| A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form | |
| by Paul Lockhart |
A Review
Wonderful Ideas in Theory, with tragic flaws in argumentAs a mathematics teacher and long-time student of mathematics, I was overjoyed to begin reading this book, finally one that attempts to explain the beauty and elegance of mathematics and to expose the way in which we are teaching it, which does not do justice to it at all. I absolutely agree with *most* of Lockhart's assessment on many points, for example, that mathematics is an art, that it should not be taught as procedures and formulas and meaningless word problems that contrive to be about "real life."
I agree, most of our math teachers do not have this kind of appreciation for mathematics, which is tragic because it means our kids will grow up scared and intimidated by math ("math anxiety") instead of awed at its power of abstract interpretation. I agree our approach needs to be completely overhauled.
My 2 star rating is due to the fact that Lockhart's analysis is strongly lacking in a historical understanding as well as pedagogical/curriculum knowledge. For example, he says that word problems should not be contrived to be about real life (I agree with this point), and that math is beautiful precisely BECAUSE it is irrelevant to real life.
As a mathematician I cannot possibly comprehend how another mathematician could possibly believe the beauty of mathematics comes from its "irrelevance" of abstraction: in fact, the reason math is SO powerful is that these abstract representations have all been historically "discovered" or "invented" (depending on what you believe math is: inherent in the world, or a human game of abstraction)--particularly in order to try to model and explain phenomena observed in "the real world."
Lockhart says math was created by humans "for their own amusement" (p. 31), but ignores that in fact all branches of mathematics in the past were created in response to actual world problems, and not only that, but now, some of the most fascinating mathematics is being created again in response to solving some of the most complex problems we have imagined, such as the mathematics behind string theory. I don't know how Lockhart could possibly consider that humans invented counting, ways to measure their plots of land and keep track of money, or ways to measure the orbits of planets (thus leading us to the current "space age") as "purely amusement"--perhaps, if LIFE is amusement in general, but really, all of these inventions had a very REAL, concrete, specific historical cultural purpose and are not "just made up" for fun!!!
In fact, math is EMBODIED in our cognitive schemas and perception, and THIS IS PRECISELY what makes it so WONDERFUL: its RELEVANCE to EVERYTHING in real life and humanity's inherent capacity for thinking about the real world in this abstract way! Math is not "just" "fantasy" (as on p. 39) (see especially Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being).
I teach functions (precalculus, AP calculus) and the main theme and point of math for example at this level is to teach kids how basically, in life, we track patterns of change in anything and everything--public health data, unemployment, polling, the stock market, baseball stats, etc. Functions are just the most abstract way to model these changing patterns over time (or some other variable) and thus give us the powerful tool of projecting into the future/past and otherwise analyzing trends. Yes, functions are abstract, but they are NOT "just fantasy play," irrelevant to the real world, or made up simply for the fun of it, in fact, quite the opposite of all of these.
Further, my (and I believe, many) students would be aghast to learn that a math teacher is suggesting an overhaul of math education based on the idea that"kids don't really want something that is relevant to their daily lives." This is the most absurd statement I have ever heard, so I am guessing Lockhart knows nothing about adolescent/child development, interest, and pedagogical literature. Learning in general is based on making connections to prior knowledge, and I have never heard any question asked more often in math class when I didn't explain the relevance in advance than "Why do I need to know this? How is this relevant to my life?" This is probably the MOST pressing question for adolescents in general..
Other examples of pedagogical tragedies in this book include Lockhart's admonitions that "you can't teach teaching," that "schools of education are a complete crock" and that teachers shouldn't lesson plan because this is somehow "not real" or authentic (p. 46-47). While I agree schools of education are not preparing our teachers well and what we need is much more systemic training in content knowledge (for example, math teachers should all have to double major in math/pedagogy or education), IT IS absolutely not true or supported by any research (except perhaps by the current corporate brand of the reform movement) that teaching is something you "have" that you don't need to "learn" and, further, that you shouldn't plan because this is inauthentic.
A plan should of course never prevent a teacher from moving in new directions as suggested by the course of the class, but coming in without a plan is certainly not considered sound practice in any theory of learning and from any angle, and in general is not a sound principle of life (i.e., just doing everything by the seat of your pants and counting on your "genius" to lead you through whatever you should have planned usually doesn't work, unless you are in a feel-good movie). Only in Lockhart's fantasy "lala land" of irrelevancy is planning a vice and not a virtue. Plus, there's so much more to "planning" than thinking about the flow of the lesson, how you will help students make connections, etc. I assess and plan hand in hand for example, so I will grade the last night's HW and that day's Exit Slip and plan the next days's and week's lessons all the while incorporating items my students did not fully understand the first time, and also while addressing specifically the mistakes they made (and each class/year of students tends to have different problems and make different mistakes so it is important to constantly plan and reflect as a teacher on what is best for your particular students NOW).
