Elkies addressed how much a piece of music needs to change before it is a different piece, rather than a variation on the original. He also talked about how much information remains when the redundancy of repeated themes in a piece is accounted for.
.. But his lecture ended with an impressive performance, from memory, of a piece made from a baroque-style repeating arpeggiation where the root of the chord changed from measure to measure based not on a conventional harmonic progression, but on the digits of π. The result was an intriguingly disorienting congress of order with randomness, evoking something like an inebriated Buxtehude.
.. Tymoczko’s objective, data-driven approach to musical practice seems to have the power to reveal other significant insights. His love and deep understanding of music was evident in the joy with which he shared his discovery of hidden canons (or rounds, as in “Row, Row, Row Your Boat”) in pieces by Bach and Marenzio.
.. As the Pythagoreans discovered, harmonious intervals in music are formed from frequencies that are the ratios of small whole numbers.
.. The highlight here was a recording of a barbershop quartet in which Wright discovered a subtle use of microtonics during a chord change. In shifting from one chord to another, three of the singers changed notes, while the fourth held on to his note across the chord change—or should have, based on the notes on paper. In practice, the sustained note shifted frequency by a small fraction of a semitone to maintain just intervals with the notes of the second chord.
These types of microtonal manipulations are part of what give barbershop quartet performances their intensely harmonious quality.
In fact, what’s needed is a different kind of proficiency, one that is hardly taught at all. The Mathematical Association of America calls it “quantitative literacy.” I prefer the O.E.C.D.’s “numeracy,” suggesting an affinity with reading and writing.
.. Calculus and higher math have a place, of course, but it’s not in most people’s everyday lives. What citizens do need is to be comfortable reading graphs and charts and adept at calculating simple figures in their heads.
.. I thought they would focus on what could be called “citizen statistics.” By this I mean coping with the numbers that suffuse our personal and public lives — like figures cited on income distribution, climate change or whether cellphones can damage your brain. What’s needed is a facility for sensing symptoms of bias, questionable samples and dubious sources of data.
.. The Carnegie and A.P. courses were designed by research professors, who seem to take the view that statistics must be done at their level or not at all.
.. Deborah Hughes-Hallett, a mathematician at the University of Arizona, found that “advanced training in mathematics does not necessarily ensure high levels of quantitative literacy.”
As Pitts began his work at MIT, he realized that although genetics must encode for gross neural features, there was no way our genes could pre-determine the trillions of synaptic connections in the brain—the amount of information it would require was untenable. It must be the case, he figured, that we all start out with essentially random neural networks—highly probable states containing negligible information (a thesis that continues to be debated to the present day). He suspected that by altering the thresholds of neurons over time, randomness could give way to order and information could emerge.
The end result is a group of students who have “succeeded” in high school calculus without really having the proper foundations, a tower built on sand. It is quite possible for students to learn the mechanics of many categories of calculus problems and to answer questions correctly on exams without really understanding the concepts. To quote the MAA’s report:
In some sense, the worst preparation a student heading toward a career in science or engineering could receive is one that rushes toward accumulation of problem-solving abilities in calculus while short-changing the broader preparation needed for success beyond calculus.
.. The college course covers the same material in a quarter of the time; students must therefore have solid skills in algebra and geometry along with good study and work habits to succeed.
.. The rush to AP Calculus has instructed students in the techniques for solving large classes of standard calculus problems rather than prepare them for success in higher mathematics.
.. But if we want to advance STEM education and continue to produce a high-quality technical workforce we must confront this issue. We need to stop the rush to calculus and focus instead on a thorough grounding in algebra, geometry and functions.
Calculus is one of the great intellectual achievements of the last 400 years; shortchanging it by reducing its beauty and utility to a list of problems to be checked off a rubric does a disservice to everyone.