The Concept and Teaching of Place-Value in Math

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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.

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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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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).

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How somethingis taught, or how the teaching or material is structured, to a particularindividual (and sometimes to similar groups of individuals) is extremelyimportant for how effectively or efficiently someone (or everyone) canlearn it. Sometimes the structure is crucial to learning it at all. A simpleexample first: (1) saying a phone number such as 323-2555 to an Americanas "three, two, three (pause), two, five, five, five" allows him to graspit much more readily than saying "double thirty two, triple five". It iseven difficult for an American to grasp a phone number if you pause afterthe fourth digit instead of the third ("three, two, three, two (pause),five, five, five").