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Topic: back to fractions - 9/13
Replies: 4   Last Post: Oct 1, 2013 3:10 PM

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Clyde Greeno @ MALEI

Posts: 220
Registered: 9/13/10
back to fractions - 9/13
Posted: Sep 28, 2013 10:08 PM
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So, we continue a years-long recurrent dialog about fractions, but from an
alternative angle. That's good ...especially since the "CCSS" version of
fractions is no better than traditional versions and badly needs to be
reconsidered!.
.
Re: Schremmer's response to 3(9ths): "That is the worst way to introduce
fractions. One of the dangers being that, sooner or later, 3/9 gets
confused with "3 out of 9"

"Worst?" ... and there is something wrong with "3 out of 9" [exactly as he
does, below]?
His claim cannot be judged without knowing what it actually says.

I suspect that Schremmer's judgment of "worst" is based on the fact that the
traditional "ordered pair" development ... number-over-number numerals for
fractions ... has been a disaster. But the fraction vocabularies that our
cultures use must be accommodated ... by some kind of conceptual
under-standing for meaningfully interpreting the traditional symbols. We
need to bridge from whatever students do understand, to the "weird" numerals
for fractions.

I cannot believe that he thinks that the quantity-scales for fractions ...
as with 0(9ths), 1(9th), 2(9ths), ... , 345(9ths), .... are "the worst."
There is strong clinical evidence (including many video tapes) that creation
of such fraction-scales is very natural for children and adults ... so long
as the "n/d" symbols are not used until *after* the conceptual framework is
functional. [Even in writing, the quantity-language matches the phonics, "3
ninths" ... as "3(9ths)", not as 3-over-9.] But I have yet to see a
curricular presentation that introduces fractions, well, as quantities. The
usual "cartwheel" and "slab" and "bar" diagrams are rarely, if ever, used
as solid footings for formulating the quantity-scales.

I will VERY interested in clinically learning if his quotient-development of
fractions also is humanly natural.

As for "3 out of 9", there are many kinds of things that can be
common-sensibly *fractioned* (a version of factoring), including the
fractioning of some clusters of things. Any confusion of the kind he claims
most likely is due to curricular failures to milk the fractions (as
quantities or as numbers), out from the material entities that they
describe.

His use of the ordered-pairs as quotients from base-ten division of 3 by 9
... 3/9 = 3 DeciDivBy 9 ... appears to call for children engage in repeated
decimation of remainders. His preference to "deal immediately with
division---which is something children have no problem with and actually
love" does not seem consistent with the traditional difficulties with
curricular long-division. So we can count on him to disclose that in "the
proper context, i.e. Dienes multiarithmetic blocks, children have no
problem with" long division?.

To be sure, his defining "3/9" to be the result from doing base-ten division
of 3 by 9 ... does circumvent the need to conclude that 3(9ths) is the
result from 3(wholes) DivBy 9 ... and to conclude that the base-ten division
algorithm actually works. [Is that like conveniently "defining" the product
of two negatives to be a positive ? ... easy to teach, even if not learned
as mathematical common sense?]

Do students actually believe that "3/9" means "DeciDiv3 by 9" ... or are
they being dictated to rely on that definition? Does it mean that all
operations with fractions must be done "calculator wise" as operations with
decimal-point numerals?

And his "In fact, "3/9 apples" should be read as the number-phrase "3 of
which
it takes 9 to exchange for one apple" where "3" is the numerator and
"of which it takes 9 to exchange for one apple" is the denominator" ...
sounds to me like "3 out of 9 , each of which equates with 1(9th) of an
apple." ??? Is it being supposed that students might connect such a
descriptor with the process of 3 DeciDivBy 9? ... with the decimal-point
quotient equating with 3(9ths-of-apples)? A bit complicated for early school
children?

Let me try to grasp that scenario. Given apples, A, and given somethings
(S) for which 9S~1A; then 3/9 A ~ 3S ... 3 being the numerator of 3S, and S
being the "denominator" (denomination?) of 3S. The A-scale and the S-scale
thus are being matched ... much like English and metric length-scales are
matched. I fail to perceive how the supposition of somethings (S) "of which
it takes 9 to exchange for one apple" is in any way advantageous over
cutting the apple into 9ths.

