• Beyond Direct Instruction

  • By Tom Sallee, Co-founder of CPM
    Professor of Mathematics, Emeritus, University of California, Davis
  • One of the most profound truths I learned in forty years of teaching college-level mathematics is that what feels like learning in the classroom frequently does not translate into actual understanding of the topic. As an undergraduate at CalTech I occasionally had the good fortune to have Richard Feynman, the great physicist and famous expositor, guest lecture in our physics classes. The lectures were unbelievably good and parts of them I can still quote 50 years after the fact. The problem was—I did not really learn what he was trying to teach me until I went back and wrestled with homework and fought through the ideas with my roommate. The regular professor, whose name I have long forgotten, was at least as good at helping me to understand what I was supposed to learn.

  • I had to relearn this lesson after about ten years of my own teaching. Everyone thought that I was a great teacher. My student evaluations were always among the highest in the department because students believed that they were learning from me and it felt really good in the short term; I believed that I was teaching and they believed that they were learning. But the next quarter, if I had them again for the succeeding course, it became clear that they had not learned what they should have.

  • At about this time I worked with a graduate student from Berkeley who was doing his thesis on an alternative way of teaching physics and I was the instructor of the control group. At the end of the quarter it was clear that students liked my teaching much much better than they liked his and my students believed that they had learned more than his students believed that they had learned, but the finals showed this simply was not true. His students learned far more than mine.

  • At this time I began to understand that my teaching was the problem. My students were serious, they were hard-working, they were doing everything that I asked, but they were not learning for the long term. And a lot of this was the fault of my teaching. I needed to do something different

  • Just so everyone is clear, I believe that direct instruction must be part of any well-conceived mathematics program. Students cannot guess what the definition of a trapezoid is, nor are they always ready to see connections among pieces of similar knowledge as they learn them. This is part of the teacher's and the curriculum's role. But direct instruction alone is not enough. A little is essential, but much less than most American students receive. Rather, students need to approach mathematical concepts in other ways to enable knowledge to be integrated into a cohesive conceptual framework and to be retained for years to come. These realities have led to the problem-centered approach of lessons that is so central to CPM's curriculum and teaching philosophy.

  • The more research that is done, the more persuasive the research becomes on this point. Roughly speaking, when you are told something, you know it in the narrow context in which you learned it, and can usually do much better on a quiz given in the next week or so, but the knowledge is both fragile and relatively short-lived. If knowledge is to be robust and usable in the long term, being told ideas and rules and then practicing many similar problems is not nearly enough to reach understanding. Students need to engage the ideas and very frequently the best way to engage these ideas is to work on problems with others. A very common misconception is that such an approach might work well for the very best students, but average and weak students need to be told the rules. This is simply not so. Data is showing that virtually all students need to engage mathematical ideas to be able to learn. For more information read the CPM Research Base at: http://www.cpm.org/teachers/info.htm.