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Differentcolor poker chips alone, as Fuson notes (p. 384) will not generate understandingabout quantities or about place-value. Children can be confused about therepresentational aspects of poker chip colors if they are not introducedto them correctly. And if not wisely guided into using them effectively,children can learn "face-value (superficial grouping)" facility with pokerchips that are not dissimilar to the face value, superficial ability toread and write numbers numerically. The point, however, is not to let themjust use poker chips to represent "face-values" alone, but to guide theminto using them for both (face-value) representation and as grouped physicalquantities. What I wrote here about the use of poker chips to teach place-valueinvolves introducing them in a particular (but flexible) way at a particulartime, for a particular reason. I give examples of the way they need tobe used to teach place-value in the text. The time they need to be introducedthis way is after children understand about grouping quantities and countingquantities "by groups". And I explain in this article precisely why differentcolor poker chips, when used correctly, can better teach children aboutplace-value than can base-ten blocks alone. Poker chips, used and demonstratedcorrectly, can serve as an effective practical and conceptual bridge betweenphysical groups and columnar representation, because they are both physicaland representational in ways that make sense to children --with minimaldemonstration and with monitored, guided, practice. And since poker chipsstack fairly conveniently, they can be used at earlier stages for childrento count individually and by groups, and to manipulate by groups. (Columnsof poker chips can also be used effectively to teach understanding aboutmany of the more difficult conceptual and representational aspects of fractions,which is another matter about teaching that I only mention here to pointout the usefulness of having a large supply of poker chips in classroomsfor a number of different mathematics educational purposes.) ()

## The Concept and Teaching of Place-Value Richard Garlikov

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When my children were learning to "count" out loud (i.e., merely recitenumber names in order) two things were difficult for them, one of whichwould be difficult for Chinese-speaking children also, I assume. They wouldforget to go to the next ten group after getting to nine in the previousgroup (and I assume that, if Chinese children learn to count to ten beforethey go on to "one-ten one", they probably sometimes will inadvertentlycount from, say, "six-ten nine to six-ten ten"). And, probably unlike Chinesechildren, for the reasons Fuson gives, my children had trouble rememberingthe names of the subsequent sets of tens or "decades". When they did rememberthat they had to change the decade name after a something-ty nine, theywould forget what came next. But this was not that difficult to remedyby brief rehearsal periods of saying the decades (while driving in thecar, during errands or commuting, usually) and then practicing going fromtwenty-nine to thirty, thirty-nine to forty, etc. separately.

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In a discussionof this point on Internet's AERA-C list, Tad Watanabe pointed out correctlythat one does not need to regroup first to do subtractions that require"borrowing" or exchanging ten's into one's. One could subtract the subtrahenddigit from the "borrowed" ten, and add the difference to the original minuendone's digit. For example, in subtracting 26 from 53, one can change 53into, not just 40 plus 18, but 40 plus a ten and 3 one's, subtract the6 from the ten, and then add the diffence, 4, back to the 3 you "alreadyhad", in order to get the 7 one's. Then, of course, subtract the two ten'sfrom the four ten's and end up with 27. This prevents one from having todo subtractions involving minuends from 11 through 18.