The topic of the week was probability and counting. Everything from flipping coins to Bayes’ theorem. A week packed with definitions and example problems. A week that went by in a blur. I’d like to reflect this week on a choice I made before the course began.
I chose to teach the course in a standard lecture format where I introduce topics, offer examples, ask questions, answer questions and then move on. I follow the text; the students know what I will cover (it’s in the syllabus); homework is assigned to challenge understanding, promote connections among material, new and old—formative assessment; quizzes and tests are given to check on more basic understanding–summative assessment.
I did not choose to teach the course with student-driven construction of concepts. I do not form in-class small groups to develop a method to solve a problem, or a theorem to consolidate results—I often present arguments in my lectures that motivate the use of a formula or theorem, but the hard work and joy of discovery are muted.
Did I make the wrong choice in choosing the standard lecture format? I don’t think so. Every course at my college is defined with an official description of content that lists topics that will be covered in the offered course. This description is not a suggestion, it’s a contract. Other schools agree to accept transfer credits based on what is covered in the course, as stated in the description of record. For this particular course, we have to cover the majority of ten text book chapters in approximately eleven weeks of class. In my judgment, I could not facilitate student-driven discovery of that much material in that little time.
Could it be done? I’ve taken a graduate-level introduction to topology that was entirely student-driven discovery. Every definition and theorem was constructed from a very small start of very few concepts. We developed proofs alone or in groups and defended them at the board (sometimes successfully, often not). Agreement that a proof was good-enough allowed us to move on. Did we make it through the description of record? Maybe three quarters of the way. The professor resorted to lecture at the end to finish the process on time. So maybe it could be done, but for undergraduates it would be challenging for many if not all of the students. You could argue that students don’t really internalize the material if it’s based on the lecture-homework-test model, anyway. But can we ever expect internalization in the first introduction to ten chapters of discrete mathematics in eleven weeks? I doubt it.