My terms "conek, loconek, hiconek" conform with my FIRST LAW OF MEASUREMENT: LET ALL YOUR NAMES BE UNIVALENT -- SPEAK WITH ONE VOICE OR MEANING!

Have you ever planted an African violet cutting? If so, and if you've a green thumb (or, at least, not a gangrene thumb), the cutting may have put down roots and begun to grow. Later, it may flower. (But you don't expect rooting without planting, as we do in just-in-time-education in our schools.)

The key-words (for my model) are CUTTING, ROOT, FLOWER. The "cutting" models "conek", which are essential concept or procedure. The "roots" model "loconek": a conek's primitive beginnings in daily life or some stage in human history. The "flower" models "hiconek": conek's extension to important applications in "hitech".

I argue: ALL CONCEPTS AND PROCEDURES ESSENTIAL TO EDUCATING OUR CHILDREN AND YOUNG PEOPLE SHOULD BE PROCESSED AS CONEKS, ALONG WITH THEIR LOCONEKS AND HICONEKS.

We plant seeds or cuttings and give them time to root and flower. Why not this for educating our young?

I conceived this many years ago. But now a second opinion.

A report (11/20/96) by the U. S. Dept. of Education -- "the largest international study of how American students perform in math and science" -- rates American students 17th in science and 28th in math! (The "leaders" were Singapore, Japan, and Korea.)

The report finds that American students spend more class time on math and are assigned more homework than students of leading countries in the survey. (The American math text shown on TV was 4 times as thick as the comparable Japanese text.)

But "the study contends that American math teachers often emphasize basic mathematical formulas rather than helping students understand the deeper concepts behind them, and try to accomplish too much too soon in class."

It was found that, in Japan, "teachers devoted more than 80% of the time to developing the concepts, not merely stating them. In Germany, teachers developed about 75% of the math topics. But, in America, "teachers develop only 20% of their lessons and only briefly address the rest of them."

Exactly what I'm promoting here, but I've a program for "developing concepts", which can be tested and improved. In terms of ideas elsewhere, the effective way may emphsize STRATEGY-OVER-TACTICS, versus the American TACTICS-AT-EXPENSE-OF-STRATEGY. For, elsewhere, I advocate division of math class time into days devoted to the STRATEGY OF PROBLEM-SOLVING (when calculators or computers are allowed to that students don't get lost in the wilds of calculation while learning strategy) and other days devoted to the TACTICS OF PROBLEM-SOLVING (when students must work with pencil and paper).

I argue that what we practice in American schools (in all subjects!) is what, elsewhere, I call JITED EDUCATION -- JUST-IN-TIME EDUCATION -- treating our children and young people as unfinished products on an assembly-line. Put a child on the line. Sand her/him with this concept. Stamp him/her in that procedure. Solder this whatsis onto her/him. Glue that on. Etc. When the student exits the assembly-line, we inspect and ACCEPT or REJECT the "product". Great quality control experts, such as W. Edwards Deming and Joseph Juran, taught us that this regimen doesn't provide good quality in the factories. And it doesn't work in the schools, either!

To illustrate, you can, elsewhere, read about a famous educational program, which I was (unwittingly) involved with, from its inception: "The New Math". I learned from that expwrience, and I argue that a "VIOLET-ED" approach to EDUCATION would AVOID such foolishness!

THE CONEK OF NUMBER What better conek to consider than number? We possess many number-loconeks, showing the beginnings, in prehistory or in practices of contemporary "primitive" peoples, of the number-concept. (A file at this Websites, "Our Alfanumeric Roots" deals with this.)

Tell children about a wolf bone dating from around 30,000-25,000 BC -- a bone with 55 cuts across the bone, collected in groups of 5. This bone must have recorded the count of something important for some prehistoric ancestor. And, as I show in my "ROOTS" PROGRAM, we can guide children to simulate various stages of basic development of the NUMBER CONEK.

Thus, I describe, in another file, BONFIRE OF THE INSANITIES, the use (only a few hundred years ago) of tally sticks to inventory the number of gold or silver pieces in the British "Treasury". I cite also use of pebbles in a bag; knots in string; etc. And we know of peoples who could only count the equivalent of "One, two, many", and we know that the word "three" derives from the Latin word "trans" for "beyond two". These, and other loconeks, help children understand the development of the number-conek.

In particular, logician-mathematician-philosopher, Bertrand Russell 1877-1970), said "It must taken eons for humans to realize that a brace of partridges and a couple of days are both instances of the number two." Show students that one word was used for a number of this and a different word for the same number of that and still a different word for the same number of something else. Etc. Thus the general number-conek slowly developed.

Then, in MATHEMATICS ARITHMETIZED, I show the "flowering" of the NUMBER CONEK in various hiconeks of different number-systems; of limits in calculus; of vectors and multivectors; and, in general, into the rich hiconeks of The Arithmetic of Clifford Numbers, which encompasses 25 or more vast systems of mathematics. From such primitive beginnings -- highly HICONEK!

Howevr, in another file at this Websites -- "BROWN-BAGGING CARDINAL AND ORDINAL NUMBERS" -- I show how FAILURE TO PLANT THE CONEKS OF ORDINAL AND CARDINAL NUMBERS and failing to show their ROOTING in basics (illustrated by the brown-bagging) can result in trauma on the HICONEK level of DIFFERENTIAL AND INTEGRAL CALCULUS in our high school and freshman college courses!