The term "algorithm" derives from the name of an Arabic astronomer, Mohammed ibn Musa al-Khwarizmi (c. 82d). It denotes A SEQUENCE OF STEPS, IN FAMILIAR OPERATIONS, STARTING FROM AN EFFECTIVE STATEMENT OF A PROBLEM, AND ENDING IN SOLUTION OF THE PROBLEM. IT IS THE EQUIVALENT OF A COMPUTER PROGRAM -- BUT USUALLY AS SLOW AS THE TORTOISE IN ÆSOP'S FABLE. (For example, ancient Babylonian priests knew an algorithm for approximating a square root -- say, (2)^1/2 -- by ITERATIVE DIVISION.)

 HEURITHM (my own jargon for a HEURISTIC PROCESS): DOESN'T ENTIRELY SOLVE PROBLEM, BUT IS AS FAST AS THE HARE. (The distinguished Princeton U. statistician, John Tukey, wrote a paper of heurithms in his field: "Quick and Dirty Methods in Statistics".)

---

OK, Hamlet, if you're not too proud to learn "a kid's way and HOPPY'S way", I'll show you how easy it is to understand the difference between ALGORITHM & HEURITHM. (It's "minimalist algorithm"!)

You see, elsewhere, I show how a kid can teach HOPPY (THE CHILDREN'S FRIENDLY LOW-SPEED DIGITAL COMPUTER) TO ADD AND SUBTRACT.

This is what I've called Householder Education, because mathematician Alton S. Householder said you may not really know something unless you can make the computer do it!

For Householder knew that THE BEST WAY TO THOROUGHLY LEARN SOMETHING IS TO TRY TO TEACH IT. Householder also knew that this, sometimes, TRAUMATICALLY EFFECTS A HUMAN STUDENT -- BUT WON'T DETER A COMPUTER!

 
1. Awkwardness of "new" teacher may irretrevably impair human student learning (but not for the computer!).
2. Nervousness of "new" teacher may incite nervousness in human student (but not in the computer!).
3. The teacher may unintentionally or intentionally omit some "step" or information, leaving the human student to make it up (but this won't work with the computer!).
4. The teacher may try to intimidate the human student -- "Of course, you understand this!" -- (and this won't work with the computer!).

So, says Householder, making a computer "do it" PROVES KNOWLEDGE OF THE SUBJECT OR PROCEDURE! Hence, we can entrust "householder education" to kids and HOPPY.

---

A kid teaches HOPPY to ADD by THE BASKET ALGORITHM (which includes a HEURITHM). Then teaches HOPPY TO SUBTRACT by THE STAIR-STEP ALGORITHM (which also includes a HEURITHM).

(Don't scowl at me, Ophelia! I didn't make up "algorithm". It's an epinome, from a 7th century Arabic mathematician. And kids of this "Computer Generation" should be exposed to this word early, because it's "computer jargon" also. Anyway, I sometimes speak of "Hoppy's Basket Trick" or "Stair-Step Trick". And you can con the kids into it. "See! See! Al. Has Al got rhythm? Yehh! See! See! Al go rhythm!")

---

THE BASKET ADDITION ALGORITHM:

 
• Given a SUM OF TWO ADDENDS -- such as 4 + 3 -- think of these ADDENDS as BASKETS CONTAINING STICKS.
• The FIRST ADDEND (here, 4) is thought of as THE INPUT BASKET OF STICKS (here, 4 sticks).
• the SECOND ADDEND (here, 3) is thought of as THE OUTPUT BASKET OF STICKS (here, 3 sticks).
• The ALGORITHM (or HEURITHM) OPERATES BY TAKING A STICK FROM THE OUTPUT BASKET AND PUTTING IT INTO THE INPUT BASKET -- STOPPING WHEN THE OUTPUT BASKET IS EMPTY.
• The COUNT in the OUTPUT represents THE SUM.

Given 4 + 3, we find:

 
1. START: 4 + 3.
2. THEN: 5 + 2.
3. THEN: 6 + 1,
4. FINALLY (reaching EMPTY OUTPUT Basket): 7 + 0.

HOPPY COUNTS THE STICKS IN THE INPUT BASKET AND FINDS 7. So, 4 + 3 = 7.

Let's "empty out" another SUM:

 15 + 13 = 16 + 12 = 17 + 11 = 18 + 10 = 19 + 9 = 20 + 8 = 21 + 7 = 22 + 6 = 23 + 5 = 24 + 4 = 25 + 3 = 26 + 2 = 27 + 1 = 28 + 0 = 28.

 (Likely, the kid would have heurithmed off at 20 + 8 = 28, or otherwhere.)

Why is this AN ALGORITHM?

 
1. BECAUSE THE OUTPUT (2ND ADDEND) BASKET WILL ALWAYS BECOME EMPTY, ENDING THE REPLACEMENT PROCESS.
2. AND THE SUM CAN BE OBTAINED BY COUNTING STICKS IN THE INPUT (1ST ADDEND) BASKET.

