Mathematician's lament.

[Schrödinger's hat]

Just discovered www.maa.org/devlin/lockhartslament.pdf and thought it worth sharing/possibly stickying.

Edited August 27, 2012 by Schrödinger's hat

[mississippichem]

I have read said lament before. Some powerful ideas those are. I think every math teacher should be required to read and summarize it before being allowed to teach.

 

+1

[Bignose]

But what about

the real story? The one about mankind’s struggle with the problem of measuring curves; about

Eudoxus and Archimedes and the method of exhaustion; about the transcendence of pi? Which

is more interesting— measuring the rough dimensions of a circular piece of graph paper, using a

formula that someone handed you without explanation (and made you memorize and practice

over and over) or hearing the story of one of the most beautiful, fascinating problems, and one of

the most brilliant and powerful ideas in human history? We’re killing people’s interest in circles

for god’s sake!

 

the above is from the 9th page.

 

I can see the author's passion for the subject, but the above is not really a fix. I'm sorry, but the kids ain't exactly clamoring over each other and stampeding into history class.

 

I am still convinced that the solution is to couch mathematics in the framework of problem solving: that mathematics provides well defined tools and problems and wholly right or wrong solutions to their problems. And in practicing problem solving under these ideal conditions, the problem solving skills will be reasonably sharpened when life throws real problems at you -- where tools are imperfect, where problems aren't perfectly stated, and where many solutions may be right, or no solutions exist.

 

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I also wanted to add:

 

mathematics isn't like art or music in some very basic ways. There is plenty of imagination, but if a student can't do the addition tables or multiplication tables, there isn't very far a student can get. And a lot of the 'symbol pushing' is just to get nomeclature down. A lot of people can weave a good tale, but if someone wants to write it down, they need to learn proper spelling, grammar, and punctuation. That is what a lot of mathematics 'symbol pushing' is -- learning the grammar of the language.

 

Maybe the author's same point can be made about English grammar as well, but I think he'd have a much harder time convincing schools to drop their grammar and spelling programs than dropping a lot of the current mathematics teaching.

Edited August 28, 2012 by Bignose

[rutski]

I have read said lament before. Some powerful ideas those are. I think every math teacher should be required to read and summarize it before being allowed to teach.

 

+1

 

I've know about that essay for years, definitely a +1 from me as well.

 

Those general problems aren't confined to the math classroom though. They exist in one form or another in English, Chemistry, History, Art, and probably even PE too. Primary schools suck the soul out of just about everything. Colleges do too, in my opinion, though to a lesser degree.

Edited September 3, 2012 by rutski

[irishredfox]

This essay is great. He's become one of my heroes. I can't help but think back to some of my art classes in high school. "OK, today we are going to learn about the Impressionist period, and we are going to learn the techniques that defined the style." They didn't sit me down and say "Let us do a hundred paintings so we can paint the exact same way." The teachers tried to show you others that had come before in order for you to develop your own style. And every formula and equation that has come to us came from creative thinkers. But we are not encouraged to study the work of Descartes, Gibbs, or Godel; instead we are just told what they came up with. It is a shame too, math isn't something you can just memorize, it's a puzzle with an answer that needs to be worked out.

[Iota]

Very accurate article.

[Vastor]

SNIP

Yep, I agree with you on how the article is not really accurate on what maths really is & how maths interested others, but lets face reality, the education happen to be (terribly) bad and mathematics got bad names because of it. The typical "I can't do Math, there's too much formula to be memorized" complains started to piss me off on the education system.

 

 

but if a student can't do the addition tables or multiplication tables, there isn't very far a student can get.

I can't agree on this, there are students who able to do those table without have any a clue what the table is about, this is the problem the writer talk about. Especially, in high school, we use calculator anyway. and I'm sure that almost every average person have enough for memory multiplication table, they 'can't get far' in math because they have no interest in the first place. Until we can spark some awe into others and make them an actively thinking person, we can't get anyone far in mathematics.

From my perception, I see mathematics as a (shortcut) Language + Philosophy with good Axiom. So yeah, it's not really an Art because it's what standardized by mathematician themselves. but teaching them through memorizing (plug & chugged the exercise and repeat) is very bad step to begin with.

