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

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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

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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

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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 shlomif@...

Home Page: http://t2.technion.ac.il/~shlomif/

Home E-mail: shlomif@...

The prefix "God Said" has the extraordinary logical property of

converting any statement that follows it into a true one.