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

From: Chen Shapira <[email-masked]>
Date: Thu, 4 Jan 2001 19:33:39 +0200

> Are you serious? I don't think that more than 10% of the kids 
> will be able to
> undertake such a curriculum... Also, what about geometry, and 
> simple algebra
> (polynoms, equations, word problems, etc.)? Are you sure that 
> children can
> fully comprehend infinitessimal calculus at the age of 12 
> (with all the "for
> all epsilon there exists a delta....")? You and I could 
> proabably have (though
> I waited till I was 15 ;)), but most kids won't :(

Just to emphesize the point I'll add that understanding material and being
able to memorize proofs is very diffrent than being able to prove something
yourself.
The former being an order of magnitude easier and the latter is what we
actually want to teach.

---

From: "Nadav Har'El" <[email-masked]>
Date: Thu, 4 Jan 2001 23:59:10 +0200

> The former being an order of magnitude easier and the latter is what we

Not necessarily (about "the latter is what we want to teach"): indeed, a
mathematician needs to be able to prove new theorems. But from a well-
educated non-mathematician I'd expect to know what a proof is, when to
expect a proof when a statement is made, and how to check a proof for
validity. But non-mathematicians don't necessarily need to be able to make
up new proofs on their own. It's a plus if they can, but it's probably not
a must.

Compare this to computer education: educated kids today need to learn the
basic of computers, how not to fear them, how to use them, and perhaps even
how to do simple programming, or understand simple programming concepts when
discussed at the dinner table. But not everybody is (or should be) expected
to become a proficient programmer.

BTW, the current schools don't even succeed at teaching kids the concept of
a proof, let alone to prove stuff on their own. I still remember vividly
one day when my younger sister (a humanist at heart) asked me for some help
before a test about solving quadratic equations. When I saw she had trouble
remember the infamous "formula", I asked her if she knew where this formula
came from, and how it was derived. Not only did she say she didn't have a
clue where this formula came from or why it was true - she also exclaimed
that she didn't care, and that I shouldn't bother telling her the proof
because it's not in the curriculum and that it'll only clutter up her mind...
That's how kids are taught - to use formulas like robots, and never to
question themselves (or anyone else) why these formulas are true. The same
kids (or adults) can later be told that "Pisces is compatible with Sagittarius"
or that "Arabs are all terrorists",and they also won't question these "facts".

P.S. If anybody is curious how the quadratic solution formula is derived,
here it is (as I said, my dad taught me this derivation when I was in 5th
grade (if I remember correctly), and I remember it since):

If the equation is ax^2+bx+c=0, transform it to:
          x^2 + b/a x + c/a = 0
then do what is called an "hashlama leribua" (this is the ingenious part):
          x^2 + b/a x + (b/2a)^2 + (c/a - (b/2a)^2) = 0
which is
                (x + b/2a)^2     + (c/a - (b/2a)^2) = 0
which is
          x + b/2a =  +/-  sqrt(-c/a + (b/2a)^2)
                   =  +/-  sqrt(-ca + b^2/4) / a
(the +/- can be -/+ if a is negative, but that is of no importance of course)
so finally,

          x =  -b +/- sqrt(b^2 - 4ac)
               ---------------------
                     2a

Not that difficult, isn't it? I don't expect every pupil to remember this
derivation - what I expect every pupil is to remember that he or she has
seen it once, and that they wouldn't accept a "formula" without seeing it
proved.

-- 
Nadav Har'El                        |      Thursday, Jan 4 2001, 10 Tevet 5761
[email-masked]             |-----------------------------------------
Phone: +972-53-245868, ICQ 13349191 |The person who knows how to laugh at
http://nadav.harel.org.il           |himself will never cease to be amused.

---

From: Shlomi Fish <[email-masked]>
Date: Fri, 5 Jan 2001 14:45:48 +0200 (IST)

> up new proofs on their own. It's a plus if they can, but it's probably not

My motivation is for children to be able to prove problems they are given.
Not necessarily theorems, but problems which they are capable to handle
using their current mathematical tools. I'd very much like to see math
riddles such as those that are given by Math Olympiads, being given to
commonplace students for their solution. Not as part of their homework but
for a grade bonus.

I believe part of the reason why it is advised to learn mathematics is to
become proficient in the mathematical way of thinking and doing things.

> to become a proficient programmer.

Again, I disagreee. I believe one day there will be 4 r's instead of three
(including the r in programming). Everybody will be expected to learn how
to program, and teachers will give programming exercises in their classes.
One cannot get through an EE degree without knowing Matlab, and depending
on your orientation one probably needs to learn C and VHDL too. And
naturally, perl is of great help. But I believe soon people without a good
grasp of programming will be considered like illiterate are considered
today.

> one day when my younger sister (a humanist at heart) asked me for some help

I don't believe there are such beasts as "humanists at heart" or "exact
scientists at heart". Usually it's a function of how those subjects are
taught at school. And IMO, a man cannot be complete without being capable
in exact science as well is in humane areas such as philosophy, history,
epistomology, writing (fiction or non-fiction) etc.

> or that "Arabs are all terrorists",and they also won't question these "facts".

When I was taught the quadratic equation at 10th grade, my teacher, a
Russian immigrant who was actually quite good, IMO, showed us how to
derive this proof. I still can derive it myself, in case of need, but I
remember it by heart anyways. 

