The coverage issue become problematic when we sacrifice understanding for the sake of getting through the long list of topics. I'm going to use the words of college students to convey the core ideas. Keep in mind that these students were some of the best in their high schools, have taken at least precalculus, and most of them have had calculus. Several have had some upper level courses.
- "I feel as if this was the first math class ever that I actually took the time to understand what exactly a graph was saying. Before graphs were just lines, or parabolas, that I plotted..."
- "I'm astounded by the fact that I only did 5 actual math problems in my middle and high school careers. I drilled procedures, and didn't really learn anything from it."
- "The idea of teaching students how to spit out answers to problems is not effective in learning. As a learner, I would like to understand why we do math a certain way and not that it is 'just done this way.'"
- "I had never known why we 'completed the square.' I understood that completing the square produced an equivalent expression or function because whatever was added was also subtracted, but it always seemed so complicated. For the first time [using algebra tiles] I really had a clear understanding of what was going on when I completed the square."
Instructors often do not have much control over how much material a course must cover. In this way, there exists a systemic issue or crisis. Our system measures coverage in antiquated ways (i.e. a list of topics) and does not include covering "practice standards," such as problem-solving ability, communication, understanding concepts, and being able explain ideas.
To IBL instructors this poses a Teaching Minimax problem -- Minimize as much as possible unnecessary content and maximize the amount of time available for students to explore why.
More on this in future posts...