A Mathematician's Lament: An essay on mathematics education

[morphism]

I've found this fantastic essay (with a very apt title) by Paul Lockhart on the (sad, sad) state of mathematics education in North America.

Here is the link: http://www.maa.org/devlin/LockhartsLament.pdf

It's definitely worth reading for all educators out there.

How many people actually use any of this “practical math” they supposedly learn in school? Do you think carpenters are out there using trigonometry? How many adults remember how to divide fractions, or solve a quadratic equation? Obviously the current practical training program isn’t working, and for good reason: it is excruciatingly boring, and nobody ever uses it anyway. So why do people think it’s so important? I don’t see how it’s doing society any good to have its members walking around with vague memories of algebraic formulas and geometric diagrams, and clear memories of hating them. It might do some good, though, to show them something beautiful and give them an opportunity to enjoy being creative, flexible, open-minded thinkers— the kind of thing a real mathematical education might provide.

 

[rasmhop]

I've only read the first 7 pages so far, but plan to save this and read the rest later. It's an amazing essay. It's just how I have come to think of math, but I have never been able to express it nearly as clearly as Paul Lockhart. I wouldn't say it's just worth a read for educators, but pretty much everyone. This should be mandatory reading in school to make sure students at least get a chance to glimpse what they may be missing.

 

[Mark44]

Very interesting. Here is what Lockhart describes as a "completely honest course catalog for K-12 mathematics."

The Standard School Mathematics Curriculum
LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not
something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.

MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures, akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are handed out, and the students learn to address the church elders as “they” (as in “What do they want here? Do they want me to divide?”) Contrived and artificial “word problems” will be introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison.
Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’ and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent preparation for Algebra I.

ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation. The smooth narrative thread that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no characters, plot, or theme. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.

GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of
students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy and distracting notation will be introduced, and no pains will be spared to make the simple seem complicated. This goal of this course is to eradicate any last remaining vestiges of natural mathematical intuition, in preparation for Algebra II.

ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.

TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and symmetry. The measurement of triangles will be discussed without mention of the transcendental nature of the trigonometric functions, or the consequent linguistic and philosophical problems inherent in making such measurements. Calculator required, so as to further blur these issues.

PRE-CALCULUS. A senseless bouillabaisse of disconnected topics. Mostly a half-baked
attempt to introduce late nineteenth-century analytic methods into settings where they are neither necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to obscure the intuitively clear notion of smooth change. As the name suggests, this course prepares the student for Calculus, where the final phase in the systematic obfuscation of any natural ideas related to shape and motion will be completed.

CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. To be taken again in college, verbatim.

 

[Troponin]

Very interesting. Here is what Lockhart describes as a "completely honest course catalog for K-12 mathematics."

The Standard School Mathematics Curriculum
LOWER SCHOOL MATH. The indoctrination begins. Students learn that mathematics is not
something you do, but something that is done to you. Emphasis is placed on sitting still, filling out worksheets, and following directions. Children are expected to master a complex set of algorithms for manipulating Hindi symbols, unrelated to any real desire or curiosity on their part, and regarded only a few centuries ago as too difficult for the average adult. Multiplication tables are stressed, as are parents, teachers, and the kids themselves.

MIDDLE SCHOOL MATH. Students are taught to view mathematics as a set of procedures, akin to religious rites, which are eternal and set in stone. The holy tablets, or “Math Books,” are handed out, and the students learn to address the church elders as “they” (as in “What do they want here? Do they want me to divide?”) Contrived and artificial “word problems” will be introduced in order to make the mindless drudgery of arithmetic seem enjoyable by comparison.
Students will be tested on a wide array of unnecessary technical terms, such as ‘whole number’ and ‘proper fraction,’ without the slightest rationale for making such distinctions. Excellent preparation for Algebra I.

ALGEBRA I. So as not to waste valuable time thinking about numbers and their patterns, this course instead focuses on symbols and rules for their manipulation. The smooth narrative thread that leads from ancient Mesopotamian tablet problems to the high art of the Renaissance algebraists is discarded in favor of a disturbingly fractured, post-modern retelling with no characters, plot, or theme. The insistence that all numbers and expressions be put into various standard forms will provide additional confusion as to the meaning of identity and equality. Students must also memorize the quadratic formula for some reason.

GEOMETRY. Isolated from the rest of the curriculum, this course will raise the hopes of
students who wish to engage in meaningful mathematical activity, and then dash them. Clumsy and distracting notation will be introduced, and no pains will be spared to make the simple seem complicated. This goal of this course is to eradicate any last remaining vestiges of natural mathematical intuition, in preparation for Algebra II.

