I would add that there was also an esthetic dimension: it seemed to us that the construction was so beautiful that, per force, it could not but be optimal for learning purposes. We had forgotten how haphazard the learning process is.
Best regards --schremmer
On Sep 16, 2011, at 1:20 AM, Clyde Greeno @ MALEI wrote:
> Examined Dolciani's books? I have all too often *taught* from her > books > (because those were owned by schools in which I taught) ... and I > later > conducted teacher-institutes for in-service teachers who were > finding her > books to be inadequate. > > Your challenge to my claim is actually about semantics -- our > differing > meanings of "abstract" ... and how my meaning differs from > "formalism". The > confusion is widespread, and so crucial to the teaching of > mathematics that > it is worth re-viewing, here. > > But let's not confuse the dimensions of abstractness and formalism > with the > dimension of appropriateness (or not) of the topics attended. The > book and > topics that you cited were written for a small minority of pre- > college, > pre-STEM majors who traditionally have had no education in the > mathematical > logic which traditionally permeates the calculus curriculum. SMSG was > committed to education in mathematical proofs ... which they > thought was how > students learned "the structure" of mathematics. That thrust did > not greatly > please many of the STE students, and no one would argue that all > high school > students should take such a course. The back-to-basics movement later > curtailed the high school teaching of mathematical logic. > > SMSG books were Sputnik-motivated and were written for pre-STEM > students. > The collateral damage came from the nationwide use of SMSG [pre-STEM] > algebra with the other 95% of American students ... enabled by MD, > because > nothing better was available. It is really unfair to condemn a > minority-oriented book for not well serving everybody else. But > apart from > MD's olden choice of topics, the dilemma of "abstractness" still > persists. > > You can read a book to a puppy dog ... and it might get messages > about you, > about your love, the about togetherness, about the smell of book, > etc. It > will not get whatever message the author put into that book. The pup's > abilities for abstracting are too low for it to ever learn to > decode the > wording. So, we do not try to teach SMSG algebra to puppy dogs. > > Send to me a poem written in Chinese, and I cannot interpret > it ... NOT > because Chinese is "too abstract" ... if Chinese pre-schoolers can > learn it, > I probably could ... but because I presently have no meanings for the > symbols ... so could not even begin to perceive the syntax. Until I > overcome the translation problem, the possible abstractness of the > poet's > ideas is irrelevant. That metaphor is not unlike the plight of > students who > could not comprehend the "mathe-canto" that SMSG brought into American > algebra-education. The foreignness of the instructional language > precludes > any considerations of possible abstractness of the contents. > > The ESL population is even more illustrative. When one has *some* > meanings > for *some* words, *some* messages might be interpreted in *some* > ways ... > but often the interpretation is not at all what the message was > intended to > say. [No need to dwell on the human strife caused by that syndrome!] > > So it was with MD and SMSG ... whose math-as-a-second-language > students (and > most of their teachers) simply were not prepared to make > mathematical sense > of such formalisms. Students usually can get superficial meanings that > enable them to parrot, but without conceptual understanding, they > cannot > *mathematically comprehend* what they are doing. > > It is NOT because the mathematics of which MD spoke was "too > abstract", but > because it was spoken in a "foreign" language for which students > had, at > best, very superficial meanings. There is no mathematical concept > or fact > within any of MD's books that is "too abstract" for students who > are led to > internally construct those abstracts. But they cannot glean such > abstracts > from the language of "mathe-canto" without already owning an adequate > language base. What "makes sense" to the mathematically advanced > often is > sheer non-sense to the rest. > > That is the major weakness of didactic instruction in > mathematics ... it > works well *only* when the presenter and the audience share the > language > that is used. Skillful teachers can pull it off ... but the > textbook-followers typically rely on parrot-training. Non-didactic > instruction often guides the learners around many of the language > barriers. > > In psychomathematical perspective, an "abstract" is a mental > picture which > portrays one or more of its generator-things that the abstract > fits. Once > the abstract is generated, *all* things that fit it are its > "examples". > Some of those examples constitute the abstractor's own *meaning* of > that > abstract. > > Abstracts are *formulated* when they are expressed in a language. One > purpose of doing so is to enable transmitters to convey those > meanings to > receivers. But the process relies on the transmitter and receiver > being > fluent in the same language ... so that the receiver's > *interpretation* of > that formulation can be a relatively accurate *facsimile* of whatever > meaning the transmitter expressed. > > All normal humans acquire their languages through naturally > abstracting from > whatever "examples" they encounter. Until they do so, as infants, they > cannot communicate through use of any kind of language. > > But that encoding -->decoding process does not work well unless the > receiver already owns enough information to be able to glean the > transmitter's meaning from the "faxed" formulation. Otherwise, a > formulation > that expresses an "abstract" within the transmitter cannot likewise > function > as an abstract within the receiver. Instead, the formulation is > received as > being at least partially gibberish ... leaving the receiver to get > only the > most it can, from whatever it can surmise. > > That is what happed with MD & SMSG. Books that were "totally > sensible" to > their authors did not have the same meanings within most of their > user-teachers ... and far less meaning for the students. What were > abstracts > within the authors were not actually taught as being abstracts ... > the books > failed to actually construct the abstracts. So what actually got > book-taught > was only the formalisms ... not as expressing abstracts, but as > formalisms > that admit to data-processing. > > The MD &SMSG books were not "too abstract" for students. But by > failing to > develop the abstracts within the students, their algebra books > were, as most > still are, largely formalistic non-sense to the students, > themselves. It is > up to the teacher to fill in the gaps. [Do "imaginary" and "real" > numbers > ... and the quadratic formula ... have any substantive meaning for > today's > students in introductory algebra? ] > > Cordially, > Clyde
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