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 dipshit 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?)

[Sankaku]

It turns out that in any sort of East Asian math book, there are no "word problems" at all in the American sense.
While I admit ignorance of other countries, the Singapore math system is very word-problem heavy. The ones I did with my daughter were generally well thought out, required thinking rather than plug-n-chug, and were appropriate for a 7 year-old. I know there are strong opinions about word-problems (I have mixed feelings), but I think the Singapore primary system was successful because of them.

I think this changes dramatically in secondary-school, though, and might be what you are referring to.

[atyy]

While I admit ignorance of other countries, the Singapore math system is very word-problem heavy. The ones I did with my daughter were generally well thought out, required thinking rather than plug-n-chug, and were appropriate for a 7 year-old. I know there are strong opinions about word-problems (I have mixed feelings), but I think the Singapore primary system was successful because of them.

I think this changes dramatically in secondary-school, though, and might be what you are referring to.

Is it true that the Singapore word problems are meant to be solved without algebra? I once looked at them and found them ridiculously hard - maybe Singapore's system is deteriorating.

[Sankaku]

Is it true that the Singapore word problems are meant to be solved without algebra? I once looked at them and found them ridiculously hard - maybe Singapore's system is deteriorating.
The grade 2 and 3 ones I have seen are not that hard - just a little challenging compared to Canadian school. What grade level were you looking at? I cannot see how you are supposed to solve anything in math 'without algebra.' That is like running without breathing.

In what sense is it deteriorating?

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

http://nces.ed.gov/timss/results07_math07.asp

http://nces.ed.gov/timss/table07_1.asp

http://nces.ed.gov/timss/figure07_2.asp


While curriculum is somewhat important, I have come to believe that a culture's attitudes toward learning are even more important. Twofish's comments about Taiwan seem to support this. Willingness to work hard and respect for achievement are just more universal than in the West.

[ideasrule]

While I admit ignorance of other countries, the Singapore math system is very word-problem heavy. The ones I did with my daughter were generally well thought out, required thinking rather than plug-n-chug, and were appropriate for a 7 year-old. I know there are strong opinions about word-problems (I have mixed feelings), but I think the Singapore primary system was successful because of them.


In mainland China, elementary school textbooks certainly have plenty of word problems. I've never gone to high school there, but the Chinese calculus textbook I have has few word problems and the Chinese university entrance exams that I looked at online have none. That said, I'll ask my cousins whether they did word problems in high school; it's better to get some info than to assume things.

[twofish-quant]

While curriculum is somewhat important, I have come to believe that a culture's attitude towards learning are even more important. Twofish's comments about Taiwan seem to support this. Attitudes toward hard work and achievement are just more universal than in the West.

I don't believe this at all, and I don't think for a second that Americans *are* particularly lazy.

When given a choice people in Taiwan can be just as lazy as Americans, it's just that people that work hard do so because they really don't have much of a choice. Once you grow up rich, it's much harder to work hard when you don't have to, but that's nothing to do with nationality. But what happens in the US is that once you have a group of immigrants become rich and lazy, you have a group of poor and hungry one's come in right afterwards.

[Sankaku]

I don't believe this at all, and I don't think that Americans *are* particularly lazy.
I don't think any culture in the world is particularly lazy. I do think that some cultures put a stronger emphasis on achievement through hard work, though. Whether those cultures work harder becuase they are not rich is a different question.

The suggestion above to read 'Outliers' is worthwhile. While I doubt that Gladwell will win any awards for scientific rigour, I think his basic premise covers a lot of this ground (including why Asian countries succeed at teaching math). It is an easy book to read and gives plenty to think about even if you don't agree with all of it.

[ideasrule]

I don't know about Taiwan, but mainland parents are certainly much more concerned about education than Western parents. Maybe this is due to the fact that in China, there's not much choice: there's no hope of getting into university without working your butt off, and there's certainly no hope of getting a tolerable job without getting into university. However, even Chinese parents in the West seem to have this kind of attitude.

This is not necessarily a good thing. Plenty of their parents force their children to study 24/7 and participate in a bunch of useless contests so they have something to show off when applying to university. People don't become good scientists/mathematicians because they're forced to study; they become good scientists/mathematicians because of their natural curiosity or thirst for knowledge, and I'm not sure China's education system is conducive to either.

[Mark44]

I don't think any culture in the world is particularly lazy. I do think that some cultures put a stronger emphasis on achievement through hard work, though.

I think the culture plays a very important rule. In the US, people tend to think of mathematical ability in binary terms: either you have it or you don't, and if you don't have it, why work at it. The East Asian cultures tend to think of this ability as being learned through effort, and that if you don't get it, you need to work harder, the same as becoming proficient at a musical instrument or in athletics. Sure, not everyone has the ability to become a mathematical prodigy, but a much larger proportion of people have the potential to become mathematically literate (or numerate, in John Allen Paulos's terminology - see "Numeracy").

