Looking backward in revision

Post date: Jun 2, 2017 2:18:35 PM

When teaching a course, we know the whole picture and in our head we have made decision on how we approach certain topics because of where they lead. The expectation is that as students engage in self-learning, they will start to see the picture picture and understand why we did certain things that might not have been so obvious. This sometimes can be especially true in pure maths, where students can resign themselves to "this is just another definition that doesn't make sense, I have no idea why we are learning this". This mean that they can miss the whole point of the course.

To counteract this, and to help them with their revisions, one of the last tutorials I did was starting from the end and working backwards. Well not quite the end.

To revise for analysis, I asked my students "Given a regulated function f, how can I find the integral of f (between a and b)?". This is a good stepping stone to cover a good chunk of the course. Before answering this question I check who knows the definition of a regulated function. This brings up the idea of left and right limits, so it is a good place to check they understand the formal definition of limits (on a function), i.e., can they turn "limit of f(x) as x goes to a is L" into an epsilon-delta definition, and vice versa.

Then comes the questions of what is the point of regulated functions. The common answer is "they can be approximated by step functions". "What do we mean by approximated?". This opens up the discussion of limits of sequences of functions. Since we only know (from earlier Analysis work) how to find the limit of real numbers, we try the "naive" approach of looking at the sequence of function at every point. Hence the definition of point-wise convergence. But after a recap of the limitation of the point-wise limit (namely, you can loose the continuity property), we can go over uniformly convergent. So finally we can answer the question "Given a regular function, f, there always exists a sequence of step functions which converges uniformly to f"

Obviously, at this point we have encountered a new definition, step functions, and we need to recap everything we know about them. Their definition (and a few pitfall about their definition), and the fact that we can define an intuitive definition of integration on them. Intuitively, "the area under the graph", but in this case, the intuition can be turn into a formal definition.

At this stage we can finally answer the question. To integrate a regulated function, f, you first find a sequence of step functions that converges uniformly to f. You look at the integral of these step functions, and take the limit of the integral to be the integral of f.

At this point, I told my students that this is what they were doing at A-level when they learned to integrate. They would approximate the area under the curve by drawing smaller and smaller rectangles underneath it. We just formalised the process of doing that, and to do so we had to introduce a lot of new definitions (point-wise and uniform convergences, step functions, regulated functions, left and right limit) and the theorems that goes with them.

In conclusion, I remarked to my students "see how a simple question can be unpacked to cover a big part of your course. If you see how each bit of the module fit together, you are more likely to understand the material". I think it is a good revision practise to start with a question that you can unpack to cover as much material as possible. Especially in first year, where a lot of the material builds on top of each other, but students can be tempted to memorise single bits, without knowing what it is built on (i.e., depends on) or what will be built on it (i.e., what was the point to learn it).