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-benTo me, that’s the difference between Singapore and traditional materials. With Singapore, you become a local in the land of math; with traditional programs, you’re a tourist.” - Laurie4b, as archived on Paula’s Archives
Ever wonder why the average American is so atrociously bad at math? So did the people behind Where’s the Math?: Washington Parents and Educators for a Mathematically Correct Curriculum, which made the video featured in today’s Youtube Monday.
When I first watched this video, I thought it wasn’t going to be anything particularly noteworthy. But two minutes into the explanation of the “alternative methods” for solving simple arithmetic problems, I found myself shocked at the kind of drivel that gets pushed into elementary school1 curricula.
Unlike the monolithic national syllabi used in Singapore, there is no standardized curricula in the United States system. Instead, groups of schools in the same “school district” implement their own decisions as to what curricula to pursue, which textbooks to buy, etc. The presenter, D. M. McDermott, does a fantastic job of showing how riduculously convoluted the methods for simple arithmetic problems can get in American curricula. (Remember that educational textbooks in the US are also ridiculously expensive.) She also ends up hawking “Singapore Math“, the American term for our mathematics syllabi.
A quick Google Search for “Singapore Math” reveals the existence of an entire counterculture of American parents and teachers exasperated with the “fuzzy math” that passes for mathematics education in the United States. Teaching Singapore Math was a topic of intense debate amongst concerned parents back in late 2004 and early 2005. Read, for example, these two pro and con opinion pieces. A lot of the interest stemmed from Singapore’s No. 1 ranking in the Trends in International Mathematics and Science Study (TIMSS).2, and there is plenty of anecdotal evidence that school districts that implement Singapore Math can rapidly exceed their peers using average curricula.
Virtually all the American reviews of Singapore Math are unaware of how the Singapore education system is structured. There is also a very absorbing podcast of a discussion on NPR, which focuses on the things stressed in the Singapore syllabus: mental math, word problems and bar charts. And while the primary school series is very well received, the divergent needs of the Singapore and American systems quick give educators trouble when trying to ape the Singapore curriculum exactly. Curiously enough, there is absolutely no mention of ‘A’ Maths anywhere on these websites.
Update 2007-01-25:
Tomorrow.sg features a recent article in the New Jersey Star. According to the newspaper,
FootnotesSingapore Math is used in 100 districts and about 500 schools across the U.S.[...] The program is most widely being used in Massachusetts.
Sandra Kase, a consultant with the New York Comprehensive Center, said they plant to launch a pilot program of Singapore Math in 10 schools in New York City in the fall.
- For my Singapore readers, elementary school is what Americans call primary school.↩
- This No. 1 ranking, unlike many others, is something that we Singaporeans can be rightfully proud of. The ranking is consistent over several years, and the study is statistically rigorous. Perhaps tellingly, the TIMSS report shows that the Singaporean students surveyed are below the world average in dealing with negative numbers and statistics and probability.↩
the americans also forget a major component of singapore math - tuition.
It’s pretty interesting to read the opinion pieces and the comments on YouTube. I think McDermott is aware that most people proficient in mathematics use the TERC method to do quick mental multiplication sums in their heads (I’m pretty sure they learn it themselves, not from anyone), but she bashes it anyway because it’s being used to teach maths and work out math problems.
I think we should be thankful for the rigorous design of our math syllabus which not only allows us to have an almost peerless foundation of math principles and reasoning, not to mention really understanding how the principles work, however, it’s true that it also hurts us in one important area - I don’t think we have any idea how to create formulas in the first place.
I know I don’t hahahaha.
