Shari sent around this article today. The whole 25 page pdf is a fairly entertaining and thought-provoking read -- I recommend it.

To tease your appetite, here is my favorite quote from p.24, when he's doing "an honest course catalogue for K-12 mathematics":

"TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and symmetry. The measurement of triangles will be discussed without mention of the transcendental nature of the trigonometric functions, or the consequent linguistic and philosophical problems inherent in making such measurements. Calculator required, so as to further blur these issues."

The central thesis is that mathematics is an art, a way of thinking about and interacting with the world driven by curiosity and a drive to elegant, satisfying logic, while the mathematics curriculum taught in the US for the past many years has been equivalent to teaching only the notation and techniques of that art. The (effective) analogies he uses are teaching music by teaching only how to read musical notation and understand music theory, and teaching art by teaching only how to understand color theory and paint by numbers.

On the one hand -- he's totally right. The math curriculum we teach our students in the U.S. today *is *artificially divorced from its history and its natural curiosity and elegance.

On the other hand, I think he's a little bit too blithe in the way he dismisses the fact that the kinds of math notation and techniques taught in our schools are actually needed for most of the high tech professions in the world today -- computer science and statistics in all it's multitudes of applications and every kind of engineering and physical science and even the life sciences as they drill down into protein structures and DNA analysis.

It's as if one attacked an English curriculum for only teaching students to read and write with correct grammar and never engaging them in literature. A valid criticism, but if you taught them all about literature without ever teaching them to read and write themselves, it would be equally unbalanced. Lockhart acknowledges the need for balance here, but if I have to choose between my kids joyfully exploring the elegance of number theory and my kids being able to do arithmetic, I chose arithmetic. Maybe that's too pragmatic, but I want them to know whether they're being cheated when their change is handed back to them just as much as I want them to be able to read the news or a sign posted in a store or directions at the airport. Or maybe I just lack the imagination to picture a curriculum that effectively accomplishes both, simultaneously, across every school in the nation.

Which brings us to my third hand (just call me Vishnu): he also readily admits that most of the math teachers in the U.S. today (including myself and most of my colleagues in this certification program who are majoring in math) were "raised" in the current system, and have not ever done much (if any) real, artistic mathematics themselves. He doesn't really propose any solution for this, but unless all the math teachers in the nation start suddenly reading Pythagoras, Newton, and Green on their summer breaks or attending boot camps taught by math professors who can bring themselves down to the secondary-teacher level, I'm not sure there's a ready remedy. (I actually think the second part of that is harder to put together than the first -- my favorite quote about teaching math came from a friend who was teaching college-level calculus for the third time: "Every time I teach this, it becomes more obvious.") Also, we'd need to rewrite the standard curricula that have become so widespread -- just as a side hobby.

So... an interesting, thought provoking article. Which brings me back to a theme that's emerging in several of our classes. It goes like this:

- Standardized testing is a bad way to test real learning, and frequently drives bad teaching that attempts to achieve good standardized test results but fails to enable good learning. (e.g. a math curriculum divorced from the art of real mathematics)
- But standardized tests exist for a few reasons: we need to be able to measure the outcomes of our education system in a way a) that does not absorb all the free resources of time and money in the world and b) allows us to objectively compare outcomes across classrooms, schools, districts, and states.
- We need to be able to measure outcomes in an objective way because it's the only way to know where we as educators are doing well and where we need to improve, at the student, teacher, and school levels.

So... what solutions exist or could be created that allow us to measure outcomes in an efficient, objective way that also measure real learning (such as the artistic ability to struggle with a math puzzle and come up with an elegant solution)? Or, should we concentrate instead of making all teachers capable of good teaching that drives real learning and also (incidentally) enables good standardized test results?

So many great thoughts swirling here -- the issue of conceptual understanding versus "formulas" is one that many in education struggle with given the enormous number of curriclum objectives. Jeff recommended blog.mrmeyer.com to me a while back -- he wrestles with concepts like the ones you discuss here regularly. As a non-math person, it gives me a new way to be excited about math.

ReplyDeleteI'm not sure I can add much here in terms of the overarching discussion. I certainly see the tension, and know that it is in some ways a fundamental one; if you could spend hours of personal time with each kid, perhaps you could always get across a deep understanding of mathematics, but that is not realistic. I do believe that people who have that deep understanding will do well on the standardized tests and keep their practical abilities better than those who don't, but getting kids there is hard.

ReplyDeleteOn a more practical note, I really enjoyed Steven Strogatz's columns in the New York Times this year. In the second column (http://opinionator.blogs.nytimes.com/2010/02/07/rock-groups/), he talked about ways to make math more concrete (heh) and elegant. He mentioned a book (http://www.blpbooks.org/books/mathlament.html) that I have not purchased, but would like to read eventually. It may have some good ideas on ways to make that elegance more obvious and speed the learning process.

Okay, I'm an idiot--I lost track of tabs and forgot to read the column. But interestingly it brought me around to the same person. I guess my suggestion is that there are ways to teach this deeper understanding of math, but perhaps we haven't figured them out very well in many cases. But, remember, this is from someone who loses track of where he is in his tabs. :-\

ReplyDeleteKristin: I love dy/dan -- thanks for the recommendation!

ReplyDeleteJames: thanks for pointing out the NYTimes column and that Lockhart wrote a whole book -- I'll have to add it to my reading list! (but maybe not until I'm done with the masters... but before I end up inflicting, I mean, teaching math :)

I think it is a rare individual who finds math an art. Such a person might become a teacher. They could impart some of their enthusiasm to their students...but most will just think they are bit loopy. It would be nice if such a view of math could spread, but I don't see it happening. For now, SohCahToa works quite well. And one might even get the chance to introduce a politically incorrect story into a math classroom...

ReplyDeleteI share many of the same concerns raised in the Lament ..... I haven't read it, yet, but hope to read it soon. It really is a shame that math is taught without any sense of its history or how it has developed as a language to describe observed phenomena. Too many kids are left seeing math as no more than a set of computational rules. There are alternative math curricula out there that take the approach of starting with inquiry and then developing theory to support the inquiry. The problem is that parents tend to operate from their own paradigm of how they learned math, so they fight such innovative curricula in their child's school.

ReplyDelete