Subject: Lightning Speed Math Teaching

From: Shlomi Fish <[email-masked]>
Date: Wed, 3 Jan 2001 09:54:14 +0200 (IST)

IMO, having to study for 12 years in order to learn the basis of
Mathematics which will enable one to acquire a technical diploma, is very
unsuitable for the information age. 

I think children of average intelligence can be taught Math much more
quickly than that. For once, Algebra should be introduced already in
Elementary School. And I think Geometry should be taught in the first
place from the basic theorems according to the High School curriculum.

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

One thing I can think of is privatizing schools. (not to start a flame
war here). Doing this will create competition between the competing
schools or school networks, and will raise teacher's salary. Right now,
the public school system has failed to prove itself in many places, except
those where the Teachers' salary is extremly high. (e.g: Switzerland)

Regards,

	Shlomi Fish

----------------------------------------------------------------------
Shlomi Fish        [email-masked] 
Home Page:         http://t2.technion.ac.il/~shlomif/
Home E-mail:       [email-masked]

The prefix "God Said" has the extraordinary logical property of 
converting any statement that follows it into a true one.

---

From: Alex Shnitman <[email-masked]>
Date: Wed, 3 Jan 2001 11:23:09 +0200

Hi, Shlomi!

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

None at all?

How about teaching the kid more about reading, writing, socializing,
perhaps drawing, building things, being creative? Math is not
everything, you know.

I think it's a common mistake that school is there to give the kid
knowledge. It isn't -- it's there to aid his development. Teaching the
kid more math will *not* aid his development (for most of the kids
anyway). Chen -- studies until 17:00?!?! Are you nuts? There isn't
even such pressure in universities.

Why do people think that the more the kid knows, the better he's off?
It's wrong. Chen, you mentioned how Russians learn a lot and Americans
don't. Now, look at the Russian immigrants from a social point of
view, and look at the Americans. Who's better off? What good does all
the knowledge do for the Russians? Zilch! None at all. It doesn't even
bring them money -- Americans are far better off there. And that
happens because of one simple thing: knowledge is not everything, and
most definitely not the most important thing in life.

-- 
Alex Shnitman                            | http://www.debian.org
[email-masked], [email-masked]   +-----------------------
http://alexsh.hectic.net    UIN 188956    PGP key on web page
       E1 F2 7B 6C A0 31 80 28  63 B8 02 BA 65 C7 8B BA

Ambition is a poor excuse for not having enough sense to be lazy.

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From: Adi Stav <[email-masked]>
Date: Wed, 3 Jan 2001 12:03:59 +0200

> even such pressure in universities.

I don't know. I agree that the most important thing to teach kids is
"thinking habits". But how do you teach those? I think mostly through
examples, and algebra in particular is good in teaching logical and
mathematical thinking and analysis skills.

> most definitely not the most important thing in life.

I see Russian immigrants in knowledge intensive industries in numbers
well beyond what could be expected from their relative part of the
Israeli population. So I'd say it did help, a great deal.

	- Adi Stav

---

From: Moshe Zadka <[email-masked]>
Date: Wed, 3 Jan 2001 20:33:24 +0200 (IST)

> unsuitable for the information age. 

You're in luck, becasue it doesn't. All the 12 years, as far as math is
concerned, are wasted. When you study math at the uni, none of this
depends on what is taught at highschool. 

If it were up to me, I'd make the current 5 units the 3 units curriculum,
and have what is now taught in first degree in math to be the 5 units
curriculum. A common student has, on average, 20 hours a year. Since
highschool is 6 years, not 3, you can do with 10 hours of math per
year. This means a mere 2 hours/day, not that unreasonable!

-- 
Moshe Zadka <[email-masked]>
This is a signature anti-virus. 
Please stop the spread of signature viruses!

---

From: Alex Shnitman <[email-masked]>
Date: Wed, 3 Jan 2001 15:24:30 +0200

Hi, Adi!

> mathematical thinking and analysis skills.

There are many ways to think. Mathematical thinking is one of them.
People of humanities, for example, often think in associative ways.
You don't have to nurture *one* type of thinking.

> Israeli population. So I'd say it did help, a great deal.

