There is no way I can say this in a nice way and be honest at the same time: American’s are dumb and they are getting dumber.
Worse, they are totally OK with that. If you think this only includes kids in poor schools, or fundamentalists who scream hallelujah on Sunday, you’d be wrong… very wrong.
Recently, the science journal, Nature, rejected an obituary article about one of the most important mathematicians of the 20th century, Alexander Grothendieck — because it was too “mathy” and technical.
That’s right, a journal that claims to be all about science rejected a freaking obituary about a mathematician for discussing too much math… his life’s work! That’s like rejecting an obit about a painter because it talked too much about painting!
The authors of said obituary had to rewrite it and dumb it down. (I have the full version of the original re-posted below.)
Here’s an excerpt of a blog post by one of the authors, David Mumford, on his experience:
John Tate and I were asked by Nature magazine to write an obituary for Alexander Grothendieck. Now he is a hero of mine, the person that I met most deserving of the adjective “genius”. I got to know him when he visited Harvard and John, Shurik (as he was known) and I ran a seminar on “Existence theorems”. His devotion to math, his disdain for formality and convention, his openness and what John and others call his naiveté struck a chord with me.
So John and I agreed and wrote the obituary below. Since the readership of Nature were more or less entirely made up of non-mathematicians, it seemed as though our challenge was to try to make some key parts of Grothendieck’s work accessible to such an audience. Obviously the very definition of a scheme is central to nearly all his work, and we also wanted to say something genuine about categories and cohomology.
The whole thing is a compromise and I don’t want to say Nature is foolish or stupid not to allow more math. The real problem is that such a huge and painful gap has opened up between mathematicians and the rest of the world. I think that Middle and High School math curricula are one large cause of this. If math was introduced as connected to the rest of the world instead of being an isolated exercise, if it was shown to connect to money, to measuring the real world, to physics, chemistry and biology, to optimizing decisions and to writing computer code, fewer students would be turned off. In fact, why not drop separate High School math classes and teach the math as needed in science, civics and business classes? If you think about it, I think you’ll agree that this is not such a crazy idea.
While I’m not all-for dropping dedicated math classes, I do believe he is correct about the root of the problem: education, or the lack there of — which has its own cause in our cultural willingness to be dumb.
A Culture of Wimps
The truth is that most math teachers are poorly prepared — which is not their fault! It’s OURS. Again, I think this is more of a deep-seated cultural problem. We have a culture full of wimps when it comes to mathematics, and that bleeds into high school, college, and teacher-prep programs.
Consider this fucked up dichotomy:
- If someone told you they had never even heard of Shakespeare, and hated all classical music and jazz, you’d probably write them off as not being very smart.
And yet, it’s considered perfectly fine to tell someone that you suck at math, hate it, and did poorly in it. No one will accuse you of being stupid.
This is seriously screwed up and back-ass-wards.
How to Fix it
According to a recent TED study, there are tangible reasons why other countries prepare both their students and their future math teachers better:
- Students get rigorous math education in high school
- University teacher programs are highly selective and demanding
- Teaching is an attractive profession with high pay
Not one of those is true in the United States.
We’re sending teachers off into the wild of math education unprepared to do their job.
That is OUR fault.
More Reasons Life Sucks 4 Teachers
Bottom line: life sucks for teachers. Here’s a few quotes from an interesting article by a scientist who does continuing science education for middle and high school science teachers (quote):
- An urban area high school biology teacher remarked that he didn’t think his students had read 6 pages in a row. Ever. In their lives. Perhaps there was some hyperbole there, but the fact that he could make that statement without any of the other teachers (from diverse districts) looking surprised, made me concerned. He reported that the majority of his students voluntarily take a 0 on an assignment to avoid reading a few pages. Not because they couldn’t do it, but because they didn’t want to.
School districts are constantly adjusting how they teach different groups of students. The definition of terms like “honors” and “low performing” vary from one year to the next, leaving teachers constantly trying to teach the same concepts to a different mix of abilities. As one 7th grade teacher put it, her students range in ability from 2nd to 10th grade.
Perhaps part of the reading problem can be traced back to the fact that many school districts in the state have gone text book free. On the surface one could argue that text books may not be the most dynamic reference text, but they have replaced them with… nothing. Teachers are now supposed to be finding their own materials, but following from the above point, that often means two or three readings on the same material, geared for different abilities. Add the fact that copy paper budgets were not adjusted to compensate, and school districts are running out of paper in February. I guess the assumption was that kids all have computer access at home? I don’t know.
The Alexander Grothendieck Obit
To do my part in perpetuating a culture of non-morons, here’s the unedited version of the Alexander Grothendieck obituary. Keep in mind that this was intended for a scientific audience, and yet it was still rejected as too technical…
What matters is NOT that you understand all of the technical stuff, but that you are able to get a feel for what he cared about. After all, obituaries are supposed to be about a real person and their real life. His real life was (in part) dedicated to the art of mathematics.
Although mathematics became more and more abstract and general throughout the 20th century, it was Alexander Grothendieck who was the greatest master of this trend. His unique skill was to eliminate all unnecessary hypotheses and burrow into an area so deeply that its inner patterns on the most abstract level revealed themselves — and then, like a magician, show how the solution of old problems fell out in straightforward ways now that their real nature had been revealed. His strength and intensity were legendary. He worked long hours, transforming totally the field of algebraic geometry and its connections with algebraic number theory. He was considered by many the greatest mathematician of the 20th century.
