Wednesday, July 21, 2010

Mathematics and Lockhart's Lament

Presumably some of you have already read this, but my mother and little brother forwarded me an interesting essay by Paul Lockhart, a math teacher, which was written in 2002. You can read a PDF of the essay here. (It's 25 pages long - so I'll forgive you for skimming it instead...). Called "A Mathematician's Lament," the essay makes the case that the way math is taught is usually boring and irrelevant, turning children off to the beauty of mathematics. Or to use his words:

"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."

He writes that instead of learning formulas, students should play games, play around with shapes, pose their own word problems based on questions that are actually interesting in real life, etc.

It's an interesting read, though I'm of mixed mind about it. I took a geometry class during the summer after 7th grade that used this conjecture method, and while it was fine, I can't say that it made me enjoy math more than the traditional geometry class I took the next year. I also think that creativity and discipline have to be closely linked. One of the most frustrating things, looking back on my own writing education, is how often teachers just encouraged me to be creative. It turns out that there are actually rules of grammar and best practices of argument and story plotting and other such things which can be enormously helpful in getting your point across!

But anyway, I'm quite curious what Gifted Exchange readers think -- does the usual method of math instruction kill all the joy of the subject?

11 comments:

Anonymous said...

I would say a teacher's attitude rather than methods has a lot more to do with it. I was definitely given some hand-on projects in every math class I took (calculus board games, fractal art projects, etc.) which I enjoyed but the teachers made the difference. The lowest point of my math education was the veteran teacher who wouldn't allow questions because she had been teaching the subject so long she knew "what questions we would ask anyway". She also marked questions with grades but didn't mark what we got wrong. It was like she didn't care what we learned.It's not a coincidence that several students wandered off the gifted math track after that. The previous 3 years our teacher had been interactive, he would demonstrate problems a variety of ways, give our parents detailed information on our progress, and offered office hours on instant messenger. Students can tell when their teacher is invested in xyr learning. I think the most important qualities are being interested in your subject, and adaptable to your students' needs.

gasstationwithoutpumps said...

I think that one has to balance fun and fundamentals. Too much of either is pointless.

See my posts on coaching a math team:
http://gasstationwithoutpumps.wordpress.com/2010/07/19/math-team/
math vs. memory work:
http://gasstationwithoutpumps.wordpress.com/2010/07/05/math-isnt-memory-work/
and live-action math:
http://gasstationwithoutpumps.wordpress.com/2010/06/29/live-action-math/

Anonymous said...

I think he doesn't understand the fact that most kids would not draw the line. Most kids are not creative mathematically. Many kids would simply not care.

Most early math these days is taught in a "creative" way through the use of manipulatives, games etc. However, at some point a kid has to memorize the formulas.

His music analogy is just the beginning. Most kids love to play around on the piano or with the recorder. Most kids have a desire to make music. However, the majority of kids end up dropping their instruments because it becomes hard, they have to follow the notes (the rules) and just don't want to practice. I think you could make math super fun and once it turned complex, many people would give up.

He is right about certain kids - gifted ones in particular. But it is not realistic to expect that all kids will see the beauty in math.

Mike Hardy said...

Someone wrote:

"I think he doesn't understand the fact that most kids would not draw the line. Most kids are not creative mathematically. Many kids would simply not care."

I don't think that's a problem. I don't think Paul Lockhart fails to understand that. I think we should simply not try to instruct everyone in mathematics. We (I'm entitled to that pronoun because I teach math) should make it know to everyone that mathematics exists; that it is a field in which new discoveries continue to be made at an immense rate; that it is a field in which challenging unsolved problems remain; that it is a major part of the intellectual mainstream; that its appeal is esthetic; that it has many uses; that some aspects of it are easy to understand and others hard; __but__ we should provide disciplined instruction in it only to those who want it. In most places in the USA that's what's done with foreign languages, and lots of pupils study one or more of those and lots don't.

