When E.D. Hirsch writes about Depth vs. Breadth in education, he suggest it's a false polarity, and that we ought to "strike a reasonable balance between deep generalizable concepts and broad based facts." The problem is, it's not reason that is guiding math content in middle-school classroom-- it is a textbook, whether on-line or paper form-- and the top names generating textbooks are necessarily known for their comprehensive coverage of topics. In a purvue of recent math texts, the top publisher normally had between 11 and 14 chapters, each covering 10 topics per chapter in up to eight contexts for each sub-topic. Just from the birds eye view of looking at the scope of the content, I was left feeling overwhelmed and scared, and this is coming from someone who performed well on the both SAT and GRE math sections.

If the argument of depth versus breadth is moot from a purely philosophical perspective on what should be, the emotional impact in actual classrooms is nevertheless tragic. When surveyed, over 80% of students had something negative to say when they were asked how they felt about math. And teachers of public schools are often torn between the cognitive preparation of their students and covering content printed in a text. With impunity and the best of motives, teachers often feel compelled to go on. 

We shouldn't spin it too far. I'm not suggesting that only a few concepts ought to be taught over the course of a year. Still, the teachers who have the most success in the classroom do not solely follow a textbook and tend toward a more limited scope of content to teach. In a 2001 study out of University of Virginia, high school physics students of teachers who did not rely on a text as their primary teaching source ended up with better grades in college, intimating that teachers who don't try to "cover the world" tend to have more success.

Personally, I don't think Hirsch is incorrect. We ought to strike a balance between depth and breadth as each impact the other. I think the real problem is that we nowhere near a balance. I'd argue that when a top selling text proudly advertises as it's number one asset a total of 671 pages for a single course in middle-school math, we're a little off kilter.