I think the standardized interational math tests only go up to grade 8 level (?) so the ‘A’ Math syllabus wouldn’t be covered, anyway. But I think the prevailing attitude of US educators is that once you start them off with good foundations the rest will follow. Unfortunately the system in US schools is so rotten that most students are entering college with only minimal calculus (excepting those who go to top public, or private schools), not to mention absymal algebra skills. And it’s completely acceptable!
coleman: If you read the discussion boards I had linked to, you would see that a significant fraction of the teachers and homeschooling parents trying out Singapore Math actually are quite aware of the need for tutoring outside the standard syllabus.
felumpfus: Yes, TIMSS stops at grade 8, but a lot of people trying out Singapore Math want to implement it wholesale for K-12. Then they get upset at how Maths Syllabus ‘D’ (E Maths) is missing all sorts of things…
I think we should be thankful for the rigorous design of our math syllabus which not only allows us to have an almost peerless foundation of math principles and reasoning, not to mention really understanding how the principles work, however, it’s true that it also hurts us in one important area - I don’t think we have any idea how to create formulas in the first place.
Erm, I don’t think the Sg math syllabus lets us understand how the principles work at all. It’s all a matter of telling us the formulas and teaching us how to use them; how does that say anything about principles? Your last sentence sums it up. Although I was always one of the better students at math, doing real proof-based math in university was a huge culture shock. And also a lot more interesting.
We don’t learn proofs here at all, so we know what to do, but not why we need to do it. Of course the first it necessary for the second, but still…
PC left wing people would say that standardised tests miss out a lot and are thus useless as an indicator of math ability =D
We are moving towards proof based mathematics in our schools. People tell me that topics like graph theory and linear algebra are taught in our JCs. It’s quite hard to avoid proofs for these topics.
Linear algebra was covered in F Maths in my time, and a little in Math C.
Now those are some alternative algorithms! But eh… all of them made sense to me, I can even tell you how the lattice method introduces place value! Hint: the lattice method is not entirely different from the standard algorithm, in fact, it’s exactly the same, just arranged differently. I must admit that the planning trips are over the top though.
And I did F-maths in JC. Linear algebra and graph theory had ZERO proofs. It was just repetitive application of the standard method you were taught to find eigenvalues and eigenvectors; graph theory is also more of the same repetition of using a standard method.
Oh come on, I don’t recall learning any mathematics in school, just more of the same ’standard algorithms’ for variations of the same problems. Utterly mind-numbing. I certainly preferred the math in Uni.
takchek,
I was also a F Maths student. We did linear algebra but without proofs. And I don’t mean proofs like how a matrix is invertible iff its determinant is non-zero. One can prove that just by memorizing the rule that the determinant of a matrix product is the product of its determinants — without real understanding. I mean things that start from the axioms that tell us what linear algebra actually is, and what a vector space is. Essentially I emerged from F maths (and Maths S-paper) thinking that maths was about calculating stuff, which is completely wrong.
I never saw, throughout all the differentiation and often rather calculationally complex integration we did up to A-levels, any mention of what a limit really is (there was some hand-waving about functions approaching a value, which is of course completely useless as a mathematical definition). It now strikes me as faintly ridiculous to do calculus without knowing that.
Okay. I did not do F math but I did math in uni and because of that my perspective may be skewed. I just find it hard to imagine how linear algebra can be taught without proofs - calculus, sure but linear algebra? Anyway, my bad.
I still agree with Diodati that Singapore math is something to be proud of. With regards to fundamental theories, I can’t speak for everyone but I do recall my A maths teacher in secondary school spending substantial time on explaining limits and calculus from first principles. Whether we could (or wanted to) absorb or appreciate it at that level, well, it’s another matter altogether.
I also see some form of bashing Singapore math for its inadequacies going on here. That’s really not the point, I feel. Compared to age- and level-matched peers, what I learned in math classes up to JC is unrivalled in terms of breath and depth to what my American-educated friends have experienced. As I mentioned before, the level of math education in the US is so uneven (due to vastly differing standards between states and public vs private schools) that there is a large degree of variation between what a top student from school A and a top student from school B might know. There is a strong case to be made in favor of standardization of math education, and I feel proud that Singapore has done it in a fairly competent manner.
