The Pythagoreans taught us that the prime numbers are the multiplicative gnomons or building-blocks of "counting numbers" (Natural Numbers -- "integers" came later).

The concept of "prime number" cannot be "planted" too soon in the thinking of elementary children. More than the "atomic" vagaries of Anaxagoras (500-428 BC) -- which provides concept but no procedure -- the prime-composite number concept and factoring procedure is the model for

• the chemical table of elements;
• types of physical energy;
• blood typing;
• genes and dna;
• other instances of basic elements.

Also, many children will work on the "information-highway" where they may use prime-number cyphers to protect their secret files.

Moderm mathematicians express the importance of primes by The Fundamental Theorem of Arithmetic for Integers: An integer can be factored into prime factors in only one way, if we ignore the order of statement of the primes.

This Theorem holds for Natural Numbers, for Rational Numbers, and for (rational and irrational) Real numbers. The fact that this Theorem breaks down for the Complex Numbers has resulted in a great deal of mathematical research in the past two centuries.

Factoring into Primes ("Row-Numbers")

Every nonzero number has at least two factors, 1 and itself. These are called it "improper factors" (meaning, "trivial"). A number such as 9 (with improper factors 1 and 9 and proper factor 3) is a composite number. A prime number has no proper factor.

The terminology of "prime and composite numbers" can, initially, be avoided by use of Bottlecap Geometry, shown at this Website for PRESCHOOLERS. We show children how to build number patterns from ROWS of bottle caps. We admit that rows can be turned into columns. But we point out that, by rotating our position, a column looks like a row. So the row pattern will be standard.

We show that "squares", such as 4, 9, 16, 25, are represented by repeated rows -- 4 as 2 rows of 2 dots; 9 as 3 rows of 3 dots; 16 as 4 rows of 4 dots; 25 as 5 rows of 5 dots; etc.

*   |  *  *   |  *  *  *  | *  *  *  *  | *  *  *  *  *  |
    |  *  *   |  *  *  *  | *  *  *  *  | *  *  *  *  *  |
    |         |  *  *  *  | *  *  *  *  | *  *  *  *  *  |
    |         |           | *  *  *  *  | *  *  *  *  *  |
    |         |           |             | *  *  *  *  *  |

We show that 10 can be constructed of two rows of five bottle caps or five rows of two botttle caps.

    *  *  *  *  *  |   *   *
    *  *  *  *  *  |   *   *
                   |   *   *
                   |   *   *
                   |   *   *   

We use the calendar to explain to the children what we mean by a "table" of elements. The columns represent days, while the rows represent weeks. A table is composed of columns of equal-numbered rows.

We now tell the children, "Any number pattern built of two or more equal rows represents A TABLE-NUMBER." (For teacher or parent, a Table-Number or composite number is represented as in the form r X c, number of rows r > 1, number of columns c > 1.)

And we tell the children, "Any number which CANNOT be built as a TABLE pattern (of EQUAL ROWS) is simply a ROW-NUMBER." (For teacher or parent, a ROW-NUMBER or prime number is represented as in the form r X 1, r > 1; or 1 X c, c > 1.)

Background

In WEB EDUCATION I argue that every "conek" (concept or procedure), such as sieving should be "a plant" with both "roots" ("loconeks") and "flowers" ("hiconeks"). I've already discussed some hiconeks of sieving. And its very name suggests a familiar loconek. You separate vegetables you have boiled from the water boiling them by a sieve. (Gold miners also pour, thourgh a wire-mesh, buckets scooped from a stream of water believed to contain gold, to sieve out any tiny gold rocks present.) Sieving is credited to the Greek mathematician and philosopher, Eratothenes of Kyrene (c. 276-194 BC), who also used "shadow-reckoning" to calculate the circumference of the earth. Sieving is related to many other algorithms, such as inclusion-exclusion.

Prime/Composite-Numbers:t-/o-Numbers:SchizMath/FlatScience

Regarding a number with primes only to first power, mathematicans label it as "square-free". By implication, the others would be clumsily labeled "unsquare-free". Drawing upon the theory of measurement scales, I speak of "square-free" numbers as "t-numbers" (example, 30 = 2 x 3 x 5), since they involve only the MEASUREMENT-SCALE of TYPE or KIND (not ORDER or DEGREE), and I label the "unsquare-free" as "o-numbers" since they also involve ORDER or DEGREE (example, 60 = 2 x 2 x 3 x 5, continaing factor or gnomon 2 to 2nd degree or power).

But the distinction has pervasive mathematical implications ("Schizoid Math and "Flatland Science") which (I argue) have STYMIED mathematical and scientific progress for 2500 years! Platonists (who dominate worldwide university mathematical departments) insist that critical math fields, such as set theory, probability theory, logic, SHALL NOT MAKE DISTINCTIONS OF ORDER OR DEGREE.

Thus, the numbers 30 = 2 x 3 x 5 and 60 = 2 x 2 x 3 x 5 have the set-theoretic representation, namely, {2, 3, 5}, since THE EXTENSIONAL AXIOM OF SET THEORY IGNORES MULTIPLICITY OF TOKEN, that is, {2, 2, 3, 5} = {2, 3, 5}. I realized in 1958 that this meant that "The New Math" -- which promoted set theory -- could not support "The Old Math", involving the factor theory of arithmetic. But I also realized that all such limited theories could be EXTENDED.

This brought me a blazing reprimand from The National Science Foundation, which which promoted "The New Math" and had sponsored the NSF Institute which I held in 1957 and therein explained the extersions. This resulted in my become a "nonperson" to the NSF. (My applications for grants were simply ignored, without explanation. All of these applications involved aids to teachers or for students, which are still not available.)

Why this position of the Platonists? Because allowing distinctions of ORDER or DEGREE in set theory would preclude the two most powerful NONCONSTRUCTIVE AXIOMS -- "proof by contradiction" and "Axiom of Choice". (In 1984, my fable about this -- "The Battle of the Frog and The Mouse", title taken from a comment of Albert Einstein -- was published The Mathematical Intelligencer, and reprinted in Pi in the Sky (1991) by John Barrows.)

Elsewhere, I've shown how this results in "Schizoid Math" and "Flatland Science", and continues to stymie scimath progress -- 2500 years after Pythagoras distinguished prime/composite-numbers. (The Platonist stymie also has legal consequences, allowing clever lawyers to "get off" clients who might, otherwise, be found guilty of crimiinal acts. The distinction I support is allowed in Scottish jurisprudence.)

Sieving

The children can practice finding ROW-NUMBERS (primes) and TABLE-NUMBERS (composites) while learning sieving. (And perhaps some of them will grow up to support my position.)