Subject: RE: [hackers-il] Lightning Speed Math Teaching

From: Chen Shapira <[email-masked]>
Date: Wed, 3 Jan 2001 10:03:23 +0200

> What ways do you have in mind in order to fasten pre-academic Math
> studies?

I often thought of it, schools waste so much time.
Russian schools are 10/11 years and they know more in the end than we do.
American schools are 12 years and they know less than us in the end.

Obviously schooling could be speeded.
The obvious way is by adding more hours, why should school end at 12:00
while it can continue until 17:00? parents would be happier too. 
Another way is having several levels since elementary school, some people
may need to spend a year learning 2+2, but shlomi can probably do it in few
weeks, infact his father taught him multiplication and division before he
even entered first grade! why should he waste his time? He can probably know
linear algebra and infi by 6th grade, after all as opposed to some of his
class mates, he never had any problems with abstract math, he loves it.

Which brings us to the worse problem: Do *you* know of any 6th grade teacher
that can teach Infi? I don't.

BTW. our vision here maybe very very skewed, every lesson is easy after you
know it. Looking back I can't tell what was so difficult with highschool
trigonometry, but I remember having lots of problems getting it. Perhaps
looking back through the years we remember things as easier than they really
were?

---

From: Oleg Goldshmidt <[email-masked]>
Date: 03 Jan 2001 11:21:17 +0200

> Russian schools are 10/11 years and they know more in the end than

Hmm. I am a graduate (many years ago) of a Russian school, albeit not
quite an ordinary one. It lasted 10 years, from 7 to 17. My experience
with the Israeli school program is limited to private tutoring (also
quite a few years ago). And dealing with Israeli school graduates at
TAU. ;-)

Maybe I can offer some quick points to ponder...

The biggest difference, I think, was in the conceptual leap between
arithmetics, i.e. concrete examples, and algebra, i.e. real, abstract
mathematical approach. The difference between abstraction levels is, I
believe, crucial. The Israeli students were taught off examples (don't
hold me to it - just my impressions as a tutor). They were shown how
to solve 11x^2+5x-3=0, then 4x^2-8x+6=0, etc. Then they were still
having problems solving 6x^2-9x-25=0. All that at what age? 14 or
something?  I don't recall.

The Russian school just started from ax^2+bx-c=0, developed a solution
with the discriminant, and told the students that *any* quadratic 
equation could be solved this way, and that depending on the sign of
the discriminant there could be 2, 1, or zero real roots. Thus, the
conceptual leap between arithmetic and algebra (read: "real math") was
done from the start. All that - at age 9 or 10. I can't say if
children had problems with the concept, but I don't recall anyone who
could not master it. Of course, 3x-12=0 was solved in the first grade,
at age 7. Yes, with x (or y, or z), i.e. algebra from the start.

For the last few years of school we had 12-14 hours of math per week
(and 6-8 hours of physics, 4 hours of chemistry, etc). This did not
include homework or all sorts of "hugim" (mathematical and physical)
that were numerous.

What can you do in 14 hous/week? Real, irrational numbers, roots, etc
by the 3rd or 4th grade, trigonometry by the 5th-6th grade, planar
geometry ended (with an exam) by the 8th grade (9th and 10th was
stereometry), non-Eucledian geometry (basics) by the 6th grade,
calculus (limits, series, trancendental functions, differentiation,
integration) by 8th-9th grade, differential equations (ordinary and
PDEs) by 9th-10th.  In the mean time, there was probability theory,
complex analysis, a bit of functional analysis, metric spaces, Fourier
analysis, numerical methods, programming (that's seventies - early
eighties), etc.

To emphasize again, the approach was really mathematical and rigorous,
and the students had to deal with it. The proper mathematical training
from the early age probably helped.

So my answer to the question, "is it possible?" is affirmative.

Was that good? Definitely. Was that free lunch? Probably not. What is
the trade-off? Well, in the worst case, one's childhood. I don't think
that was the case with me. There were books, trips, movies, etc. There
was soccer 4-5 times a week as well :-). Nor am I sure that this is
worse than video games as a major time hog for an inquisitive kid.

On the other hand, I was not very much impressed by the humanities /
social studies curriculum. A big reason for that was ideology and
some sort of social xenophobia. E.g. literature was strictly Russian /
Soviet. The world literature was represented by Shakespeare, Molier,
Goethe, and Byron. 45 minutes allocated for all of them together. Not
very impressive. Economics was essentially nonexistent. History and
geography were pretty decent, though I made a point of reading Western
history books when I came to Israel. Literature was not a problem, but
was outside of curriculum, i.e. at home.

So maybe one tradeoff is, supplementing advanced math or humanities /
social studies outside of the curriculum. 

Another point. I cannot say which method of education - Russian or
Israeli - is "better" observing people who got beyond the MSc level.
Thus, advanced math at an early age is probably not essential. 

And another point, very important: the math curriculum that I
described above was by no means representative of a typical Russian
school.

We can pick the topic up on Friday...

-- 
Oleg Goldshmidt | Comgates Ltd. | [email-masked] 
"... We work by wit, and not by witchcraft;
 And wit depends on dilatory time." [Shakespeare]