[atyy]

The grade 2 and 3 ones I have seen are not that hard - just a little challenging compared to Canadian school. What grade level were you looking at? I cannot see how you are supposed to solve anything in math 'without algebra.' That is like running without breathing.

In what sense is it deteriorating?

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

http://nces.ed.gov/timss/results07_math07.asp

http://nces.ed.gov/timss/table07_1.asp

http://nces.ed.gov/timss/figure07_2.asp


While curriculum is somewhat important, I have come to believe that a culture's attitudes toward learning are even more important. Twofish's comments about Taiwan seem to support this. Willingness to work hard and respect for achievement are just more universal than in the West.

Instead of algebra, one is supposed to use "bars" or something weird, isn't it? http://www.nychold.com/art-hoven-el-0711.pdf.

I find the bars method ridiculously hard, so I can't do the problems, so I conclude the Singapore maths system is deteriorating (of course I could also conclude my maths skills are inadequate :smile:).

[atyy]

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?)

Hey, these are great! What about something for kids?

[mgb_phys]

Hey, these are great! What about something for kids?

http://www.snopes.com/humor/question/mathtest.asp

[atyy]

http://www.snopes.com/humor/question/mathtest.asp

:bugeye: Don't you think #1 is badly phrased? :tongue2:

[Sankaku]

Instead of algebra, one is supposed to use "bars" or something weird, isn't it?
Those exercises are visual training for kids to prepare for algebra; they are not 'instead' of algebra.

If you ever teach math to children, you will find physical blocks ('math manipulatives') to be very helpful in conveying concepts. This is just a paper version of making the question out of blocks. Remember the age level of the kids this is aimed at.

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

Once kids can do the algebra, you don't need stuff like this any more. If a teacher is saying that you 'can't use algebra,' that is a problem with the teacher.

[atyy]

Those exercises are visual training for kids to prepare for algebra; they are not 'instead' of algebra.

If you ever teach math to children, you will find physical blocks ('math manipulatives') to be very helpful in conveying concepts. This is just a paper version of making the question out of blocks. Remember the age level of the kids this is aimed at.

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

Once kids can do the algebra, you don't need stuff like this any more. If a teacher is saying that you 'can't use algebra,' that is a problem with the teacher.

Why not just teach algebra, wouldn't that be easier - ie. why not teach the easy way right from the start, instead of teaching them the hard way first? Wouldn't a kid pick up bad habits by thinking in blocks rather than algebraically? (Or is this preparation for powerful diagrammatic methods like Feynman diagrams and graphical models?)

[Sankaku]

Why not just teach algebra, wouldn't that be easier - ie. why not teach the easy way right from the start, instead of teaching them the hard way first?
Do you teach algebra to 8 year-olds? If you do, then I want to learn your secrets.

This IS teaching them algebra. It just takes a little time to transition from concrete examples (blocks) to abstract reasoning (x). There are other ways of teaching algebraic concepts to young children, but they almost all use something concrete as a bridge to 'doing it right.'

We could ask this: Why don't we just teach primary school kids about fields and rings? Well, in fact we are - using ideas that they understand and building up to more abstraction as they have enough experience to make sense of it all.

[atyy]

Do you teach algebra to 8 year-olds? If you do, then I want to learn your secrets.

This IS teaching them algebra. It just takes a little time to transition from concrete examples (blocks) to abstract reasoning (x). There are other ways of teaching algebraic concepts to young children, but they almost all use something concrete as a bridge to 'doing it right.'

We could ask this: Why don't we just teach primary school kids about fields and rings? Well, in fact we are - using ideas that they understand and building up to more abstraction as they have enough experience to make sense of it all.

I haven't taught algebra to 8 year old kids, so maybe if those methods work they're ok. I have to say I'm skeptical though, why not hold off until they can do with a method they will still use find useful as adults (or do many adults still use bars in everyday life?). There must be easier problems they can do before that that don't require bars. My own personal experience is that I found word problems really, really hard to do until my elementary school teacher taught me algebra.

[Sankaku]

I have to say I'm skeptical though...
You are thinking either bars or algebra. This IS algebra. You are just not seeing the connection for some reason.

[mgb_phys]

I haven't taught algebra to 8 year old kids, so maybe if those methods work they're ok. I have to say I'm skeptical though, why not hold off until they can do with a method they will still use
You start with, what number do you have to add to 3 to get 5
Then, what number do you have to write in the empty box, ____ + 3 = 5 to make the sum correct
Then you ask, x + 3 = 5, solve for x

There is a percentage of the population that will stare at you blankly once you mix letters and numbers - they grow up to be managers

[Chewy0087]

while yes, at my school/college (UK), there is alot of "it's given to you in the exam, don't bother to learn it" in regard to formulae etc i can't believe half of the stuff said here about the US system.

regularly we go through conceptual as well as numbered question, have numerous practicals/demos etc, and the work is rarely if ever boring. i must admit while i can't remember much of the GCSE maths (2 years ago) in terms of lesson time, it certainly put me in good stead to step up to a-level maths & further maths....

