*think*they understand. In the fall, I saw my students doing this almost automatically as they wrote brief essays over the course of the semester; many chose to write about their own understanding of mathematics (specifically calculus), what doing math means, and how math is used in the wider world. (More on these essays later.)

This spring, I want to make the metacognitive aspect more explicit in my classes. I will be teaching a multivariable calculus class and an “advanced calculus” class (which will be focused on the geometry, topology, and calculus of manifolds in Euclidean space of arbitrary dimension). Each class will have a distinguishing feature that lends itself to self-reflection. In multivariable calculus, I will be using standards-based grading (SBG) in place of traditional number-and-letter grades, in order to better focus students’ efforts towards improving specific skills, rather than a single, not very helpful grade. In advanced calculus, students will complete biweekly projects that tie together ideas from lectures and prod them to explore new ideas on their own.

How could I use these to encourage “thinking about thinking”? In the first case (multivariable calc), I’ve been considering also having brief, weekly self-reflections in which students will describe what they think the main ideas for the week were, and what they found confusing. (I’ve borrowed this idea from some other source, which I can’t find at the moment.) It seems like it would be easy enough to put the relevant standards on the self-reflection form and have them mark where they think they stand. In the second case (advanced calc), I am trying to follow this principle in creating projects:

It’s OK for new ideas to be abstract, as long as they’re familiar in some way. And it’s OK for them to be unfamiliar, as long as they’re concrete. If I, as a teacher, try to introduce an idea that is both abstract and unfamiliar, then it is likely most students will be lost.(Paraphrased from a comment made by Steven Strogatz about the expository essays he has written for NYTimes.) I plan on stating this principle outright to my students: when you start a project, you should expect to find ideas that are either familiar or concrete. Look for those, and build your understanding around them. Even better, take that principle and use it to transform your approach to learning. If an idea is unfamiliar, find a way to make it concrete (often by looking at specific examples). If an idea is abstract, connect with something familiar.

I think these two ways of dealing with metacognition are appropriate for the different levels of the classes. Multivariable calculus remains among the set of math classes that are not always taken for the sake of math itself; often students approach it with the paradigm of “mathematics as tool”, and so their reflection on their understanding itself needs a concrete form and guidance. This advanced calculus course, on the other hand, is a transitional course into higher areas of mathematics and is more likely to have students who are interested in math for its own sake; these students should be practicing independent learning and encouraged to make self-reflection a part of that process. These ideas are relatively recently formed in my own head, and I plan to fill them out more over the next two weeks as I prepare for the start of the semester.

One final credit: I first encountered the idea of “metacognition” during a presentation by a representative of McGraw–Hill about one of their online products. While I have not had a chance to explore this product in depth, I am grateful to the presenter for sharing this idea.

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