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Rn Direct You Mathlearndirect L Math Math Learn Direct Group Ro 1 Math Learn Direct The Concept and Teaching of Place-Value in Math

Rn Direct You Mathlearndirect L Math Math Learn Direct Group Ro 1 Math Learn Direct

searcho Math tsearcho Direct esearch7 Learn Math a Direct od Direct Mathlearndirect 1 Math 90 Group c Direct t%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%BD%D0%B0%D1%8F%20%D0%B3%D0%B5%D1%80%D0%B0%D0%BB%D1%8C%D0%B4%D0%B8%D0%BA%D0%B0%20%D1%81%D0%BA%D0%B0%D1%87%D0%B0%D1%82%D1%8C%20%D0%B1%D0%B5%D1%81%D0%BF%D0%BB%D0%B0%D1%82%D0%BD%D0%BEgrsearchz Math ssearchw Learn asearch h Learn searchasearchlchampakonline.blogspot.com ic Group esearchsi Direct g Math y b Direct trsearchc Learn msearchd Math l Mathlearndirect Math f Group m Direct lsearchi Math i Mathlearndirect it Direct n Learn m Group e Math s searchssearchn Learn You bj Math c Math s Mathlearndirect rsearchpi Group t Learn rdecorte%2C%20e.%2C%20greer%20b.%20%26%20verschaffel%2C%20l.%20(2000).ssearch asearchdinsearchl Math dsearchssearchmn Direct isearchnoKannada+hudugi+kama+kategalu searchh Learn o Learn ekenya's%20math%20worksheets I tsearchin Direct m%20e45645ws Learn a Learn p Mathlearndirect o Group r You a You esearch-dmelike%20nikbayf Learn ersearchnt Math c Direct l You rsearchp Mathlearndirect ksearchr You chipbl%20calculus%20worksheetssearch-whzebrareeks%20cryptoc h Direct Kannada+hudugi+kama+kategaluosearchn Group s Direct u Direct tsearch searchesearchc Math n Learn esearcht%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA%20%D1%81%D0%B5%D1%80%D0%BF%D0%B8%D0%BD%D1%81%D0%BA%D0%BE%D0%B3%D0%BE%20javaalsearchy searchi Mathlearndirect ilr Group t Mathlearndirect searchgypi Math n Group h Math e Math ogysearchhsearchc Direct -- Learn n Mathlearndirect whic Math Direct d Learn fferent looking "marker" is used to represent tens. And he says "Using a different-looking ten marker may help some children --particularly those of low ability-- to bridge the gap between highly concrete size embodiments and the [next/last] relatively abstract model [involving relative position of markers]."

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

Further, Baroody labels all his categories as kinds of "trading", but he does not seem to recognize there is sometimes a difference between "trading" and "representing", and that trading is not abstract at all in the way that representing is. I can trade you my Mickey Mantle card for your Ted Kluzewski card or my tuna sandwich for your soft drink, but that does not mean Mickey Mantle cards represent Klu cards or that sandwiches represent soft drinks. Children in general, not just children with low ability, can understand trading without necessarily understanding representing. And they can go on from there to understand the kind of representing that does happen to be similar to trading, which is the kind of representing that place-value is. But with regard to trading, as opposed to representing, it is easier first to apprehend or appreciate (or remember, or pretend) there being a value difference between objects that are physically different, regardless of where they are, than it is to apprehend or appreciate a difference between two identical looking objects that are simply in different places. It makes sense to say that something can be of more or less value if it is physically changed, not just physically moved. Painting your car, bumping out the dents, or re-building the carburetor makes it worth more in some obvious way; parking it further up in your driveway does not. It makes sense to a child to say that two blue poker chips are worth 20 white ones; it makes less apparent sense to say a "2" over here is worth ten "2's" over here. Color poker chips teach the important abstract representational parts of columns in a way children can grasp far more readily. So why not use them and make it easier for all children to learn? And poker chips are relatively inexpensive classroom materials. By thinking of using different marker types (to represent different group values) primarily as an aid for students of "low ability", Baroody misses their potential for helping all children, including quite "bright" children, learn place-value earlier, more easily, and more effectively. (Return to text.)

