In 2001, I joined a math department that included one of that year’s PAEMST winners (Presidential Awards for Excellence in Mathematics and Science Teaching). It is the nation’s highest honor for math teachers. “Awardees serve as models for their colleagues, inspiration to their communities, and leaders in the improvement of mathematics and science education.”  To top that off, Jerry Young, our PAEMST-winner, is just a heck of a nice guy.
In retrospect, I should have noticed my co-workers tippy-toeing around the honor. People quietly congratulated Jerry, but no one suggested even taking him out to lunch. Suffering from social tone-deafness, I took it upon myself to email the math department invitations to a little celebration. We all gathered for lunch at a nearby restaurant. Everyone had to sit somewhere at the table, but the half that sat furthest from Jerry actively ignored the rest of us.
Math Wars. Wikipedia defines Math Wars as the debate in modern mathematics education over traditional mathematics and reform mathematics, philosophy and curricula.  Ladieees and gentlemen… At this end of the table, we have Jerry Young, passionate advocate of reform mathematics. And, at the far end we have the traditionalists, firm believers in time-honored methods.
What is the difference and who is right? In the simplest terms, traditional methods rely on direct instruction. Teachers provide full and explicit guidance accompanied by practice and feedback. Students do not “discover” what they must learn.  Reform methods, on the other hand, challenge students to make sense of new mathematical ideas through explorations and projects. Teachers do not explicitly tell students what or how to think. Teachers provide support and guidance as students develop new ideas by building on what they already know.  Sage-on-the-stage versus guide-on-the-side. Though either side may take exception with this generalization: traditional curricula tends towards more breadth, reform tends towards more depth. The debate over which is better has raged for more than 20 years.
Did you hear the one about the two ants? I digress here, but hang with me. Two ants stand at water’s edge arguing about how to build a bridge to cross the river. Ends up, ants have a remarkable ability to build bridges out of their collective live bodies. One of the ants has experience with live-body bridges, the other wants to use the more standard grass-blade technique. Soon, the argument gets physical and the ants are trying to sever each other at their skinny little waists. Zoom out from the scene… back a bit further… the river is a mile wide and an inch deep.
The River. Many teachers will recognize the phrase “mile wide and inch deep.” It is often used to describe math curriculum in the United States, particularly when compared to other top achieving countries. “Mile wide” refers to the large number of topics taught at each grade level; “inch deep” refers to the unavoidably shallow understanding that results from covering too much, too fast. It matters little how you teach when the time spent on each topic allows for only rote or algorithmic understanding of the problems. (Note: The most traditional of teachers maintain that K-12 requires a mile-wide river to insure their view of “math literacy.” For them, the goal is to memorize and use as many different algorithms as possible.)
Many reasons account for the creation of our mile-wide river. Among the first stands the breadth of topics in No Child Left Behind high stakes tests, which were based on NCTM Standards . Fortunately, as we move onto the new Common Core Standards, those days are behind us. Or are they?
Common Core has something for everyone, reformists and traditionalists, alike. And therein lies the danger. Few would question that Common Core calls for deeper understanding. At the same time, for high school math, it also addresses breadth with a vast number of standards.* Do we now have a river that is 0.8 miles wide and 0.8 miles deep?
How wide and how deep is the river? With the number of school days pre-set, breadth and depth are interdependent. In practice, whichever gets established first becomes the limiting factor for the other. In policy-making, that interdependence can slip by forgotten or ignored. The language and rationale that go with Common Core seem to prioritize depth; however, the number of high school standards provides a breath-taking width. In practice, since depth of understanding can be difficult to quantify, hitting all those standards often becomes the driving factor – at depth’s expense. Once again, we find ourselves gazing across the River of Perpetual Return.
Teachers need policy-makers to clearly define depth and specify a compatible breadth in the number of standards. Until then, it really does not matter if one teaches reform or traditional. Neither can be fully effective.
* I cannot speak to the number of standards in K-8.
 https://www.paemst.org viewed 01/02/2104.
 http://en.wikipedia.org/wiki/Math_wars viewed 01/02/2014.
 http://www.aft.org/pdfs/americaneducator/spring2012/Clark.pdf viewed 01/02/2014.
 http://mathematicallysane.com/reform-mathematics-vs-the-basics/ viewed 01/02/2104
 I use the phrase “NCTM Standards” for the National Council of Teachers of Mathematics recommendations for K–12 curriculum contained Principles and Standards for School Mathematics (2000).