Grothendieck was born in Berlin on March 28, 1928 to an anarchist, politically activist couple — a Russian Jewish father, Alexander Shapiro, and a German Protestant mother Johanna (Hanka) Grothendieck, and had a turbulent childhood in Germany and France, evading the holocaust in the French village of Le Chambon, known for protecting refugees. It was here in the midst of the war, at the (secondary school) College Cevenol, that he seems to have first developed his fascination for mathematics. He lived as an adult in France but remained stateless (on a “Nansen passport”) his whole life, doing most of his revolutionary work in the period 1956 – 1970, at the Institut des Hautes Etudes Scientifique (IHES) in a suburb of Paris after it was founded in 1958. He received the Fields Medal in 1966.
His first work, stimulated by Laurent Schwartz and Jean Dieudonne, added major ideas to the theory of function spaces, but he came into his own when he took up algebraic geometry. This is the field where one studies the locus of solutions of sets of polynomial equations by combining the algebraic properties of the rings of polynomials with the geometric properties of this locus, known as a variety. Traditionally, this had meant complex solutions of polynomials with complex coefficients but just prior to Grothendieck’s work, Andre Weil and Oscar Zariski had realized that much more scope and insight was gained by considering solutions and polynomials over arbitrary fields, e.g. finite fields or algebraic number fields.
The proper foundations of the enlarged view of algebraic geometry were, however, unclear and this is how Grothendieck made his first, hugely significant, innovation: he invented a class of geometric structures generalizing varieties that he called schemes. In simplest terms, he proposed attaching to any commutative ring (any set of things for which addition, subtraction and a commutative multiplication are defined, like the set of integers, or the set of polynomials in variables x,y,z with complex number coefficients) a geometric object, called the Spec of the ring (short for spectrum) or an affine scheme, and patching or gluing together these objects to form the scheme. The ring is to be thought of as the set of functions on its affine scheme.
To illustrate how revolutionary this was, a ring can be formed by starting with a field, say the field of real numbers, and adjoining a quantity satisfying . Think of this way: your instruments might allow you to measure a small number such as but then might be too small to measure, so there’s no harm if we set it equal to zero. The numbers in this ring are with real a,b. The geometric object to which this ring corresponds is an infinitesimal vector, a point which can move infinitesimally but to second order only. In effect, he is going back to Leibniz and making infinitesimals into actual objects that can be manipulated. A related idea has recently been used in physics, for superstrings. To connect schemes to number theory, one takes the ring of integers. The corresponding Spec has one point for each prime, at which functions have values in the finite field of integers mod p and one classical point where functions have rational number values and that is ‘fatter’, having all the others in its closure. Once the machinery became familiar, very few doubted that he had found the right framework for algebraic geometry and it is now universally accepted.
Going further in abstraction, Grothendieck used the web of associated maps — called morphisms — from a variable scheme to a fixed one to describe schemes as functors and noted that many functors that were not obviously schemes at all arose in algebraic geometry. This is similar in science to having many experiments measuring some object from which the unknown real thing is pieced together or even finding something unexpected from its influence on known things. He applied this to construct new schemes, leading to new types of objects called stacks whose functors were precisely characterized later by Michael Artin.
His best known work is his attack on the geometry of schemes and varieties by finding ways to compute their most important topological invariant, their cohomology. A simple example is the topology of a plane minus its origin. Using complex coordinates , a plane has four real dimensions and taking out a point, what’s left is topologically a three dimensional sphere. Following the inspired suggestions of Grothendieck, Artin was able to show how with algebra alone that a suitably defined third cohomology group of this space has one generator, that is the sphere lives algebraically too. Together they developed what is called etale cohomology at a famous IHES seminar. Grothendieck went on to solve various deep conjectures of Weil, develop crystalline cohomology and a meta-theory of cohomologies called motives with a brilliant group of collaborators whom he drew in at this time.
In 1969, for reasons not entirely clear to anyone, he left the IHES where he had done all this work and plunged into an ecological/political campaign that he called Survivre. With a breathtakingly naive spririt (that had served him well doing math) he believed he could start a movement that would change the world. But when he saw this was not succeeding, he returned to math, teaching at the University of Montpellier. There he formulated remarkable visions of yet deeper structures connecting algebra and geometry, e.g. the symmetry group of the set of all algebraic numbers (known as its Galois group Gal ) and graphs drawn on compact surfaces that he called ‘dessin d’enfants’. Despite his writing thousand page treatises on this, still unpublished, his research program was only meagerly funded by the CNRS (Centre Nationale de Recherche Scientifique) and he accused the math world of being totally corrupt. For the last two decades of his life he broke with the whole world and sought total solitude in the small village of Lasserre in the foothills of the Pyrenees. Here he lived alone in his own mental and spiritual world, writing remarkable self-analytic works. He died nearby on Nov. 13, 2014.
As a friend, Grothendieck could be very warm, yet the nightmares of his childhood had left him a very complex person. He was unique in almost every way. His intensity and naivety enabled him to recast the foundations of large parts of 21st century math using unique insights that still amaze today. The power and beauty of Grothendieck’s work on schemes, functors, cohomology, etc. is such that these concepts have come to be the basis of much of math today. The dreams of his later work still stand as challenges to his successors.
I’ll leave you with a final quote from Mumford:
The sad thing is that this was rejected as much too technical for their readership. Their editor wrote me that ‘higher degree polynomials’, ‘infinitesimal vectors’ and ‘complex space’ (even complex numbers) were things at least half their readership had never come across. The gap between the world I have lived in and that even of scientists has never seemed larger. I am prepared for lawyers and business people to say they hated math and not to remember any math beyond arithmetic, but this!? Nature is read only by people belonging to the acronym ‘STEM’ (= Science, Technology, Engineering and Mathematics) and in the Common Core Standards, all such people are expected to learn a hell of a lot of math. Very depressing.
Depressing indeed, brother.
Now go lift something heavy,
(hat tip: n-Category Cafe)