ALGEBRA II. The subject of this course is the unmotivated and inappropriate use of coordinate geometry. Conic sections are introduced in a coordinate framework so as to avoid the aesthetic simplicity of cones and their sections. Students will learn to rewrite quadratic forms in a variety of standard formats for no reason whatsoever. Exponential and logarithmic functions are also introduced in Algebra II, despite not being algebraic objects, simply because they have to be stuck in somewhere, apparently. The name of the course is chosen to reinforce the ladder mythology. Why Geometry occurs in between Algebra I and its sequel remains a mystery.

TRIGONOMETRY. Two weeks of content are stretched to semester length by masturbatory definitional runarounds. Truly interesting and beautiful phenomena, such as the way the sides of a triangle depend on its angles, will be given the same emphasis as irrelevant abbreviations and obsolete notational conventions, in order to prevent students from forming any clear idea as to what the subject is about. Students will learn such mnemonic devices as “SohCahToa” and “All Students Take Calculus” in lieu of developing a natural intuitive feeling for orientation and symmetry. The measurement of triangles will be discussed without mention of the transcendental nature of the trigonometric functions, or the consequent linguistic and philosophical problems inherent in making such measurements. Calculator required, so as to further blur these issues.

PRE-CALCULUS. A senseless bouillabaisse of disconnected topics. Mostly a half-baked
attempt to introduce late nineteenth-century analytic methods into settings where they are neither necessary nor helpful. Technical definitions of ‘limits’ and ‘continuity’ are presented in order to obscure the intuitively clear notion of smooth change. As the name suggests, this course prepares the student for Calculus, where the final phase in the systematic obfuscation of any natural ideas related to shape and motion will be completed.

CALCULUS. This course will explore the mathematics of motion, and the best ways to bury it under a mountain of unnecessary formalism. Despite being an introduction to both the differential and integral calculus, the simple and profound ideas of Newton and Leibniz will be discarded in favor of the more sophisticated function-based approach developed as a response to various analytic crises which do not really apply in this setting, and which will of course not be mentioned. To be taken again in college, verbatim.

Due to particular circumstances (being an ignorant dipgarbage until my mid-twenties), The only course here that I've taken was Geometry in 9th grade (standardized school testing placed me in the "upper level" of math maturity after that, which put me in the AP math section, which were no longer mandatory...so by flaw of the system, I never took another math course).

Looking over this...many things make sense.
Realizing in my mid-twenties that math wasn't just putting numbers into a calculator, I self studied up through Calc I.
So, I started my "official" math education in Calc II at the University.

What is said in this quote is very interesting to me. Seeing the strange looks when I derive a trig identity that I can't remember, trying to explain why we don't actually have to calculate something because we can just picture the graph and see that the derivative has to be zero *here*, or arguing (to refusing minds) that you can see why the derivative of 1/x is ln(x) by just calculating out the change in 1/x to see that you get the value of ln(x) all fit in perfectly if what is said here is true.

There have been a number of times where my lack of formal education has "bit" me to some extent, but reading this makes me feel a whole lot better about myself.

Sorry to turn the thread into a "look at me!" post, but I had to post after reading Mark44's post.
I hope I can help my children see that mathematics doesn't have to be "that way."

 

[mgb_phys]

He misses out the bit where an attempt will be made to teach 'modern maths' by having kids draw Venn diagrams. They will then stop in order to get back to 'proper maths'. This process will be repeated later when they will be expected to remember which of a bunch of weird symbols means which kind of set - but will basically be the Venn diagrams again with more complex typos

 

[twofish-quant]

It's a lament on mathematics education in the North America. Schools in East Asia have *VERY* different mathematical education systems.

One thing that I find weird is that when it's been noted that students in the US generally do worse on math tests than students in East Asia, you'd think that the logical thing to do is to translate East Asian textbooks, find some East Asian math teachers to give talks, and basically change the system to work like the system in East Asia... But no...

What seems to end up happening is that people look at the low test scores of US math education, and conclude that the thing to do is to teach the system that doesn't work, even harder...

 

[Phrak]

The vast majority of us have no use for but a small fraction of the education provided. How academia is so ignorant of this is beyond me.

 

[mgb_phys]

One thing that I find weird is that when it's been noted that students in the US generally do worse on math tests than students in East Asia

It's an old complaint, in 1900 British scientists were complaining that British education in science,particularly chemistry and engineering was so far behind Germany's that British industry would never be able to compete.

A century later it's clear that British chemical and car industries have nothing to fear from Germany's. America is in the same boat - it was able to invent the atomic bomb and go to the moon without relying on foreign education.

