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Friday, December 17, 2010

A Mathematician's Lament

This article written by Paul Lockhart in 2002 moved me nearly to tears, amidst laughter. It is humorous, but full of a passionate rage about what's deeply wrong with our approach to technical education:
http://www.maa.org/devlin/LockhartsLament.pdf

Just an except:
...on  the other side of town, a painter has just awakened  from a similar nightmare…
I was surprised  to find myself in a regular school classroom— no easels, no  tubes of paint. “Oh we don’t actually apply paint until high  school,” I was told by  the students. “In  seventh grade we mostly study colors and applicators.” They showed me a worksheet. On one side were swatches of color with blank spaces next to them. They were told to write in the names. “I like painting,” one of them remarked, “they tell me what to do and I do it. It’s easy!”
After class I spoke with  the teacher. “So your students don’t actually do any painting?” Iasked. “Well, next year they  take Pre-Paint-by-Numbers. That prepares them for the main Paint-by-Numbers sequence in high school. So they’ll get to use what they’ve learned here and apply it to real-life painting situations— dipping the brush into paint, wiping it off, stuff like that.
Of course we track our students by ability. The really excellent painters— the ones who know their colors and brushes backwards and forwards— they get to the actual painting a little sooner, and  some of them even  take the Advanced  Placement classes for college credit. But mostly we’re just trying to give these kids a good foundation in what painting is all about, so when they  get out there in the real world and paint their kitchen they don’t make a total mess of it.”
“Um, these high school classes you mentioned…”
“You mean Paint-by-Numbers? We’re seeing much higher enrollments lately. I think  it’s mostly coming  from parents wanting  to make sure their kid gets into a good college. Nothing looks better than Advanced Paint-by-Numbers on a high school transcript.” 
...
Sadly, our present system of mathematics education  is precisely  this kind of nightmare. In fact, if I had to  design  a mechanism for the express purpose of destroying a child’s natural curiosity and  love of pattern-making, I couldn’t possibly do as good a job as is currently being done— I simply wouldn’t have the imagination  to  come up with  the kind  of senseless, soul-crushing ideas that constitute contemporary mathematics education.
Lockhart identifies as the core problem the "systematized" mathematics curriculum, in which prepackaged facts are presented to the student in an arbitrary order, instead of allowing them to experience the joy of  investigating interesting phenomena through the application of intuition, and trial & error, which is what real mathematics is all about.
“The area of a triangle is equal to one-half its base times its height.” Students are asked to memorize this formula and  then “apply” it over and over in  the “exercises.” 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.
What brought me close to tears was the recognition that I myself routinely deprive students of experiencing the joy, pain, and frustration of the creative act in my engineering classes, particularly the more mathematically oriented ones, by merely presenting arbitrarily arranged sets of pre-packaged facts and techniques, and testing them on their short-term retention of these facts and techniques. However nicely I package and present these facts, ultimately they do not matter to students who have not sought or found them on their own.

This part cracked me up:
In  practice, the curriculum is not even  so much a sequence of topics, or ideas, as it is a sequence of notations. Apparently mathematics consists of a secret list of mystical symbols and rules for their manipulation. Young  children are given  ‘+’ and  ‘÷.’ Only  later can  they  be entrusted  with  ‘√¯,’ and  then  ‘x’ and  ‘y’ and  the alchemy  of parentheses. Finally, they  are indoctrinated  in  the use of ‘sin,’ ‘log,’ ‘f(x),’ and if they are deemed worthy, ‘d’ and  ‘∫.’ All without having had a single meaningful mathematical experience. 
How true!

If you care at all about education, particularly in areas like mathematics and engineering, I beg you to read this article in its entirety and think about it. We cannot hope to reform our truly broken system of education until revolutionary ideas like Lockhart's become mainstream.

***

Here's a brief biography of Lockhart:

Paul became interested in mathematics when he was about 14 (outside of the school math class, he points out) and read voraciously, becoming especially interested in analytic number theory. He dropped out of college after one semester to devote himself to math, supporting himself by working as a computer programmer and as an elementary school teacher. Eventually he started working with Ernst Strauss at UCLA, and the two published a few papers together. Strauss introduced him to Paul Erdos, and they somehow arranged it so that he became a graduate student there. He ended up getting a Ph.D. from Columbia in 1990, and went on to be a fellow at MSRI and an assistant professor at Brown. He also taught at UC Santa Cruz. His main research interests were, and are, automorphic forms and Diophantine geometry.
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, where he says "I have happily been subversively teaching mathematics (the real thing) since 2000."
        - Keith Devlin, Devlin's Angle, MAA Online, March 2008

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