in reguard to the link posted earlier by mgb_phys, i have no idea what examination board that teacher was teaching on, but practically the whole of that article is baloney. (certainly in my experience)

[Moonbear]

You start with, what number do you have to add to 3 to get 5
Then, what number do you have to write in the empty box, ____ + 3 = 5 to make the sum correct
Then you ask, x + 3 = 5, solve for x

There is a percentage of the population that will stare at you blankly once you mix letters and numbers - they grow up to be managers

We pretty much did that since kindergarten. We learned our shapes at the same time, because instead of a blank, we got a triangle or circle to fill in. When I got to algebra and was told all we were doing was substituting a letter for the circle or triangle or blank, I was surprised at how easy it was, and at the very beginning of the course, even felt a bit insulted that we were wasting time doing kindergarten level math!

After reading the comments in this thread, I think a lot of people really lack understanding of childhood development. Children are taught a lot of things by rote and with very clearly defined rules because that is the stage of development they are in. As you mature, you begin to understand more of generalizable concepts and processes and problem solving, but if you just jumped in with that too early, it would lead to horrible failure. Schools that are large enough have an advantage in that they can split up students into different level groups for lessons. This means that those who are developing faster can get more advanced material before they get bored, and those who are slower can continue to be given material at a slower pace to avoid overwhelming them. When you have classrooms filled with students of different levels, it's hard to teach to all of them without either losing the top or bottom of the class to boredom or confusion, respectively.

[atyy]

You are thinking either bars or algebra. This IS algebra. You are just not seeing the connection for some reason.

Hmmm, perhaps I should listen to my mother then ... she used to teach from a syllabus similar to Singapore math, and I complained to her, and she said pretty much the same things you did! (She also told me I'd fail maths exams in Singapore!)

[twofish-quant]

I think a lot of people really lack understanding of childhood development. Children are taught a lot of things by rote and with very clearly defined rules because that is the stage of development they are in. As you mature, you begin to understand more of generalizable concepts and processes and problem solving, but if you just jumped in with that too early, it would lead to horrible failure.

There are different theories and approaches to elementary education, and there is a lot that is far from settled.

When you have classrooms filled with students of different levels, it's hard to teach to all of them without either losing the top or bottom of the class to boredom or confusion, respectively.

Depends on the educational approach. If you follow the educational theories of Lev Vygotsky, you want classrooms with students of different levels and different abilities, because the students and interact and teach each other, and the students that are more advanced can help the students that are less advanced. One thing that I like about Vygotsky's theories is that it very closely approaches how I saw students learn physics at MIT, and the successful way that I've seen college students learn Algebra.

Also schools in Taiwan don't generally split students into different math groups.

[jasonRF]

I missed this post last month (november was a bad month at work!).

I have two daughters, 5 and 8. Lockharts essay is a charge to us parents! It is a wakeup call for me to do right by them by thinking up / finding such "problems" that should encourage their natural mathematical curiosity. We cannot expect the schools to do it all - we need to do our part.

Do any of you educators know of any resources to help some of us parents? I can probably do okay on my own, but resources would help!

My 8 year old loves number problems, for the same reason my wife likes crosswords - they are a challenge and fun to solve. She naturally thinks scientifically, trying to understand why things work the way they do. Recently she was telling me about "proper" and "improper" fractions, just like Lockhart mentioned. I had no idea what the difference is, even though I do technical work for a living (electrical engineer with a phd). Her teacher is forced to teach this stuff. My daughter actually hates her math homework - I have convinced her to rush through it, so she can do the things she does naturally. She is recently into sewing - the geometry of the simple doll-clothes she puts together is great! If she has cut out and sewn the pieces, and needs to make the pants-legs longer for the doll, should she let out the seem or take it in? Without thinking my first guess was wrong (although the right answer only works if the doll legs are skinny enough), but she figures it out, sometimes after mistakes, of course. She is great with numbers, thinks negative numbers are cool (think about borrowing from future allowance!) and sees patterns in numbers a lot, but filling out her math worksheets over and over is dreadful to her. Her current sewing kick is more mathematical than her math homework! I am worried that her education will stifle her amazing enthusiasm for learning and doing!

My 5 year-old is very artistic - I have a painting in my office she did when she was 4 that constantly amazes me. She is now starting to get interested in shapes, numbers, etc., and truly understands what addition and multiplication mean. It turns out arithmetic comes soooo easy to her. I fear for what 1st grade will do to her!

Yes, as their father I am biased!