 

Furthermore, there's too lack of mathematical reasoning discussion in the so-called 'Maths Class'. Yet, even on these kind of chapter, (I got a) Teacher told us, "if it's a Statement, u need to answer with 'if x then y, blabla' to get the MARKS". teachers are moving in wrong direction when they start taking care about marks than teaching math which sometime make me even forgot either I am still in a Math Class. They just sampling the answer and put in our mouth, tadaaa, mathematics!

 

The effect? Well, I got an older sister who got good mark in both Math and Add. Math, A+ or so in the SPM (O-lvl exam), but now she start decide to pick a course that use least math possible & stay away from it. Because there are people who can memorized all of the formula but end up knowing (almost) nothing what they are actually doing. Whether the exam is too standardized or 'Math Class' is just a fraud or both.

 

 

The only 'benefit' from these education system is less competition for a mathematician, duh...

Edited September 7, 2012 by Vastor

[Bignose]

Maybe I didn't communicate it well. What I meant was that a student can sit there and come up with clever ways to calculate the area of a trapezoid, for example, but if they can't add correctly, it doesn't do any good to identify a trapezoid as a rectangle and two triangles.

 

More to my point: while I wholeheartedly agree that the teaching of mathematics is largely motivated poorly, I don't think that is any excuse to remove exercises and work from the students. In fact, in general, I think that most students do not do enough repetition of the exercises. When one is studying algebra, completing the square should be second nature. And the only way that happens is by practicing it over and over -- like most anything, people only get better by practicing. No matter how much artfulness a student may have, if they don't have mastery of machinations (e.g. the grammar) of the subject, they aren't going to get far.

 

As an example, I think that a fair number of the threads in the Speculations forum are very creative, very artful, and very imaginative. But the vast, vast majority of them have authors who are misunderstanding or misinterpreting some of the very basics of the sciences they are trying their hand at -- and yes, there are even a few threads in there about mathematics. Practice makes perfect, and to do a fair amount of mathematics, you do need well honed skills to do them efficiently and correctly.

 

Again, I am certainly going to agree that that practice needs to be much better motivated, explained, and taught. But, I am also saying that unless the student is an absolute savant, there is no replacement for practice.

Edited October 20, 2012 by Bignose

[PeterJ]

I also feel that the problem is much wider than just mathematics. Having watched three kids go though school I felt nothing but sympathy for the utterly brain-dead manner in which most subjects are presented. Personally I just didn't go to lessons for years. they were so boring, but I suppose we can't all do that.

 

 

[overtone]

Maybe the author's same point can be made about English grammar as well' date=' but I think he'd have a much harder time convincing schools to drop their grammar and spelling programs than dropping a lot of the current mathematics teaching.[/quote']

 

I also feel that the problem is much wider than just mathematics. Having watched three kids go though school I felt nothing but sympathy for the utterly brain-dead manner in which most subjects are presented.

 

I am in more or less complete sympathy with Bignose's demurrals concerning the OP, but I fear he may be too optimistic regarding the teaching of grammar and spelling. They have in fact, in realistic assessment, been dropped from serious consideration in many American high schools.

 

And the manner of their dismissal bears on the issue here, and the second quote above: they have been relegated to specialty sections, isolated drill and kill rote memorization stretches of brief duration, no grounding or connection to anything else (such as reasoning, philosophy, historical development of the language, rhetorical influence, anything), no subsequent repetition or application. They are taught, if they are, as technicalities of no relevance to the "content" of one's writing.

 

Whatever students write for history, "social studies", art, philosophy, science, even English other than specifically grammar and spelling class, is not evaluated or corrected or considered for its grammar and spelling. That is for grammar class. In real life, it is apparently assumed (often explicitly), the "word processing" program will fix all that anyway. The intellectual development of the child is presumed to proceed unaffected, or perhaps even freed and abetted, by removing such inherently minor technical obstacles and delays from work of more important "content".

 

The parallels here, with multiplication tables and algebra and the like, seem clear to me.

 

No normal human being can learn grammar and spelling that way, any more than algebra. And as we see, they don't. But the best response is probably not to abandon grammar, or even spelling, and focus on the art and beauty and larger brighter world of literature.