# On Numbers and God

Arbtirary thoughts on nearly everything from a modernist poet, structural mathematician and functional programmer.

## Wednesday, March 5, 2008

### On Abstraction and Education

I had started writing a rather in-depth post a while ago, and sort of lost the drive... Here's how far I got. It is a complete thought (mostly), but it isn't everything I wanted to talk about... Ce'st la vie.

Abstraction...
Abstraction is at the heart of knowledge, at the heart of learning, and thus, education. It works as follows:
Start with something we have a concrete understanding of. Now, decide there are certain properties of this object that we are interested in, and everything else is unimportant. Now create (in your mind) a conceptual version of that object with only those properties you are interested in. This is the abstraction step. The beautiful thing is that you now can analyze those properties that are interesting. The other beautiful (and perhaps more useful) thing is that you can now find another concrete object which shares the interesting properties, and you will now know things about it. It doesn't matter whether any other property is different, there will be properties of this new object which you have already analyzed.

This is how education works. Abstraction. Take something the student knows and abstract the object so that you only pay attention to the properties which will tie it to the next thing they need to learn. Through this abstraction, the student has a lens through which to view the new, unknown object. A good educator will not only make an abstraction clear (not necessarily explicit), but will show the abstract relation between the new object and multiple known objects.

There are, of course, other very important things in education. I realized a few of these tonight as I was helping Danielle learn to program. Perhaps more important than clear abstraction is do not let the student take a passive role. This is reiterated everywhere, and I'm sure my readers (all 2 of you) will have heard it somewhere. But it tends to be meaningless babble if it isn't explained. So I will explain, using my experience this evening as an example.

First, you must force the student to think. This is, perhaps, the hardest part. It is especially difficult when the student is frustrated with the subject material (as is often the case when tutoring someone). A frustrated or disinterested student will not want to put in the mental effort required to draw connections to previous understanding (read: to abstract). If you can force the student to think about the material, she will move from the known ideas to the abstract without you, and you can focus on shaping the lens, and pointing out key differences between the previous and new material. So, how do you get someone to think?

An answer (among others) is the Socratic Method. Pose questions which lead the the student toward the knowledge you are trying to pass on. If she is stuck, either explain a little more information that is related to the answer, and re-ask the original question, or ask a new, more specific question. In this way, you provide them a context and framework with which to understand the new knowledge. Looking at our example: At the beginning of the night, Danielle was not only lost, but not even trying. As the night continued on, I first began explaining what I was doing to change what she had done (basically copied), and then began to ask her questions, leading her to write her own code, in her own style.

By the end of the night, she was finishing her code by herself with almost no prompting by me, and I was able to focus on correcting small details and providing notes that would make her code cleaner or more efficient. By forcing her to actively think, my position as instructor changed from "I'll show her what I'm doing" to "I'll provide additional knowledge to help her do this better." In essence, I changed from a position of carrying her along, to a position of pointing out paths for her. Granted, she still has a lot to learn, but I was helping her for less than 2 hours, and she went from knowing nearly nothing, to anticipating what I was going to tell her. The amazing thing is that by putting in that extra care and effort, it actually made it easier for me and in the end, easier for her. It was harder for both of us at the beginning but once she started to pick it up, it went very smoothly and efficiently. She accomplished more in the last 10 minutes than in the first 45.