 

[hamster143]

It's a lament on mathematics education in the North America. Schools in East Asia have *VERY* different mathematical education systems.

And yet Americans are consistently in the top 3 by the number of gold medals on IMOs. So are Russians, whose education is very similar to what's described here.

But we must be doing something wrong, according to Lockhart. We must drop all the useless trigonometry and quadratic equations and make mathematics an elective course. We must definitely stop teaching elementary school students to count. We can make it a high-school course (after all, numbers were regarded only a few centuries ago as too difficult for the average adult) and offer a one-year course of using the calculator.

This reminds me of another essay:

http://www.youtube.com/watch?v=fsDuL4jTkz0#lq-lq2-hq

 

[Phrak]

What is IMO?

 

[twofish-quant]

It's an old complaint, in 1900 British scientists were complaining that British education in science,particularly chemistry and engineering was so far behind Germany's that British industry would never be able to compete. A century later it's clear that British chemical and car industries have nothing to fear from Germany's.

What does that have anything to do with any of the topics under discussion here? There are some very good things about the US educational system, but primary/secondary math education isn't one of them.

Someone point out that if you went to 1900 and tried to find the best universities in the world, *none* of them would be American. American universities became great, by looking at the excellent universities that Germany had, and copying everything then could.

And the facts are wildly wrong. Quick name three German car brands. VW, Mercedes, Audi. Quick, name three British car brands? Ummmm... Ahhh.. Rolls-Royce (owned by BMW). The German chemical industry is twice the size of UK. The German auto industry is three times that of UK. And remember that this is *after losing two world wars*.

America is in the same boat - it was able to invent the atomic bomb and go to the moon without relying on foreign education.

Whoa! That's an incredibly silly statement.

Fermi? Einstein? Von-Braun? Von Neumann? Bethe? Szilard? Von Karman? Edward Teller? Felix Bloch? J. Robert Oppeheimer was US-born but studied in Germany. Let's go down the names of the people that were involved in the Manhattan and Apollo projects, and see how many of them were foreign educated. Also go to any 1950's monster movie, and note that the stereotypical crazy scientist had a German accent. If you go into any research lab in the US, you'll find that most of the people there are Indian or Chinese.

The US has benefited hugely from foreign education, since it has an open door to scientists and engineers that were educated with school systems that are a lot better than the US for teaching math, but that doesn't mean that the US wouldn't be better off it it improved the school system.

Relying on foreign immigrants to do US science and engineering is good, but in the long run, it's not sustainable. As the economies of China and India improve, people are going to be much less willing to come to the US for education, and people in the US will be looking at going back. Right now, China is looking at the US back and forth, up and down to look at ways that it can improve it's educational system. If the US also doesn't do the same thing, then all of the improvements will be one way.

 

[rasmhop]

And yet Americans are consistently in the top 3 by the number of gold medals on IMOs. So are Russians, whose education is very similar to what's described here.

The American system is pretty good at helping individuals who are already motivated or naturally sees through the stark presentation. The problem are the people who don't really find math as presented interesting.

Also is the IMO really a good measure of the kind of ability an average student should have? I remember when I was at some training sessions for IMO (not the American team) there were a few people who spent so much time focusing on learning useless inequality techniques devised explicitly to work on IMO problems, but never apply to anything of real interest. This time could be so much better spent learning more general problem-solving techniques or more advanced theory, but this wouldn't give as good a score.

But we must be doing something wrong, according to Lockhart. We must drop all the useless trigonometry and quadratic equations and make mathematics an elective course.

This isn't what he's arguing. The most important part is that it shouldn't be forced on the student as a rule set in stone, but rather as the beautiful end-product of an interesting process. Consider solving quadratic equations. If I ask a student how to solve a quadratic equation they will reply:
(-b +- sqrt(b^2-4ac})/2a
But this is a spoonfed formula. Why not teach it to the students by giving them some examples to try, get them to come up with the formula themselves. As for trigonometry it's also taught wrongly. Mnemonics should play no part in the teaching of math. They should understand trigonometry, not remember the trigonometric identities. The problem isn't that they know this stuff, but that they know the formulas, but doesn't understand them.

We must definitely stop teaching elementary school students to count. We can make it a high-school course (after all, numbers were regarded only a few centuries ago as too difficult for the average adult) and offer a one-year course of using the calculator.

