And I believe teaching involves more than just letting students (re-)inventthings for themselves. A teacher must at least lead or guide in some formor other. How math, or anything, is taught is normally crucial to how welland how efficiently it is learned. It has taken civilization thousandsof years, much ingenious creativity, and not a little fortuitous insightto develop many of the concepts and much of the knowledge it has; and childrencan not be expected to discover or invent for themselves many of thoseconcepts or much of that knowledge without adults teaching them correctly,in person or in books or other media. Intellectual and scientific discoveryis not transmitted genetically, and it is unrealistic to expect 25 yearsof an individual's biological development to recapitulate 25 centuriesof collective intellectual accomplishment without significant help. Thoughmany people can discover many things for themselves, it is virtually impossiblefor anyone to re-invent by himself enough of the significant ideas fromthe past to be competent in a given field, math being no exception. Potentiallearning is generally severely impeded without teaching. And it is possiblyimpeded even more by bad teaching, since bad teaching tends to dampen curiosityand motivation, and since wrong information, just like bad habits, maybe harder to build from than would be no information, and no habits atall. In this paper I will discuss the elements I will argue are crucialto the concept and to the teaching of place-value.


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