It depends what your goal is. Let me tell you, from personal
acquaintance, that many of these people are *not* happy. Not many of
them have a life to speak of.

A printer consists of three main parts: the case, the jammed paper tray
and the blinking red light.

---

From: Oleg Goldshmidt <[email-masked]>
Date: 03 Jan 2001 15:44:15 +0200

> them have a life to speak of.

Many do, I think, and certainly quite a few like their jobs. I don't
know if there is any positive or negative correlation between happy
life in the hi-tech world and the level of mathematical education at
school. A more important question is, how many people actually use the
skill in calculus or differential equations in their professional or
everyday lives. I do, but I suppose I am an exception rather than the
rule. Can you guess it when a child is 8 years old? 13? 17?

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

---

From: Adi Stav <[email-masked]>
Date: Wed, 3 Jan 2001 16:00:37 +0200

> You don't have to nurture *one* type of thinking.

You need to encourage as many as you can. Education should be broad. I
think good teaching of understanding history, for example, should not
come in place of understanding mathematics. They are both very
important and should be unbderstood rather than learnt by heart. But
that would be utopia :)

> them have a life to speak of.

But that is the stereotype of the entire "hi-tech" workers in
Israel. How do you measure happiness, compare it, encourage it? I
don't think there can be even partial answers to these questions. We
are all eventually responsible for our own happiness and assessment of
happiness, and that is why I believe any attempt to affect the
happiness of a large group of people is doomed to fail on its own
terms. So I think schools should give students the opportunity to
succeed, and leave "happiness" to the students and to their families.

(That said, I agree that school as a "misgeret" that gives children
help, counsiling etc is important and should be given more
resources. I just don't think it has anything to do with teaching.)

	- Adi Stav

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From: Alex Shnitman <[email-masked]>
Date: Wed, 3 Jan 2001 16:42:21 +0200

Hi, Adi!

> important and should be unbderstood rather than learnt by heart. But

Exactly. Knowing more maths will not help anybody. If you decide to
have more studies (an arguable decision on its own), you have to learn
more of everything.

> don't think there can be even partial answers to these questions.

You'll just have to believe me here. I'm a Russian immigrant myself,
and I have some sociological insight from the point of view of Russian
immigrants and Israelis. Russian immigrants, especially the older
people, do *not* have a healthy psychology, and the education race has
a lot to do with it.

> We are all eventually responsible for our own happiness and assessment of

School is the place where the kid turns from almost nothing to
everything he is. The person is built up in school. I think that not
the knowledge makes the man, but a developed, multi-faceted, healthy
psychology. Such a person will go far and gain all the knowledge that
he needs. Such a psychology is created with the right mixture of
studies and education in all the subjects, a supporting framework and
love from the school and the parents, and lots and lots of social fun.
That's how I'd like my child to grow up, if I ever have one. :-)

"Xenix is the pinnacle of modern UNIX design, and will be used for
many years to come."
	-- Xenix OS API manual

---

From: "Nadav Har'El" <[email-masked]>
Date: Wed, 3 Jan 2001 23:36:52 +0200

Hi fellow Kackers!

> place from the basic theorems according to the High School curriculum.

Many people have already replied to this thread, and gave many valid points,
both for and against teaching more math in school.

But I think there was a logic flaw which was common to many people's posts:
Say math is taught for 6 hours a week for 12 years. Many people seemed to
assume there are only two options:
  1. Teach 12 hours of math a week, be done with the current material by 6th
     grade, and in 7th grade start teaching advance material.
  2. Leave the situation as is, because math is not the only important subject.

I think both options are bad. Option #1 is bad because most of the kids
will not be able to handle the extra hours of math (or want them - as people
pointed out, many kids show inclination to humanistic subjects). Option #2 is
also bad, at least as far as my experience in school.