Furthermore, it makes perfect sense that proofs and conceptual appreciation is taught in university — students are much more equipped to process that level of understanding at that point. In university, students have the luxury of specialization, giving more time to pursue their majors of interest in depth. Considering that the average secondary school student is studying 6-8 highly diverse subjects simultaneously, we can’t really fault the system for not being able to impart to students the full understanding of everything they are learning.
F Maths is already phased out of the MOE curriculum anyway.
I think our curriculum is awesome for the elementary school. After that it kind of loses steam as it gears up toward pre-specified ‘O’ and ‘A’ level syllabuses which may or may not reflect the skills needed in modern jobs.
Formal mathematical reasoning by proofs and all is very rewarding, but the mental faculties for such are not well developed until a very late age. It seems possible to introduce formal reasoning at the high-school level, but not earlier.
I’m not opposed to learning by example. I believe formal theories are best learned when peppered with concrete examples from which students can be guided into abstractions. But in the Singapore system, there is almost exclusive focus on algorithms, i.e. “how to do it right”, and then practice ad nauseum until like it or not, you know how to do them.
We do a good job laying the foundations for mathematical reasoning, but we sure as heck don’t make it a point to point the way to abstraction. Who would have thought that determinants would become such monstrosities when framed in the context of linear operators?
Actually, it’s possible to do proof-based mathematics as early as secondary school. You don’t even need to do it in math classes. Proof-based reasoning can be easily introduced in simple set theory (e.g. prove that the union of an open set with another open set is also an open set) or very simple number theory (e.g. prove that the there is an infinite number of primes). Another example: Euclid’s proofs for geometry used to be taught in British grammar schools.
One advantage of learning formal reasoning is that it promotes logical thinking. I think it is a skill that people can at least take away from their schooling experience. For those of the more practical bent, logical reasoning is fairly important in the study of algorithms in computer science.
I don’t think formal reasoning is necessarily any harder than computation-based math in secondary schools. Computation-based math can be as hard anyone wants to make it. For example, computing the integral $\int^{1}_{0} dx \frac{x^{2}\ln x}{\sqrt{1-x^{2}}$ or inverting a 4 by 4 matrix by hand can be quite challenging but one does not learn very much from doing it. I recall most of my JC math C (the only math I studied in school) lessons was about doing all kinds of complicated integrals quickly and nothing about formal reasoning. What a waste of time given that Mathematica was invented by then.
The Singapore system can probably be more holistic if it taught some kind of very basic formal reasoning a la Euclid’s proofs or even simple truth tables.
Two more comments:
1. It would be hard to teach formal reasoning in mathematics in secondary schools and JC because not all math teachers are have degrees in mathematics.
2. Americans may be lousy in mathematics but one cannot then readily infer that Singapore has a world-beating high school math curriculum. From what I know, HK’s A-level mathematics exams are way more challenging that Singapore’s. The French and Germans have a highly mathematical preparation in their technical streams in high schools. I’ve been told Malaysia’s STPM is considerably more difficult than Singapore’s A-levels.
What Fox said.
Some of the integrals I did in F Maths were more difficult than any of the calculations I had to do in undergrad physics (in undergrad maths there are, naturally, no calculations). I don’t see why we have to learn how to compute ridiculously tedious integrals that scientists would just plug into Mathematica, and I don’t think computing those is any more demanding on one’s intelligence than doing calculus by proofs.
I also agree about the benefits of proof-based mathematics for logical reasoning. I grade the homeworks for a lower-level calculus course at my university and the number of elementary logical fallacies students commit is astounding. Open any editorial page in the newspaper and you are guaranteed to find plenty of arguments based on logical fallacies, which people lap up willingly as long as they support their ideologies. So this is not exactly a socially irrelevant issue. Of course this may be better served by making everyone take elementary logic, but that would probably be an even harder thing to sell to the pragmatists.
Re: math teachers’ qualifications — this is what happens when MOE goes completely nuts and takes on engineers and accountants as math teachers. *shrug* On the other hand the supply of qualified mathematics majors probably doesn’t match their demand as teachers.
If we taught reasoning and proofs in school wouldn’t students just memorize them? That seems to be the commonest approach to studying in Singapore, regardless of the subject.
