Ummm...don't look now but your article contradticts itself. How is deep conceptual understanding "indefinable and immesaurable" when its put foward as an objective when researchers arguing in support of an algorithmic approach can provide research saying their way "develops conceptual understanding".
Fostering conceptual understanding of the material absolutely should be a goal. Arithmetic mistakes will happen no matter how much drilling you do. Conceptual understanding increases the chance that a ridiculous answer can be recognized and discarded, while at the same time fostering reasoning skills sorely missing when a student moves on to advanced mathematics.
I've noticed that a dead give-away to a smug, sneering, pedantic post is when one begins with "Ummm..."
And, another perception that I have had, all of those who use "dr." in their avatar are sorely in need of one.
But enough chit-chat.
This has been quite a long thread, and everyone has expressed his/her viewpoint.
But- (not 'ummm') the 800 lb. gorilla in the room, or in this case, in the argument, is the fact that scores have fallen precipitously since liberals, 'progressives,' have taken over the school system.
"Americans have been dissatisfied with their public schools for several decades. There are
many reasons for this. It is well known, for instance, that the achievement of American students,
when compared with that of students from other industrialized nations, is consistently near the
bottom (166-167).1 Employers complain that many high school graduates lack even the most
basic skills in reading, writing and mathematics. Colleges and universities must offer remedial
work for large numbers of incoming students before they are prepared to do college level work."
http://www.macalester.edu/~reedy/h2.pdf
So, (not 'ummm") aside from the minutiae, how do we account for the seemingly related factors of the installation of progressive theory, and falling grades?
Coincidence? Artifact of statistics?
How many decades are necessary before we scrap these new and trendy theories, and return to those that have actually shown results?
I'd agree to mitigate some of the criticism of progressive education, by stipulating that the progressive direction in society is even more to blame.