In what ways is high school math education inadequate?

[Mépris]

I was under the impression that with my math background (A-Level aka, AP Calc + AP Stats + the pre-reqs to these courses), I should be able to do fine. After a few recent posts in the "Who wants to be a mathematician..." thread, I was surprised as to how horrible I am. I knew that I had a fair bit of work to do before I could take on a rigorous math text but I thought that I was pretty good with algebra. It turns out I'm not and the odds are, most A-Level/IB students are not. It's fairly ironic that I can solve a first order differential equation but I can't prove that "sqrt2 + sqrt5 is irrational".

I understand that the situation is quite dire in the States as well but considering I have no first hand experience, I cannot comment any further. I also stumbled upon something called "New Math" and have heard mathwonk and a reviewer on Amazon speak about an "experimental high school math class in the 60s". Could some light be shed on this?

With A-Levels and International Baccalaureate, mathematics is mostly a "math methods" class, with very superficial understanding of the material involved. Relatively advanced topics like complex numbers and differential equations are covered as well.
I gather that some European systems (Belgium, Holland, France, Germany, Russia) involve a lot of higher math and topics are covered with more rigour but again, I do not know for sure. After a few conversations with undergraduate and graduate Indian mathematics/physics students, I've come to conclude that while they cover a lot of material, more emphasis is placed on memorisation rather than on understanding. Rote learning is a practice that is adopted here as well, with the British A-Levels. However, according to older people I've spoken to, the curriculum was much harder in their day, i.e ~50 years ago, but I don't know what "harder" entails.
Note that I'm not from the UK but from a country who uses the same high school qualifications as them.
If anyone here has been through or is a student of other educational systems, I'd appreciate if you could contribute to this.

So, when did things get dumbed down? Why? What could be done to better the situation? Or do you think things are fine as they are?

 

[jedishrfu]

Check wikipedia before the lights go out:

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

I went thru this form of math it stressed concepts over arithmetic skill. Some kids thrived others really needed the old way. You could tell when they couldn't do a division or multiplication without some errors. Also parents couldn't help their kids with homework because they didn't understand what the kids were doing. You know the line: 'thats not the way the teacher did it...'

A friend of mine who was educated in Taiwan could do lightning fast calculations without the need for a calculator because of the rote learning they had. I would struggle but once I got to Algebra things got easier.

Its like the difference between learning the rules of soccer before playing vs playing the game and learning the rules as you go along.

Rote learn your tables, gain lightning recall doing problems then get taught the concepts behind them
seems like the best strategy.

 

[fact713]

I am a high school student from Russia.
Nowadays many people claim that our formal education system has degraded considerably since the USSR collapsed.

Mathematics has always been my number one subject, but there is something wrong with the way we are taught it: it now seems like teachers think that the only one aim of studying math in high school is to pass USE (somewhat of a SAT equivalent).

Well, in some schools they study really rigorous math- they have university profs and cover university-level material (http://en.wikipedia.org/wiki/Saint_Petersburg_Lyceum_239).

Probably it'd be interesting for you, that's what we study in 10th & 11th grade:
Real numbers, numerical functions(not sure about the translation), trigonometric functions, trig equations, trig expressions transformation, complex numbers, derivative(a huge part), combinatorics&probability - it was 10, now 11:polynomials, powers&roots, power functions,
exponential & logarithmic functions, antiderivative & integral, statistics & probability, (simultaneous) equations & inequations.


Feel free to ask any question.

 

[Mépris]

It sounds like the same material we cover but again, I don't know how further in depth you guys would go. If you have copies of the exams you do, that would be interesting to see. I'm guessing they're in Russian?

At what age do students typically finish high school? Is grade 11 the last year of high school? Sixteen? If so, that would explain the 6-year long course at the MITP! Do you intend on going there?

 

[Obis]

I recommend reading these:

http://www.maa.org/devlin/devlin_06_10.html

http://www.maa.org/devlin/LockhartsLament.pdf

 

[fact713]

Mépris, here is the PDF demo (B=medium level, C=advanced level)
http://mathege.ru/diagnostics/2011/march/5-8.pdf
And yes, they're in russian.

They typically finish at 16-17-18. Yes, 11th is the final grade. I guess the 6-year long course at phystech (MIPT's unofficial name) is what was called "specialitet", before we switched to Bologna, and is approximately equal to 4 years of undergrad studies+2 years of graduate studies, sometimes it is considered more advanced, but I am not 100% sure.

I applied to Waterloo, U of T and McMaster, but if I had to choose a Russian university, MIPT would be the one I would really strive to enroll.

By the way, there is a famous anecdote about MIPT(adopted to English):
Once an examiner asked a university entrant a question:
"o,t,t,f,f,s - continue the row"
- "s", answered the pretendent
- well, your are either a genius or a fool, but in any of these cases, this is not the place for you

(hint: One, Two, Three, Four, Five, Six, Seven)

In Russian it seems to be more complicated)

 

[Intervenient]

Without overcomplicating it, I think it's a combination of non-passionate mathematics teachers, a late introduction of calculus, and the attitude of "I'll NEVER use this stuff" that teachers allow to perpetuate.

 

[nonequilibrium]

From Obis' first link:

The answer is, not well. In an international survey conducted in 2003, students from forty countries were asked whether they agreed or disagreed with the statement: "When I study math, I try to learn the answers to the problems off by heart." Across all students, an average of 65 percent disagreed with this statement - which is encouraging since it is a hopeless way to learn math - but 67 percent of American children agreed with it!

That is scary... Makes me sad. Does anyone have the feeling that this declining can be reversed?

 

[Polymathiah]

I say the main culprits are standardized testing, which shifts the focus of the classroom from learning concepts and information to learning how to ace tests (so that the school may receive funding), and the need for parity in results across the board.

America is big on the blank slate idea of school success, which posits that all children are inherently capable of succeeding in school to the same degree. The only way this parity is possible is to chop down the brightest students by dumbing down the entire curriculum.

 

[jedishrfu]

Actually GH Hardy always said the tripos test in England put English math back a 100 years from the rest of Europe. Students studied to the test, had tutors... because they realized it established their hierarchy in the academic world.

 

[mathwonk]

Here is a reprint of a 100 year old American algebra book that I learned about in an article called "The decline of American mathematics textbooks" written by my son's high school math teacher, the late, great Steve Sigur. You might try this one.

https://www.amazon.com/dp/1177320452/?tag=pfamazon01-20the irony of being able to solve a linear de but not simplify a fraction, reminded me of my harvard russian language class after which I could say "thermal heat capacity" in russian but not "shoes".

 

[mathwonk]

by the way, as i recall, (homogeneous) linear diff eq's have form P(D)y = 0 where P is a polynomial, D is the basic differential operator y-->Dy = y', and the solutions are all linear combinations of the basic ones: t^k.e^(rt), where r is a root of multiplicity > k of the polynomial P(t) = 0.

So far this is a trivial subject. (But can you prove these are the only solutions? That part is often omitted from first courses.)

And note this only reduces the problem to solving the polynomial equation P(t) = 0, hence brings us back to knowing algebra.

Of course the non homogeneous problem P(D)y = g, is harder, and perhaps you know also how to do this.

A less trivial basic subject is solving a first order non linear diff eq. Then beautiful iterative methods come into play.

But do not be discouraged. Knowing where you need improvement is a good thing. I did not even know trig when I went to grad school! I could prove a function of bounded variation has graph of measure zero but could not integrate sin^2(x).