Are there any education systems anywhere in the world that teach logic and reasoning to pre-tertiary students as a matter of course?
Felumpfus,
Memorising particular proofs will not help you to construct proofs for a general class of problems. When I grade calculus homeworks I can easily tell when students are attempting to regurgitate memorised techniques and when they are actually reasoning. In the latter case you can see the actual thread of reasoning, the advancement from premises to conclusion. In the former case nothing makes sense — they make arbitrary connections between aspects that may be related in their memorised proof but not in the proof required for this problem, or they assume the conclusion and prove the assumptions, etc.
As for logic and reasoning in other countries, I have no idea, but I think Sg would be better served spending the time used to teach students how to compute difficult integrals to teach them basic reasoning skills instead.
I actually agree that at the primary to lower secondary level the Singaporean method is probably the way to go. However it makes no sense to continue that mind-deadening strategy when students already have the basic apparatus for manipulating abstract entities.
Back to the video:
It seems to me that the solution to McDermott’s dissatisfaction with Washington state’s math curriculum would be to use old junior school math textbooks from 10 or 20 years ago or even Canadian textbooks. There’s nothing special about Singapore Math.
Well, most of what I have to say about Singapore’s math syllabi has already been said, especially re: computing difficult integrals endlessly.
I agree that for the majority of Americans, the math education doesn’t serve them at all.
But I majored in math in college, and my peers were damn bloody good at the subject! And I don’t just mean the type who attempts to prove Fermat’s Last Theorem or the Riemann Hypothesis at the age of 16. And the majority of them did not come from expensive private schools or Thomas Jefferson. One guy hated math in high school, was set on a biochemistry/pre-med major when he first came to college, and by the time he graduated, he won multiple awards in the department, received A+s for graduate level classes, and is now on a prestigious scholarship to a top graduate school. Another girl nearly did not attend college, flunked out of Calc 1 in her freshman year, but for some reason decided that math was the major for her. She went on to ace the graduate algebra class that I spent an entire semester cursing and swearing at, and is now also at a top graduate program.
Where do these people come from?
And the great thing about the American math system is that the motivated can take college classes at their local college. The honors calculus classes at my alma mater are regularly attended by local high school students. Michigan State has an accelerated math program for promising middle schoolers. One of my friends went through this program. He was from a working class background with no one to advise him on educational matters. Now he has a doctorate in engineering and has landed a damn good job (in terms of work, pay and prestige) at a national lab.
On average, the Singapore math system produces better students than the American system. But there is something about the American system that produces extraordinary students.
l’oiseau rebelle: Maybe the US has just so many people that they naturally have more extraordinary students on an absolute basis?
The motivated JC students can take classes at NUS now. I think NTU also.
Fox:
“Actually, it’s possible to do proof-based mathematics as early as secondary school.”
I can see where you’re coming from. However, like you, I am concerned about whether our current crop of Mathematics teachers are up to the task. Formal proofs can be very hard to teach, also because there (almost always) isn’t a single definitive solution. Also there is the danger of going too far and ending up with something similar to the New Math fiasco of the 60s and 70s.
I remember in my GEP days trying to “prove” that the sum of angles in a triangle add up to 180 degrees. The result of that exercise never seemed to lead up to the notions of formal reasoning, but rather some kind of exhortation to go look up the answer in a math book…
“logical reasoning is fairly important in the study of algorithms in computer science.”
If anything, I believe you understate the case. The entire field of algorithms can more or less be boiled down to manipulations of axioms in universal computing machines.
Wowbagger:
“I don’t see why we have to learn how to compute ridiculously tedious integrals that scientists would just plug into Mathematica, and I don’t think computing those is any more demanding on one’s intelligence than doing calculus by proofs.”
You should repeat that to the people who teach quantum field theory and graduate statistical mechanics.
felumpfus”
“If we taught reasoning and proofs in school wouldn’t students just memorize them?”
