Monday, July 30, 2012

Bare Necessities of Math

My wife of nearly 3 years, at one point in a certain job interview here in the US, was required to produce her college transcripts for review.  This itself was expected, since she'd received her degree in India.  In the course of a cursory review, one of the comments she received was that there were no fundamental algebra courses on her transcript.

Her response?  "Well, of course not!  It's a college transcript."

The very idea of basic algebra being a college level course was both shocking and horrifyingly appalling to her.  As well it should be.  I'm a product of the public schools in this country, and I recall that I had to be pushed two years ahead of the standard schedule to get to the point where I was taking algebra through middle school and high school.  And even then, the limiting factor was the schools, which simply didn't offer anything beyond basic differential and integral calculus in high school (and they limit you to 2 years ahead so that you at least have a math course every year of school).  You had to go at least to a community college to get anything beyond that.  Although I wasn't there, I can only imagine my wife's mouth must have been wide agape for several seconds in shock at the idea that the U.S. considers algebra a college-level subject.

And then today I read a little op-ed piece on the NY Times, that espouses doubt on the value of making algebra a necessary math course.
I sighed...  and I placed my face in my palms, and experience much pain.

Algebra??  ALGEBRA??!?  It's pretty much foundational, and yet this guy thinks it's superfluous?

The main crux of his argument is that there are tons of people dropping out of school the moment real mathematics comes into the fray.  Students in the U.S. generally do well compared to other countries when looking at math on the scale of basic arithmetic and such.  It fails utterly when you get into real "math" math.

Now I'm not about to say that the writer of the article is wrong to argue that there are a wide range of professions which do NOT directly apply any sort of algebra, though it is a misconception to presume that those which have no direct usage lack any sort of indirect value thereof.  There is the whole point of quantitative reasoning skills being advanced because algebra is pretty much the first class where kids learn about relationships between quantities rather than operations on the raw numbers themselves.  Conveniently, Andrew Hacker, who wrote the op-ed acknowledges this, but brushes it aside as not the important matter on which he wishes to discuss.

The real argument he tries to make lies with the student drop-out rate.  He quotes some figures from states such as South Carolina (which just so happens to be the state that ranks lowest in education overall), where high percentages of people drop out just as they hit some real math.  He talks to teachers who seem to say the algebra classes are the cause.  Less than half of the people who enter college actually graduate, and everybody blames higher math, apparently, because it seems to be the major hurdle that students can't surmount.  Yes, all this is true...  but the problem is partly a correlation-causation fallacy, but even more than that, it's just bad aim when looking for a target to blame.  Why on earth are none of these people looking at the teaching of math that's making it hard for people to grasp?  Well, for one, you friggin' talked to teachers!  As if they're going to say something that negates their own profession!  Instead they blame the subject itself, as if it's somehow beyond kids to understand it.  If that was actually true, kids in other countries would have about the same failure rate.  They reason it isn't so is because you suck, and they don't...  but I suppose it's way too much to ask of teachers to actually be self-critical.

It's not really a matter of teachers being just plain bad, but about being used to doing things with a downright ineffective approach.  One of the best examples I can point out in this regard is the teaching of basic differential calculus.  Most any of the students who do poorly in a calculus class will have significantly less trouble in a calculus-based physics class.  Differential calculus is pretty basic when studying classical mechanics.  Likewise, integral calculus becomes pretty fundamental in the study of electricity and magnetism.  So what's different?  The difference is that students are given a real context here.  They're not just seeing x's and y's, they're seeing speed, direction, distances, forces, etc.  Things that almost anyone can picture in their heads and give real substance to.  Is it so impossible to approach algebra in a similar way?  I mean, that is how algebra was first created, after all.  And no, I don't mean something like "Johnny buys two more than 4 times the number of apples that Angela bought" types of problems.  Something that will really grab kids' interests.  We can, for instance, make a fundamental trigonometry problem out of finding the distance to a nearby supernova (I know YECs won't like this because one of the simplest examples of this shatters their beliefs).  This is connecting something that kids normally have no other context than a bunch of triangles and waves and showing how it applies to some real-world cosmology.  Of course, it will probably offend the YECs, but they deserve to be offended anyway ;-).

