Edward Frenkel recently resurrected an old complaint in his Los Angeles Times op-ed, “How our 1,000-year–old math curriculum cheats America’s kids.” He observes that no one would exclude an appreciation for the beauty of art or music from the need to build technique. Why do we do that in mathematics? As I said, this is an old complaint. Possibly no one has voiced it more eloquently than Paul Lockhart in A Mathematician’s Lament, the theme of Keith Devlin’s 2008 MAA column, “Lockhart’s Lament.” Enough time has passed that it is worth my while to bring this lament back to the attention of the readers of MAA columns. I also want to respond to Frenkel’s post. I have two problems with what he writes.
The first is the suggestion that we spend too much time on “old” mathematics and not enough on what is “new.” I share Frenkel’s disappointment that too few have any appreciation of mathematics as a fresh, creative, and self-renewing field of study. Frenkel himself has made a significant contribution toward correcting this. In his recent book, Love and Math, he has opened a window for the educated layperson to glimpse the fascination of the Langland’s program. But I disagree with Frenkel’s solution of devoting “just 20% of class time [to] opening students’ eyes to the power and exquisite harmony of modern math.” There is power and exquisite harmony in everything from early Babylonian and Egyptian discoveries through Euclid’s Elements to the Arithmetica of Diophantus and the development of trigonometry in the astronomical centers of Alexandria and India, all of which were accomplished more than a millennium ago and are still capable of inspiring awe.
In fact, I believe that one of the worst things we could do is to create a dichotomy in students’ minds between beautiful modern math and ugly old math. We must communicate the timeless beauty of all real mathematics. The challenge of the educator is to engage students in rediscovering this beauty for themselves, not outside of the standard curriculum, but embedded within it. The question of how to accomplish this leads to my second problem with Frenkel.
Frenkel makes the implicit assumption that what we need is a wake-up call, that it is time to recognize that mathematics education must do more than create procedural facility. In fact, the need to combine the development of technical ability with an appreciation for the ideas that motivate and justify the mathematics that we teach goes back at least a century to Felix Klein and his Elementary Mathematics from an Advanced Standpoint. It is front and center in the Practice Standards of the Common Core State Standards in Mathematics. It was a driving concern of Paul Sally at the University of Chicago, who we so recently and unfortunately lost. It continues to motivate Al Cuoco and his staff engaged in the development of the materials of the Mathematical Practice Institute. It lies at the root of Richard Rusczyk’s creation of the Art of Problem Solving. It permeates the efforts of literally thousands of us who are struggling to enable each of our students to encounter the thrill of mathematical exploration and discovery.
As we know, it takes more than good curricular materials and good intentions to accomplish this. It requires educators who understand mathematics both broadly and deeply and can bring this expertise to their teaching. Many are working to spread this knowledge among all who would teach mathematics to our children. This is the inspiration behind the reports of the Conference Board of the Mathematical Sciences on The Mathematical Education of Teachers. It is a goal of the Math Circles, in particular the Math Teachers’ Circles that reach those who too often are unaware of the exciting opportunities for exploration and discovery within the curricula they teach.
The mathematician’s lament is still all too relevant, but it is neither unheard nor unheeded. I am encouraged by the many talented and dedicated individuals and organizations working to meet its challenge.