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.
- -------------------------------------------------- From: "Alain Schremmer" <email@example.com> Sent: Friday, September 27, 2013 2:31 PM To: <firstname.lastname@example.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 **************************************************************************** * To post to the list: email email@example.com * * To unsubscribe, email the message "unsubscribe mathedcc" to firstname.lastname@example.org * * Archives at http://mathforum.org/kb/forum.jspa?forumID=184 * ****************************************************************************