I don't think you've really gotten the point. The point isn't to suspend all teaching or dumb it down, but to make it appear more natural. Why not let the young kids devise an optimal strategy for playing tic-tac-toe? That would teach them something about how to split a problem into cases, how to experiment, how to THINK. Sure they should learn addition and multiplication, but try to postpone it a bit and see if you can get them to come up with algorithms themselves (maybe just for simple cases such as 2-digit + 2-digit number). Get them involved in the thought process. And for god's sake don't waste several years of their time trying to teach them to follow algorithms efficiently. I don't think there is an easy answer as to how math should be taught, but I think Lockhart makes a good point that something should be done and he points at the areas where something really is needed. I remember my math teacher in 7-grade who often set an hour aside for the purpose of letting us work on a problem. I remember working out the formula for an arithmetic progression myself, establishing some facts about angles in general polygons using induction (though we didn't name it induction), etc. This is the kind of thing more math teachers should do. Still today I distinctly remember the experience of working out the formula for an arithmetic progression; the whole class found it a fairly enjoyable experience even though few succeeded, but most had guesses, or ideas, or computations.

What is IMO?

An abbreviation for the International Mathematical Olympiad; an international contest for high school students.

 

[hamster143]

The point isn't to suspend all teaching or dumb it down, but to make it appear more natural.

I really don't think that it's his point.

It is very useful for children's minds to study math, to memorize formulas, to solve problems. Yes, math is monotonous for someone who grew up on Spongebob. That does not mean that we should replace it with song and dance. Vegetables are good for you, but, if you're used to snack on cheetos and chocolate bars, you won't appreciate vegetables. That does not mean that we should give up trying to feed children vegetables and replace them with vegetable-shaped sweets.

 

[twofish-quant]

And yet Americans are consistently in the top 3 by the number of gold medals on IMOs. So are Russians, whose education is very similar to what's described here.

Ummmm... I don't see this

And China is #1

http://www.imo-official.org/results.aspx

In 2009, the US ranked #6, and three of the people on the US team had Chinese names...

One of the purposes of the mathematical education is to produce large numbers of technically trained factory workers, and it does a very good job at that. One thing that even non-math majors from East Asia note is how trivially easy college math is.

But we must be doing something wrong, according to Lockhart. We must drop all the useless trigonometry and quadratic equations and make mathematics an elective course. We must definitely stop teaching elementary school students to count. We can make it a high-school course (after all, numbers were regarded only a few centuries ago as too difficult for the average adult) and offer a one-year course of using the calculator.

No, because that's not how math is taught in East Asia.

US has a wonderful system of higher education, but just like the US financial system is being propped up by Chinese money, the US system of higher education is being propped up by Chinese and Indian brains. That's not going to last for more than another ten years.

 

[twofish-quant]

It is very useful for children's minds to study math, to memorize formulas, to solve problems.

The trouble is that you end up with large numbers of people without the ability to solve problems. One thing that is really interested in East Asian math education is that there is hardly any memorization of formula.

Yes, math is monotonous for someone who grew up on Spongebob. That does not mean that we should replace it with song and dance.

I think this is the "no pain, no gain" principle. Math must be painful, because if it isn't painful then obvious it's not good for you. The trouble is that this principle is total nonsense. It's possible for a system of education to be *both* painful and useless. One reason that I don't think very highly of the US primary/secondary math system is that I've taught Algebra 101 at the University of Phoenix, and I've found myself to be half therapist trying to undo the damage of badly taught math courses and taught some basic skills that everyone in Taiwan would have learned by the 7th grade.

If math is monotonous then it's being badly taught.

Vegetables are good for you, but, if you're used to snack on cheetos and chocolate bars, you won't appreciate vegetables. That does not mean that we should give up trying to feed children vegetables and replace them with vegetable-shaped sweets.

Another useless random analogy. What's any of this got to do with math education?

 

[hamster143]

In 2009, the US ranked #6, and three of the people on the US team had Chinese names...

#3 in '02, '03 and '08, #2 in '01, '04 and '05. The three people with Chinese names are probably third-generation immigrants from Taiwan whose grandparents immigrated in the 60's.

Some people are fundamentally incapable of solving problems. It's not the matter of the right or wrong approach. They will get some benefit from rigorous math treatment, but they won't become good technically trained factory workers, even in Taiwan. Dumbing down the subject per Lockhart's suggestions won't help them much, but it will lower the bar even further for the ones capable of abstract thinking.

One thing that even non-math majors from East Asia note is how trivially easy college math is.

I heard the same thing about school students coming to the United States from Russia. American school mathematics has already been dumbed-down to a remarkable level (in many cases, using exactly the methods Lockhart describes, to make it more "intuitive") and that is definitely something that needs to be reversed.

