A Mathematician's Lament: An essay on mathematics education

[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 because 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.aspWhile 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 ).

 

[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

Don't you think #1 is badly phrased?

 

[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 a lot 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.