[For the deci-division: cut each of 3 apples into 10ths, get 30 tenths,
divide by 9, get-3tenths as a place-qotient, and 3tenths remainder ...
decimate those into 30hundredths, divide by 9, etc. Even knowing how to do
that process does not establish that the deci-division process gives the
actual quotient for dividing 3 by 9 ... even if "3/9" is so-defined as
meaning 3 DeciDivvyBy 9.]
&&&&&&&&&&

Re: his response to the alphabetics of decimal-points, "... I think that
*that* is not natural. "
Seemingly quite "unnatural" ... until you think about it for a while,
outside the realm of arithmetic. Alphabetics is one place where the content
of arithmetic overlaps the language/library arts. Actually, that is how
all persons who understand the vocabulary of decimal points do compare
their "fraction parts" ... lexicographically, just as in the dictionary ...
*independent of length*. Because the [length-dependent] alphabetics of the
simple Arabic numerals is somewhat different, students have undue difficulty
with ordering the "fraction parts." A quick survey of alphabetics ...
*without* concern for numeric meanings [e.g. by use of text characters or
various base-number vocabularies] can circumvent much confusion about the
various lengths of the decimal-point numerals.
&&&&

Re: His comment about the 13W(ashingtons), "Yes, there is something very
wrong: you were not able to *represent* this "real world collection" in the
"paper world" without involving the exchange of ten one-dollar-bills for one
ten-dollar bill. "

?? Since we *can* write it by "13W", we must presume that he meant something
else ... entailing conversion of the ("decimal phrase"?) quantity, 13W , to
the equi-valued 1H+3W combination. Perhaps he meant, "... you were not able
to represent [13W] by an Arabic numeral without ...." He might thus have
been suggesting that, in writing, "13" can be construed *only* as meaning
the combination 1T+3S (1ten+3singles).

That assumption has long prevailed in curricular mathematics. It is an
*adult*, vector-analytic, interpretation of the Arabic numerals ... a bit
like locating a point of a plane by imposing a coordinate system to locate
that point as a pair of components.

For children, the more natural meaning for "13" is simply "what comes next
after "12" ... which is next after ...." Very naturally, they
*alphabetically* construct the line of Arabic numerals, long before they can
make sense of "place values." By recognizing *places*, and by learning the
denominations (names) for those places, children readily learn to pronounce
the first 10,000 Arabic numerals ... and to chant short successions, while
knowing very little about number-ness, much less about "place values."

Because regarding "13" as meaning 1T+3S is NOT a natural interpretation of
"13" by early-age children, that "combination" myth has created a major bog
for generations of teachers and children in the early grades. It also is
largely responsible for mistaken traditional beliefs that young children
must understand "place values" ... long before they are developmentally
mature enough to do so.

That myth ... that "13" can *only* mean such a combo ... probably comes
from the phonics. The number, 345, is normally pronounced as a combo:
3H(undred),4T,5 ... meaning 3H+4T+5S, a vector-analysis of 345. But even
that pronunciation does not require knowing that the numeric value of the
third place is 1H; that the numeric value of the second place is 1T and that
the numeric value of the first place is 1S. Even adults talk and use the
alphabetics, places, denominations, and conversion factors ... but use
"place values" mostly with measurement systems (such as digital clocks).

Schremmer points out (without using my words) that, by using material
objects, very young children easily learn to reduce/carry "improper"
quantities (13W) to "proper" combos (1H+3W). In fact, "piggy bank"
mathematics is perhaps the best "lab" medium for comprehending the
mathematical nature the "borrowing" and "carrying" conversions ... *prior*
to instruction in the operations. [In the base-ten equivalence classes, the
Arabic numerals are the *proper* base-ten numerals. As often seen in
subtraction or in short-division, an *improper* case has at least one entry
that is higher than 9. cf
http://sections.maa.org/okar/papers/2005/greeno1.pdf ]

For terminology: "borrowing" and "carrying" can be accepted from the
traditional curriculum. Legend has it that the words came from the Arabian
traders' use of column-grooved boards ("counters") and stones ("calculi")
which got moved around in the grooves. In today's monetary cultures,
"making" and "breaking" coins, bills, etc. are more appropriate terms.

The role of "carrying" is to *reduce* the total number of components in the
combo: 13W has 13 parts; 1H+3W has only 4 parts. [Likewise, 3(ninths) has 3
parts, but reduces to 1(3rd), which has only 1 part.] The term "improper"
also is accepted from the traditional curriculum with fractions. A "proper"
combination is one that has the least number of parts, for the same value.
>From the cashier, 3$ + 47pennies is not proper change (50 parts); proper
change is 3$+1Q+2D+2C (8 parts).

I confess to using some terms in ways that are not commonly done in
traditional curricular literature. Far more important to use a language
that is maximally effective with students.

