Mathematics Education: What in the World Is the Point?

There seems to be some question, in the minds of some (many?), about the value of teaching mathematics in middle and high school, and even whether we should, as a society, continue to institute such education.  Being a science- and mathematics-trained philosopher (and, in some attenuated sense, an historian), I usually find myself defending the humanities against the pervading scientism of, “what’s the point of all these poems, stories, and philosophizing; what does it do, from a practical standpoint?”  When I hear the question, “why should we teach the general populace mathematics beyond elementary school?” I become thoroughly disconcerted —can’t we see the value in any intellectual activity?  I came across the TEDx video by John Bennett (below), in which he says, ‘I am a middle school and high school math teacher, but I have to tell you something: I don’t think what I teach is very important.  In fact, if it were up to me, I would no longer require math to be taught in middle school or high school.’

What concerns me, more than just the fact that someone is audacious enough to give a TED talk on why we should not teach math, but the fact that this individual went into the noble profession of educating young minds without having a very good reason for doing so.  Surely, when one goes into a chosen profession, they have a thorough grasp of why they are doing it and the general merit entailed therein, or at least some value for it, especially when it comes to something like teaching science and mathematics in high school.  Sadly, this deplorable human being is part of the problem, the problem of way too many people moving into high school education , hoping to get a nice salary and summers off, and not even seeing the big picture of what it is that we are collectively trying to be accomplish.  If you don’t know the point of what you are teaching, how is it that you will inspire a youthful mind?  Education does not occur in such an environment, for as Horace Mann said, ‘The teacher who is attempting to teach without inspiring the pupil to learn is hammering on cold iron.’  I would add, in snide (and qualified) agreement with Bennett, that, when you cannot inspire a mind —because you don’t, yourself, substantially value what you are teaching a young mind—, then that discipline might as well not be taught, be it mathematics or otherwise.  What I will proceed to do is subvert Bennett’s negative, nihilistic project, by supplying a positive word about the value of mathematics.  While only the tip of the iceberg will avail itself, it shall cast a shadow upon Bennett’s small-minded and deplorable opinion.

One of Bennett’s claims, in his TEDx talk, is that ninety-nine percent of us don’t need mathematics.  However, it seldom occurs to folks espousing this sort of claim that there are two components to mathematics, the content that directly deals with the mathematical objects of that particular branch of mathematics (e.g., statistics, algebra, geometry, etc.), and the general reasoning ability of dealing with relations, regardless of the relata (i.e., the mathematical objects specific to a particular branch).  There is quite a bit of evidence that mathematical ability does correlate very closely to, or even equates to, the general ability to reason.  It has been my experience that philosophers, among all students taking the Graduate Record Exam (GRE), score rather remarkably well on the GRE; further, unlike other students in the humanities, philosophers tend to score very well on the quantitative section of the exam.  Rough comparisons of score by discipline have been made on Philosiology’s blog (click here) and Physics Central’s buzz blog (click here), but somewhat more accurate analyses have been performed the American Philosophical Association (APA) (click here) and Discover (click here).  Incidentally, despite the numbers given, I have not actually met a philosophy grad student at any mediocre (or better) program who has scored less than 160 of 170 on the quantitative reasoning section.[1]  The point I’d like to make is that students in graduate philosophy programs tend to have done more coursework in math or logic (the latter is a requirement for philosophy undergraduate programs), the latter dealing with relations similarly, as undergraduates.[2]  For instance, while an undergraduate, a colleague of mine at Stanford University’s philosophy department chose to take linear algebra, out of the blue, compiling a record of two math courses (linear algebra and, I think, graph theory) and logic (and, I think, a philosophy of mathematics course).  He was one of few of such colleagues to go on to graduate philosophy study.  I would not doubt that the majority of philosophers, who were able to meet the greatly demanding minimum requirements of graduate philosophy programs, had either taken a math course in college, in addition to a logic requirement, or, if not having taken a math course in college, took math up through calculus in high school.  The point has even been made that the discipline of philosophy, as compared to all other humanities, produces the most scientists (click here).  There is a reason Plato wrote above the entrance to his Academy for philosophers, ‘Let none but geometers enter here,’ and I believe that is because, even in those times, a sound manipulator of relations was capable of manipulating mathematical and more general relata alike; but it is the orderliness and necessity of mathematical deductions that instill this ability in the mind, making necessary study of geometry, algebra, calculus, etc., pace Plato’s requirement.  Historically, a number of wise individuals, Bennett not numbered among them, saw the extent to which mathematics teaches the mind how to reason.  Abraham Lincoln was one of them, studying Euclid’s Elements for the sake of learning how to reason in general, that he might soundly reason in the court of law.  Though it may not be completely clear how this works precisely, that mathematics should teach the mind to reason, there is plenty of evidence that it does.  More importantly, there seems to be no thoroughgoing efficacious way to inculcate the mind so as to heighten capacity of general reason other than by formal study of mathematics, and the thought of minds like Descartes (developer of analytic geometry) and Leibniz (notable contributions in computing, modal logic, and mathematical codification of kinetic energy) may be adduced to further this point, as they were powerhouse metaphysicians[3] who were, coincidentally, trained in mathematics.  The best (verbal?)[4] reasoners, historically and contemporarily, have been philosophers, and so citing the field as an example, I hope, has made the argument clear.

