

Compatibility
Ross/Regents Math


	Ralph Abraham
Visual Math Institute
POB 7920
Santa Cruz, CA 95061
Ph. 408-425-7436
Fx. 408-425-8612
abraham@vismath.org
http://www.vismath.org


Introduction

Nowhere in the school curriculum is the Sheldrake principle more important that in the math element. For many studies have shown that math anxiety, math failure, and the math avoidance syndrome endemic in the US are traceable to two main roots:
ˇ	lack of visual representations, and
ˇ	premature presentation.
The latter is usually due to either of two faults:
ˇ	violation of the Sheldrake principle (incorrect sequence of topics), or
ˇ	presentation at too early an age,
or both. Of all this my main concern has been the sequence fault, and the math curriculum plans I have presented to the Ross School have been carefully screened to avoid this fault. The historical sequence (as opposed to the sequence of formal logic that underlies the math curriculum of the most public schools in the United States) is the fundamental skeleton of integrity of our plans.

The map prepared by Michael Schneider (19 August 1996) of the conventional (NY Regents) math curriculum for grades 5 through 8 Đ with topics in violation of the cultural historical time line of the Ross School core curriculum brought out in boldface type Đ indicates a small number of conflicts. In this report I will address these one at a time, and (I hope) resolve them all. Here they are.
Conflicts to be resolved

Excepting a few very minor items, the topics of the Regents curriculum which are out-of-order in our view, are the following.
5th Grade, 1750 ą 1000 BC
ˇ	Operations with numbers in decimal notation (current ca 1200 AD)
ˇ	Graphs with coordinates (invented ca 1600 AD)
6th Grade, 1000 BC ą 400 BC
ˇ	Operations with numbers in decimal notation (as above)
7th Grade, 400 BC ą 800 AD
ˇ	Operations with numbers in decimal notation (as above)
ˇ	Graphs with coordinates (as above)
ˇ	Equations and inequalities in symbolic notation (from 1500 AD)
8th Grade, 800 AD ą 1500 AD
ˇ	Graphs with coordinates (as above)
ˇ	Equations and inequalities in symbolic notation (as above)
In toto, without repetitions, in temporal order
#1. Operations with numbers in decimal notation (current ca 1200 AD)
This occurs in the Regents 5th, 6th, and 7th grades, where it is 
Ross-anachronistic, and in the 8th, where it is kosher.
#2. Equations and inequalities in symbolic notation (from 1500 AD)
This occurs in the 7th grade, anachronistically, and in the 8th, where
it is borderline or OK.
#3. Graphs with real coordinates (invented ca 1600 AD)
This occurs in the Regents 5th, 7th, and 8th grades, where it is
Ross-anachronistic until the 9th.
That is all: three conflicts, which recur. 

The resolutions
Here s a compromise plan for each of these three conflicts.
#1, Decimal notation.
Let us admit that in learning to read, and in daily reading of newspapers, magazines, etc., that 1-2-3 is as familiar as A-B-C. Thus there can be no gain in the avoidance of the Hindi numerals as such. And they can be used for reference in studying all historical numerical symbolic schemes: cuneiform, hieroglyphic, alphabetical, etc. They are also very helpful in understanding the counting systems on bases other than decimal: binary, octal, hexadecimal, etc., which occur in our curriculum in all grades. 

In short, I recommend using decimal notations throughout the grades, just as we will use the alphabet in studying prehistory in the 4th grade.

#2, Symbolic notations.
These could be introduced anytime, actually, as long as the proper preparation is made in advance. In our curriculum, the proper preparation for algebraic notations is the geometric algebra of Euclid, Books I and II. According to our current plan, this is presented in the 6th grade. Thus symbolic notations could be introduced in the 7th grade, as recommended by the Regents, if there was a pressing need for this. It must be done with great care, as it is one of the special stumbling blocks of the US curriculum. Then it would still be anachronistic in the Ross time frame for the 7th grade, but it would not be a violation of the Sheldrake principle on historical sequence. This is an important distinction: historical time and historical sequence are not the same thing.

But my preference here would be a compromise: hold off symbolic notations until the 8th grade.
#3. Coordinate graphs.
This is easier. Coordinate graphs were used by builders in ancient Mesopotamian and Egyptian civilizations. There is abundant archeological evidence for this, ad the actual methods are described in detail in Tod Brunes, Ancient Geometry. To be historically correct, one would use grids marked out by nonnegative integers at the beginning, in the 5th grade.

What must be avoided until the 9th grade is the real coordinate grid introduced by Descartes. This means, especially, to scrupulously avoid symbolic notations such as:
	(x - 1)^2 + (y - 2)^2 = 4
for the circle centered at (1, 2) of radius 2.

Thus, we have no conflict with coordinate graphs.
Conclusion
In sum, we have no problem with decimal notations, nor with coordinate graphs. We recommend holding back symbolic notations until the 8th grade, in violation of the Regents' recommendation, but we do not insist.