One scary thing is that Lockhart's concern doesn't go high enough. In college I remember when I was short on time I could use my "math skills" to punch through problem sets without really learning the subject at hand! Intermediate micro-economics was this way for me - I could maximize a profit function subject to constraints (simple Lagrange multiplier stuff that I could do flawlessly every time, although I didn't actually understand why Lagrange multipliers work!) without really understanding the ecomonics! The econ department was bamboozled into thinking this turning-the-crank "math" was economics. I treated the engineering courses that I didn't like the same way - my "good math skills" allowed me to learn nothing in them when I chose. Of course, the classes I loved I really worked on to really understand, and played with ideas on my own. Electromagnetic theory really caught my imagination and being lucky enough to take four consecutive semesters was like a dream! And the semester after that I took three courses had serious electromagnetic theory content! I was depressed as a grad student TAing the only required electromagnetics course for majors. The professor I TAd for didn't ever ask the students to really think about or understand anything! He would do an example problem in lecture, force me to work an almost identical one in recitation section, put another one on the homework, and then ask it again in the exam. Dreadful.

[ideasrule]

(relating to previous comments about abstract reasoning) I don't think children are inherently unable to use abstract reasoning. They can understand concepts like friendship, love, revenge, randomness, time, space, etc, none of which are physical entities that can be touched or seen, yet they can't grasp the concept that "x" stands for an unknown number? That's like saying they can't understand why "Alice" and "Bob" can stand for arbitrary people.

[i_emanuel]

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....

wholeheartedly agree with this.

[mgb_phys]

wholeheartedly agree with this.
Remember schools are to education what lawyers are to justice.

Schools are there to look after the kids while parents go to work, to give local politicians something good and wholesome to support and to give teacher's unions a reason to exist.

If the children learn anything it's pretty much a happy accident.

[Mathnomalous]

I have read Lockhart's essay twice as well as this thread in its entirety and I'm still do not know what a "good mathematics education" is. I do know that what I am now is what Lockhart calls a "trained monkey", that is, I can identify what I'm being asked to solve and find the right method(s) or formula(e) to solve the problem; basically, I can follow the steps. Sometimes I understand why the method works, other times I don't. These gaps in my understanding frustrate me because I've become very passionate about mathematics simply because I do not understand mathematics.

I think I know why 1/2 + 1/4 = 3/4, why x^2 + y^2 = r^2 is a circle, or why the graph of y = x is a line but after reading through all this I have more doubts than ever.

[EvilKermit]

I read the essay, and although mathematics education has its flaws in the United States, the essay emphasizes education should be learning for learning sakes. Interesting it points out how many of the subjects taught in secondary education is not used by the majority of the public. It then concludes that mathematics is really a pointless subject except for a few people and therefore should be taught for enjoyment rather than practicality.
American public education is extraordinary expense, so I expect that the money that taxpayers pay is an investment. Such output from education like economic progress as a whole, lower crime rates, and a chance of economic mobility are all outcomes that are worth paying for. A chance for students to 'learn for enjoyment' is not worth paying for and it's just plain impractical.

But let’s say for one moment that education is ‘learning for enjoyment’. There are many joys and hobbies that people have. There are hundreds maybe even thousands of activities or hobbies that somebody might do. It can be something traditional taught in school like writing, or mathematics, not traditionally taught in schools like martial arts or learning to play the guitar, or something that might require any skills or learning like watching television. If learning is supposed to be a recreational activity how are we suppose to determine which subjects are to be taught? How is ‘mathematics’ any more important than any other hobby, or at least an ‘important hobby’? Perhaps those hobbies that a significant number enjoy can be free to the public. However, I do not seem mathematics being on the list that a significant number enjoy. Furthermore, if we adhere to ‘learning for learning sakes’ there is no way that grades should even be considered, because I cannot think of any recreational or even a competitive activity that involved grading.
To look at a true example of mathematics done for fun there exists academic clubs (math team, science Olympiad, fine arts group, etc.), and academic camps. However these groups do not adhere to the same formula as compulsory education. There are no grades but sometimes competitions, if any teaching it is usually done informal, social interactions are a more important factor, and there is often food and other activities done aside from the main pursuit. The guys model can be used for these types of clubs and camps, but not for school.

The fact is, that in the applied mathematics and applied sciences, sometimes it’s not necessarily important to understand ‘why’ or the ‘beauty’ behind why ‘area of triangle= .5*b*h’. It is more practical to know the formula. It becomes increasingly more complex to ‘show the beauty’ behind certain mathematics concepts.

We are just training others and should be training others. Few will ever enter pure mathematics or even into a job one truly loves. I don’t mean that everyone hates their job. I’m just stating that not everyone will love it. How much somebody likes their job is also factored into how much they make. Those with ‘fun’ jobs will have low salaries, because a high supply of people would want them. Likewise those with ‘boring’ jobs will have high salaries, because a low supply of people would want them. I would pay you to eat cake, but you would have to pay me a decent price to drink urine.

[rasmhop]

American public education is extraordinary expense, so I expect that the money that taxpayers pay is an investment. Such output from education like economic progress as a whole, lower crime rates, and a chance of economic mobility are all outcomes that are worth paying for. A chance for students to 'learn for enjoyment' is not worth paying for and it's just plain impractical.
You don't think people (of all backgrounds) learning to enjoy academic pursuits and being given the opportunity to pursue this is worthwhile? I would argue that this may very well help improve social mobility and eventually reduce crime rates.

