Along a certain dimension, learning styles can be categorised into roughly two types: top-down learners and bottom-up learners. Today I witnessed a small anecdote on the distinction of the two, and thought I should write it down.

Scene in class

Someone (Linus, actually) said, “Talk is cheap. Show me the code.” I, for one, would think that programmers in general, would be top-down learners, and would just ask for the (preferably authorative, succinct and intuitive) definitions of things, stare at it for a while and understand it — as opposed to, you know, asking for examples first.

During network defense class, we were being told some RSA stuff.

Euler’s totient function (or Euler’s phi function), denoted as φ(n) or ϕ(n), is an arithmetic function that counts the positive integers less than or equal to n that are relatively prime to n.

(The above definition is actually copied from Wikipedia by me; the lecturer gave something of a similarly maths-y wording style.)

phi(3) = 2
phi(4) = 2
phi(5) = 4
phi(6) = 2
phi(7) = 6
phi(8) = 4

Everyone’s like “Er… what?”. Someone asked, “Isn’t there a short, simple way to explain it?”. After some more failed attempts at trying to explain the thing using only a few short phrases, the lecturer decided to go through a full example:

Say we want to calculate phi(5). What are the numbers from 1 up to 5? — 1, 2, 3, 4, 5.
gcd(1, 5) = 1
gcd(2, 5) = 1
gcd(3, 5) = 1
gcd(4, 5) = 1
gcd(5, 5) = 5

We notice 4 “1”s, hence phi(5) is 4.

When the lecturer counted “1, 2, 3, 4” for the “1”s, half the class went “OOHHHH!! I GET IT NOW!”


During class when whoever-it-is that asked for a “short, simple” explanation, I turned around, pointed to the slideshow and said, “well the definition is written right there, and it’s only, like, a sentence long.” I now realise I must’ve sounded condescending or mean; perhaps I shouldn’t have said that.

People do learn differently! When half of the class went “Oh!!!”, I was shocked that, even among programmers, there are so many people who seem to be bottom-up learners — to prefer examples over definitions during first-contact to some piece of knowledge. Personally, I paid much more attention to the definition than to the examples.

Perhaps my a-priori, subjective impression was very wrong in the first place. Perhaps this sort of “to understand it, first stare at the definition, hard” mindset would be more common among maths students than among programmers. Having just taken a maths paper (mostly about formal proof techniques too!) last semester, maybe I was the most maths-y one in the audience today.

There’s probably a common view that, specifying things succinctly, using a formal language, is an elegant and efficient thing to do. I came across an example just this week while reading a paper about TLS where they tried to specify and implement TLS using functional programming, with a goal of reducing ambiguity. For a given concept, say “Foo”, examples and definitions can both tell you what sort of thing would count as a Foo, but only definitions can (reliably) tell you also what would not count as a Foo — it’s something examples fail miserably at.

Sometimes/Often, clearly specifying the problem actually yields a solution. This strongly reminds me of functional-style declarative programming.

“How do you find, for some given natural number n, the number of natural numbers from 1 to n inclusive that are co-prime with n?”

“Well, (duh,) you just count the number of numbers from 1 to n for ones that are co-prime.”

“The factorial of a natural number n is the product of multiplying together 1, 2, 3, …, n.”

“Well then, to calculate the factorial of n, just multiply together the first n natural numbers.”

As I write the above two dialogues, they started to seem too tautological… I’m not sure how well this “solution via problem definition” works in general. It feels very good solving problems this way though!