I came across the following on Twitter (original tweet). It's the prologue to a 1914 book called "Calculus Made Easy," and it's gold:
Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.
Some calculus-tricks are quite easy. Some are enormously difficult.
The fools who write the textbooks of advanced mathematics (and they are mostly clever fools) seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.
Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.
The insightful point here is that every complex topic is composed of simple ideas, with a few tricky ones thrown in. Learning becomes intimidating only because we can't see the easy parts.
But why can't we see the easy parts? Because we don't know how it all fits together. Here's what I mean:
In truth, advanced math is just composed of a bunch of related ideas stacked on top of each other. Some are quite easy to grasp, given enough context. Mastering them one by one would be time-consuming, but not especially strenuous.
An "expert" is thus someone who is able to see the whole stack, and how each idea leads to the next. But in becoming an expert, something is lost along the way.
By becoming an expert, and ascending the big stack of ideas, it's easy to forget the simple bits at the bottom. You don't need to talk about those anymore, as you spend most of your time dealing with the ideas at the top of the stack.
You may even forget the simple ideas exist, as they've become so intuitive. You begin to speak in a jargon that only makes sense for people who understand the most advanced parts.
This makes your life easier, of course, because it's an act of compression. You compress the stack of simple ideas into a few dense ones.
The role of the teacher
The role of the teacher, by comparison, is expansion. The teacher must break the stack apart.
In other words, the teacher's role is to remember the easy bits. This is what this quote captures so well.
The teacher's job is to show you which parts are easy, which parts are hard, and how they all fit together.
And (often most difficult), the teacher must remember that it is no great feat to master the stack, and that any fool can do this. They must be an expert without an ego.
The role of the learner
If you're learning on your own, you face the additional difficulty of breaking down the stack on your own. You have to take all those advanced concepts and wrestle them down to their component parts.
This is grueling work, but it is also incredibly valuable. Assembling the stack of knowledge by yourself means you have a far deeper understanding of it.
The most important thing to remember here is one particular point, the one made so elegantly made in this prologue: that any fool can do it.
The expert stands on top of a mountain of ideas that seems impossibly tall. Once we start climbing it, however, we realize it is really just a long, gentle slope. All we have to do is put one foot in front of the other.