If in our research series we come across something that our mind does not want to understand well, we should stop and refrain from further unnecessary investigation of subsequent things.
Descartes, “Rules for the Direction of the Mind”
How does this relate to teaching mathematics?
Descartes’ words fit very well with teaching mathematics. At the very beginning, let’s distinguish between two situations: one where you are panicking and rushing for an exam you have tomorrow morning, and one where you simply want to learn calmly and move on to other things.
My whole post refers rather to situation number 2.
While learning, we encounter difficult things that we don’t understand. That’s natural. If it were otherwise, it wouldn’t be called learning. It’s not good to “skip over” such things. It’s much more efficient to stop at such a thing and make every effort to understand it.
Ask someone. Look through a few books. Google it.
In mathematics, very often one thing follows from another. If you don’t understand what a sequence limit is, you won’t understand what a function limit is (according to Heine at least, but let’s not complicate). If you don’t understand what a function limit is, you won’t understand what a derivative is. If you don’t understand what a derivative is, you won’t understand what an integral is. And so on.
Learning math will become a grim torture, torturing yourself by memorizing hundreds of transitions and formulas.
By “understanding” I don’t mean strictly memorizing definitions, but “feeling” what something is, describing it in your own words. For example, sequence limits can be understood intuitively and handled excellently. With such understanding, the formal definition seems obvious to us.
