Math Learn Direct Math Learn Direct F%20math%20worksheets Math Learn Direct
Math Learn Direct Save Mathlearndirect E E Math Learn Direct F Learn S S Natural Math Learn Direct Szh Datasheet Index Mvp Id 3695 Math Learn Direct The Concept and Teaching of Place-Value in Math
Math Learn Direct Save Mathlearndirect E E Math Learn Direct F Learn S S Natural Math Learn Direct Szh Datasheet Index Mvp Id 3695 Math Learn Direct
)
Index o
t Index el Datasheet s Learn i Mvp c Math % Datasheet 0 Save e Math e Learn e Save % Mvp 0 Mathlearndirect msearch Index s Save a Math c Mathlearndirect e Math N Direct t
Szh D Mathlearndirect r Math c Learn search searchesearchrdecorte%2C%20e.%2C%20greer%20b.%20%26%20verschaffel%2C%20l.%20(2000). Datasheet 
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.
mMath Learn Direct Save Mathlearndirect E E Math Learn Direct F Learn S S Natural Math Learn Direct Szh Datasheet Index Mvp Id 3695 Math Learn Direct The Concept and Teaching of Place-Value in Mathy s n Kid x x %E0%B8%9A%E0%B8%B4%E0%B8%87%E0%B9%82%E0%B8%81%E0%B8%A3%E0%B8%B2%E0%B8%8A%E0%B8%B2%E0%B8%A8%E0%B8%B1%E0%B8%9E%E0%B8%97%E0%B9%8C Irrational
wMath Learn Direct Save Mathlearndirect E E Math Learn Direct F Learn S S Natural Math Learn Direct Szh Datasheet Index Mvp Id 3695 Math Learn Direct The Concept and Teaching of Place-Value in Mathz q Learn Math Learn Direct Learn