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The Need to Grind Concrete Examples Before Jumping Up a Level of Abstraction

justinmath.com

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Re: The Need to Grind Concrete Examples Before Jumping Up a Level of Abstraction

#71

Post by gowld »

xg15 wrote: post #41
schneems wrote: post #40

A concept I learned about in Knowledge Based AI (gatech ONSCS) is called “version spaces” where instead of starting at specific examples and moving to be more general or the other way around you do both as a kind of knowledge bidirectional search. I feel humans work that way too. We need both specific examples and generic models to help us converge to a deeper understanding of topics.

True. I feel, this is why sometimes, the notation a_0 + a_1 + ... + a_n is easier to understand than the shorter sum(i = 0 to n) a_i . But on the other hand, 2 or 3 elements of the sum are usually enough, i.e. you probably wouldn't improve understanding by writing out the first 10 elements or so.

Neither of those forms is sufficient information to solve a problem, except for a "guess what I'm thinking" puzzle.

You need:

   sum(i = 0 to n) *f*(i)
and you'd like:

   Example: f(0) = *a_0*, f(1) = *a_1*, f(2) = *a_2*
where the stuff in *italics* is given as a concrete substitution.

Re: The Need to Grind Concrete Examples Before Jumping Up a Level of Abstraction

#72

Post by gowld »

pitched wrote: post #15
erikerikson wrote: post #14

The author is generalizing their preference here to say it's "right". Some brains need this, others need the abstraction before the examples provide the most benefit. As noted by others this preference influences how one learns code effectively too. It's a pretty basic trait. The author's stated preference is most common but it is not the only one.

Not sure I’m following here. Happen to have a concrete example of when abstracting first is more helpful?

The abstraction can clarify the boundaries and scope.

For example:

  I'm thinking of a *number* between 1 and 10, for example, 3 or 7.
Is 1 allowed? Is 10 allowed? non-integers? Irrationals?

A definition and theorems can seem trivial, if you latch onto the first trivial example and treat it as the canonical situation. This is very common difficulty when learning intro abstract math.

Re: The Need to Grind Concrete Examples Before Jumping Up a Level of Abstraction

#73

Post by gowld »

fiddlerwoaroof wrote: post #18
danielmarkbruce wrote: post #13

Earlier quoted context omitted.

It's not just for you. It's the most effective way. I know several people who all crushed college etc and only later realized maybe they aren't so smart, maybe they just stumbled into the right way(by an order of magnitude) to study... hell, I'd redo problems and find depth or nuance in the same problem that I didn't see the first time.

I think a lot of people do college too young: if you can find a reasonable “real” job, a gap year or two will give you some life experience and a chance to mature a bit, which puts you in a much better place for actually learning.

A real job is usually not learning academics, so it won't make people better at the raw learning. It might help in other, social ways.

Re: The Need to Grind Concrete Examples Before Jumping Up a Level of Abstraction

#74

Post by nhlx2 »

klysm wrote: post #44
nhlx2 wrote: post #43

Similar to grinding scales on a music instrument.

That feels different to me. Grinding scales helps with muscle memory and technique. There’s certainly aspects of that with math, especially with algebraic manipulations. Doing math problems can yield a deeper understanding of the underlying concepts and how they behave. Thinking you understand doesn’t cut it

Individual scales are concrete examples of diatonic scales, pentatonic scales etc.
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