The history and traditions of American curricular "education" in "mathematics" are often used for attempting to excuse the "teaching", to children, of "mathematics" before it possibly can make mathematical common sense to them. That practice is contrary to the nature of mathematical thinking ... and to how children *naturally* learn mathematics. It also is unhealthy ... being largely responsible for the nationwide epidemic of MLD (Math-Learning Distress: anxiety, alienation, depression, phobia, etc) ... and for the majority of adults being dysfunctional at the level of HS algebra.
The statement that "Often, in pedagogy, just the mechanics is mostly the goal, for the time being" actually is a serious condemnation of the ranks of teacher-educators ... who never bothered to clinically study how humans naturally learn rational mathematics ... and why they have so much difficulty with it. Tradition is no excuse for ignorance-based malpractice.
Re RH's: "The blunder of these researchers was saying that the students ?understood? place value, not that students could identify that numbers have places, labeled one?s, ten?s, hundred?s etc. Sure, it?s not mathematical, it?s syntactical, but still required, before you get to the mathematical meaning of it."
In "the whole journey", the focus on successive places actually is quite mathematical ... and is the basis of all considerations of sequences and their series. The users choose to attach specific *denominations* (names: from de-nomen-ate) to specific places according to the needs at hand ... as with (milk, bread, coffee) and with (year/month/day/ hour: minute: sec) and with ($ilvers, halves, quarters, dimes, nickels, cents, mils) and with countless kinds of inventory systems. The semantics and phonics of such denominations do not automatically entail any particular syntactic combinations. Neither do they automatically entail any "place values" ... i.e. numeric values for the *units* that identify with those places. [cf. the place-values posted on the shelves in the grocery stores.]
In the special context of simple Arabic numerals ... i.e. the ten single-digit ones and those multi-digit ones that start with non-0 entries ... those are (ordinal-) alphabetically ordered. That is the basic "mathematical meaning" of the Arabic numerals. Despite the widespread mis-education of American educators (and their committees), that basic meaning in no way depends on the places having numeric values. Rather, the "values" of the units are logical consequences of the alphabetic ordering.
In the English phonics for the first thousand Arabic numerals, all multi-place numerals are read as combinations of *quantities* ... each being the combination of a digit/number and a spoken denomination ... except that the denomination for the first-place digit is normally S(ilent). But the ability to pronounce 2345 in English does not imply, and does not depend on, owning numeric values for the denomination-units. However, the cardinal meanings of those terms have misled educators to think that students are/should use numeric meanings for those words, even long before children own own those numbers.
The more natural "journey" is for the very young to begin with non-numeric meanings for the first four Arabic denominations ... perhaps first as color-coded poker chips, and later as play-money. In the MALEI Clinic, we often use some of the standard bridge-deck of playing cards ... along with the letters Y, H, T, and S. Thus, by using card-suits for place-denominations, the "2345" succession is read "2 thousand, 3 hundred, 4 tee, 5" ... and read/written as "2Y + 4H + 4T + 5S." ["Y" comes from the Old English "th", as in "Yousand" ... H(undreds), T(ees), and S(ingles).]
Even though the denominations have no numerical meanings, such material, phonic, and written "combos" are very "mathematical." In fact, the general theory of vector algebra is all about such "linear" combinations ... which are *not* required to have numeric values for their denominations.
Very young children easily learn the place-denomination phonics for line, 0-though-9999, without having numeric meanings for the four places/units. The meanings are very mathematical, but the only "syntactic" connection is that, when placed as successive digits, the place-quantities are read in sequence. Childhood readiness to attach numeric values to denominations comes at about age 7.
Until children are fluent in making change with genuine coins, any earlier "instruction" about place *values* is contra-productive ... especially if teachers are administratively "compelled" to indulge in that kind of malpractice.
-------------------------------------------------- From: "GS Chandy" <firstname.lastname@example.org> Sent: Sunday, December 29, 2013 8:55 AM To: <email@example.com> Subject: [SPAM]Re: Kids understand multi-digit numbers as early as age 3 ??? Researchers' blunder.
> Robert Hansen (RH) posted Dec 25, 2013 3:32 AM > (http://mathforum.org/kb/message.jspa?messageID=9349936) - GSC's remarks > follow: >> >> On Dec 24, 2013, at 1:27 PM, Clyde Greeno @ MALEI >> <firstname.lastname@example.org> wrote: >> >> > It is bad enough that the Achieve Corporation's >> mathematics "standards" for American schools start >> t the teaching of "place values" at primary-school >> levels ... so prompting under-informed schools to >> inundate the very young with mathematical meanderings >> that are beyond their abilities to comprehend. >> >> Whoa now. There is nothing wrong in talking to these >> things during the normal course of school work. It is >> an absolutely necessary activity. Remember, these >> notions grow on you and they don?t start as notions >> because they can?t start as notions without any >> substance to base them on. Wayne once pointed out >> something that stuck with me. And I will paraphrase: >> Often, in pedagogy, just the mechanics is mostly the >> goal, for the time being. The blunder of these >> researchers was saying that the students ?understood? >> place value, not that students could identify that >> numbers have places, labeled one?s, ten?s, hundred?s >> etc. Sure, it?s not mathematical, it?s syntactical, >> but still required, before you get to the >> mathematical meaning of it. >> >> Study the whole journey Clyde. Not just one moments >> worth. >> >> Bob Hansen >> > Whoa, now, RH. Try and study what the person is actually saying/claiming. > Not just what you wrongly think he is doing. > > GSC > ("Still Shoveling! Not PUSHING!! Not GOADING!!!") >