Subject: Guidelines for fast teaching of pre-academic mathematics

From: Shlomi Fish <[email-masked]>
Date: Sun, 7 Jan 2001 19:57:29 +0200 (IST)

This is the first draft of this document. Those that are interested in
this matter (and do not think pupils today are taught in a manner that is 
efficient enough) are welcome to read it, comment or add their own
material. I hope to post it to Slashdot or something like that one day,
but meanwhile this forum will do.

I'll establish a web-page for it soon enough.

Regards,

	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.). Floating point arithmetic is a convenience more than a necessity and should 
be treated as such.

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. 

----------------------------------------------------------------------
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.