I know that many social projects aim to reduce crime by giving children with a troubled background something they find worthwhile to do.

But let’s say for one moment that education is ‘learning for enjoyment’. There are many joys and hobbies that people have. There are hundreds maybe even thousands of activities or hobbies that somebody might do. It can be something traditional taught in school like writing, or mathematics, not traditionally taught in schools like martial arts or learning to play the guitar, or something that might require any skills or learning like watching television. If learning is supposed to be a recreational activity how are we suppose to determine which subjects are to be taught? How is ‘mathematics’ any more important than any other hobby, or at least an ‘important hobby’? Perhaps those hobbies that a significant number enjoy can be free to the public. However, I do not seem mathematics being on the list that a significant number enjoy.
People do not enjoy mathematics due to the way it's taught (i.e. only something resembling math is taught). Math should take precedence over playing the guitar because the skills learned are universally applicable. Learning how to spot patterns, think critically, simplify, generalize, form abstractions, think deeply about a concept, etc. are all skills that should be taught in math and which are widely applicable to other fields. However where I went to school it was possible to take classes on playing the guitar (well music with a focus on an instrument), and physical education (i.e. playing various sports) was mandatory. People should be encouraged to pursue their hobbies.

Furthermore, if we adhere to ‘learning for learning sakes’ there is no way that grades should even be considered, because I cannot think of any recreational or even a competitive activity that involved grading.
You cannot think of any sport whatsoever where people are ranked? Or perhaps graded on a scale 0-10? If you watch for instance the olympics quite a lot of the disciplines are graded. Also school is in itself a competitive activity (who has the highest grade? Who got into the best school?) Also the science olympiads which you mentioned yourself are very competitive and graded. I really see no reason why you can't grade someone who is learning for learning's sake. The important thing is not to let the test taking and grading scheme decide how the subject is taught (for instance teaching how to write grammatically correct English is good, but teaching how to receive a high score on the SAT is stupid).

To look at a true example of mathematics done for fun there exists academic clubs (math team, science Olympiad, fine arts group, etc.), and academic camps. However these groups do not adhere to the same formula as compulsory education. There are no grades but sometimes competitions, if any teaching it is usually done informal, social interactions are a more important factor, and there is often food and other activities done aside from the main pursuit. The guys model can be used for these types of clubs and camps, but not for school.
Why can't this approach be used? Only do infrequent evaluations (these are also used at academic clubs aiming to compete because the best to represent the school have to be chosen).

Were I live it used to be policy not to grade students until after 8 years of study, and in my opinion this worked well (to clarify: people were still given feedback by teachers, but the teachers were forbidden to use a standarized scale and the results were not used for anything except to tell the student how good he was doing). Recently these policies have been changed due to political pressure and everyone pretty much agrees that the change is for the worse since people are to some extent taught how to take tests, not how to understand the subjectmatter.

[Sankaku]

...I cannot think of any recreational or even a competitive activity that involved grading.
You mentioned martial arts yourself. However, HOW to grade things is a big issue of its own...

[Klockan3]

People do not enjoy mathematics due to the way it's taught
You are delusional if you really believe in this. It might apply to a small minority but in my experience most actually likes the pseudo maths taught early better than the real maths taught later, people in general do focus on the real world and the further away something is from it the less people likes it.

[ephedyn]

I disagree with some of the things said about math education in Asia. I won't quote exactly which, but I'll just dispel some myths, or clarify some details. I speak from an educational background in Singapore, and having competed in some mathematical competitions, done my SATs etc.

If you take a look at the math notes for high schools in Singapore, you'll find them heavily condensed, to the point that they are essentially formula lists that could just as well be the appendix of a physics textbook. These notes cover much depth and breadth, but don't do anything to cultivate one's interest in mathematics per se. Asians do very well on aptitude tests not because of a better education system, but simply because much more time is spent on rote memory and computational exercises (or 'mechanizing' the problem solving process in math).

We have something called "ten year series (http://en.wikipedia.org/wiki/Ten_year_series)" down here. Most high school students complete all of these papers within 1-2 months before the final examinations. On top of that, each high school will come up with a 'preliminary' examination paper with predicted questions based on the trends from previous years. The high schools then share their preliminary papers among each other: and it's the student culture here to finish all of these papers. So you have high school students doing at least 15+ sample examinations before the actual thing. That's our weekly homework: each examination is set for a 6 hour sitting. Besides these, many of my schoolmates went for biweekly lessons at tuition agencies to supplement their classes; these are very similar to the "cram schools" in Japan. "Spend at least 30 to 40 hours outside of school studying." That's the advice given by a teacher to my friend's class in another school. At least. And this really happens. If you ever have a chance to land in Singapore past 12 midnight local time, take a look at the cafés at the airport. Many of the seats will be occupied by high school-ers in their study groups

China, India and the rest of SE Asia send many of their top scholars to Singapore too, and I can tell you from being in the same classes as these people for more than 4 years that they have the same formulaic way of doing well at math.

