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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.) ()

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I do not believe that his categories are categories of increasinglyabstract models of multidigit numbers. He has four categories; I believethe first two are merely concrete groupings of objects (interlocking blocksand tally marks in the first category, and Dienes blocks and drawings ofDienes blocks in the second category). And the second two --different markertype and different relative-position-value-- are both equally abstractrepresentations of grouping, the difference between them being that relative-positional-valueis a more difficult concept to assimilate at first than is different markertype. It is not more abstract; it is just abstract in a way that is moredifficult to recognize and deal with.


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In any case, Jews are news, and when the news seems bad, Jews are blamed—everywhere and always—from the Black Plague to crises in the global economy & international terrorism.
It would be best for us to unify, it would be best for the world to support that unification directly rather than clubbing us as helping the goose to lay golden eggs is a smarter policy than killing it. Smashing a loose, flickering bulb will not just black-out the bulb, but the whole room. This is why anti-Semitism is known as “the Socialism of Fools.” It starts with the Jews, but never ends there.
Are we Jews really responsible though, or just wronged? By Jewish Law we are responsible. If one just passes an uncovered pit in the public thoroughfare that the person didn’t dig, the person is not responsible for any damage to say, an ox that stumbles into it. However, if one decides to be noble and board up the pit—but subsequently decides to take the boards back, then that person is now liable for future damage. The boards are our ancient, national commitment to be a role model of mutually responsible unity.