In any case, such acceptance of dogma is indeed a bad habit to learn in
the educational system. However, as opposed to mathematicians, engineers
are less concerned with proofs of the theorems they learn. I still only
remember the general idea of the Taylor Series proof, and I still think
they should not have let us learn it by heart for the Hedva 1m exam,
because learning it in class was enough.

But kids in school should be taught that everything in mathemtics can be
proved or demonstrated (except for such theorems which a proof was not
yet found), and parents should make sure a child understands that dogma
has no foundation in reality.

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: Sun, 7 Jan 2001 03:57:17 +0200

Hi, Nadav!

On Thu, Jan 04, 2001 at 11:59:10PM +0200, you wrote the following:

> The same kids (or adults) can later be told that "Pisces is
> compatible with Sagittarius" or that "Arabs are all terrorists",and
> they also won't question these "facts".

Actually I'd think it's mathematical thinking that brings people to
right-wing politics ("they attack us, so we should attack them too,
therefore lets roll tanks into Ramalla" ;-), but that's a whole
different flame war. :-)

-- 
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

Personally, I never finish my

---

From: "Nadav Har'El" <[email-masked]>
Date: Sun, 7 Jan 2001 10:46:23 +0200

> therefore lets roll tanks into Ramalla" ;-), but that's a whole

The worst kinds of extremism are caused when a large part of the population
(or just the army) stops thinking, and accepts the claims made by the person
in power (political, religious, or miltary) as truths, regardless of how
stupid or horrible they actually are. Whether it's right wing (e.g., Hitler
and Mussolini) or left wing (e.g., Stalin and Paul Pot), when people stop
thinking for themselves and leave all the thinking to their
pseudo-intellectual leaders, disaster occurs.

Also, you can't reduce mathematics to arithmetic, or Newton's "for each action
there's an equal and opposite reaction" law. Mathematics also has the
concept of axioms (and you might have ones like "thou shalt not kill"), and
complete theories like game theory (which shows that "games" are not
necessarily zero sums - "players" can cooperate and it is not true that what
one side gains the other side loses). But obviously, I'm not claiming that
the entire morality can be based on a mathematics-like thinking of gain
and loss (a la utilitarianism). In fact I've been thinking recently on the
the origin and nature of morality - but I still haven't come to a final
conclusion (I have a few ideas, if anybody wants to open a philoshopy-il
chapter of hackers-il ;))

I was talking about right-wing extremism because currently (and this has not
always been the case), violent left-wing extremism is virtually non-existent
in Israel, and because personally I'm much more bothered by the killing of
human beings than by the political control over this or that neighborhood or
shrine (but let's not get into that - it's now completely off-topic).

-- 
Nadav Har'El                        |        Sunday, Jan 7 2001, 12 Tevet 5761
[email-masked]             |-----------------------------------------
Phone: +972-53-245868, ICQ 13349191 |I have a watch cat! If someone breaks
http://nadav.harel.org.il           |in, she'll watch.

---

From: Shlomi Fish <[email-masked]>
Date: Sun, 7 Jan 2001 10:52:00 +0200 (IST)

> therefore lets roll tanks into Ramalla" ;-), but that's a whole

Actually, that is not mathematical thinking. That is a very simplistic and
primitive thinking, which is not good for mathematics or for anything
else. Do you think more mathematicians are right wing than those that are
not mathematicians? I don't think so.

BTW, contrary to what you said about music there was a research that
showed that many music composers had a strong tendency towards
mathematics, and vice versa. I don't know how able a mathemtician David
Bowie is, but even if he isn't I'm sure there are others who are.

It's very sad to know that there are people who think humanities might be
spoiled by having mathemtical thinking. But the facts show otherwise: many
important philosophers were also competent mathematicians: Descartes,
Leibniz, Pascal, not to mention all the Greek philosophers.
BTW, my Literature teacher did not know Leibniz was a mathemtician who
co-invented Infi. with Newton. Nor do many poeple know that Newton was a
mathemtician as well as a physicist. But on the other hand, many Israelis
also believe Maimonides was only a phisician. Oh well.

Regards,

	Shlomi Fish
 
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---

From: Oleg Goldshmidt <[email-masked]>
Date: 07 Jan 2001 11:10:54 +0200

Shlomi Fish <[email-masked]> writes:

> mathematics, and vice versa. 

Some history could help. As late as in the 16th century (just re-read
Montaigne) 4 parts of mathematics were distinguished: arithmetic,
geometry, astronomy, and music.

> Nor do many poeple know that Newton was a mathemtician as well as a
> physicist.

He was deep into alchemy and chemistry, history of Western
Civilization, chronology. Theology and philosophy (prophecies of
St. Daniel and the Apocalipse) were among his main occupations through
his lifetime. Probably not many people know that he devoted a special
study to Solomon's temple as described in the Bible.

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

---

From: Shlomi Fish <[email-masked]>
Date: Sun, 7 Jan 2001 16:26:06 +0200 (IST)

> study to Solomon's temple as described in the Bible.

I know that Newton was a man of many talents but he was less prominent in
them than he was in Math and Physics. Leibniz on the other hand, mastered
all the fields of knowledge that were available at his time (and was
considered the last man to do so).

Regards,

	Shlomi Fish