But there's a third option, to speed up learning math while not making it
harder. Sounds impossible? I think it isn't. Remember how you were learning
math for the first 7 years? First, it wasn't really math, but rather
arithmetic. Second we spent years and years _doing_ stuff rather than
_learning_ stuff. _doing_ takes a lot of time and effort (so kids complain math
is hard) but doesn't teach anything. What do I mean? Consider how much time
you spent practicing long-division in school. After, say, a month, of such
practice most kids should get the idea, and spending a year on it can do only
little to improve the kid's proficiency in division (a proficiency which will
be forgotten by the time he is 18), teaches the kid nothing new, and is
actually hard at the same time it is futile (try doing 20 division excercises
right now. Even though you know how to do it, all that boring technical work
isn't easy to get right all the time). The arithmetic taught at elementary
school is almost pure sisyphic work.

Imagine instead being taught natural numbers and positive small rational
numbers in first grade, as well as addition and subtraction, then in second
grade being taught negative numbers, general rational numbers, multiplication,
and division. In third grade you start geometry (simple stuff at first, but
right from the start emphasise the idea, and importance, of proof), and very
simple linear equations. In forth grade you continue geometry, do some word
problems, and some simple number-theory (prime numbers, GCDs, etc.), and
learn to work with concepts of algebra (polynomials, binom, etc.). In fifth
grade you learn quadratic equations, systems of equations, and complex
numbers, and do more advanced geometry. In sixth grade you continue geometry
(finishing the current corriculum) learn about functions and their various
features (roots, asymptotes, discontinuty, etc.), preparing for starting (but
only starting) calculus at the end of the year - giving motivation for the
concepts of derivative, integral and limit.

So far I outlined how what currently takes 10 years to study can be done
in only 6, while not making it harder on the pupils or spending more time
studying math. I think that 90% of the kids will be handle this "faster"
corriculum (I don't think more than that can handle the current corriciulum!)
and those who can't, for some reason, will be given extra help until they
reach some mimimum level (say, the aforementioned pre-calculus) at which
they are allowed to stop taking math (and take other subjects instead).

When I think back on my school days (only 9.5 years... long story...),
I remember that even though I was very interested in math (in 5th grade I
was curious about quadratic equations, so my father taught me the formula
to solve it and _how to reach it_,  and then when I inquired, complex
numbers), I really didn't like the math school work: I didn't like the
repetitive work of doing hundreds of calculations, all of them exactly
the same but with different numbers, I didn't like the childish "replacements"
for real arithmetic (bdidim, cutting and pasting stuff, etc.), and I
didn't like solving equations. But I loved the little pieces of inteligent
stuff we were allowed to do from time to time (e.g., in 8th grade I was
sent to find a proof to Pythagoras's theorem, and to find a method to extract
a square root using a straightedge and compas for extra credit). But I didn't
think I'd go into math (I have a MSc in math, BTW) until I got to 11th grade
(which I never finished - again, the long story...), and got a great (although
very strict) math teacher, which taught us (some of it in after-school hugim)
calculus, group theory, number theory, and riddle solving ("math olympics").
Only then did I really understand how interesting math could be, and that
the math you learn at school has almost nothing to do with the math you
learn at the university.

Contrast the way you study arithmetic for 6 years, and then as an adult rarely
use it (I don't remember the last time I actually did long multiplication
without a calculator!), to the way you study reading for 1-2 years, and then
use it all your life. I doubt that arithmatic is really that much harder
than reading and writing.... (Reading, Righting, and Rithmatic are known
as the three R's :))

> those where the Teachers' salary is extremly high. (e.g: Switzerland)

And who's going to pay that increased salary? To be able to do that you'll
need to either lower the number of teachers/hours and pay more to each (I
don't think this is a viable solution), or increase the funding. This should
be done with tax money, not the parent's money. Kids should not be given
worse education if their parents are poor. Parents should be allowed to
send their kids to private schools with "extra" features (gold-plated desks 
and 5 kids in a class), but all children should get some minimal (but
still very high) standard of education. I think that being someone who pays
a lot of taxes and doesn't have any kids, I can say this without anyone
thinking I'm biased.

I studied in the 4th and 5th grade in a public school in New Jersey. The
school was _lightyears_ away from schools I went to in Israel, in every
conceivable manner. 20 kids in a class, each one at a desk, large classrooms
with carpets and sinks, hot meal, studying until 3pm, place to keep books
at school (rather then carry them around all the time). I also remember
how we were taught Latin in 5th grade, in a very interesting way without
being tested or graded (the teacher said that she wanted us to get interested
in this subject, and didn't care for us to learn by heart words, or stuff
like that). This was a public school. Obviously not all schools in the U.S.
are like that, but at least it shows what can be done with proper funding.
BTW, the math (arithmetic) there was not much better than in Israel, but not
worse.