If the discussion is put into context, and we exam the purpose of the public education system, especially with respect to the year following fifth and sixth grade, the attention shifts from developing functional skills that an adult will need (basic reading and counting) to creating high quality citizens.  What society needs from its citizens —i.e., a criterion that makes a citizen high quality— is the ability to vote for holders of public office, which further requires the individual to arrive at well-reasoned conclusions about public policies, etc.  Much of high school is about information content acquisition, such as learning about our nation’s history and foreign cultures, but I think it has been wrongly assumed that mathematics is necessarily about the content, when, really, it is the ability manipulate relations and deduce conclusions.  To my mind, the pervasive lack of appreciation for the content is as sad as the same lack of appreciation for a great book, like Steinbeck’s East of Eden; but it is the capacity for reason that supersedes the value of the content, so far as producing high-functioning citizens is concerned.

One consideration must necessarily be addressed, namely, the reason that so many people seem to hate mathematics, as Bennett is so quick to point out.  I do not dispute this point, but I do think professor Edward Frenkel of UC Berkeley has some important things to say about this matter.  In essence, he says, what if we took an art class and were only taught to paint a fence, but were never shown the works of the great masters?  If, at a later time, we come across the works of the master painters, we will not admit to disliking art or having been bad at painting, but that we were not good at painting the fence.  A big part of the problem, Frenkel thinks —and I agree— is that people are not being exposed to the masterworks of mathematics, nor the inherent beauty of the discipline.

Beyond Frenkel’s points, there is virtually no area of thought that isn’t related to mathematics, and inspiring people is contingent upon linking people to mathematics through other areas of interest, as well as the magnificent theorems and relations of mathematics.  Here are a couple of examples.  Van Gogh’s paintings, in which the post-impressionist technique of impasto is used, very closely simulates turbulence, a phenomenon that is understood mathematically (click here).  Jackson Pollock’s paintings contain a mathematical physics concept, called “fractals,” which are extremely interesting (click here).  These are instances in two areas that interest me, art and physics, inspired me to learn more math.  For one, I was most assuredly interested in others intellectual pursuits (physics, for instance, in eighth grade) that drew me into mathematics.[5]  Then there are books with mathematical puzzles of varying degree for high-school level and early-college level mathematicians, such as Ian Stewarts[6] Game, Set, and Math, an example of the former, and Slicing Pizzas, Racing Turtles, and Further Adventures in Applied Mathematics by Robert B. Banks, an example of the latter.

If anything, mathematics and the ability to reason it lends the mind is needed more now than ever.  Rather than going astray, embracing Bennett’s[7] suggestion of ditching mathematics, altogether, modes of inspiration are necessary to light young minds ablaze.  There is no recipe for inspiration, which is why we need teachers that are passionate about their discipline, not ones that feel that their discipline is purposeless and have taken up the profession to get summers off (or superficially think, without further reflection, that the discipline is “cool”).  Mathematics, like the sciences and humanities, constitutes a rich and autonomous human intellectual heritage that goes back to the ancient Chinese, Babylonians, Egyptians, and Greeks, and has many functions, over and above its formal content, as I have already suggested.

[1] Most likely, this is because of the exclusivity of graduate philosophy programs, which is to say, that most actual philosophers (M.A. or Ph.D.) probably score nearly as well as those students in the sciences on the quantitative reasoning, more than the statistics would suggest.  Many philosophy undergrads go into medicine, law, M.B.A., and other Ph.D. programs.  My experience has been that the majority of philosophy undergraduates who go to grad school go into a field other than philosophy, just to give the reader some idea of the exclusivity.  The Splintered Mind blog has quite a bit of detail on this matter of how difficult it is to get into philosophy graduate programs.

[2] Of course, the relation between math and logic is so strong that there has been a push in the twentieth century to unify the two by reducing the former to the latter.  The view that this is possible is called “logicism.”

[3] I take it for granted that metaphysics is considered a discipline of pure reason.

[4] I qualify, only on the basis that my friends in the sciences may take insult, though comparing is verbal and quantitative reasoning may be like comparing apples to oranges.  It makes no difference to me, being so trained, but I do have an opinion.

[5] My opinion, which goes beyond the scope of this little blog post, is that abstraction taught abstractly ends in disaster, and this is precisely the prescribed mode of teaching mathematics.

[6] I highly recommend Ian Stewart’s books.

[7] The only element of value that I see Bennett bringing to the table is that of analytic enterprises.  Being beyond the scope of this blog post, I will not include my opinion, here.  I will say that analytic games, while in some respects similar to mathematics, are fundamentally of a different, though very important and useful, nature.  My views on K-12 education, especially in the latter half of the progression, are quite radical, and I do not wish to enter into an exposition of and apologia for my views.  However, I will say that the value of analytic puzzles, especially, chess and word games (like Scrabble), has gone virtually unnoticed, being that they are taken to be “mere play,” as if that somehow mitigated their value.

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