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.