Yes, the quantitative reasoning you use in the field is very different from the classroom.  That is an inimitably stupid reason to forgo the subject.  In the just about 2 decades that I've worked in graphics, I've hardly ever had to factor a polynomial.  That doesn't mean it was a waste of time.  Having the ability to think about relationships between quantities rather than just quantities themselves is really a difference of perspective on math, not just a facet of calculation.  Algebra merely has the distinction of being the most basic of all branches in which you start to get exposure to this perspective.  I would also point out that it is quite important to note that algebra is not a very high minimum bar to set.  Any engineering discipline, for instance, tends to have at least differential equations as the minimum bar, if not beyond.  Trying to remove certain courses just in order to get a higher rate of graduation is not exactly improving education.  You can't make education work better by removing stuff so that more people get all the way through.  It's not an education anymore!

Hacker doesn't seem to get that there is more to algebra than just doing something challenging.  It is considered fundamental for a reason.  Instead, he hurls the accusation that all the programs out there simply throw it onto the curriculum to look tough.  Bullcrap.  That sort of thinking betrays a mighty ignorance of how ALL scientific fields work.  The fact that you think calculus isn't important in the medical field means you don't even get that there's more to medicine than merely diagnosing disease.  Whatever happened to clinical research?  Or is that not part of medicine?  Even the doctors who don't take part in trials at least have to comprehend the data when they read the publications out there...  and that doesn't require any advanced mathematics?  Maybe it would be better if doctors didn't stay up to date with the latest findings.  That's good enough, right?

Why, oh, why, Hacker...  do you consider algebra of all things to be a lofty bar for the likes of poets and dancers and philosophers?  Especially philosophers who need to have a pretty in-depth understanding of discrete math...  Heck, they need to be every bit as comfortable with it as any computer scientist.  I'm sorry, but you are dead wrong to think that algebra is a high bar to set.  It's just one notch up from arithmetic, but it's a whole other approach to looking at problems, and that's what makes it so difficult for people to grasp after a number of years of just rote number-crunching.  If nothing else, we have a problem trying to just leap into this new perspective with the same mechanisms that were applied to arithmetic.  More importantly, exposing students to another perspective on problem-solving is every bit as important in terms of making someone open to exploring other pathways.  Without that, people don't even begin to see STEM fields for what they really are.  You cannot tell me that this isn't every bit as real as exposing an artistic talent to different media.  He's got it wrong because he simply has no concept whatsoever of the real significance of any mathematical field. 

All the while, Hacker waxes emotional about the poor pitiful American student having to be subjected to something so mentally taxing as the most basic of pure math subjects.  If only we got rid of math, we'd not be killing off all the creative minds out there!  It's because Hacker is easily clueless in every way about every engineering field that he doesn't even attempt to acknowledge that there is a creative element to it, too.  More importantly, that getting there requires a strong foundation.  Likewise, he doesn't seem to realize that a field like music is extremely mathematical, and I don't mean simple arithmetic -- statistics, Markov processes, ZF set theory, and so on all come up in the study of music.  Drawing, sculpture, painting, all of them require a good firm structural foundation.

To be fair to Hacker's piece, he does get a number of basic realities correct, and his statistics are indeed accurate.  Yet he overlooks nearly ALL of the finer points of the game here, and as a result, makes the most horrifying interpretations of these facts.  The problem isn't that algebra is hard.  It's that we don't teach it well.  It's not that putting math on the table kills artistic talent.  It's that arts aren't given much of any priority in the first place.  You can perhaps argue that multivariable calculus isn't essential for a sculptor (though one can make a few arguments to the contrary), but that's not the same level as something so basic.  If this guy seriously thinks that it's the subject that's too hard, he needs to take it over again, and again...  and again.  If anything, we need to push it down to grade school.

Yes, America is way behind other developed nations when it comes to math and science education.  Yes, we have a high dropout rate.  Trying to fix the latter by willfully weakening the former is not helping to make the country competitive again.  We already have enough issues with religious fundamentalists trying to make independent thought illegal.  We have a country where 90% of science teachers at the grade school and middle school level have never taken a college-level science course.  And now you see stats that show that the teaching of math is another source of problems...  so your answer is to give up on it?  What the hell good does that do?