Teaching Math In 1950
A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price. What is his profit?

Teaching Math In 1960
A logger sells a truckload of lumber for $100. His cost of production is 4/5 of the price, or $80. What is his profit?

Teaching Math In 1970
A logger sells a truckload of lumber for $100. His cost of production is $80. Did he make a profit?

Teaching Math In 1980
A logger sells a truckload of lumber for $100. His cost of production is $80 and his profit is $20. Your assignment: Underline the number 20.

Teaching Math In 1990
By cutting down beautiful forest trees, the logger makes $20. What do you think of this way of making a living?

Topic for class participation after answering the question: How did the forest birds and squirrels feel as the logger cut down the trees?
(There are no wrong answers.)

Teaching Math In 2005
El hachero vende un camion carga por $100. La cuesta de production es....

 

[mgb_phys]

The logger bit isn't a joke anymore - see;
http://www.wellingtongrey.net/articles/archive/2007-06-07--open-letter-aqa.html

The problem with counting medal winners is that there is a small difference between medal winners and average.
African-americans win most athletics gold meals - this does not translate into African-Americans having the best health in the population.
Similarly, as described in another thread on health care, the USA has more surgeons and MRI machines than any other country - it does not mean that there are no problems in US health care.

ps. two-fish, read the sig!

 

[Ben Niehoff]

The logger bit isn't a joke anymore - see;
http://www.wellingtongrey.net/articles/archive/2007-06-07--open-letter-aqa.html

That's downright shocking! My high school physics course was terrible, but it at least involved equations!

 

[ideasrule]

#3 in '02, '03 and '08, #2 in '01, '04 and '05. The three people with Chinese names are probably third-generation immigrants from Taiwan whose grandparents immigrated in the 60's.

That's simply insulting. Do you have ANY evidence at all to back up your claim? You're so sure that their grandparents came from Taiwan, not from Hong Kong or the mainland? You're so sure they're not one of the many children born in China (or Hong Kong, or Taiwan) who come to America and find the math insultingly easy? When was the last time you attended school? I'm attending school RIGHT NOW and I can tell you that the best math students are recent Chinese immigrants, not third-generation immigrants.

 

[ideasrule]

One thing that is really interested in East Asian math education is that there is hardly any memorization of formula.

Where did you get this information? How many American high school students know how to express tan (x/2) in terms of sinx and cosx? Be honest: do you know? How many know the half-angle formulas ? All Chinese students who intend to study the sciences (as opposed to the humanities) are required to know these off the top of their heads. I don't know about the situation in Japan, but I assume it's similar.

Math involves a negligible amount of memorization, considering the variety of dates, events, names, and implications that a history or philosophy student has to remember. I really don't think students are having trouble because they can't memorize a few formulas; they're having trouble because they can't understand them.

The difference between China's math education and that of the West is that there's very little plug-and-chug in Chinese math classes: all the problems are difficult and take a lot of thinking to solve. There's also a huge, almost unbearable amount of homework because the competition for admission to university is intense. When I was in grade 6, I did about an hour of math homework every evening, plus half an hour of math at lunchtime. And how many hours does the typical American sixth-grader do?

 

[ice109]

Where did you get this information? How many American high school students know how to express tan (x/2) in terms of sinx and cosx? Be honest: do you know? How many know the half-angle formulas ? All Chinese students who intend to study the sciences (as opposed to the humanities) are required to know these off the top of their heads. I don't know about the situation in Japan, but I assume it's similar.

Math involves a negligible amount of memorization, considering the variety of dates, events, names, and implications that a history or philosophy student has to remember. I really don't think students are having trouble because they can't memorize a few formulas; they're having trouble because they can't understand them.

The difference between China's math education and that of the West is that there's very little plug-and-chug in Chinese math classes: all the problems are difficult and take a lot of thinking to solve. There's also a huge, almost unbearable amount of homework because the competition for admission to university is intense. When I was in grade 6, I did about an hour of math homework every evening, plus half an hour of math at lunchtime. And how many hours does the typical American sixth-grader do?

how much thinking does that involve?

i tutored at a place called kumon learning here in the states, which is based on japanese methods, and this is from their website:

"This method involves repetition of key mathematics skills, such as addition, subtraction, multiplication, and division, until mastery is reached."

so although i don't have good information my impression is that asian mathematics education is actually terrible, as far as real mathematics is concerned.

 

[ideasrule]

how much thinking does that involve?

i tutored at a place called kumon learning here in the states, which is based on japanese methods, and this is from their website:

"This method involves repetition of key mathematics skills, such as addition, subtraction, multiplication, and division, until mastery is reached."

so although i don't have good information my impression is that asian mathematics education is actually terrible, as far as real mathematics is concerned.

