4) representation (of groupings)

The answer is not nine, but "still just three, because the ship willrise with the tide." This does not demonstrate respondents do not understandbuoyancy, only that one can be tricked into forgetting about it or ignoringit.

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Math learning does not have to go in some particular arithmeticalorder only, at some particular age. There are all kinds of mathematicaltypes of things that children can do at various ages. There is more tomath than just algorithmic arithmetic; and children can do the "more" evenin some cases where they cannot yet do the algorithmic arithmetic. Childrencan reason; they just sometimes need some help or practice or feedback,or they sometimes need a reasonable or reasonably channeled challenge,in order to hone their reasoning skills. ()

5) specifics about representations in terms of columns.

2) Simple Addition and Subtraction

If you don't teach children (or help them figure out how) to adroitlydo subtractions with minuends from 11 through 18, you will essentiallyforce them into options (1) or (2) above or something similar. Whereasif you do teach subtractions from 11 through 18, you give them the optionof using any or all three methods. Plus, if you are going to want childrento be able to see 53 as some other combination of groups besides 5 ten'sand 3 one's, although 4 ten's plus 1 ten plus 3 one's will serve, 4 ten'sand 13 one's seems a spontaneous or psychologically ready consequence ofthat, and it would be unnecessarily limiting children not to make it easyfor them to see this combination as useful in subtraction. ()

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Unfortunately, when formal systems are learned incorrectly or when mistakesare made inadvertently, there is no reason to suspect error merely by lookingat the result of following the rules. Any result, just from its appearance,is as good as any other result.

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Arithmetic algorithms are not the only areas of life where means becomeends, so the kinds of arithmetic errors children make in this regard arenot unique to math education. (A formal justice system based on formal"rules of evidence" sometimes makes outlandish decisions because of loopholesor "technicalities"; particular scientific "methods" sometimes cause evidenceto be missed, ignored, or considered merely aberrations; business policiesoften lead to business failures when assiduously followed; and many traditionsthat began as ways of enhancing human and social life become fossilizedburdensome rituals as the conditions under which they had merit disappear.)

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In a thirdgrade class where I was demonstrating some aspects of addition and subtractionto students, if you asked the class how much, say, 13 - 5 was (or any suchsubtraction with a larger subtrahend digit than the minuend digit), yougot a range of answers until they finally settled on two or three possibilities.I am told by teachers that this is not unusual for students who have nothad much practice with this kind of subtraction. ()

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When Iexplained about the need to practice these kinds of subtractions to oneteacher who teaches elementary gifted education, who likes math and mathematical/logicalpuzzles and problems, and who is very knowledgeable and bright herself,she said "Oh, you mean they need practice regrouping in order to subtractthese amounts." That was a natural conceptual mistake on her part, sinceyou do NOT regroup to do these subtractions. These subtractions are whatyou always end up with AFTER you regroup to subtract. If you try to regroupto subtract them, you end up with the same thing, since changing the "ten"into 10 ones still gives you 1_ as the minuend. For example, when subtracting9 from 18, if you regroup the 18 into no tens and 18 ones, you still mustsubtract 9 from those 18 ones. Nothing has been gained. ()