Taking a Left Turn

A contemplation of non-linear improvement and its connection to (micro)management.

A little exercise:

Think of one thing that characterizes poor teaching (by whatever measure).  Now, think of one thing that characterizes good teaching. Is there a straight path from one to the other?

In many examples I can think of, you can get from one to the other, but not by a straight path.  For example, poor teaching is often disorganized, while good teaching is engaging, but I don’t think that just becoming more organized (the straight path from disorganized) will necessarily make you engaging. As one goes from poor to good, from disorganized to engaging, they have to take a turn somewhere. But where should this happen?

Now we get to talk some math.  There’s an optimization algorithm called Steepest Descent (SD) in which you go off in a straight path searching for the best spot along the path. The key is picking the direction and for SD, you choose the one that gives you the best initial improvement. So, from our example, if you are disorganized, your initial direction is one that increases your organizing skills.

In nice cases it is easy to see how SD eventually finds the optimal solution, however even in simple cases it can be shown that the optimal solution is only reached in the limit, i.e. after an infinite number of steps. You move towards the optimal solution, but taking a zig-zag path.

We can bring this back to our question of improvement, as there are two ways to ‘fix’ SD relevant to ‘taking a turn’. The first in Conjugate Gradient (CG) and in simplest terms, you put the ‘turn’ in the choice of the initial direction by turning it so that it doesn’t ‘go back’ in the previous directions. In our example, that would mean after you optimize all you can by becoming more organized, you figure out your next area of improvement, but make sure that you minimize any aspects that have to do with organization.

CG works when SD works, and in nice cases CG can find the solution in a finite number of steps. Even in less than nice cases, it works better than SD.

The other fix uses some sort of Dogleg (DL) process, and is typically for non-linear problems. In this case you start your step in the SD direction, but then at some time you change your path towards another point. The key is that the path choices are built not on the original problem, but on an idealized approximation which makes the base calculations easier. The final answer is based on the original problem. In our example, this would mean that you would start becoming more organized, but based on education literature or other’s experiences, you’d at some point turn towards some theoretical ideal. You’d be using your real experience to measure the success and stop at the best point along that path.

So why do I bring all this up? I’ve been involved in various efforts to ‘improve’ things, and when those efforts have been successful it has been when the overall goal is clear and the action taker is also a key decider on the thing. I’d say this works because the improvement uses a CG or DL type approach. For example, if someone is trying to improve their teaching, when they know what sort of broad results they want, and are involved in identifying the issues and the solution, then there is a higher likelihood for real improvement. In the case where the action taker is responding to someone else’s choices or someone else has the broad vision, there is usually much more frustration and little improvement. I’d argue that outside management (micromanagement) takes a more SD type approach, as they work from only a snapshot of the situation and usually focus on places that give the quickest response.

There are no real surprises here. It seems most modern leadership/management models tend toward a collaborative/shared-governance perspective which captures the broader or holistic view inherent in the CG and DL type approaches. But it is good to be on the look-out for the less effective, zig-zag approach inherent in a SD type approach.