Long after conceiving my Activithm idea and its name, I discovered the following in J. R. Searle, The Philosophy of Language, 1978: "[W]hat counts as reality ... as a glass of water or a book or a table ... is a matter of what categories we impose on the world .... Our concept of reality is a matter of our [linguistic] categories."The "Activithm" Strategy means that knowledge comes only from fallible action which sometimes creates patterns-in-patterns-in-patterns-in-..., when you
- impose a pattern, P upon "whatever";
- see if pattern P has subpatterns, S.
- If so, then see if subpatterns S are conserved under a transformoation group G;
- If so, then label this as the pattern P conserved by group G.
Such a case is my mathtivity which I label Colored Multiplication Patterns, Colored Conservation Laws. In fact, it after creating this MATHTIVITY and looking at it thoroughly that I realized the inherent STRATEGY within it which I labeled "activithm", meaning "activist heurithm", where "heurithm"is another word I created. The word "algorithm" has long been used to denote "an effective problem-solving procedure". But the term "heuristic" was developed for "a rocedure which sometimes works, but is quicker to implement". To provide linguistic parity with algorithm, I changed the "-istic" to "-ithm" to achieve "heurithm". (Elsewhere, I have a file which proclaims the HOMOLOGY -- ALGORITHM: HEURITHM:: THE TORTOISE: THE HARE.)
(Perhaps the most important use of The Activithm, to date, is the imposition of physical dimensions, which I discuss with older students.)
Note: The well known property of serendipity is activithmic. The word "serendipity" originated from the 18th century fairy tale by Horace Walpole about the Princes of Serendip, who rode out seeking adventures AND knew how to recognize an adventure "on sight".
Note also that Win-Win Patterning Strategy, which arises within a Figure-&-Ground Strategyy. For, if our Figure of pattern P, fits the Ground of "Realtiy", then we WIN; otherwise, some unnoticed aspect of "reality" is exposed, which also makes us a winner.
COLORED MULTIPLICATION PATTERNS, COLORED CONSERVATION LAWS To initiate this mathtivity, I first tell children two true (but ignored) stories.
An archaeologist traveled, as airplane passenger, over terrain he knew at ground level. From this perspective the archaeologist saw, outlined on the ground below, the buried fragments of walls and buildings. Later, diggers discovered ruins dating from the Roman invasion of Britain (around 120 AD), during Emperor Hadrian's reign. People had lived upon these ruins for centuries without being aware of them.
I explain this to children: "Those people were too close to see buried patterns. Same way, you're too close to arithmetic to see the number patterns!"
The second story teaches the children "how to rise above the number-blob".
In 1839 German biologist, Theodore Schwann, discovered (in the kind of pattern-searching I emphasize that the basic plant and animal unit (gnomon!) is the cell, which reproduces (subdivides) during the process called "mitosis" to exhibit subpatterns. STAINING the cell highlights the subpatterns.
These subpatterns became labeled by two Greek words: "chromo" for "colored"; "soma" for "body" -- hence, produced the word "chromosomes". (Later chromosomes were identified as hereditary carriers, with genes of "DNA".)
I exhort the children: "STAIN the ARITHMETIC BLOB -- to highlight its number subpatterns!"
To do so, you create 10 x 10 grids as representation of decimal numeration. Then you label COLUMNS with the numbers, "0-9"; and you label ROWS with the numbers "0-9, 10-19, 20-29", etc.
10 has a PRIME FACTOR, 2, so (implicitly) BY IMPOSING THE PATTERN OF TEN, WE ALSO (BY INERITAMCE) IMPOSED A PATTERN OF 2 AND ITS MULTIPLES AND POWERS.
Children INTERACTIVELY DISCOVER THIS by coloring multiples of 2 or multiples of 4 (2 x 2) or multiples of 8 (2 x 2 x W), etc. The children find similarly repeating Thus, conservation laws follow. And, in general, a number written in DECIMAL NUMERALS is a multiple of an n-fives-product if, and only, if its last n digits forms such a product. But 10 has another PRIME FACTOR, namely, 5. Ah! IMPLICITLY, WE IMPOSED A PATTERN OF FIVENESS AND MULTIPLES OF FIVE -- ANOTHER INHERITANCE!
Children INTERACTIVELY DISCOVER THIS by coloring multiples of 5 or multiples of 25 (5 x 5) or multiples of 125 (5 x 5 x 5), etc. The children find similarly repeating patterns upon this replacement of 2's by 5's. Thus, conservation laws follow. And, in general, a number written in DECIMAL NUMERALS is a multiple of an n-fives-product if, and only, if its last n digits forms such a product.
