A Mathematician’s Lament

A mind revolving piece on Lockhart’s views on the mathematics education and curriculum in our educational system.

In my times as a student spent in school math classes, I always have my common complaint while struggling in rote memorization, drills, lectures, and exercises: when I will ever going to use these in my life again?

Apparently, this is a question that goes across many other students’ minds before, even now, and probably still will be for a while. Our educational system is not inspiring enough, even somehow erasing the few sparks that remained in the young’s heart towards exploring, amusing and imagination, with math and other subjects as well.

What is mathematics? Is it something useful for us? Or should it be useful? Why do we have to learn it? How we should learn? Is it be that boring honestly?

In Paul Lockhart’s book A Mathematician’s Lament, he discussed the cruel facts existing in our math classes, and also he explained why he loves math, as an art, an imaginary place, a landscape of elegant, fanciful structures, inhabited by wonderful, imaginary creatures who engage in all sorts of fascinating and curious behaviors.

Truth be told, schools are not exactly a place for thinking and creating. It is merely about training children to perform and gain enough facts so that they can be sorted in society. Lockhart strongly responded: it’s no shock to learn that math is ruined in school; everything is ruined in school.”

Lockhart gave several amusing examples of the mathematical aesthetic in his book (which I enjoyed with great laughter), to try to explain mathematical structures are designed and built not so much by us, as by our proofs. Rather by exploration and imagination just like poetry and music.

One of the examples given in the book — a triangle inside a rectangle box.

Wonder how much of the box the triangle takes up—two-thirds maybe? If we chop the rectangle into two pieces, we can see that each piece is cut exactly in half by the sides of the triangle. That means that the triangle must take up exactly half the box!

Skip through this “unnecessary” process, what we have been given in classes is this:

Triangle Area Formula: area=1/2 * base * perpendicular height

“Gone is the thrill, the joy, even the pain and frustration of the creative act. There is not even a problem anymore. The question has been asked and answered at the same time—there is nothing left for the student to do.”

Lockhart’s core point is captured well by the first two paragraphs of this book:

A musician wakes from a terrible nightmare. In his dream, he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made — all without the advice or participation of a single working musician or composer.

Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics, and are generally put off until college, and more often graduate school.

As you might guess, music is an analogy for how Lockhart argues we now treat math. It has been reduced to an exercise in rote learning that bears no resemblance to the practice of mathematics as understood by a mathematician. To Lockhart, math is not a mechanical tool requiring rote learning and endless practice. It’s a creative exploration of ideas and rules. We can create rules arbitrarily, but then have to faithfully follow the rules of our creation when analyzing their properties.