Does this feel familiar: “I thought I understood thing X, but then I learned something new and realized that I’d never really understood X?”
For example, consider a loop in some programming language:
var i = 0; while (i < n) { i = i + 1; }
If you’re a programmer, you probably understand it just fine. How it works, in what order the lines are executed, how the variable changes over time… But have you ever noticed that the simplest way to compile such a loop to machine code involves two jump instructions—one conditional and one unconditional? (Try doing that with only one jump, it won’t work.)
Now you might feel that your “understanding” of loops has become slightly closer to “proper understanding”.
Or not!
An alternative view is that understanding is translation. It’s a two-place word. Your understanding of loops, in the sense of translating them to execution histories, was perfectly fine. But your understanding of loops, in the sense of translating them to machine code, was slightly lacking.
When you see that pattern once, you notice it everywhere. A middle-schooler can understand numbers, in the sense of translating them to amounts of apples and such, but doesn’t immediately translate the expression “x > 5” to a half-open ray on the number line. A self-taught singer can translate from heard notes to sung notes, but can’t translate either to notes on a staff; a self-taught guitarist is missing a different subset of those skills. A bilingual person can translate a Japanese sentence with the word “integral” to English, without knowing what integral means. You can be good at translating other people’s facial expressions to emotional states, but lousy at translating them to pencil sketches; your friend is the opposite; which of you “understands” human faces better? There’s no answer, or many answers. Don’t ask whether someone understands X. Instead, ask if they can translate X <-> Y.
That has implications for teaching. If you walk into a classroom intending to make students “understand” X, you’ll fail at teaching. (I’m speaking from experience here.) But if you find some Y, already understood by the students, that can be translated to X—and drill them repeatedly on both directions of translation—then they will begin to “understand” X.
Understanding is translation
Does this feel familiar: “I thought I understood thing X, but then I learned something new and realized that I’d never really understood X?”
For example, consider a loop in some programming language:
If you’re a programmer, you probably understand it just fine. How it works, in what order the lines are executed, how the variable changes over time… But have you ever noticed that the simplest way to compile such a loop to machine code involves two jump instructions—one conditional and one unconditional? (Try doing that with only one jump, it won’t work.)
Now you might feel that your “understanding” of loops has become slightly closer to “proper understanding”.
Or not!
An alternative view is that understanding is translation. It’s a two-place word. Your understanding of loops, in the sense of translating them to execution histories, was perfectly fine. But your understanding of loops, in the sense of translating them to machine code, was slightly lacking.
When you see that pattern once, you notice it everywhere. A middle-schooler can understand numbers, in the sense of translating them to amounts of apples and such, but doesn’t immediately translate the expression “x > 5” to a half-open ray on the number line. A self-taught singer can translate from heard notes to sung notes, but can’t translate either to notes on a staff; a self-taught guitarist is missing a different subset of those skills. A bilingual person can translate a Japanese sentence with the word “integral” to English, without knowing what integral means. You can be good at translating other people’s facial expressions to emotional states, but lousy at translating them to pencil sketches; your friend is the opposite; which of you “understands” human faces better? There’s no answer, or many answers. Don’t ask whether someone understands X. Instead, ask if they can translate X <-> Y.
That has implications for teaching. If you walk into a classroom intending to make students “understand” X, you’ll fail at teaching. (I’m speaking from experience here.) But if you find some Y, already understood by the students, that can be translated to X—and drill them repeatedly on both directions of translation—then they will begin to “understand” X.