What do I think about this education setting? It's definitely great for basic applied math skills: that's why our physics and chemistry papers can afford to be heavily quantitative; and give any average student here something like the SAT II and he/she will slaughter it. 11% of us have a perfect score on the SAT II Math paper.

But if you give the average student something like the AMC papers, or any Olympiad paper, you'll find that he/she will underperform, because the questions on these don't have formulaic, or formulaic sequences of, solutions. As someone has pointed out, that's why many Asian countries' Olympiad teams have poorer medal tallies than USA's consistently. (Note: China's performance on the IMO, on the other hand, should be attributed to its huge population rather than the strength of its education. They have the resource pool to pick the few dozen people who can be bothered to go home, on top of their heavy workload from school, spend hours cracking problems on AoPS (http://www.artofproblemsolving.com/).)

Now, what's the point I'm driving at? I think that no one should be deceived that implementing an 'Asian' education in US/Europe will solve the problems laid out by Lockhart: these problems are just as pertinent over here.

It's true for the high schools in the States that a more rigorous/intensive curriculum that requires more time devoted to homework/rote memory will indeed grease the cogwheels of the economy and industry to meet growing competition from China. But this doesn't address the problems of declining appreciation for math as an art, language, field, or community that has a rich history. Even at the tip of USA's Olympiad program, many high school students are merely taking part to earn them a place in the most prestigious colleges - this for example, can't be solved by implementing a more challenging syllabus. This growing challenge besets all of today's educators in math, not just in the USA.

[mal4mac]

There's a lot of research on what makes a hobby/job fun - see the positive psychology literature. That 'fun jobs' will not be well paid is just plain wrong. Surgery has been found to one of the most fun jobs, it has a nice balance of challenge and novelty that neatly matches improving skills, given some talent. Surgery, performed by a skilled practitioner, is perfect for achieving the state of flow that is essential to fun in a job or hobby. But don't worry, physicists, you don't need to become doctors. Solving mathematical problems can provide flow, as can designing and setting up experiments. But, then again, so can almost anything, as long as it is suffciently challenging, but not too challenging - from yacht building to reading a novel. So which to choose? (Nice problem!)

[rasmhop]

You are delusional if you really believe in this. It might apply to a small minority but in my experience most actually likes the pseudo maths taught early better than the real maths taught later, people in general do focus on the real world and the further away something is from it the less people likes it.

Maybe I'm delusional, but until something different is attempted we really can't determine whether that is the case.

When people prefer "pseduo math" to "real math" I feel that this too is because they have never been taught how to understand math. They still see math as a mechanical activity, except now it's a mechanical activity together with a bunch of "tricks". People may memorize entire proofs word by word, or remember very specific tricks. A good student will when he encounters something he couldn't think of himself stop and ask himself what kind of motivation went into this and what the general thought behind it is, and then when later asked to reconstruct a proof he can reconstruct it on the fly because he understands all the techniques needed.

Take something such as integration by parts. People learn mnemonics for it, write it on cheat sheets, etc. and in my experience it's one of those formulas that people in calculus 1 are most scared of. However people should realize that it's really just the product rule of differentiation being integrated, and few people seem to have trouble with the product rule. If people just understood this they wouldn't need to remember a proof since it would be obvious.

Similarly in linear algebra people set great store by finding a formula for a linear transformation given the corresponding matrix, and conversely finding a matrix given a linear transformation. The important part is not the procedure, but that there is an isomorphism between the space of linear transformations F^n \to F^m and the m x n matrices with entries in F. Once you know this the procedure should be pretty obvious (for instance just evaluate at the n basis elements F to get the columns of the matrix). People are presented with this viewpoint, but due to exercises and exams testing them on the procedure not the theory behind it, often people only remember the procedure and not why it works or what's so important about it.

A math freshman at my college mentioned to me how he was confused on an algebra exam because question 1 said "Give an argument showing that ..." and question 2 said "Give a proof for the fact ...", but he was unsure exactly what to do. Basically he wondered how rigorous an argument has to be compared to a proof. The answer is of course that in math an argument and a proof is exactly the same thing. There is no such thing as a somewhat rigorous argument. There is a correct argument (aka a proof), incomplete arguments and motivation. I feel many people have this notion that a proof is something very mathematical and when they are asked to give one they start thinking what kind of proofs there are (contradiction, contraposition, direct, by cases, etc.) But ask the same person why tic-tac-toe can't be won against a good opponent, and he would just start experimenting and probably start with something like "well if I place my first piece in a corner, then, ..., if I place it in the center then, ..." this is actually a proof by cases, but it comes intuitively to them, and often a few arguments by contradiction are also used. People need to see math in the same intuitive manner, and not think of it as symbol manipulation even if at a technical level that's a valid viewpoint.