Now, GIVEN ALGORITHM TO BACK UP HEURITHM, THE KID CAN TRANSFORM ANY SUM INTO A KNOWN ONE!

Where's the HEURITHM?

Suppose the Kid (or HOPPY) REMEMBERS THAT 5 + 2 = 7. Since 5 + 2 "CAME FROM 4 + 3", IT "FOLLOWS THAT 4 + 3 = 5 + 2 = 7". So, THE PROCESS HAS A SHORTCUT. AN ALGORITHM REPLACED BY A SHORTCUT IS A HEURITHM. But YOU DON'T HAVE THE CERTAINTY ABOUT THIS THAT YOU DO WHEN YOU END WITH "7 + 0"!

---

Now, consider THE STAIR-STEP SUBTRACTION ALGORITHM:

 
• Given a DIFFERENCE OF TWO NUMBERS -- such as 7 _ 4 -- THINK OF THESE AS KIDS POSITIONED ON STEPS OF A STAIR.
• The MINUEND (here, 7) is thought of as a kid ON THE SEVENTH STEP OF THE STAIRS.
• The SUBTRAHEND (here, 4) is thought of as a kid ON THE FOURTH STEP OF THE STARIS.
• After "START", the "kids" DESCEND STEP-BY-STEP IN CADENCE.
• WHEN THE LOWER ONE IS AT THE BOTTOM OF THE STAIRS ("ZERO"), THE COUNT OF THE STEP OF THE UPPER ONE IS THE DIFFERENCE.

  To illustrate:

   
  1. START:7 - 4.
  2. ONE CADENCED STEP DOWN: 6 - 3.
  3. ANOTHER DOWN: 5 - 2.
  4. ANOTHER DOWN: 4 - 1.
  5. ANOTHER DOWN and FINALLY one kid at the BOTTOM: 3 - 0.
  Then we find the DIFFERENCE: 7 - 4 = 3.

  Let's "step down" another DIFFERENCE:

  23 - 17 = 22 - 16 = 21 - 15 = 20 - 14 = 19 - 13 = 18 - 12 = 17 - 11 = 16 - 10 = 15 - 9 = 14 - 8 = 13 - 7 = 12 - 6 = 11 - 5 = 10 - 4 = 9 - 3 = 8 - 2 = 7 - 1 = 6 - 0 = 6.

  (Likely, the kid would have heurithmed off at 20 - 14 = 6 0r 10 - 4 = 6.)

  Why is this an ALGORITHM?:

   
  1. BECAUSE THE "LOWER KID" (SUBTRAHEND) WILL ALWAYS REACH "BOTTOM" (ZERO).
  2. AND THE POSITION OF THE "HIGHER KID" (MINUEND) WILL REPRESENT THE DIFFERENCE.
  It's HEURITHM is that HOPPY MAY "REMEMBER" ONE OF THESE PARTIAL DIFFERENCES -- say, that 5 - 2 = 3 -- AND SHORTCUT THE PROCESS.

  But again, this doesn't have the certainty of the ALGORITHM has by "reaching bottom".

  Given ALGORITHM TO BACK UP HEURITHM, THE KID CAN TRANSFORM ANY PRODUCT INTO A KNOWN ONE!

  ---

  In 1963, I tested these at the Campus School at Inter American University of Puerto Rico, San Germán, Puerto Rico. I taught HOPPY'S BASKET ADDITION TRICK to three 3rd Graders.

  Upon learning it, Susan Custer danced about singing, "Now I won't have to ask Momma to help me add!"

  After I taught HOPPY'S STAIR-STEP SUBTRACTION TRICK, Alvin Muckley jumped into the air, shouting, "Now I can do any subtraction in the world!" ALVIN MUCKELY -- GREATEST SUBTRACTER IN THE WEST!)

  ---

  "The Tortoise and the Hare" is a Fable told by Æsop, the Greek slave who lived on the Island of Samos, where Pythagoras was born. Perhaps Pythagoras, as a child heard Æsop tell this particular Fable. Note, further, that a variation of this "Tortoise and Hare" Fable is the famous "Achilles and the Tortoise" Paradox, formulated by Zenon (490-430 BC) of Elea, to attack the Pythagorean Philosophy of Mathematics. This Paradox was not resolved until the invention of the LIMIT concept, gnomon of "the differential and integral calculus of functions".

  MATHEMATICS ULTIMATELY NEEDS ALGORITHMS AND RIGOROUS PROOFS. BUT SPEEDY HEURITHMS SUFFICE TO GET US THROUGH LIFE ALIVE.

  We START WITH HEURITHM. Sustained by it -- in many results -- we then learn the ALGORITHM. And "we're in business"!

  Yet!!! Guess what? NEITHER ONE IS TAUGHT TO MOST CHILDREN! As a Teacher-of-Teachers, I must have gone through over a hundred elementary textbooks. I can't remember any that STATED THIS HEURITHM. Kids just memorize and recite additions BY ROTE -- like good little tape recorders!

  So, kids who may hear "The Fable of The Tortoise and the Hare", may never know about "The Algorithm Tortoise" and "The Heurithm Hare"!

  Please help me teach this!