At 08:38 AM 7/2/2014, Louis Talman wrote: >I was certainly so trained. I was taught to identify mathematics with the >sort of arithmetic that RH boasts of being able to teach: Learn the >algorithms and perform them without any reasoning.
I don't know how greatly overblown this is but those algorithms were NOT taught that way in my one-room country school although, I'm sure, sometimes "learned" that way and my teacher was prepared by one college year (Normal School). As examples, borrowing and carrying (by any math ed "expert's" name) were explained - probably repeatedly - - and understood completely. Shifting the partial product with each digit in multiplication by a multidigit number ("numeral" but let's not be picky) was certainly explained repeatedly and why writing all those 0's would be possible but redundant, just keep things halfway straight. In division of one natural number by another with remainder, it was pointed out that - and why - this is simply repeated subtraction done as efficiently as possible within the (historically critical) opportunity of positional arithmetic and how the remainder could be written as an ordinary fraction and why. (The immediate answer being correct by the distributive law - and multiple applications of critically important but less transparent others - was omitted and, I believe, rightly so.) Extending to decimal fraction arithmetic, not only were we taught to insert little carets in parallel fashion, we were taught why this was simply converting all we had learned about the (yes THE) division algorithm (as opposed to the strikingly misnamed mathematical one that isn't even an algorithm) for natural numbers simply by multiplying numerator and denominator of the obviously equivalent fraction by the appropriate power of 10 so that the denominator=divisor becomes a natural number. Similar remarks could he made for ordinary fraction arithmetic but I won't bother.
Part of my eternity in Hell will be appropriate reward for helping prospective teachers of this material who don't - and never will - understand these algorithms even to that level of sophistication to become certified as teachers, a few in Michigan and hundreds (more than 1000?) in California. I'll be further back in line than those who just give the kids calculators and tell them - and their more apprehensive; i.e., less-gullible parents - that in our sophisticated era, those old-fashioned algorithms are unnecessary and just a relic of the primitive past.
> (Could this be why so many students can't do word problems? Do > they expect there to be a bit reasoning-free algorithm for each problem?)
No it couldn't but it does make a nice fairytale. The primary reason is the growth of the education industry - as an independent entity - as opposed to a service organization to its component academic areas of responsibility such as mathematics, science, language arts, and history/political "science".
Once upon a time, math books had lots more word problems, mine certainly did in elementary school and I still have the Algebra I book I taught from in high school 50 years ago. Much smaller, much more focused, and lots of word problems including little algebraic (analytic=coordinate) geometry proofs and problems in anticipation of a more formal synthetic geometry treatment (with occasional embedded analytic geometry) the following year. For example, an algebraic proof of the Pythagorean Theorem was included in the text discussion with another form for students to follow the lead in the exercises. Having decision-makers (teacher preparation, textbook selection, new faculty selection, etc.) made by "education experts" have managed to wreck a quality system. Most of my courses were mathematics in the mathematics department, the lab school was closely tied to and informed by the mathematics department (my geometry guy held a joint appointment and was happily anticipating each (then) new volume of Bourbaki). On graduation, I was hired entirely within the high school department of mathematics and the meeting with the principal and associate superintendent entirely pro forma (to explain benefits and the like), we chose our textbooks, and made our own decisions. We decided democratically that we wanted a departmental final for Algebra I which we wrote (none of this SBAC or PARCC "authentic" nonsense). As committee chair (low man on the totem pole?), my biggest problem was denying each faculty member his (or a few her) favorite word problems and keeping the exam to be reasonably meaningful instead of a test to determine the next math genius.
>Geometry partially opened my eyes,
Mine, too, but, in my case, completely. My instructor was good enough to recognize the limitations of the ancient Greeks and, even more important, his own recognition that a proof he presented was not magic because he had presented at or the author had or the solutions manual had and that a clever student in the class such as myself could well improve on any of them and was happy to see such and share it with the class. (A home run or touchdown pass would've felt better but that wasn't gonna happen.)
>but I had a teacher who did her very best to keep that from >happening by insisting that the only way to approach the subject was >to memorize the proofs of the theorems in the textbook; we were not >allowed to give our own proofs of those theorems.
That this bad as it gets and, in that situation, it is probably better to do nothing than to stifle the creativity undergirded by (appropriately compromised) formal deductive logic. That was the point at which I decided to become high school mathematics teacher because it was as far past the end of the corn row as my limited horizon could see. For me, personally, the axiomatic undergirding all of formal mathematics had yet to come and I was convinced and, in retrospect, accurately that the formality of the New Math most especially, the Illinois project, were too much too soon both for students and for faculty. That was exactly why the industry God, Polya, vociferously refused the invitation - overt begging is closer - - to participate when Ed Beagle was holding his seminal SMSG conference at Stanford.
My impression was always that SMSG was pretty close to getting it right from the perspective of strong students under the leadership of strong teachers but I thought it was beyond most teachers of mathematics and now they're much worse. That said and in spite of strong NCTM support, I knew that it could not work at the suburban high school just outside of Chicago where I was teaching high school. Regrettably, the New New Math has taken the mathematics out along with any vestige of formal deductive logic.
Calculus didn't do it for me, it was too easy and drinking and dating were so much more pleasant that I didn't get back to formalities until grad school but I have never lost my conviction that at least a semester of well-taught (appropriately compromised)deductive logic in the context of Euclidean geometry is a great place to start.
>It wasn't until I got to calculus that I began to understand that I'd been >sold a bill of goods, and that only about five percent of what my teachers >had called "mathematics" was really that. > >- --Louis A. Talman > Department of Mathematical and Computer Sciences > Metropolitan State University of Denver > > <http://rowdy.msudenver.edu/~talmanl>