Actually I feel that in early grades analysis of various games is an excellent way to both excite young students and teach them what math is about, not what technical machinery has been introduced through the ages. Even people not interested in math seem to enjoy learning about a good game. For instance in the tic-tac-toe example there is a great deal of symmetry which will teach students how to exploit this kind of structure in an argument. It's also easy to miss explaining how you deal with a particular case and this kind of error will teach the students to think through carefully whether their proof really works.

Maybe your experiences differ, but in my opinion people enjoy a good thought-experiment as long as you remember not to call it a thought-experiment or math. They have learned that they do not enjoy math, but they can often enjoy a discussion which is essentially of a mathematical nature, but with the machinery removed.

Ask a senior in high school how to add two positive integers of 2-5 digits and he can probably do it and has probably done it hundreds of times, but ask him to explain why to procedure he uses is correct and he will probably not be able to give you a satisfactory answer. He will probably not even try, but what should be the case is that he should immediately start experimenting and thinking. Maybe start by thinking of the case where there are no carries, or maybe just consider 2-digit numbers. We need to add some creativity into math, because until college (and to some extent in many intro-college courses) it is severely lacking.

[cristina02]

As a mathematics teacher, this essay help me at this moment. I’m currently developing a set of intervention and extension exercises for grade 6 students. And now I feel so unclean, knowing that I’m not teaching maths at all, in fact I’m burying the maths under a method and calling this mathematics.

[marcusesses]

Remember schools are to education what lawyers are to justice.

Schools are there to look after the kids while parents go to work, to give local politicians something good and wholesome to support and to give teacher's unions a reason to exist.

If the children learn anything it's pretty much a happy accident.

How would you suggest students learn mathematics?

I'm not disagreeing with you, but it seems that if schools are not (or are unwilling to) educate students, that job would fall to the parents, who may not have the requisite knowledge to teach math.

Are you suggesting a less-rigorous, more informal curriculum would work (i.e. learn to solve problems as they come up, but allow students more freedom to learn what they want)? If so, I lean towards agreeing with this, although it would cause problems for teacher's unions, universities, working families...

[Mu naught]

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.

I wonder how many of the people who made those things possible were "pure" Americans?

[bignum]

I think the fundamental problem is not the students (on some level, it is), but it is the teacher's fault and the media.

Media says doing Math is for geeks, delinquents thinks it is true, and they hate math.

Incompetent Math teacher discouraging a talented Math student, Math student hates teacher, Math student hates Math.

[thrill3rnit3]

http://www.stmarksschool.org/academics/mathinstitute.aspx

Take a look. I think his method is pretty interesting.

[eof]

I remember having read Lockhart's essay a while back. I think my biggest issue with math is that we do not tell kids why the boring stuff we are learning should actually be appreciated. With this I mean Alfred Whitehead's famous quote:

Civilization advances by extending the number of important operations which we can perform without thinking about them.

A case in point would be addition. It would have been great if in later classes someone would have e.g. told me how this boring procedure that was done over and over might not have always been so trivial. With this I mean the following observations:

1. The reason why addition is simple is because of the base-10 system in which we write numbers (any other base would do too, so the key idea was the invention of based numbers).

2. Try adding by using e.g. Roman numerals.

3. When doing (2) you will actually cheat, because in your head you add in base-10.

4. The name of numbers in our language are in base-10 which makes them easy to add. Some ancient languages had random names for different numbers and could only count easily to some limit (i.e. up to where they had words for these numbers). We can easily add e.g. 2+3 but if the numbers where e.g. 27+35 and all our numbers had distinct names without a pattern say up to 100, could you do it?

[Astronuc]

Those exercises are visual training for kids to prepare for algebra; they are not 'instead' of algebra.

If you ever teach math to children, you will find physical blocks ('math manipulatives') to be very helpful in conveying concepts. This is just a paper version of making the question out of blocks. Remember the age level of the kids this is aimed at.

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

Once kids can do the algebra, you don't need stuff like this any more. If a teacher is saying that you 'can't use algebra,' that is a problem with the teacher.

Why not just teach algebra, wouldn't that be easier - ie. why not teach the easy way right from the start, instead of teaching them the hard way first? Wouldn't a kid pick up bad habits by thinking in blocks rather than algebraically? (Or is this preparation for powerful diagrammatic methods like Feynman diagrams and graphical models?) I loved the Cuisinaire rod set. My school in Australia introduced students in grade 2, possibly grade 1. That was back in 1964-1965, when I was about 6-7. It was helpful to visualize addition, multiplicatio, subtraction, division, factorizing, equivalence of 3x2 and 2x3. My parents even bought a set for my sister and younger brother.

I was surprised when I came to the US that they were essentially unknown, at least where my family was living. I found the US method of teaching math somewhat archaic and unspiring. I skipped half of 3rd grade because of the move, and found 4th grade quite easy, especially when it came to arithmetic as it was called. I had my math workbook confiscated because I preferred to do math problems during music lessons, and the 4th grade teacher was a bit upset that I was way ahead of the class. I also got marks off because I did the math in my head, and I didn't have to cross out numbers when carrying.

