Feeding abstraction with understanding
Abstractions empower and accelerate. As usefully encapsulated nuggets of understanding, the creation of novel abstractions drives a field’s progress, but their invention is possible only with deep understanding of present ideas. So I declare: if we are to master a field, we must accept none of its abstractions as magic. Rather, we should yoke them as automations of what we already understand.
This concept separates the random, slowly-reinforced walk of genetic evolution from the directed, rapid exploration of scientific evolution. Understanding provides not only a compass bearing but also course correction, without which extended progress is impossible.
For instance, if you always cook from recipes (without understanding why you’re executing each step), you’ll be poorly equipped to fix a dish that’s gone wrong, improve an average one, or synthesize something new altogether. The same limits apply to parochial understandings of abstractions in musicianship (“minor chords sound sad”), writing (the five-paragraph essay), mathematics (the Pythagorean theorem), and engineering (caching systems).
Say you accept this idea. We’re then faced with a practical problem: how can you ever learn to do anything useful without endless prerequisites? Algebra classes are particularly afflicted.
- TEACHER: Today we’ll learn how to factor quadratics of this form.
- STUDENT: Why do I need to learn that?
- TEACHER: You could use it to find where parabolas hit the x axis.
- STUDENT: Why do I need to learn that?
- TEACHER: You could use that to find the trajectories of point masses.
- STUDENT: What?
- TEACHER: Believe me, you’ll be glad you learned this someday.
To foster real understanding, we must format concepts so that students can derive solutions for themselves, not encourage them memorize some procedure. And of course, they must be motivated to find those solutions by a problem they actually care to solve.
Microworlds may form the basis of one solution. The idea is to create a tiny, self-consistent sandbox in which the student can explore some concept, given little instruction beyond what he already knows. The sandbox doesn’t hide detail so much as focus. One can imagine, then, constructing a successively expanding universe of these microworlds—each itself useful—so that hops between neighbors aren’t too vast.
Imagine trying to learn to play the keyboard. You could start with one which only allows the student to play in the pentatonic scale in C. Five keys aren’t so intimidating. The others are there, but dimmed out. You can still play all kinds of melodies. You can explore rhythm. You could play chords, though many won’t realize it. Then you could switch over to a hexatonic blues scale mode: also self-contained; also not too intimidating. Then you could start displaying the notes the student plays on a staff above the keyboard. And so on. The skills learned on these limited keyboards are not abandoned when moving to more complex ones—they’re fruitfully employed!
Each microworld will naturally engage different sorts of students, and that’s okay: they should be free to roam between any within reach. The goal is to provide a sandbox—not a syllabus—for experimentation and the formation of understanding. Understanding is marvelous because it’s so readily a feedback loop: we can use it to make more of it.