Footnote 8. Remember, written versions of numbers are not the same thing as spoken versions. Written versions have to be learned as well as spoken versions; knowing spoken numbers does not teach written numbers. For example, numbers written in Roman numerals are pronounced the same as numbers in Arabic numerals. And numbers written in binary form are pronounced the same as the numbers they represent; they just are written differently, and look like different numbers. In binary math "110" is "six", not "one hundred ten". When children learn to read numbers, they sometimes make some mistakes like calling "11" "one-one", etc. Even adults, when faced with a large multi-column number, often have difficulty naming the number, though they might have no trouble manipulating the number for calculations; number names beyond the single digit numbers are not necessarily a help for thinking about or manipulating numbers.

Karen C. Fuson explains how the names of numbers from 10 through 99 in the Chinese language include what are essentially the column names (as do our whole-number multiples of 100), and she thinks that makes Chinese-speaking students able to learn place-value concepts more readily. But I believe that does not follow, since however the names of numbers are pronounced, the numeric designation of them is still a totally different thing from the written word designation; e.g., "1000" versus "one thousand". It should be just as difficult for a Chinese-speaking child to 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 Chinese child 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 than nine of a given place-value, you have to convert the "extra" into a higher place-value in order to write it (e.g., you can say "five one hundred's and 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 children to write "concatenated" columns --columns that contain multi-digit numbers-- when we teach them the borrowing algorithm of subtraction; we do write a "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 to write "five hundred twelve" as "512" instead of "50012", where the child writes down the "500" and puts the "12" on the end of it).

But there is, or should be, more involved. Even after Chinese-speaking children have learned to read numeric numbers, such as "215" as (the Chinese translation of) "2-one hundred, one-ten, five", that alone should not help them be able to subtract "56" from it any more easily than an English-speaking child can do it, because (1) one still has to translate the concepts of trading into columnar numeric notations, which is not especially easy, and because (2) one still has to understand how ones, tens, hundreds, etc. relate to each other so that one can trade between higher and lower column-name-designations; e.g., between thousands and hundreds or between millions and hundred thousands, etc. And although it may seem easy to subtract "five-ten" (50) from "six-ten" (60) to get "one-ten" (10), it is not generally difficult for people who have learned to count by tens to subtract "fifty" from "sixty" to get "ten". Nor is it difficult for English-speaking students who have practiced much with quantities and number names to subtract "forty-two" from "fifty-six" to get "fourteen". Surely it is not easier for a Chinese-speaking child to get "one-ten four" by subtracting "four-ten two" from "five-ten six". Algebra students often have a difficult time adding and subtracting mixed variables [e.g., "(10x + 3y) - (4x + y)"]; is it going to be easier for Chinese-speaking children to do something virtually identical? I suspect that if Chinese-speaking children understand place-value better than English-speaking children, there is more reason than the name designation of their numbers. And Fuson points out a number of things that Asian children learn to do that American children are generally not taught, from various methods of finger counting to practicing with pairs of numbers that add to ten or to whole number multiples of ten.

From a conceptual standpoint of the sort I am describing in this paper, it would seem that sort of practice is far more important for learning about relationships between numbers and between quantities than the way spoken numbers are named. There are all kinds of ways to practice using numbers and quantities; if few or none of them are used, children are not likely to learn math very well, regardless of how number words are constructed or pronounced or how numbers are written.  (Return to text.)

Footnote 9. Because children can learn to read numbers simply by repetition and practice, I maintain that reading and writing numbers has nothing necessarily to do with understanding place-value. I take "place-value" to be about how and why columns represent what they do and how they relate to each other, not just knowing what they are named. Some teachers and researchers, however (and Fuson may be one of them) seem to use the term "place-value" to include or be about the naming of written numbers, or the writing of named numbers. In this usage then, Fuson would be correct that --once children learn that written numbers have column names, and what the order of those column names is -- Chinese-speaking children would have an advantage in reading and writing numbers (that include any ten's and one's) that English-speaking children do not have. But as I pointed out earlier, I do not believe that advantage carries over into doing numerically written or numerically represented arithmetical manipulations, which is where place-value understanding comes in.

