Learning - context and details

My mathematics lecturers often try and introduce a new topic with a motivating example. I think this is a great idea. I am a slow learner and find that an initial concrete example is the best way to approach a new abstract concept. I realise that this can sometimes pose challenges further on, i.e. in generalising from the concrete to the abstract, but this way seems to work the best for me.

Unfortunately, there are many times that the motivating example is not that motivating. I view teaching as often building a bridge from the student's current position to where you want to take them. To achieve that, requires providing an initial high-level context in terms that the student can relate to.

As an example, earlier this year I studied introductory linear algebra that included eigenvectors. Kudos to the lecturer for trying to motivate with a modern example (search engine page rank). Even though I am a professional programmer, this example didn't really work for me and I finished the subject with only a vague idea about them.

Recently I read this article which introduced them with simple physical examples. The second paragraph of the current wikipedia page nicely related it back to matrices and vectors and it all fell into place.

Some mathematical ideas are represented as a sequence of steps, e.g. calculating Maximum Likelihood Estimators in statistics. Once a good motivating example is done to provide context, things can proceed between two extremes.

1. Present high-level steps and then dig down into the details of each step


2. Work through the details of each individual step and bring it all together at the end

In the latter approach, I sometimes miss the high-level steps if I get lost in all the detailed algebra and reasoning underlying each step. Can't see the forest for the trees. I prefer the former approach as it provides more purpose and is therefore more motivating. It also makes it easier to isolate difficult sections that can be returned to later, so as not to get bogged down and lost for the remainder of the discussion/lecture.

I wish I had utilised these ideas better in my Intro to Functional Programming talk a few weeks ago.