Subject: Proposition for a better Math-Teaching Scheme - Revision 2

From: Shlomi Fish <[email-masked]>
Date: Tue, 13 Feb 2001 17:02:52 +0200 (IST)

It is included below for your comments and suggestions.

Disclaimer:

1. This article is pro-Mathematics but it is not anti-Humanities. If you
believe Mathemtics in any way hinder Humane skills, please ignore this
thread.

2. I do not believe Math Teaching today is at its possible optimum and the
suggestion is based on the assumption that children of average
intelligence or higher can be taught better within the same hours
dedicated to it (perhaps with a little more).

Well, here goes.

	Shlomi Fish

Algebra
-------

Linear Equations should be taught from as early as the first or second grade.
Throughout their studies, children learn to solve equations such as 
[Square] + 5 = 6, [Square] - 10 = 5, 6 - [Square] = 2, but are not introduced
to the general solving mechanism presented by algebra. Thus, they are unable
to solve such equations as 2 * [Square] + 9 = 3 * [Square] + 6. Showing them
the algebraic way of adding, substracting, multiplying or dividing both sides
by a certain value, will both ease their lives, and give them important
mathematical intuition. 

A balance can be used to demonstrate these concepts, with various unmarked
weights as variables.

Later on, the students should be introduced to a set of linear equations:
x + y = 6 and x - y = 4, and further. The methods of adding or subtracting two
equations should be taught, as well as the method of isolating a variable
and substituting for it in the other equations. A discussion should be held on
how to solve an equation set most quickly assuming a certain variable is
the one requested.

Second-grade equations should be shown a small time after the power
and the root are introduced. Students should be explained that the key to
solving x^2+A*x+B is finding two numbers whose product is B and whose sum is
A. It should be practiced a little, but the general solution for quadratic
equation should be presented with its derivation, soon afterwards.

Geometry
--------

Geometry should be taught from the beginning from the Euclidean basics 
while teaching everything that is essentially nowadays taught starting at
Junior High and High school. The concept of a proof should be emphasized, and
used. The children should be given problems to prove as homework.

I believe a geometry proof is more of a directed acyclic graph than a
sequence. Thus, I think kids may find it easier to write each sequence of
conclusions on one place on a big sheet (say A3), and connect those sequences
with arrows. Later on in their education, they can be taught that it should be
presented as a sequence. But I think it's important to show that mathematical
thinking is in no way completely sequential.

Perhaps it's possible to write a computer program that will help students
arrange their proofs that way, and maybe even verify that the deduction is
correct. The students should not depend on it though.

Arithmetics
-----------

From early on, children should be introduced to the concept of the zero number
as indicating "nothing", and negative numbers as indicating numbers on its
opposite side. A line can be shown with one number written every 10
centimeters, the zero at the middle and the negative numbers at its left side.

One half can be introduced as the solution to the equation 2*[Square] = 1.
Fractions are simply the result of dividing two integers by each other. As is
the common practice nowadays, splitted circles can be shown in order to
visualize fractions, but the students should be reminded that a fraction can 
be a part of any quantity. Sometimes it doesn't make sense (like 1/5th of a person, or
a table) but it always exist.

I believe children of tender age are able to grasp multi-digit arithmetics, so 
the move from single-digit numbers to two-digit, three-digit and beyond should 
be made  as swift as possible. Al-Khuarizmi's algorithms' and the multiplication 
board should be  taught of course, but I believe they can be done with by the 
second or third grade.

The numbers at the right side of the floating digit can be introduced as an 
extension  similar in spirit to the negative numbers. Children can be shown that 
100 = 1000/10, 10 = 100/10, 1 = 10/1, and that 0.1 = 1/10 (and 0.01 = 0.1/10 
etc.).

The connection between floats such as 1/3 or 1/7 and their floating-point 
representation should be shown as soon as students have grasped the concept 
of floating-point arithmetics. Children should understand that every rational 
number has a repetetive part at the end, but it may not necessarily constitute 
of the full non-full number. 

Programming
-----------

Programming can be a useful tool because it puts algebra to practice. There
are many math-related problems which Children can learn by trying to implement
them in a computer program. 

