she had leaned over and whispered just three little words.
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Once children have gained facility with counting,and with counting by groups, especially groups of 10's and perhaps 100's,and 1000's (i.e., knowing that when you group things by 100's and 1000'sthat the series go "100, 200, 300, ... 900, 1000; and 1000, 2000, 3000,etc.), I believe it is better to start them out learning about the kindof representational group values that children seem to have no troublewith -- such as colors, as in poker chips (or color tiles, if you feelthat "poker" chips are inappropriate for school children; poker chips arejust inexpensive, available, easy to manipulate, and able to be stacked).Only one needs not, and should not, talk about "representation", but merelyset up some principles like "We have these three different color pokerchips, white ones, blue ones, and red ones. Whenever you have ten whiteones, you can exchange them for one blue one; or any time you want to exchangea blue one for ten white ones you can do that. And any time you have tenBLUE ones, you can trade them in for one red one, or ."Then you can show them how to count ten blue ones (representing ten's),saying "10, 20, 30,...,90, 100" so they can see, if they don't already,that a red one is worth 100. Then you do some demonstrations, such as puttingdown eleven white ones and saying something like "if we exchange 10 ofthese white ones for a blue one, what will we have?" And the children willusually say something like "one blue one and one white one". And you canreinforce that they still make (i.e., represent) the same quantity "Andthat then is still eleven, right? [Pointing at the blue one] Ten [thenpointing at the white one] and one is eleven." Do this until they catchon and can readily and easily represent numbers in poker chips, using mixturesof red, blue, and white ones. In this way, they come to understand grouprepresentation by means of colored poker chips, though you do not use theword representation, since they are unlikely to understand it.
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explains how the names of numbersfrom 10 through 99 in the Chinese language include what are essentiallythe column names (as do our whole-number multiples of 100), and she thinksthat makes Chinese-speaking students able to learn place-value conceptsmore readily. But I believe that does not follow, since however the namesof numbers are pronounced, the numeric designation of them is still a totallydifferent thing from the written word designation; e.g., "1000" versus"one thousand". It should be just as difficult for a Chinese-speaking childto learn to identify the number "11" as it is for an English-speaking child,because both, having learned the number "1" as "one", will see the number"11" as simply two "ones" together. It should not be any easier for a Chinesechild to learn to read or pronounce "11" as (the Chinese translation of)"one-ten, one" than it is for English-speaking children to see it as "eleven".And Fuson does note the detection of three problems Chinese children have:(1) learning to write a "0" when there is no mention of a particular "column"in the saying of a number (e.g., knowing that "three thousand six" is "3006"not just "36"); (2) knowing that in certain cases when you get more thannine of a given place-value, you have to convert the "extra" into a higherplace-value in order to write it (e.g., you can say "five one hundred'sand twelve ten's" but you have to write it as "620" because you [sort of]cannot write it as "5120". [I say, "sort of" because we do teach childrento write "concatenated" columns --columns that contain multi-digit numbers--when we teach them the borrowing algorithm of subtraction; we do writea "12" in the ten's column when we had two ten's and borrow 10 more.] (3)Writing numbers normally without "concatenating" them (e.g., learning towrite "five hundred twelve" as "512" instead of "50012", where the childwrites down the "500" and puts the "12" on the end of it).