Inside the inverted proofs class: Guided Practice holds it together

In the last couple of posts on the inverted transition-to-proofs course, I talked about course design, and in the last post one of the prominent components of the course was an assignment type that I called Guided Practice. In my opinion Guided Practice is the glue that held the course together and the engine that drove it forward, and without it the course would have gone a little like this.

So, what is this Guided Practice of which I speak?

First let’s recall one of the most common questions asked by people learning about the inverted classroom. The inverted classroom places a high priority on students preparing for class through a combination of reading, videos, and other contact with information. The question that gets asked is — How do you make sure your students do the reading? Well, first of all I should say that the answer is that there really is no simple way to “make sure”. We can only make it easier or harder for them to do so, and then measure the results indirectly.

But insofar as there’s a direct answer to that question — let’s first explore how not to get students to do the reading. That would be, unfortunately, the approach that most of us take: Give students a reading assignment and tell them to do the reading. You can even add threats and injunctions on top of the reading assignment — “If you don’t do the reading, you’ll be lost!” and so forth — and I can almost guarantee you that most students will still not do the reading. This is even true for watching videos. Why? Are students that lazy?

Well, no, not most of them. It’s simply because they don’t know how to read. I don’t mean that students are illiterate, of course. What I mean they don’t know how to make sense of a text — especially a dense, complicated, and often poorly-written mathematical text. Like time and task management, which I mentioned in the previous post in this series, this is not something they’re born knowing how to do. They have to be taught this skill — by us.

When expert learners in mathematics sit down with a text or an article, certain intellectual activities are going to take place besides just “reading”. We are going to try to place the reading into context with what we already know. We are going to instantiate new terms with examples and nonexamples before moving on. When we read a proof, we are going to pursue understanding the main argument and understanding the technical details as two interrelated but separate tasks. When we reach gaps in the exposition, we are going to pause and try to fill in those gaps. When you begin to unpack just what it takes to be a productive mathematical reader, it starts to become clear that expecting novices simply to “just read the book” is something like placing a brand new pilot in the cockpit of an F–16 and asking them to “just fly the plane”.

That’s where the notion of Guided Practice comes in. The fact is that we do need students in the inverted classroom to do the reading, and we can’t wait to have some sort of minicourse on “How to Read a Mathematical Text” first. So the idea is to provide on-the-job training on precisely those actions that expert mathematicians employ when they read.

In our course, Guided Practice was assigned before almost every class and had the following parts:

• An overview of the content the students were about to encounter. This was just a paragraph that tried to place the new material in context with the older material. Those connections are very important.
• A bullet-point list of specific learning objectives for the material. Specifically this makes it clear to the students exactly what they needed to be able to do prior to coming to class, so the expectations are clear.
• The list of reading and videos. With the videos, I always included the running time of each and a total of the running times, to make sure students knew the time I was requiring of them. (In past classes where I encountered more resistance to inverting the classroom, having the total running time printed on the assignment was a way of reassuring students I was not requiring any more time from them than a normal in-class lecture would.)
• A list of 2–3 exercises that focused on instantiating new concepts with examples and non-examples, asking students to identify the main argument in a theorem, fill in a gap in the exposition in the book or video, or ask a high-level conceptual question about the material.
• A link to a Google form where students could turn in any questions they wanted to discuss about the material in the Q&A time at the start of class.
• Finally, a review of what students needed to turn in and when.

Here’s an example of a Guided Practice document for the transition-to-proof class. Here’s another that I’m using in Linear Algebra now, which although that course is not 100% inverted like the proofs class, I still want students to do some preparatory work. The main idea is that I am not just giving students a reading assignment and telling them to do it. We are extending the active learning concept outside of class to the preparatory phase, so that if students work closely with the Guided Practice, they will not only do the reading and learn some things prior to class, they will also learn the habits of mind that expert learners practice. I kind of see Guided Practice like training wheels on a bicycle. They help keep you upright while you learn the basic motions; and eventually they should come off.

I mentioned I am using Guided Practice in linear algebra now. I have made one change to the original setup, and that’s to break the list of learning objectives into two — one list for what you need to know how to do prior to coming to class, and a second list that should be mastered after class, with more practice. I want to stress to students that we’re not asking them to teach themselves linear algebra — just the bits of linear algebra that they can handle independently, and the reading and sense-making skills that university-educated students need to have.