Mathematical Thinking
---------------------

It is a well-known fact that a person cannot force another person to think. 
What can be done is that the Teacher will make thinking for the student more
enjoyable. To do that, the teacher cannot assume that the student doesn't 
know tools which he is capable of knowing, such as algebra or Euclidean
Geometry taught in the Euclidean way. 

There are enough problems which are hard, challenging and enlightening that
can be solved while (theoretically) knowing all the math that has been invented 
so far. It is my belief that imposed ignorance cannot be made a tool in the 
hands of the pedagogue.

In order to develop mathemtical thinking, the teacher should expose the
students to challenging problems, such as those given in mathematical
contests, or in riddle books. The corresponding way of telling students to
think of their own challenging problems is also very beneficial.

----------------------------------------------------------------------
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: Gaal Yahas <[email-masked]>
Date: Tue, 13 Feb 2001 18:03:34 +0200

> believe Mathemtics in any way hinder Humane skills, please ignore this

I hope this disclaimer isn't addressed to me. (Because I don't believe
anything I've said gave you grounds to think I "believe Mathemtics"
hinders "Humane skills".)

I'm still eager for your reply on the previous subthread, by the way.

> dedicated to it (perhaps with a little more).

I've no quarrel with the assumption, but I think that in order to
make a convincing case with this proposal, you need to supply more
justification--*why* this or that method is effective--for every
assertion. Claiming "X should be Y" alone is unwarranted speculation.

Education is an immensely complicated process. It almost obvious that
it is not optimum now. But you're presenting an alternative without
explaining how yours is better. I would expect from a proposal such as
you are attempting to either identify specific weaknesses of the current
scheme (and not "oh, it's too slow and people don't understand Math well
enough". Choose a scope you can responsibly address) and solve those,
or to give a good definition of the goals and to proceed from there to
construct a method, expaining why this construction is to be favoured.

-- 
Gaal Yahas <[email-masked]>

---

From: Shlomi Fish <[email-masked]>
Date: Fri, 16 Feb 2001 05:23:40 +0200 (IST)

> hinders "Humane skills".)

Actually, it's more adressed towards AlexSh, because he at least seemed to
think so.
 
> I'm still eager for your reply on the previous subthread, by the way.

I'm not going to answer it because I don't see much point. You madea lot
of small points, but when reading your post I couldn't get what the bigger
picture was. 

> construct a method, expaining why this construction is to be favoured.

Ahh - the burden of proof (Habeas Corpus). Naturally, I was not going to
avoid it altogether, but the objective of the post was to get some
feedback from the other members of the list and see what they think about
it.

Best regards,

	Shlomi Fish

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From: Gaal Yahas <[email-masked]>
Date: Fri, 16 Feb 2001 17:23:29 +0200

> of small points, but when reading your post I couldn't get what the bigger

That's unfortunate, as I did try to be clear about that. Please
consider this revision. In my first post in the thread, I wrote:

GY> Although I share some of Shlomi's feelings regarding the
GY> unreasonableness of education, some of the claims he presents here
GY> just don't make sense, or else are in dire need of qualification. I
GY> also disagree strongly with some of his conclusions.
GY> 
GY> The strongest methodological move Shlomi makes is the distinction of
GY> the humanities and the exact sciences. He thinks that the humanities
GY> are less "crucial" to modern life, and implies that underlying
GY> any successful humanistic work are skills that come from the exact
GY> sciences.

My main point was to quesiton your distinction of the humanities
and the exact sciences, and the inferences you make regarding how
education ought to take place _based on that distinction_. Along the
way, I challenged some of your assertions, or asked you to "tighten"
them (specify their scope more clearly), since as they stood they
didn't make sense, to me.

You are proposing a theory of education. I am claiming that as it
stands, I can't accept it, because as I see them, the assumptions
on which it is based are either wrong or nonsensical. I raised many
issues; the point of "a lot of small points" is that unless they are
plugged, the holes they make in the theory cause it to sink.

> feedback from the other members of the list and see what they think about

Here's my feedback: I think there's more at stake than "mere"
correctness here. If you not only *state* your case, but *prove* it
too, your proof can serve as explanation: we can actually *understand*
your proposal better if you explain why you think it is a good one.