In my last post, I described three of five aspects of an IBL instructor’s role: managing expectations, managing emotions, and keeping the students engaged. In this post, I take up the remaining aspects.

**Finding out what students know**is an ongoing task. For those who follow such things, this is also called formative assessment, and it is a critical part of a successful teaching-and-learning experience. There are a couple of purposes for formative assessment. One is to formulate responses as the instructor that will help students move forward in understanding the mathematics. Another is to identify opportunities where specific students may benefit from working on particular problems, or to find opportunities for students to share what they know at a time when it will benefit the class. One of the great benefits of teaching via IBL is that there are so many opportunities to hear from students and to develop a picture of where they are in their mathematical development. By listening to discussions between and among students in pairs or groups, and during presentations and the ensuing discussions, the instructor should have a good idea of when students might have something especially productive to contribute, or when a discussion from one group should be shared with the whole class, for instance. Notice that while formal quizzes or exams remain a source of information, as an IBL instructor the opportunities to find out what students are thinking go far beyond this, and are embedded in the everyday tasks of the class. Also notice that grades are not really a purpose of formative assessment. The focus is on student learning, and how to enhance it.

**Fitting the problems to the students**is a task that begins before the semester, but continues to occur through the semester. Before the semester, the major task of an IBL instructor is to determine the main course content goals, which could be particular theorems, skill with specific kinds of problems, or facility with certain techniques. Sources for beginning this work on your first attempt with a class might be the department course syllabus, and/or standard textbooks. From these, the instructor’s job is to put a priority on the central ideas. Then, the instructor works on developing a sequence of problems, lemmas, etc., that will carry the students from their anticipated starting point through to the goal results. As the semester gets underway, the IBL instructor works (1) to find problems to engage particular students (often the highest students or the ones struggling the most), or (2) to use to the students' advantage what they know and are thinking about, and to respond with a set of problems that provide an alternate path to the results, and (3) to modify the difficulty of the problems as the students may be more or less advanced than anticipated and more or fewer lemmas are needed between the main results to keep the majority of the class moving in a positive direction.

I hope this captures at least some of the key ingredients in the recipe for a successful IBL course. Let me know your thoughts.

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