Math courses are taught in a variety of ways, and even within the IBL community there exists numerous differences. This is something that should be expected, because environments and goals differ across institutions. Just as we would not expect Michelin star restaurants to be identical across the world, we should expect that successful instruction will be different and varied to suit the needs across different institutions. This honors the diversity of humanity, and allows a teacher to be true to her or his personality
Here's the model I have in mind:
In this model I use an idea I call the "Sense-Making Continental Divide." A key feature that defines IBL instruction is that students regularly are encouraged to do the sense-making tasks, including validation of solutions or proofs, understanding statements of problems, and working from definitions and first principles. IBL instructors set up courses to get students over the Sense-Making Continental Divide, where students are regularly doing activities that require students to think, decide, explain, evaluate, and reflect. You have crossed over the SMCD if your students are (a) deeply engaged in rich mathematics and (b) have opportunities to collaborate and discuss ideas and solutions. (This succinct characterization of IBL is from Sandra Laursen, University of Colorado Boulder.)
It's easy to see there are numerous options for implementing sense-making activities. From the palette of teaching options is derived the multiple variations of IBL. In my perspective, this explains why IBL comes in so many different forms. We have more choices, and the "correct" teaching decision depends on real-time conditions in the specific class setting an instructor is in.
What typifies traditional instruction is that the instructor does the processing and sense-making through presentations. "This is the proof of theorem 3.6..." Students do not get many opportunities in class to do the structuring or validation. In such classes, students might be unintentionally encouraged to memorize facts rather than make sense of the ideas.
The Hybrid IBL zone contains the different forms of IBL methods that are often a result of practical limitations instructors face. There may be a required syllabus, or a course may be predominantly procedural in nature (e.g. calculus). A course may have large enrollment, or an instructor may not have the requisite skills or experience to comfortably run a full IBL course.
It could also be the case that the (full) IBL class instructor shares a solution on occasion in the event that students are floundering, and moving ahead would be more beneficial mathematically for the students. Flexibility and adaptability are key traits of effective instruction. An IBL course may change during the term to adapt to specific needs.
Is your course an IBL course of some kind? One way to see is if your students are regularly over the Sense-Making continental Divide.
For more about your personal IBLishness, see also this post on IBL Levels.
