can charge a higher price for Chinese silk than for American wool, or that a man who can write makes in general more than one who cannot - or to propounding the most absurd theories - as when they suggest a government with no actual money and a vast debt can improve a nation's status by borrowing still vaster sums of money.  (If by some miracle - which I doubt - that last is accurate, the absurdity of the theory is not reduced, but the absurdity of the world may be greater than we like to admit.)

All of the evidence suggests that we have very little idea what we are doing when we attempt to alter things, that it is easier to disrupt normal lives than to improve them intentionally, and that the most unfailing of natural laws, outside of the law of gravity, which attends human activity is the one of unintended consequences.  All of this I take as evidence supporting the presupposition of modern so-called conservative thought: that the government should do nothing that is not necessary.  The difficulty of course is that it is hard to tell what is necessary; it is harder still to restrain humanity from attempting much more.  Any fan of baseball recognizes the sinking feeling that arrives when you realize that some journeyman outfielder with half a bat is determined to swing for the bleachers rather than go through the trouble of working the count.  Of course he occasionally comes through, but most of the time - there's the dejected walk back to the dugout after the third strike, or the half-hearted jog down the first base line as the ball floats into the night sky for an easy fly ball out.

This is all to say that when I call myself a conservative I mean that I start with the self-evident proposition that the government does not in general know what it is doing or what the sum of the consequences will be; I draw the conclusion that it is best to restrict the government.  This results in the conservative saying to the anonymous man in pain, "I do not know what it is best to do for you; I certainly do not know whether what is good for you is also good for your friend down the street; therefore I refrain from issuing directives."  The liberal may or may not agree with the premise; however, he says, as allegedly FDR said, that doing anything about a problem is better than doing nothing; that it even makes more sense to try a bunch of things at once and hope some of them work.  The conservative restricts government because government is human and therefore fallible; the progressive tries to work through government because government has power and therefore the possibility - which most progressives see as a likelihood, if not an inherent tendency - to do good.

The conservative view then ends up seeming clearly the more cynical, the less (with apologies to Bush II) shallowly compassionate, the more, as I titled the post, apparently hard-hearted.  And yet, it is also more realistic: all of the programs and redistributions of empowered government have not ended poverty, or reduced familial traumas, and there is no evidence to suggest they ever will.

So we return to necessity.  (I am coining a word: I am a necessistist.)  What is necessary?  There we can begin a debate; but at the moment we do not have such a common starting point.  We have a progressive movement advocating the use of power "for good" and then the doing of anything that seems to be a good idea; and a reactionary conservatism, feeling itself under pressure, retreating even past its foundations to the (almost equally absurd) point of fighting to do an actual nothing.


Suspicious Looks at "Gay Marriage"

I have three main objections to homosexual "marriage", so called.

The first is that I believe homosexual activity is (theologically) a sin and (biologically) quite different from heterosexual intercourse - you know, in the part where it can't result in children.  The church institutional, I am confident, will continue to recognize (and even publicize) the distinction, but any time there are significant differences between the proscriptions of one part of society, and society officially as a whole, trouble tends to follow.

So stemming from that, the second is that I worry that, if the government recognizes homosexual couples as "married", the forces of Caesar will quickly be called on to try to force the church into line.  This possibility was always present, but has become infinitely harder for activists to disavow with the current administration's dedication to running roughshod over religious (and other) freedoms in the name of "women's reproductive health".

The last is that the language involved makes no sense.  If marriage is the recognition of a formal (and spiritual) union between a man and a woman, then talking about "homosexual marriage" is ludicrous, because of the bit where there's not a woman (or a man).  If marriage is merely the formalization of a sexual relationship between two people - which, to follow the modern fads, is not actually morally binding and can of course be dissolved at any point by divorce - then what is the point of marriage at all?  If you appeal to "family" - which is to say, children - that is not a point in favor of the homosexuals, because homosexual sex does not result in children.  True, such a couple could adopt or use some other method to, ah, acquire them.  But that doesn't need a marriage, that I am aware of.

In short, I am against homosexual marriage because the only possible purpose of such a policy would be to attempt to legitimize something I do not think is legitimate, and - even more of a concern to my logical mind - equate two things I do not think are equal.


You Are Taking My Money

Or, "Well, not that different I guess."

One of the things I find most perplexing about the recent HHS debacle is how, after an absolute minimum of pull-back by the administration, most liberals I know went right back to the "oppressed women" screed, despite the fact that objectively (as opposed to formally or technically) nothing has changed with regards to the issue of freedom of religion.  To forestall some comments, yes; I posted before saying I found the manner of the legal protection of religious freedom potentially problematic, logically speaking.  I also added subsequently that intentionally and contextually, it would be hard to improve on the amendment as written.  And even if I did find it objectionable, the fact is that it remains the first law of the land (and favorable to me), and so should not be run over roughshod by a bureaucracy empowered by an (itself probably unconstitutional) law - much as I, despite the significant amounts of alcohol consumed in my life, and the absurdity of most US drinking laws, still would do my best to stay within those laws.

But let us pretend for a moment that the Health Services contraception mandate was not opposed on the basis of any religious objections or Constitutional concerns.  Let us pretend that some of the forms of contraception and infanticide covered by it do not touch on any vexed moral questions - and however clear your own opinion, what else are we supposed to call, politically speaking, an issue where half the country disagrees with the other half?  Let's say, for the sake of argument, that I don't even object to what you are doing myself and am just being a mean old Scrooge.

I still get a say in the law.  I can still object, with perfect justice, to something I dislike.  "How so?" you scream in outrage.  "It's my body!" Well, of course it is.  I don't care what you do with your body.  (I mean, obviously, within limits.  I don't want you and your body to kill me, or steal from me, or... I could go on for a while.)  But the part I get upset about is when you try to use my money to do something I don't want to happen.  Oh, it's not my money?  It's just a government handout?

Where do you think the government get the money?  That's right, me - and a bunch of other people.  (And you - but if you're just spending "your share", why does the government need to have it in the first place?)

So in conclusion - forget you and your panicking about "rights" you need somebody else to pay for.  Your rights are not in danger.  No one is trying to tell you you can't do it yourself.  But you are taking my money, and I don't want you spending it like that.  I need exactly zero justification beyond that to present a perfectly reasonable objection.