neither do educators, textbook authors, publishing companies,
and sadly, neither do most of our math teachers. . . "
- Paul Lockhart
By C.J. Westerberg
I was blown away by this remarkable (and strangely empowering) critique about math education: how we view it as a culture; how teachers are teaching it (or not teaching it); how and why some students struggle with it; how some students who apparently "get it" don't; how parents perceive it; how testing may not be showing us what we want to know, and how we can change math education for the better. We are thankful to author Paul Lockhart who let us share this work from his book, A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form.
This post begins with what we call "The Dream" (which is really a nightmare), followed by a few paragraphs addressing math as art. I read the following chapter to my tween daughter who became spellbound by this new idea of Math as Art. Her question, "Why hasn't anybody talked about math like this before?" opened up a conversation unlike any other we've had about Math.
With that being said, Lockhart shares a most spectacular ride into those hidden corners of "education truths" - complete with screenplay moments tossed in - making each "chapter" come to life, as he must do in his math classroom. Math is widely used in poker games and while playing poker online in online poker websites. Have a look at the list of poker books from this website to learn about math in poker.
At the end of this post is the link to an extensive excerpt which is a masterpiece which includes a pretty scathing take on K-12 math curriculum. Every parent, teacher and administrator should check it out.
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 for the primary and secondary schools, their mission is to train students to use this language-- to jiggle symbols around according to a fixed set of rules: "Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way."
In their wisdom, educators soon realize that even very young children can be given this kind of musical instruction. In fact it is considered quite shameful if one's third-grader hasn't completely memorized his circle of fifths. "I'll have to get my son a music tutor. He simply won't apply himself to his music homework. He says it's boring. He just sits there staring out the window, humming tunes to himself and making up silly songs."
In the higher grades the pressure is really on. After all, the students must be prepared for the standardized tests and college admissions exams. Students must take courses in Scales and Modes, Meter, Harmony, and Counterpoint. "It's a lot for them to learn, but later in college when they finally get to hear all this stuff, they'll really appreciate all the work they did in high school." Of course, not many students actually go on to concentrate in music, so only a few will ever get to hear the sounds that the black dots represent. Nevertheless, it is important that every member of society be able to recognize a modulation or a fugal passage, regardless of the fact that they will never hear one.
"To tell you the truth, most students just aren't very good at music. They are bored in class, their skills are terrible, and their homework is barely legible. Most of them couldn't care less about how important music is in today's world; they just want to take the minimum number of music courses and be done with it. I guess there are just music people and non-music people. I had this one kid, though, man was she sensational! Her sheets were impeccable-- every note in the right place, perfect calligraphy, sharps, flats, just beautiful. She's going to make one hell of a musician someday."
Waking up in a cold sweat, the musician realizes, gratefully, that it was all just a crazy dream. "Of course!" he reassures himself, "No society would ever reduce such a beautiful and meaningful art form to something so mindless and trivial; no culture could be so cruel to its children as to deprive them of such a natural, satisfying means of human expression. How absurd!"
Meanwhile, on the other side of town, a painter has just awakened from a similar
nightmare . . . (snip)
Everyone knows that something is wrong. The politicians say, "we need higher standards."
The schools say, "we need more money and equipment." Educators say one thing, and teachers say another. They are all wrong. The only people who understand what is going on are the ones most often blamed and least often heard: the students. They say, "math class is stupid and boring," and they are right.
Mathematics and Culture
The first thing to understand is that mathematics is an art. The difference between math and the other arts, such as music and painting, is that our culture does not recognize it as such. Everyone understands that poets, painters, and musicians create works of art, and are expressing themselves in word, image, and sound. In fact, our society is rather generous when it comes to creative expression; architects, chefs, and even television directors are considered to be working artists. So why not mathematicians?
Part of the problem is that nobody has the faintest idea what it is that mathematicians do. The common perception seems to be that mathematicians are somehow connected with science-- perhaps they help the scientists with their formulas, or feed big numbers into computers for some reason or other. There is no question that if the world had to be divided into the "poetic dreamers" and the "rational thinkers" most people would place mathematicians in the latter category.
Nevertheless, the fact is that there is nothing as dreamy and poetic, nothing as radical, subversive, and psychedelic, as mathematics. It is every bit as mind blowing as cosmology or physics (mathematicians conceived of black holes long before astronomers actually found any), and allows more freedom of expression than poetry, art, or music (which depend heavily on properties of the physical universe). Mathematics is the purest of the arts, as well as the most misunderstood.
So let me try to explain what mathematics is, and what mathematicians do. I can hardly do better than to begin with G.H. Hardy's excellent description:
of patterns. If his patterns are more permanent than theirs,
it is because they are made with ideas.
After several years teaching university mathematics, Paul eventually tired of it and decided he wanted to get back to teaching children. He secured a position at Saint Ann's School (in Brooklyn), where he says "I have happily been subversively teaching mathematics (the real thing) since 2000."
He teaches all grade levels at Saint Ann's (K-12), and says he is especially interested in bringing a mathematician's point of view to very young children. "I want them to understand that there is a playground in their minds and that that is where mathematics happens. So far I have met with tremendous enthusiasm among the parents and kids, less so among the mid-level administrators" . . ."
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