Anonymous said...

I think this discussion is getting a bit off track. Our mathematics education system is terrible. I doubt too many people would say I am wrong about this. On the other hand, the key issue in this discussion is if math should be taught differently. Furthermore, we should be asking if our current math educational system is bad for gifted children.

Like Mr. Lockhart, I am a trained mathematician. Like him, I have spent time in a top-notch elementary (in my case, one in Los Angeles with API scores above 900). Like him, I am shocked by the amount of repetition and mindless focus on arithmetic drills.

Yes, we all need to know the basics of arithmetic, but there is absolutely no excuse to keep kids away from interesting topics and fun problems. I have taught elementary graph theory to kids as young as 7 and 8. I taught them how to model fluids and traffic flow with graphs. Did the kids "tune out?" No, they did not. In fact, they figured out how to find the shortest time routes and the paths with the largest transport capacity. A few kids could not do it, but the majority was able to handle it. Graphs required them to think while practicing addition, subtraction, and multiplication. They even learned the representation of geographical maps as graphs and studied simplified versions of the four color theorem.

Bear in mind we did this in a regular classroom. We did not even try this within the confines of GATE. So, why can't we teach our kids in a more fun, interesting, and useful way? Everybody needs to learn the basics. However, the basics show up in many different ways. Why not use those "ways" to create interesting problems?

Finally, mathematics should not be taught via lectures. Math is learned best by doing problems and asking what-if questions. One learns math by doing it. This is the way I learned math. This is the way every good mathematician I know or know of learned mathematics. Problems, problems, problems, and more problems.

Anonymous said...

Yes, problems and more problems, but please not 50 of the exact same kind of problem for homework for gifted kids! The ones who need the repitition need to do lots of problems *with coaching*, together, not on their own at night. And relating the problems to something the kids care about rather than just shoving a page of problems at them wouldn't hurt either.

Anonymous said...

Here's a follow-up that includes various criticisms of Lockhart, along with his responses.
http://www.maa.org/devlin/devlin_05_08.html

hschinske said...

"I think we should simply not try to instruct everyone in mathematics."

But the problem starts long before real math. The problem is kids who don't even get the point of basic arithmetic, who have so little number sense that they don't understand why multiplying by one-half is the same as dividing by two, and so forth. You can't call someone educated at all if they can't handle stuff like that (barring the occasional person with a specific learning disability, of course).

Mike Hardy said...

"But the problem starts long before real math. The problem is kids who don't even get the point of basic arithmetic, who have so little number sense that they don't understand why multiplying by one-half is the same as dividing by two, and so forth."

I think that that _is_real_math_. And we should not enforce instruction in it to those not interested. If that means some people will not be "educated", then what else is new? Education should not be forced on people.

The result of trying to force everyone to learn math is that the present math curriculum consists mainly of lies. Courses get changed into drilling pupils on things that everyone can do even if they don't want to understand anything, and pupils get told that that's what math is. That's a lie.

hschinske said...

"I think that that _is_real_math_. And we should not enforce instruction in it to those not interested."

But at what age do you decide? Basic fractions are introduced to eight- and nine-year-olds. Surely you wouldn't cut them off from instruction *then*, when they've hardly even started?

In any case, such a system seems to me to be intended to reinforce the Matthew effect: those who have, get more, while those who have less, keep losing even that which they have. I have no objection to some people learning math faster than other people, nor to some people (not entirely the same ones) ending up knowing much more math than other people, but to change slow learning to no learning seems to me to be indefensible, and to be a system that would be enormously subject to abuse.

Helen Schinske

Anonymous said...

I am a teacher of gifted students and also a mom of gifted kids. In my school the focus seems to be on the big "TEST" in the spring of each year, thank you NCLB! especially for grades 4-6. I do think that kids have no idea what is out there in the real world that requires math! I also think they need to be shown that math can be really exciting!!!