I created "conek, loconek, hiconek", as special forms of the CHARLOTTE-CONNECTION and to conform with my FIRST LAW OF MEASUREMENT: LET ALL YOUR NAMES BE UNIVALENT -- SPEAK WITH ONE VOICE OR MEANING!
Humpty-Dumpty told Alice: "I pay my words every Satusday night, and they mean what I want them to." Same here, especially "conek, loconek, hiconek".
Have you ever planted an African violet cutting? If so, and if you've a green thumb (or, at least, not a gangrene thumb), then the cutting may have put down roots and may have begun to grow. Later, it may have put forth flowers.
The key-words (for representing my model) are CUTTING, ROOT, FLOWER. The "cutting" models "conek": 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".
What better conek can we consider than number? For we possess many number-loconeks, showing the beginnings, in prehistory or in the practices of more contemporary "primitive" peoples, of the number-concept.
We can tell children about a wolf bone dating from around 30,000-25,000 BC -- a bone displaying 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, elswhere, I sketch a program to guide children through the simulation of various stages of human development of the NUMBER CONEK.
For example, the use (only a few hundred years ago) of tally sticks to inventory the number of gold or silver pieces in the British "Treasury". Also 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 will help children understand the development of the number-conek.
In particular, the logician-mathematician-philosopher, Bertrand Russell (x-y), said "It must taken eons for humans to realize that a brace of partriges and a couple of days are both instances of the number two." Kids can be shown that one particular 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. So the general number-conek was slow to develop.
And, herein, 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 richness of hiconeks in The Arithmetic of Clifford Numbers, which encompasses 25 or more vast systems of mathematics. From such primitive beginnings -- highly HICONEK!
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?
And this study of ALGEBRA calls for the "other kind of flowering", as in the rose: SEED → ROOT → STEM → FLOWER.
In MATH-HISTORY, this involved "embedding the metalanguage in the language". Those of you familiar with programming languages or, say, HTML, know about "metalanguage": CONTROLS FOR THE TERMS IN THE LANGUAGE. Those unfamliar with this, but who have suffered through English classes, have also seen this. PUNCTUATION is part of the METALANGUAGE, say, of English, TO CONTROL PARSING OF THE LANGUAGE.
A few hundred years ago, this happened in TRIGONOMETRY. In proto-trigonometry, you could not describe negative terms or rotation or orientation, etc. You had to think this out outside the math-language -- in the metalanguage. As TRIGONOMETRY flowered, these CONEKS passed from the METALANGUAGE into the LANGUAGE: MATHPOWER!
Thanks to Grassmann and Hamilton and Clifford, the "flowering" has continued. Coneks that could not be expressed in "scalar language" or even the standard "Gibbs-Heaviside vector-language" can be articulated in MULTIVECTOR-LANGUAGE. Not only african violets, but also roses flourish!
For example, you can describe LINE SEGMENTS in trig, but not DIRECTED LINE SEGMENTS. By DECLARING VECTORS TO COMPOSE A DIRECTED TRIANGLE, we DERIVED INNER PRODUCT from trig's LAW OF COSINES. On the other hand, starting from MULTIVECTOR THEORY, we can DERIVE ALL OF TRIGONOMETRY FROM OUTERPRODUCT. (The daily LOCONEKS of these CONEKS should be obvious.)
We cannot express ORIENTATION BETWEEN LINE SEGMENTS in TRIGONOMETRY, but we can use OUTERPRODUCT and BIVECTORS to find PARALLEL and OTHOGONAL components of one VECTOR with respect to another.