Sadly enough, I cannot deny that this appears to be the most likely response. Then people will try to outdo themselves mugging the shortest proofs, or most difficult proofs, or most creative proofs…
Wowbagger’s point about it being pointless to mug proofs is perfectly reasonable. Unfortunately, I doubt it will stop people from trying. Especially if you don’t get it and really, really want that A…
“Are there any education systems anywhere in the world that teach logic and reasoning to pre-tertiary students as a matter of course?”
There are specialized math and science high schools in the US. Two of which I am aware of are University High School in UIUC and the Illinois Mathematics and Science Academy.
Anonymous:
“It seems to me that the solution to McDermott’s dissatisfaction with Washington state’s math curriculum would be to use old junior school math textbooks from 10 or 20 years ago or even Canadian textbooks. There’s nothing special about Singapore Math.”
Hell no! I can only assume that you are completely ignorant of New Math and the devastating effects it has had on an entire generation of curriculum planners.
I agree there’s nothing special about Singapore Math. However it is unique among available US curricula in its no-nonsense apprach to the basics.
l’oiseau rebelle:
“On average, the Singapore math system produces better students than the American system. But there is something about the American system that produces extraordinary students.”
That just about sums up the differences betwen a Singapore education and a US education, n’est ce pas? Our lowest denominator is higher, but this appears to come at the expense of flattening out the potential of our best and brightest…
Also, what Agagooga said.
[...] - Formal Arithmetic at Age Ten, Hurried or Delayed?“; it touches on a topic brought up on an earlier post about when would be considered a suitable age to learn mathematics. It appears that until the [...]
This is very weird. I’ve gone to school in Washington all of my life and we always learned the way she showed first. I’m glad too because that cluster stuff is way too much work for something so simple.
This is indeed a very interesting forum. I am one of the top maths students in a top school and we do learn how to proof in our maths olympiad classes. I think the Singapore Maths System is fabulous and most of the kids in school know how to proof perfectly well. Singapore does produce its fair share of math geniuses (:
As a Singaporean citizen who is also a mathematics undergrad, I am very disappointed with the mathematics education I received during elementary, middle and highschool. (Primary, Secondary and JC in Singaporean terms)
What we were taught were these 3 things:
1) Memorizing formulas
2) Solving problems/equations
3) Calculating very fast
I was trained to think that mathematicians are people who do very fast calculations in their head.
As a result of this, many Singaporeans come to hate mathematics and believed maths to be a “useless” field of study. As any Singaporean and they will tell you that doing mathematics is the same as doing the 3 things above and that it is a waste of time since we can do those things using a computer. When it comes to proofs, the typical Singaporean will tell you proving is about manipulating equations, usually involving complicated trigonometrical identity.
Come university, I felt short changed by the system. I should have been taught REAL mathematics : logical thinking, mathematical ideas and proofs. Teaching REAL maths instead of blind memorization and calculations.
Real maths is fun! Real maths is about logical thinking, which is far more useful than memorizing tons of formulas and working out tedious stuff by hand.
R:
I think math up to the end of freshman year is heavily algorithmic no matter where you go. There is ample evidence to show that most people lack the mental maturity to appreciate more abstract thought until a certain age has been attained. Also, one has to know the basis in order to appreciate how things work.
Now as to whether the Singapore system focuses too heavily on the algorithmic aspects, that’s a different topic altogether, and one that is far more interesting.
I can see where all those other algorithms are coming from but the standard algorithm is still the one to learn. I usually use the cluster algorithm for multiplication because there are usually easier ways to break up a number than by their digits. But the standard algorithm is the one that made me understand the logic behind double / triple digit multiplication.
As for where the great people in Singapore go, there isn’t room in Singapore for greatness. In Singapore you are much more likely to get a pat on the back for getting 100 marks on a test than to have invented the telephone or something. It’s better to be a family man having a steady job and earning a 5 digit salary than to invent a vaccine for AIDS.
You don’t get great people coming out of Singapore because greatness entails out of the box thinking and that is not appreciated here.