By colored patterns, the children discover that DECIMAL PATTERNING also exhibits interesting SUBPATTERNS for 9 and 11.Why? Because 9 and 11 are the immediate neighbors of the base or gnomon of ten, making easy to detect. (Yes, other DECIMAL SUBPATTERNS exist, such as that for 7, but it is just too subtle to intuit and perhaps difficult to learn.)
You color (say) brown the 9-multiples: 9, 18, 27, 36, 45, ... , 81, 90, 99. Result: a diagonal pattern, going from upper grid-right to lower left.
Why? Because 9 = 10 - 1. Adding 10 drops you one row; subtracting 1 shifts one column left -- diagonally. Furthermore, digits of 9-multiples sum to 9: 18, 1 + 8 = 9; 27, 2 + 7 = 9; 36, 3 + 6 = 9; 45, 4 + 5 = 9; ... 99, 9 + 9 = 18, 1 + 8 = 9. Etc.
A number's digital root is the repeated sum of its digits. Thus, 4 is the digital root of 12478 since 1 + 2 + 4 + 7 + 8 = 22 and 2 + 2 = 4. Equivalently, "casting out the nines" in 12478 -- outing 1 and 8, 2 and 7 -- yields digital root 4. Equivalently, permuting (rearranging, as in anagrams of words) the digits of 12478 -- say, 12478->87421 or 12478->48217 or 12478->87421->48217 -- conserves its digital root, 4.
Gradually, the child can be guided to see that THE DIGITAL ROOT IS AN INVARIANT UNDER THE DIGIT-PERMUTATION TRANSFORMATION! Also, this means that 12478, 87421, 48217, and all its other permutations, yield remainder 4 when divided by 9.
A complementary pattern arises from multiples of 11, because 11 = 10 + 1. It is a codiagonal pattern.
This is another instance of THE ACTIVITHM. For, all the properties I have so far described arise from the ACTION of imposing the decimal pattern on collections of ones, that is, collecting ones in tens and tens of tens, etc.. Now, in the nines-pattern, we find that the digital root of a "decimal" number is conserved by the transformations of summing its digits; by the transformations of casting out 9's; by the transformations of permuting its digits; by the transformations of dividing by 9.
Thus, starting in the 10x10 grid with the number 7, we find that 7 + 9 = 16, 1 + 6 = 7. 16 + 9 = 25, 2 + 5 = 7. 25 + 9 = 34, 3 + 4 = 7. Etc. (This is also yields the algorithm known as "The Bookeeper's Check", which I'll remind you about, later.)
Starting over, you color (say) purple the 11-multiples: 0, 11, 22, 33, 44, 55, 66, 77, 88, 99. Result: a counterdiagonal pattern, running upper left to lower right -- or counterdiagonally to 9-pattern. Why? Well, 11 = 10 + 1. Adding 10 drops one row; adding 1 shifts one column right. Note: The "alternating sum" (alternately adding and subtracting) of digits of eleven-multiples equals zero. Hence, the algorithm of "casting out elevens". Consider 176: 1 - 7 + 6 = 0. 176 is an 11-multiple: 176 = 11 x 16. (I created the label,"codigital root", for the 11-pattern.)The codigital root of a "decimal" number is conserved by the transformations of alternate-sum of digits; the transformations of casting out 11's; the transformationsof dividing by 11.
In 6 class-hours, I taught 24 third-grade children to determine if a 10-digit number has factor 144 = 16 x 9, using the 16-rule and "casting out nines". Children pasted walls with colored patterns. If a 10-digit number's last 4 digits had a colored pattern, the number has factor 16; all 9s casting out indicates factor 9; if both cases, 144. Try this on (a) 1729306348; (b) 1525906448; (c) 1727906448. Solution: (a) 9, not 16; (b) 16, not 9; (16) both, hence 144. Each child had a different 3-problem test. Each answered correctly.
Children also experiment with eight-grids ("octal" numeration in computer science), wherein sevenness assumes the diagonal pattern of nines in ten-grids; nineness assumes the counterdiagonal pattern of elevens in ten-grids -- because seven and nine are immediate neighbors of eight (as are nine and eleven for ten).
Thus, as articulated in THE ACTIVITHM, the subpatterns depend upon the ACTION of imposing a particular pattern!
This mathtivity teaches useful arithmetical tricks (extending to basis or gnomon "x" in numerical algebra! I'll remind you of this latter.). It prepares for conservation laws in physics, as I'll remind you, later.And it echoes the religious faith (evoked by so many mathematicians and scientists) articulated in the words of a cherished old hymn, "Oh, Thou Unchanging!".
A familiar example of THE ACTIVITHM is our imposing three dimensions of space and one of time upon our "world". Albert Einstein said of this, "... time and space are modes in which we think and not conditions in which we live." Albert Einstein, Theoretical Physicist, A. Forsee, Macmillan, New york, 1961, p. 8.