What I'm going to do is narrow in on a small slice of this topic and look at this pragmatically for teaching. The main idea in this post is giving ourselves room to be flexible.
A common (but not the only) situation that causes tension is when faculty find that their students need more time on a topic AND things are going slower than the scheduled plan. We all get in this situation, no matter how well we plan. So we needed to get to the end of section 2.4 today, but students have questions, which takes time.
Tension comes from the fact that we have a plan and a syllabus and we need to "cover" a prescribed set of materials. If we go too slow, we won't hit our mark. It's like the feeling of being late for a flight. When we are behind schedule, it can feel stressful. If we view teaching through the lens of information transfer, then getting behind means we are not achieving our goals as teachers.
If instead we view teaching through the lens of maximizing learning, then we can view the slowing as a signal that important learning opportunities have arrived, and the time spent now will be highly productive. With this teacher mindset, then we can create mental room for being flexible.
Mental room for flexibility means it's okay for students to get stuck, and it's okay to be off schedule or script (at least for the time being). If we take this a step further and build in the ability to be flexible when needed some of the time, then we're capable of staying on schedule while still having the flexibility to address the necessary speed changes in learning. Learning is nonlinear.
Mental room is not enough. We need to meet the goals and expectations of the course. So how do we balance these competing forces without changing the entire culture of education? One of the obstacles is coverage, and coverage is one of the biggest albatrosses in math teaching. A common sentence I hear is, "I wish I could do that, but I don't have time." That's understandable. I get it, and I'm not suggesting anyone toss their department syllabus aside. A point I want to get to is that we don't have to throw up our arms and give up. We can up our game and plan better to at least make some room to be flexible. For example, not all topics need to given the same amount of time or emphasis. Some ideas are more central than others. Instructors can also learn about certain topics that students typically struggle with and plan for it, and instructors can use flipped strategies to move some topics outside of clas.
A reasonable solution could be as simple as asking students to read a handout or section of a book before class, and using presentation slides to go over the basic ideas, instead of writing them on the board. We can use email to send logistics info instead of the first 5 minute of class time. Now we just set aside 5-10 minutes daily that can be used to add in another activity or spend more time on an existing activity. Multiply by the number of class meetings per term, and that's a significant number of extra activities or time we can get into a course. This is like moneyball (from baseball). The idea is to push small edges in the long run, and does not require a wholesale change in teaching methodology.
Each course has topics that we know are more challenging, and even coordinated courses have syllabi that accommodate for this. These are good spots for incorporating flexibility-time in our classes. It is already understood that students find these topics more challenging. We can plan to have students work on say 4 problems, but be willing to spend more time on the first 2 allowing students to get up to speed. Careful selection of problems and having good starter problems and lead-in problems/questions/prompts that hit on core concepts or strategies can be highly useful.
The above are a couple of feasible examples of how to build in flexibility. We should touch on an aspect of what happens if we aren't flexible. If an instructor regularly gives the impression that slowing down can't be done, then students may get the impression that questions are not valid. While this is certainly not the intent, if students are sitting there with questions and know there is "no time" to answer them, then that sends a message. This message may even feed into minds with fixed mindsets, further reinforcing negative beliefs. (e.g. the attitude"If I have a question and there's no time or importance to address it, I must be dumb...") Is it really surprising then, if we come across students in math classes that are reluctant to chime in and ask questions?
Being flexible isn't about slowing down when questions arise. Being flexible is seeing opportunity in questions and taking advantage of them. Further it's being willing to use available tools and teaching practices creatively and efficiently to create time for being flexible.
Flexibility is not a reactive teaching idea. To me it is fundamentally a proactive teaching idea, where an instructor's ability to be flexible is largely established before class time. And if we build flexibility in as part of our planning process, then it frees us and more importantly our students to ask, engage, and learn!