I don't get your point. My point was that Chinese students memorize more formulas than American students, not less, and the fact that the Chinese memorize the formula for tan(x/2) is an example of that. How difficult it is to derive the formula is irrelevant.

 

[ice109]

Where did you get this information? How many American high school students know how to express tan (x/2) in terms of sinx and cosx? Be honest: do you know? How many know the half-angle formulas ? All Chinese students who intend to study the sciences (as opposed to the humanities) are required to know these off the top of their heads. I don't know about the situation in Japan, but I assume it's similar.

Math involves a negligible amount of memorization, considering the variety of dates, events, names, and implications that a history or philosophy student has to remember. I really don't think students are having trouble because they can't memorize a few formulas; they're having trouble because they can't understand them.

The difference between China's math education and that of the West is that there's very little plug-and-chug in Chinese math classes: all the problems are difficult and take a lot of thinking to solve. There's also a huge, almost unbearable amount of homework because the competition for admission to university is intense. When I was in grade 6, I did about an hour of math homework every evening, plus half an hour of math at lunchtime. And how many hours does the typical American sixth-grader do?

I don't get your point. My point was that Chinese students memorize more formulas than American students, not less, and the fact that the Chinese memorize the formula for tan(x/2) is an example of that. How difficult it is to derive the formula is irrelevant.

i would call recalling lots of formulas exactly "plug and chug." but if by difficult to solve means recalling lots of formulas to you then i guess i misunderstood you.

 

[twofish-quant]

Some people are fundamentally incapable of solving problems. It's not the matter of the right or wrong approach.

And East Asian schools don't have this philosophy at all. *Everyone* that goes through the school system is assumed to be able to understand what would be extremely basic skills in the US.

They will get some benefit from rigorous math treatment, but they won't become good technically trained factory workers, even in Taiwan. Dumbing down the subject per Lockhart's suggestions won't help them much, but it will lower the bar even further for the ones capable of abstract thinking.

It's not "dumbing down" but rather "teaching math correctly." If you go to any school in Japan, Mainland China, Taiwan, or Hong Kong, pick some random students, you'll find that even the slowest students can do math problems that most American college students find difficult. This means that the colleges don't have to spend their time teaching remedial math. You just don't find technical schools or colleges in East Asia trying teach algebra 101.

I heard the same thing about school students coming to the United States from Russia. American school mathematics has already been dumbed-down to a remarkable level (in many cases, using exactly the methods Lockhart describes, to make it more "intuitive") and that is definitely something that needs to be reversed.

Again, it's not dumbing down, but teaching math correctly. If you keep hitting your head against a brick wall and it doesn't budge, then maybe you should think about getting a sledgehammer rather than hitting the wall harder.

 

[twofish-quant]

All Chinese students who intend to study the sciences (as opposed to the humanities) are required to know these off the top of their heads.

My wife was an elementary school teacher in Taiwan. Chinese students who intended to study in the sciences are expected to *know* formula off the top of their heads. They aren't expected to *memorize* the formula. I'm willing to bet that if you ask most Chinese science students how to derive the formula, they can.

Math involves a negligible amount of memorization, considering the variety of dates, events, names, and implications that a history or philosophy student has to remember. I really don't think students are having trouble because they can't memorize a few formulas; they're having trouble because they can't understand them.

The way that math is taught in East Asia requires a negligible amount of memorization. Students in the US have problems with math because they are taught memorization and not understanding.

all the problems are difficult and take a lot of thinking to solve. There's also a huge, almost unbearable amount of homework because the competition for admission to university is intense. When I was in grade 6, I did about an hour of math homework every evening, plus half an hour of math at lunchtime. And how many hours does the typical American sixth-grader do?

Exactly. And if you have lots of trouble, then you go to a "bushi-ban." Being more "intuitive" doesn't mean "easier." What really annoys me about American math is that people see (correctly) that East Asian students work very hard at math, and so assuming that you can get the US system to work by just spending more effort on a system that don't work.

Also there is a huge investment in good math teachers.

 

[twofish-quant]

Also if you really want to improve things, you have to get some people from East Asian with hands-on math teaching experience and bring them to the US. If you try to create a program from second or third hand information, you'll just end up with a "cargo cult" airplane that won't fly. You can exchange them for American writing teachers, since I do think that American schools are much better at teaching high school students to write than Asian schools.

I do know of a few parents from Taiwan that end up sending their kids back specifically because they want them to get a better math education.