And I do not believe it is any sort of real advantage at all, since I believe that children can learn to read and write numbers from 1 to 100 fairly easily by rote, with practice, and they can do it more readily that way than they can do it by learning column names and numbers and how to put different digits together by columns in order to form the number.

When my children were learning to "count" out loud (i.e., merely recite number names in order) two things were difficult for them, one of which would be difficult for Chinese-speaking children also, I assume. They would forget to go to the next ten group after getting to nine in the previous group (and I assume that, if Chinese children learn to count to ten before they go on to "one-ten one", they probably sometimes will inadvertently count from, say, "six-ten nine to six-ten ten"). And, probably unlike Chinese children, for the reasons Fuson gives, my children had trouble remembering the names of the subsequent sets of tens or "decades". When they did remember that they had to change the decade name after a something-ty nine, they would forget what came next. But this was not that difficult to remedy by brief rehearsal periods of saying the decades (while driving in the car, during errands or commuting, usually) and then practicing going from twenty-nine to thirty, thirty-nine to forty, etc. separately.

Actually a third thing would also sometimes happen, and theoretically, it seems to me, it would probably happen more frequently to children learning to count in Chinese. When counting to 100 my children would occasionally skip a number without noticing or they would lose their concentration and forget where they were and maybe go from sixty six to seventy seven, or some such. I would think that if you were learning to count with the Chinese naming system, it would be fairly easy to go from something like six-ten three to four-ten seven if you have any lapse in concentration at all. It would be easy to confuse which "ten" and which "one" you had just said. If you try to count simple mixtures of two different kinds of objects at one time --in your head-- you will easily confuse which number is next for which object. Put different small numbers of blue and red poker chips in ten or fifteen piles, and then by going from one pile to the next just one time through, try to simultaneously count up all the blue ones and all the red ones (keeping the two sums distinguished). It is extremely difficult to do this without getting confused which sum you just had last for the blue ones and which you just had last for the red ones. In short, you lose track of which number goes with which name. I assume Chinese children would have this same difficulty learning to say the numbers in order.  (Return to text.)

Footnote 10. There is a difference between things that require sheer repetitive practice to "learn" and things that require understanding. The point of practice is to become better at avoiding mistakes, not better at recognizing or understanding them each time you make them. The point of repetitive practice is simply to get more adroit at doing something correctly. It does not necessarily have anything to do with understanding it better. It is about being able to do something faster, more smoothly, more automatically, more naturally, more skillfully, more perfectly, well or perfectly more often, etc. Some team fundamentals in sports may have obvious rationales; teams repetitively practice and drill on those fundamentals then, not in order to understand them better but to be able to do them better.

In math and science (and many other areas), understanding and practical application are sometimes separate things in the sense that one may understand multiplication, but that is different from being able to multiply smoothly and quickly. Many people can multiply without understanding multiplication very well because they have been taught an algorithm for multiplication that they have practiced repetitively. Others have learned to understand multiplication conceptually but have not practiced multiplying actual numbers enough to be able to effectively multiply without a calculator. Both understanding and practice are important in many aspects of math, but the practice and understanding are two different things, and often need to be "taught" or worked on separately.

Similarly, physicists or mathematicians may work with formulas they know by heart from practice and use, but they may have to think a bit and reconstruct a proof or rationale for those formulas if asked. Having understanding, or being able to have understanding, are often different from being able to state a proof or rationale from memory instantaneously. In some cases it may be important for someone not only to understand a subject but to memorize the steps of that understanding, or to practice or rehearse the "proof" or rationale or derivation also, so that he can recall the full, specific rationale at will. But not all cases are like that. (Return to text.)

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