Cordially,
Clyde

- --------------------------------------------------
From: "Alain Schremmer" <schremmer.alain@gmail.com>
Sent: Friday, September 27, 2013 2:31 PM
To: <mathedcc@mathforum.org>
Subject: [SPAM]Re: [SPAM]Re: percents

>
> On Sep 27, 2013, at 2:04 AM, Clyde Greeno @ MALEI wrote:
>

>> As "code" (a numeral), "3/9" normally is first learned as the quantity,
>> 3(9ths) ... numerator, 3, denominaTION, 9ths ... as in "pie slices."

>
> That is the worst way to introduce fractions. One of the dangers being
> that, sooner or later, 3/9 gets confused with "3 out of 9".
>

>> In that evolution, it can be quite a jump to recognize the theorem
>> that 3(9ths) is the quotient from dividing 3(wholes) by 9.

>
> I agree. So, the obvious conclusion is: avoid that "evolution" and deal
> immediately with division---which is something children have no problem
> with and actually love.
>

>> Another underlying theorem is that the quotient from 3 divided by 9 can
>> be got by carrying out the "long/short division" process.

>
> That is exactly what I was talking about: 3/9 is just code for "carrying
> out the "long/short division" [of 9 into 3] process". And, again, in the
> proper context, i.e. Dienes multiarithmetic blocks, children have no
> problem with it.
>

>> We can bully students into believing it, or guide them to conclude it.
>> Only by putting those two under-standing theorems together can one
>> conclude that "3/9, say, is just code, an instruction to divide 9 into
>> 3."

>
> Absolutely not: just forgo the pizza image.
>

>> Apart from calculations, there is something to be said for approaching
>> the decimal-point symbols, alphabetically (or "library- wise") ... as is
>> done also with the construction of scaled tapes/ rulers. Begin with the
>> "primary school" scale, [0,1,2,3,4,...] Then insert the 1-decimal-place
>> "codes" , [0, 0.1, 0.2. ... 0.9, 1, 1.1, ... ] ... then the
>> 2-decimal-place codes, etc.
>>
>> The alphabetized family of all such (finite) decimal-points is dense.
>> Allowing also the infinite ones yields a continuum. All of that can
>> easily be done *without* regarding the decimal-point codes as
>> representing "numbers." The cognitive gain is that students thus can
>> perceive the alphabetic ordering of those points *prior* to development
>> of the decimal numbers. [A one-time viewing of a simple video should
>> suffice.]

>
> I think that *that* is not natural even though one could probably
> introduce it as a "formal game" (Disclosure: I am a card-carrying
> Platonist.)
>

>> As for 13 one-$ bills, there is nothing "wrong" with 13W(ashingtons) ...
>> other than it is an unnecessarily cumbersome stack of bills.

>
> Yes, there is something very wrong: you were not able to *represent* this
> "real world collection" in the "paper world" without involving the
> exchange of ten one-dollar-bills for one ten-dollar bill. And this is,
> after all, what all this is about.
>

>> That is why it is an *improper* quantity ... whose *proper combination*
>> is 1H(amilton)+3W. The 13W --> 1H+3W

>
> I don't see what is "improper": most collections cannot be represented
> without involving exchanges as mentioned above.


> "carrying" conversion of the decimal-currency vectors is far from
> "meaningless."


> Of course. My point was only that the terms "carrying" and "borrowing"
themselves are utterly misleading. (Who invented them?)

>Note: what Greeno is calling "the decimal-currency vector" is
different from what I called "decimal number phrase" in my previous
post but it is the same as what I called "combination (aka vector)".
In fact, though, I finally decided, for a variety of reasons, to call
it a "list".

> Just like conversions among equivalent fractions,
> carrying and borrowing among equivalent money-vectors are fundamental
> "reduction" operations of vector arithmetic.


>In fact, "3/9 apples" should be read as the number-phrase "3 of which
it takes 9 to exchange for one apple" where "3" is the numerator and
"of which it takes 9 to exchange for one apple" is the denominator.

>Finally, here is the way to implement with children what I described
in my previous post: Use Dienes' multiarithmetic blocks and suggest
the following game: represent various bunches of blocks for the
purpose of communication. In base TEN, we can represent three blocks
of the same shape by a number-phrase but not thirteen blocks (of the
same shape). For the latter, an extra process is necessary: exchanging
ten of the thirteen blocks for one of the next higher denomination. I
had no trouble with ten-year olds and was about to try with much
younger children when I left France for the US. But, I did try again,
in Philadelphia, with third graders, lowest track, and it went again
like a charm.

Regards
- --schremmer
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