 

[atyy]

Also if you really want to improve things, you have to get some people from East Asian with hands-on math teaching experience and bring them to the US. If you try to create a program from second or third hand information, you'll just end up with a "cargo cult" airplane that won't fly. You can exchange them for American writing teachers, since I do think that American schools are much better at teaching high school students to write than Asian schools.

I do know of a few parents from Taiwan that end up sending their kids back specifically because they want them to get a better math education.

My bias is to think you are right about Chinese maths - I'm a biologist and a biologist friend of mine from China can do derivations that US engineer friends of mine marvel at. But my bias is also to suspect you are wrong about American writing education. At least in biology, I've often been recommended Strunk and White, who have tin ears for style.

 

[twofish-quant]

But my bias is also to suspect you are wrong about American writing education.

If you've ever tried to help someone from East Asia write an essay, you'll find what a painful process it is. I don't mean the mechanics of grammar and spelling, but rather the essay form. In most East Asian education, written consists of taking the words of people in authority, showing that you've memorized what they've said in great detail, and then coming to the correct conclusion.

There are political at work here. Suppose you are a high school student in Mainland China, are you seriously going to wrote an original essay on what you really think about government policy? If you are an official in the Ministry of Education, are you seriously going to develop a curriculum in which high school students are encouraged to write about and discuss government policy? Of course not. Even things like history are a landmine. There is an officially approved history. Your grade is determined by how well you can repeat the official history. If you have any alternative ideas about history, you best keep them to yourself.

This causes huge problems with students from East Asia come to the US, because what is bad in East Asia (coming up with original thoughts and ideas) is good in the US, and what is good at East Asia (copying the words of established authorities verbatim) is considered plagarism in the US. In the US, you are supposed to write using your own words and your own style, whereas this is generally considered a bad thing in most East Asian schools. You are supposed to quote authorities and use the precisely approved sentences and phrases to come up with the officially correct conclusion.

This also has other implications. When someone in the US writes an essay, it's assumed that they are writing what they believe. When someone in China writes an academic essay, then the opinions expressed in that essay may have nothing at all close to what they actually believe.

Also while East Asian students do spend more time at math, students (at least in Taiwan) don't spend that much time writing.

I don't want to give people the mistaken impression that I think that education system in Taiwan is "better" than the educational system in the US. The US has a weakness in primary and secondary math education, but it has a lot of strengths also.

 

[atyy]

If you've ever tried to help someone from East Asia write an essay, you'll find what a painful process it is. I don't mean the mechanics of grammar and spelling, but rather the essay form. In most East Asian education, written consists of taking the words of people in authority, showing that you've memorized what they've said in great detail, and then coming to the correct conclusion.

There are political at work here. Suppose you are a high school student in Mainland China, are you seriously going to wrote an original essay on what you really think about government policy? If you are an official in the Ministry of Education, are you seriously going to develop a curriculum in which high school students are encouraged to write about and discuss government policy? Of course not.

This causes huge problems with students from East Asia come to the US, because what is bad in East Asia (coming up with original thoughts and ideas) is good in the US, and what is good at East Asia (copying the words of established authorities verbatim) is considered plagarism in the US.

Also while East Asian students do spend more time at math, students (at least in Taiwan) don't spend that much time writing.

But how about essays like "On the importance of gravity"?

 

[ideasrule]

My wife was an elementary school teacher in Taiwan. Chinese students who intended to study in the sciences are expected to *know* formula off the top of their heads. They aren't expected to *memorize* the formula. I'm willing to bet that if you ask most Chinese science students how to derive the formula, they can.

I agree: Chinese students will almost certainly know how to derive it. They are also required to memorize it, but they probably use it so much that it becomes impossible not to remember it.

About American math: is it really focused on memorization? How many formulas can there possibly be to memorize?

 

[Sankaku]

...you'd think that the logical thing to do is to translate East Asian textbooks...

It has been done for the primary grades and is called Singapore Math:

http://en.wikipedia.org/wiki/Singapore_math

http://www.singaporemath.com/

It is very good and I have used it with my daughter. It definitely moves faster than the Canadian curriculum. That said, it is still not the perfect system and requires a skilled teacher.

 

[twofish-quant]

Hamster143's response is the sort of total utter non-sense when ever I bring up teaching math. I'm sorry to be harsh about this, but it has to be said.

It turns out that in any sort of East Asian math book, there are no "word problems" at all in the American sense. It's all conceptual. Conceptual does not mean easy. It is true that East Asian math classes are "harder" than US math classes, but if you make hard classes with stupid brain dead word problems, you aren't helping anyone at all. Why the hell are we talking about "wood". There is this concept called "x". And there is all of the non-sense about how wonderful things where in the past. US math education was never very good, and in the case of higher education, the US ended up just copying the Germans.