On the other hand, I was terrible in English (literature, writing, . . . ) and especially poetry. I found reading fiction stories excruciating because my mind was elsewhere. :uhh: I got into trouble for not reading fiction books, which I didn't because I wanted to read books on geography, rocks and minerals, rockets and spacecraft, . . . . , and occasionally history, or other non-fiction.

And I got D's in hand writing because I just didn't write well. Writing cursive was painfully slow and my hand couldn't keep up with my mind.

[Sankaku]

And I got D's in hand writing because I just didn't write well. Writing cursive was painfully slow and my hand couldn't keep up with my mind.
I decided, for some reason, that I hated cursive writing. I stopped suddenly in something like grade 5 and refused to do it for ever after. I print to this day...

Just an update to the algebra vs blocks idea. My daughter is now in Grade 4 and starting to grasp basic algebraic concepts. However, making something very abstract (choose R for the red apples and G for the green apples and then manipulate the symbols around) is still too much of a leap for her.

There is an interesting algebra game we play that is building some of the concepts, but going straight to paper is not quite happening yet. Maybe within a year - each child is a bit different.

This is the kind of hands-on algebra game...

http://www.borenson.com/AboutHandsOnEquations/WhatIsHandsOnEquations/tabid/1003/Default.aspx

Except we just made our own with some counters (for x), different coloured dice (positive and negative numbers) and a hand-drawn balance. There is a lot of stuff you can do with the basic idea of "do the same thing to each side of the balance" that young kids get fairly intuitively.

I like games that have a mathematical connection. The card-game "Set" is one of my favorites and kids pick it up very quickly.

http://www.setgame.com/set/index.html

[Chubigans]

I've been thinking this since I began my mathematics education. I'm glad this is a known problem. Thanks for the enlightening, inspiring, and fun paper!

[SonyAD]

The "lyceum" (a high school with math & comp. science curriculum) I went to didn't teach me a lot of stuff I wanted to know (rotation equations, rotation around an arbitrary point, translation, the math of texture mapping, fish eye projection, camera tracking etc.) which I found out on my own. I was taught some stuff that's served me well, though, some stuff I already knew, some that I've never needed (integration and derivation, abstract algebra).

The math manual my teacher chose didn't even touch on rotation equations. She was, is, rather, a calculus and abstract algebra nut. I had gotten quite good at the algebra, not because I liked it, and have since forgot everything because it was all abstract silliness with no cool/immediately apparent application. You didn't need to remember as much clutter as with calculus, though.

I barely, barely passed calculus before the year was out. I didn't have internet access at the time and my folks hired a tutor. Even so, I barely scraped by. I didn't have any interest in learning all the calculus tables by heart and I would pretend to study while researching what I liked: geometry.

Specifically triangles, geometrical and numerical relationships of triangles. I built up a catalogue of formulas for lots of stuff, from the radius of the inscribed and circumscribed circles to the length of a bisector and the projection of a vertex on the opposite side, fish eye projection

Screen_{x} = CenterScreen_{x} + FOV{const} \cdot \frac{ R_{x} }{\sqrt{ R_{ x }^{2} + R_{ y }^{2} + R_{ z }^{2}}}

Screen_{y} = CenterScreen_{y} + FOV{const} \cdot \frac{ R_{y} }{\sqrt{ R_{ x }^{2} + R_{ y }^{2} + R_{ z }^{2}}}

, etc. Lost most of them over time, though.

In 10th grade I coded a software wireframe engine and made some scenes in Excel for it.

I showed the math teacher and she was like: You made that yourself? Bye.

You bet I would have liked a say in the curriculum and educational process. Or choosing whose class I was in.

On the other hand, my computer science teacher was brilliant. She used to train the olympic lot. But her great sin was she was in the bad habit of flunking kids who refused to think (some of whom did really good at math, wouldn't you know), which is all you really need to program, and grading kids fairly. So, naturally, the parents rallied with pitchforks and torches and pressured the head master to have have her replaced so in came the dumb airhead.

Most math teachers I've come across have seemed somewhat dull. They're good at regurgitating and applying arid, abstract math but can't develop new stuff on their own and crickets chirp if I ask them to help with something of practical value. To be fair, crickets chirping whom ever and wherever I ask to help me with something math related.

[tete-a-tete]

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?

memorizing equation seems is the only way i do maths, i try to understand materials in western textbook, usually have large extend of texts combine with graphs, equations. The eastern way of doing maths require lot of hard work and practices. As long as you are keen on the subject and you know how much effort you willing to spend on the subject, thats all it about.And surely you will find the most comfortable way for your study afterwards.

[Diffy]

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?)


I'm not sure about University Algebra, but that is not really the point of an article that addresses math education in K-12.

The author addresses the attempt of math teachers trying to make math questions relevant to students and explains why this makes math boring for them.

Anyone who has been in front of a classroom of 9th or 10th graders would tell you these types of questions would immediately put your class to sleep. These students are simply too apathetic.