-- 
Nadav Har'El                        |      Wednesday, Jan 3 2001, 9 Tevet 5761
[email-masked]             |-----------------------------------------
Phone: +972-53-245868, ICQ 13349191 |The software said it requires Windows 95
http://nadav.harel.org.il           |or better, so I installed Linux.

---

From: Oleg Goldshmidt <[email-masked]>
Date: 04 Jan 2001 10:07:16 +0200

"Nadav Har'El" <[email-masked]> writes:

> Remember how you were learning math for the first 7 years? First, it
> wasn't really math, but rather arithmetic. Second we spent years and
> years _doing_ stuff rather than _learning_ stuff.

<the rest of very interesting (for me) description snipped for
brevity>

Nadav, this is exactly my impressions from tutoring Israeli kids,
only expressed much better. What you are suggesting is very similar
to the general idea behind elite Soviet schools (of old).

---

From: Adi Stav <[email-masked]>
Date: Thu, 4 Jan 2001 20:23:55 +0200

> school is almost pure sisyphic work.

I have a feeling that people at many levels of the Israeli education
system are not really concerned with teaching and education, but more
with maintaning the status quo, getting by, and making sure they have
as few as possible problematic students and incidents, where
problematic is defined as anything they are not used to handle. They
would much prefer to teach mediocricy to mediocre students rather than
ask themselves whether their student's reasonable grades really do
represents their students' knowledge and thinking habits, and they
would *much* prefer to have a mediocre student than an undisciplined
one. Sure, teachers like good and intelligent students, but mostly
because that means less problems to take care of.

I first became aware of that when my maths teacher (who was otherwise
a good teacher) became dissappointed with me when I confessed to her I
did not really know the multiplication table for 5 and 9 like I was
supposed to, and that I was getting along for several months then by
multiplying by 10 and then subtracting or halving the result. Recently
I was reminded of that when a relative told me of her 16 year old
daughter's trouble with English; she was failing constantly, until her
mother began giving her special help with her homework. And suddenly
the daughter began to receive 100s and 95s. Her English teacher sent a
note back to the mother asking to talk with her. When the mother
called, and the teacher told her that her daughter was having trouble
with English, The mother asked why didn't the teacher say anything in
several months of failures in tests, but only asked to talk with her
now that her daughter was actually getting good grades. The teacher's
reply was "Well, she was starting to get very good grades all of the
sudden, so I gathered she must be cheating".

Now, these are only anecdotes and don't show anything, but maybe they
will remind people of their own impression of schools, and I'll get to
know if it's just me who feels that way. So I don't think giving more
money to schools is either necessary or sufficient; we need to find
ways to reward teachers, principles etc for getting the most out of
each students rather than for avoiding trouble. The "histarutiut" of
the education system might be one of the big problems here.

So a curriculum that makes children spend months over months
practicing long division can be another symptom of the problem. The
children learn, eventually get good at it, there is a very low failure
rate, and so everything is just fine, isn't it?

> concepts of derivative, integral and limit.

I don't know... What we all want, really, is for school to minimize
the amount of brute force memorizing and practicing and teaching, and
focus instead on abstract thinking right from the start. Easy for us
to say, us in our 20s and 30s! Abstract thinking is one of the hardest
things there are to do and to teach! Children are having terrible time
with it, and many adults, too. Maybe the best way to understand
something abstract and complex IS through examples and practice --
hundreds if necessary. Also, to say that hackers-il represents an
average cut of the population would be to border on surreal. I doubt
it is possible to gather a group of people whose preferences of
education deviate any more from the norm than hackers-il. So people's
experiences of getting inappropriate education really doesn't say much
about general education for the general population.

Yes, I know, I am exactly contradicting everything I said in my
earlier chunk of paragraphs. But I really don't know -- maybe there is
no way to make education any better than it already is?

   - Adi Stav