There was an excellent article in the New York Times a on June 16 called The Faulty Logic of the ‘Math Wars’. It is a must-read for anybody who even cares about what happens in public education in this country.
Having been a 32-year math teacher myself, I can attest to the fact that there have been massive changes in the way mathematics is intended to be taught. I say "intended" because there has always been a hope, on my part, that teachers can handle the new wave and still manage to sneak in some real and true math instruction. Unfortunately, as time passes, more and more of our teachers will have been products of the new era, almost to the point that pencil-and-paper calculations become anachronistic, and algorithmic thinking becomes a rarity.
Failing to teach students a good dose of algorithmic skills, especially in arithmetic, forces them to fall into one, or possibly both, of two pits: that of calculator dependency and that of failure to achieve progress because they have to constantly reinvent everything that was discovered before.
Alfred North Whitehead once said "Civilization advances by extending the number of important operations which we can perform without thinking of them." I could change that and add that basic human life operates the same way. Do you believe that a good driver thinks his/her way through a road trip? Or has the process been, in essence, "built in" to the brain? Now imagine how that process was embedded, and you have little picture of how people acquire competences.
Thinking and reasoning are massively important in life, but people should not need to rely on them continuously and constantly. There should be a significant portion of our life's activities that we can do competently and successfully without having to think and re-think every action.
Musicians who are successful have taken complete unknowns and turned them into second nature habits. Especially those who read music. The brain power used to develop the skills that a classical pianist exhibits is the same brain power used developing a world-class mathematician. Leibniz once said "Music is the pleasure the human soul experiences from counting without being aware that it is counting.” A case could be made that the deaf Beethoven wrote his works using mathematical thinking.
I truly wish our educational system would look at how people like Beethoven, Whitehead, and Leibniz succeeded, and adjust our schools accordingly. Instead they listen to the business world, if anybody.
Much more on this to come....