In the MIDDLE SCHOOL section are found "transgroups", involving ALL THE TRANSFORMATIONS POSSIBLE ON, say, 4 LETTERS or 4 COLORS or 4 MUSICAL TONES or 4 DANCESTEPS or 4 DRAMATIC SCENES. This MATHTIVITY requires TEENS to fully understand and manipulate.

But PRESCHOOL Kids can be guided to recognize THE GROUP STRUCTURE in ACTIVITIES THEY REGULARLY TAKE PART IN.

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FINGER GROUPS

Start with THE THUMB GROUP on only one hand (left or right, as accustomed). Let's denote this by "T" (for thumb). How many ways can you count it?:

	T 

And that's all there is to THE GROUP OF ONE MEMBER, especially in counting it.

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Now, for the thumb and forecfinger, denoted T, F. How many ways can you COUNT this GROUP OF TWO MEMBERS?

	TF		FT 

Just two ways. Just TWO TRANSFORMATIONS ON THE GROUP OF TWO MEMBERS.

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Now, we adjoin the middle finger to thumb and forefinger, obtaining A GROUP OF THREE MEMBERS. Denote the middle finger by "M"", and we have:

	TFM	MFT
        TMF     FMT
        MTF     FTM 

That's it. SIX TRANSFORMATIONS IN THE GROUP ON THREE MEMBERS.

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Now, let's adjoin to these the ring finger, denoting "R", for THE GROUP OF FOUR MEMBERS. How many ways can we count four finger, which is saying, "How many TRANSFORMATIONS on THE GROUP OF FOUR MEMBERS?" We find:

        TFMR	RTMF	MTFR	RMFT	FMTR	RFTM
        TFRM	TRMF	MTRF	MRFT	FMRT	FRTM
        TRFM	TMRF	MRTF	MFRT	FRMT	FTRM
        RTFM    TMFR	RMTF	MFTR    RFMT    FTMR  

24 WAYS OF COUNTNG 4 FINGERS! THE GROUP OF 4 MEMBERS HAS 24 TRANSFORMATIONS!

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Finally, let's adjoin the "little finger" to the other fingers, denoting it "L", for THE GROUP OF TRANSFORMATIONS ON 5 MEMBERS.

I won't write this out, because you might get mixed up following it through. Why? BECAUSE IT HAS 120 TRANSFORMATIONS! Yes! And here's what else that means. 5 kids can form 120 DIFFERENT BASKETBALL TEAMS! Think of that when you say, "There's nothing to do!".

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And you can run through the above results with 1, 2, 3, 4, 5 bells. With 3 bells, you have a GROUP OF 6 ORDERINGS OF DING-DONGS. With 4 bells, you have a GROUP OF 24 ORDERINGS OF DING-DONGS. 5 bells allow for 120 ORDERINGS OF DING-DONGS!

Actually, grownups have done this for centuries in "the cathedral towns of England": RINGING ALL THE CHANGES ON SO MANY BIG CHURCH BELLS, WITH TWO RULES: YOU CAN ONLY SHIFT ONE BELL FOR ONE POSITION IN EACH CHANGE. AND YOU HAVE TO END UP WITH AN ORDERING THAT TURNS INTO THE FIRST ORDERING. BUT THAT'S EXACTLY WHAT WAS ACCOMPLISHED ABOVE WITH THE FINGERS!

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If you substituted COLORS for FINGERS or BELLS, you would obtain SIMILAR GROUPS OF COLORS. ETSETTERY.