Let's Say:
A Child's First Calculus--notes to Foreword

Note 1:

This observation was made by my daughter Noel Zethmayr, a teacher specializing in reading.

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Note 2:

Dedekind (rhymes with "made a hint")--some big university professor, right? Nope. High-school teacher.

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Note 3, Notation and Views "make no difference?"

Recent correspondence with Richard Shoup has alerted me to the need to qualify this "no difference" claim, even at this preliminary stage.

Although all three notations in question support and express the identical theoretic construct, there is indeed a big practical difference between standard (parenthesis-style) pancake on the one hand, and Side View pancake and the original notation of "the Laws" on the other: Standard pancake notation is one-dimensional in the same sense as numerical algebraic notation, Boolean algebra, and the usual symbolic logic notations.

This makes standard pancake the obvious choice for use in computerized resolution systems. An evaluation algorithm is readily available --readily available to me, anyway-- only for the parenthesis-style pancake notation: the Poppertwitch Algorithm.

The form of Boolean notation usually seen in digital circuit documentation is a curious harkback to two-dimensionality: the complementing overbar displaces parentheses as a grouping mechanism wherever it occurs. This is almost no different from the cross ("the Laws"), or from the pancake in side view.

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Note 4, Concrete materials:

Paper and pencil are, however, quite as good for initiating students. For 6th or 7th graders, concrete materials can be quickly dropped after the initial "lure and fascinate" stage (my own observation). For high-schoolers, the concretions are best regarded as an extra tactic, for someone who is having difficulty (my guess without observation).

Since, after all, we aim to cultivate the abstracting faculty, we cannot realistically presume it is mature in our students. Concretions will of course be left behind. But there need be no hurry.

Prolonged concrete play may be quite appropriate for kindergarten thru third or fourth graders. There's plenty of application facility and algebra to be learned from the game of Huh?, which is interesting at many levels of sophistication and is easy to adjust for level. Any group of players spontaneously decides when to graduate from concretions to paper-and-pencil.

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Note 5:

You can relegate Karnaugh mapping to a historical footnote, or just forget it (like I forgot how to say "an historical").

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