The other thing that I think is total non-sense is that idea that math is only for smart people. The important thing about East Asian math classes is that *everyone* is expected to learn the material, and if students can't then the curriculum and the teacher gets changed. The thing to notice is not how well the scientists and engineers do, but how well people that go through the Chinese equivalent of community college do. The only reason that we have this idea that math is for smart people in the US, is because of how badly it's taught, and how little resources go into math teaching.

And one more thing... It's crucial in the global economy to be multilingual. If you go into any Chinese bookstore, you see *tons* of excellent math preparation books, and it's just a fact that if you want the world's best primary and secondary textbooks, you are better off being about to read Chinese, Japanese, or Korean. If you can read Chinese, and then get any Chinese math textbook, it quickly dawns on you how horrible US math education is, and how telling students to work harder at the wrong thing is just not going work.

 

[twofish-quant]

It is very good and I have used it with my daughter. It definitely moves faster than the Canadian curriculum. That said, it is still not the perfect system and requires a skilled teacher.

That's one problem with East Asian math, is that it requires tons and tons of very skilled teachers, which means a huge number of normal schools, which means the type of massive state bureaucracies which Americans tend to distrust.

One other problem is that as the economy improves it becomes harder to get skilled teachers. In the early 1970's in Taiwan, you could pretty easily get ambitious young women from the countryside to go to boarding schools, which were run something like military training camps. You really can't do that now, so most of the older teachers think of the younger ones as "soft".

However, the fact that both Mainland China and Taiwan got so far so fast is pretty amazing. I have older relatives in which someone was considered extremely highly educated because they graduated elementary school and could read. Getting from 20% literacy to 90% literacy inside a generation was not a small thing to do, but it turns out to be essential for economic growth.

 

[twofish-quant]

About American math: is it really focused on memorization? How many formulas can there possibly be to memorize?

It's worse than that. American textbooks tend to have people try to memorize specific processes. Memorize how to calculate this type of problem. Memorize how to calculate this other type of problem. Memorize how to calculate this other type of problem. One of the first thing that you notice about East Asian textbooks is how thin they are in comparison to American ones. That's because they focus on teaching a few concepts rather that a hundred processes.

The problem is that if you have people memorize 20 different rules which are actually part of the same concept, you are making things more difficult for the student and wasting their time and yours, but anyone that complains about this is accused of "dumbing down" the curriculum. It's actually the opposite. Because the entrance exams in East Asia are so tough, you do everything you can to make things simpler, because if you make things needlessly complicated then you are doomed.

Also the way the US does standardized testing makes things worse. It's not that standardized testing is a bad thing (after all, people in East Asia go through this trouble to pass the entrance exams), but the details how how the standardized testing is set up increases this bad aspect of US math teaching. One other thing that you quickly find about education is that it's political in a bad way. What happens is that certain styles of teaching are associated with certain political ideologies so what people really are arguing about is politics and not math. (It's true.)

It's not that education is less political in East Asia, but you have *different* politics.

 

[twofish-quant]

Also problems about piles of wood are pretty stupid. No one gives a damn about piles of wood, and if you talk to a professional logger they'll tell you that the problem is bogus anyway. Since I've taught at the University of Phoenix if I have to come up with a word problem it would be something like.

1) You just lost 40% in your 401(K) last year. Assuming that your employer doesn't/does match contributions this year, how much do you have to contribute to reach your retirement goals assuming the Dow goes to 12000, 10000, 8000, 5000?

2) If you were to get laid off tomorrow, how much money in the bank do you need to survive for X months?

3) What increase in salary do you need to make your tuition in UoP a positive investment?

4) How much money will you save if you pay off your credit cards?

All you have to do is to mention three or four of these sorts of questions, and then Algebra 101 is no longer boring or montonous, and at that point you focus on concepts so that people have the skills to answer those questions, and other questions which life throws at you. Again, if the US had a decent math education system, my students would have learned all of this in the 7th grade, but better late than never.

I sometimes wonder if the reason that Chinese are huge savers is that most educated Chinese in China can do basic algebra whereas a huge fraction of Americans can't. If you can't do math, you are going to have to rely on the bank to do the math for you, and you are likely to get screwed since the guy at the other end of the table knows how much money he can squeeze out of you, and you don't.

(How much is this adjustable rate mortgage going to cost me, if interest rates go up to 8%? How much do house prices have to go down before I'm underwater?)