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A-hah! learning and Grind learning - Elf M. Sternberg
A-hah! learning and Grind learning

I have two modes of learning, and one I enjoy, and one I resent mildly.


“A-hah!” learning happens from time to time while I’m working through a larger problem in my head. One might call it the holistic learning moment; I’m reading a textbook or paper and suddenly I get it. I not only understand what the author is trying to say, but I get the ramifications of it. This happens a lot when I’m reading about type systems and it clicks for me that a type system is an additional language, with it’s own strengths and weaknesses, that helps the developer say what the program means, and more importantly, lets the programmer talk about the genericity and applicability of his code. These come in a flash, as if several different bundles of neurons that had been slowly growing toward each other suddenly connected and something suddenly makes a lot of sense.

The Grind

“Grind” learning happens when I have to learn something that should have been an “a-hah” event but wasn’t, and I have to repeat the lesson over and over until I internalize and understand it. A recent example is in my Language in 20 Minutes exercise. the hardest thing for me to get was the difference between dynamic and lexical scoping from within the interpreter. I had to write, rewrite, and rewrite again the way a functions’s free variables are informed by its lexical scope (the closure), and the way its bound variables are populated by values found in the current execution scope.

In some ways, the second problem was informed by the idea that I didn’t even have the notion of “bound” and “free” variables in my mind until I started reading up on the lambda calculus and trying to grok what it means and how it works. But even though I do have those terms now, it’s been difficult for me to internalize the ways of lexical and dynamic scoping.

It doesn’t seem to be a matter of problem size. The LangIn20 project is very elegant and doesn’t have many moving parts. SparQL is a large project that has a lot of moving parts; both were grinds to master.

I wonder what, in particular, contributes to a problem being an “A-hah!” versus a grind, and if I can turn more of the latter into the former.

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From: ext_1628379 Date: April 7th, 2015 12:18 am (UTC) (Link)


As a math teacher, many of my students expect to have all of their learning come in "a-hah!" moments. I am convinced that even these moments are actually grind moments in disguise.

You said,
These come in a flash, as if several different bundles of neurons that had been slowly growing toward each other suddenly connected and something suddenly makes a lot of sense.
I think it is the "slowly growing" neurons that evolve during the grind, and then they connect in a flash of insight(?), and your thoughts go skipping down the chains of interconnections you have just developed.

An anecdotal example: I am teaching calculus 3 this semester, and I am talking about multiple integrals (double and triple). I have a small class (~20 students) and several of them asked me about drawing the regions for integration, which means sketching a two or three dimensional set of equations to find a bounded region. I told them to use their computer graphing programs to do this, as it has taken me years of visualization to get to the point where I recognize regions from a set of equations. Someone in the front row pipes up and asks if they _have_ to use the computer. The answer is of course not, but are you sure you have the region correct if you don't? So I did an example of a triple integral that could be done any of six different ways, and told them to figure it out any way they wanted, the answer is 0.1

Now I _know_ that there are no particular benefits to doing this integral in any particular order but they don't (I had done it all six ways in my notes). So they are asking all sorts of questions about how to set it up, and then grinding through the iterated integrals. But the longer they spend working on the problem, the easier the next one will be. At some point, they will develop an intuition about the order of integration - whether the region or the integrand matters, whether the region is oriented in a particular way makes a difference, and so on. But the sudden ability to do this will only come once they have the mechanics of the idea mastered and they have "ground" on the earlier problems for long enough...

I once heard a similar story about a martial arts class, and "kicking from the knee". If you don't know that one, I can paraphrase it too. I hope you can find a way to shorten your grind phase.
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