Dig? You just CHRISTENED "Fumb" as "One", and "Lil Finner" as "FIVE".
Later, you may repeat your counting-finger-routine by starting with "Lil Finner", working to "Fumb": "One, Two, Three, Four Five".
"WAHHH! You said Lil Finner is NAMED 'Five'. Now you say Lil Finner is NAMED 'One'. You CHANGED Lil Finner's NAME!"
Names are very important to small children. Sometimes names are the first words children learn. And YOU committed NAME-ABUSE! (Would you take away the child's name?)
On the other hand (or finger), mebbe the kid won't protest. Even worse! CONDITIONING THE CHILD TO ACCEPT NAME-CHANGING AS AN EVERY-DAY AFFAIR! Teaching MATH AS A PROCEDURE FOR NAME-CHANGING!
No! I'm not talking "New Math"! (Anyway, much of that brought back "Old Math" excised by early 20th century "progressive educators".) No, The COUNT-PARADOX concerns NUMBER NAMES GOING BACK MANY CENTURIES!!!
Names such as "One, Two, Three, Four, Five" are CARDINAL NAMES OF NUMBERS. And CARDINAL names have names COMPLEMENTARY to them -- THE ORDINAL NAMES -- correspondingly, "First, Second, Third, Fourth, Fifth". Remember??? CARDINALS and ORDINALS represent TWO VERY DIFFERENT NUMBER PATTERNS! (Yes, Hamlet, you're also committing PATTERN-ABUSE!) Elsewhere at this Website, in "Brownbaging the Numbers", I show how kids can be guided in constructing cardinal number model and ordinal number models, as two different patterns of brown bags.
Returning to our problem,THE CORRECT WAY TO COUNT THE FINGERS IS ORDINALLY: "FIRST, SECOND, THIRD, FOURTH, FIFTH" -- THEN, CIRCLING THE CHILD'S FINGERS WITH YOUR HAND, YOU CARDINATE:"FIVE". Dig? (As we'll see below, ORDINATION MEASURES CARDINATION!)
IT'S THE FINGER-SET WHICH POSSESSES FIVENESS -- FIVE-CARDINALITY. Neither Lil Finner nor Fumb possesses FIVENESS!
Whether you count LilFinner-to-Fumb or Fumb-to-LilFinner, your (cardinal) climax IS THE SAME: "FIVE". This means that, IN THE FINITE(!) DOMAIN, CARDINALITY IS INVARIANT UNDER ORDINATION -- for example, any way you CORRECTLY count the fingers YIELDS THE SAME CARDINALITY! It makes for TOGETHERNESS OF CARDINALITY AND ORDINALITY -- in the FINITE(!) DOMAIN. (But be prepared for a shock, below.)
You can explain to the child that THE ROLE OF LIL FINNER IN THIS PLAY CALLED "COUNTING" MAY CHANGE FROM "FIRST" TO "FIFTH". But THE SCRIPT OF THIS PLAY DOESN'T CHANGE!
Likely, the kid has played the role of "The Pig-with-the-house-of-straw" in one staging of "The Three Little Pigs and The Big Bad Wolf" -- and the role of "The Big Bad Wolf" in another staging of it. That makes sense. AND DOESN'T CHANGE MATH INTO TRANMIGRATION OF SOULS!
No, Ophelia, I wasn't taught counting in this way. I'd heard the ordinal and cardinal names. And thought them quaint. Alas! my ignorance caught up with me when, in my first year at Columbia University, I learned that THE TOGETHERNESS OF CARDINALITY AND ORDINALITY IN THE FINITE SUMMING PROCESS DOES NOT ALWAYS EXTEND TO TRANSFINITE SUMMING!!! This lesson so traumatized me that I dazedly failed Semester Two of "Calculus" and was ready to quit college and give up "higher education", despite the maximum support I could receive from "The G. I. Bill" for World War II veterans. My teachers hadn't warned me of this. And teachers still don't warn about it!
Before explaining this, I'll try to convince you of the seriousness of the whole matter by explaining how "standard counting" ("One, etc.") implants in the child's experience something close to ONE OF THE MOST VICIOUS PARADOX WE'VE YET KNOWN IN MATHEMATICS! And this is COMPOUNDED BY CREATING AN EDUCATIONAL CUL-DE-SAC, ISOLATED FROM PROFOUND NOTIONS NEEDED LATER ON!
Briefly, I remind you of the famous syllogism of Aristotle:
- All men are mortal.
- Socrates is a man.
- Therefore, Socrates is mortal.
You can represent the class of "men" (or humans) by a circle embedded in the circle-of-mortals. You can represent Socrates as a point or dot in the circle-of-men. Then, obviously, this point is contained in the circle-of-mortals. OK?
Now, one of our funny ancestors decided to PARODY Aristotle's syllogism, thus:
- The Apostles were twelve.
- Peter was an Apostle.
- Therefore, Peter was twelve.
Yuk! What went wrong?
The property of "mortality" is a property of the set of men and it also DISTRIBUTES itself over the members of this set. The property of "twelveness" is a property of the set of Apostles , but TWELVENESS DOES NOT DISTRIBUTE ITSELF OVER THE MEMBERS OF THIS SET!
Similarly, the property of "fiveness" is possessed by the set of fingers on a hand, but FIVENESS DOES NOT DISTRIBUTE ITSELF OVER THIS SET!
Yet that's precisely what you've IMPLIED by this method of COUNTING. Let's formulate this as a "syllogism":
- The fingers of a hand are five.
- Fumb is a finger of the hand.
- Therefore, Fumb is five.
The most famous paradox in the history of logico-mathematics is popularly known as "The Paradox of The Spanish Barber". This "Counting Paradox" is very close to the viciousness of "The Barber Paradox".
The greatest harm invoked by this way of starting counting is that it cuts the child off from SOME OF THE MOST PROFOUND AND USEFUL CONCEPTS WE POSSESS, which should be taken up later on. In the language of my "African-Violet Model for Education" (also on this Website) these "cuttings" cannot be planted preschool for future "rooting" and "flowering" . No, Hamlet, they are not merely for studying math. Without one of them, -- to teach you about CHOICES and ROLE-PLAYING -- you'll be limited as a citizen in a democracy or consumer in a market economy.
Happily, once you start the little child in this ORDINAL-CARDINAL COUNTING procedure, you can then allow a SHORT-CUT. Logicians and mathematicians often say, "We can allow the abuse of language that ....", naming a short-cut. Once the child understands that the proper counting names are "First, second, etc.", then, for simplicity, the process can be "abbreviated" to "One, two, etc."
The ORDINAL-CARDINAL "cutting" has been planted for future "rooting" and "flowering". See? Another WAMPING between the extremes of PEDANTRY and PAP!
The child who suffers subliminally from the paradox of counting may become a mathephobe, like Maisy, the scaredy ostrich. But the child who has been taught to bypass the count paradox may become a mathephile, like Daisy, the brave ostrich.
Poor Maisy, the sad mathephobe, In earth to her neck she will probe To escape the Hobgoblin Of a mathematics problem -- Exposing her tail to the globe. See Daisy, the brave mathophile. Don't think you can cause her to reel At mathematics problems. Instead she will gobble 'em. She eats Math for breakfast, core and peel!