Abstraction in Education

2019-09-12T02:18:47Z

Sadly, The Atlantic is shuttering its members' forums. Before everything I've written there has disappeared, I wanted to copy this particular post to my blog for posterity.

This post was written in response to this article in The Atlantic, which posits that American elementary education has gone wrong when it starts to over-value the general concept of "reading comprehension" (which allows kids to learn for themselves, in theory) above the idea of simply teaching kids about a wide variety of things:

Many teachers have told me that they’d like to spend more time on social studies and science, because their students clearly enjoy learning actual content. But they’ve been informed that teaching skills is the way to boost reading comprehension. Education policy makers and reformers have generally not questioned this approach and in fact, by elevating the importance of reading scores, have intensified it. Parents, like teachers, may object to the emphasis on “test prep,” but they haven’t focused on the more fundamental problem. If students lack the knowledge and vocabulary to understand the passages on reading tests, they won’t have an opportunity to demonstrate their skill in making inferences or finding the main idea. And if they arrive at high school without having been exposed to history or science, as is the case for many students from low-income families, they won’t be able to read and understand high-school-level materials.

My post in response was:

I wonder if part of the problem here is related to the way we value abstraction. In particular, I’m tempted to relate this to something I see when teaching mathematics at a tertiary level.

Modern higher math is greatly influenced by formalist philosophy. Formalist mathematics had its most influential proponent in David Hilbert, who is famously quoted as having remarked that, in geometry, “One must be able to say at all times – instead of points, straight lines, and planes – tables, chairs, and beer mugs.” Which is to say, the form of a mathematical statement, along with its logical structure, is considered more important than the actual content. To be maximally content-neutral, one specifies as little as possible in order to make ones logical deductions. Accordingly, those deductions will then be able to be applied to multiple situations at once, provided that those situations satisfy the given (minimal) specifications.

Confused? Probably. My statement is in desperate need of an example. Here’s one.

We quite like “undoing” functions. E.g. we can “undo” multipling by 3 by dividing by 3. We can “undo” squaring a positive number by taking its square root. But some functions can’t be “undone”. For example, if we’re allowed to have negative and positive numbers, then we can’t “undo” squaring, because we don’t know if the original number was positive or negative before we squared it. We would have got the same answer either way (e.g. (-2) squared is 4, just like 2 squared), so there’s no way to know which number we ought to go back to.

We can generalise this. If the thing we did sometimes gives the same answer from two different start points, then we can’t undo it reliably. This is the abstract statement; the case of squaring both positive and negative numbers is just one example.

Having found such an abstract statement, we are in a potentially powerful position. We can now talk reliably about things that we know very little about. For example, if you tell me that when you beer mug a table the result is exactly the same as when you beer mug a chair then I can tell you that the action beer mug cannot be mathematically “undone”. I don’t even need to know what it would mean to beer mug a chair in order to confidently come to that conclusion.

This is fun and powerful and also wildly, wildly overrated.

Let me quote myself again, from above:

To be maximally content-neutral, one specifies as little as possible in order to make ones logical deductions. Accordingly, those deductions will then be able to be applied to multiple situations at once, provided that those situations satisfy the given (minimal) specifications.

This is an abstract statement. And if you’re not well-versed in formalist mathematics, it probably looked like gibberish to you when you first read it above. It might even still read like gibberish to you. In theory, this abstraction is “superior” to the single example that I gave afterwards. After all, it is applicable to many millions of other examples, all at once. But actually, if you haven’t seen any of those examples, it’s a pretty useless statement.

Understanding an abstraction is a more powerful than understanding a single example. But abstractions cannot replace examples. Indeed, I personally find it impossible to understand an abstraction even superficially unless I already know at least one example. Deep understanding requires many examples. So it’s not enough to simply teach the abstractions, no matter how reliable and beautiful they may be.

Mathematicians tend to think that abstractions can be true in isolation, even without the presence of examples, and they may even be right about that, but human beings still generally find that abstractions cannot be understood in isolation. Failing to understand this distinction can lead to some incredibly bad mathematics teaching at the tertiary level.

To return to teaching reading to elementary schoolers…

It’s a wonderful skill to be able to learn things from reading. In particular, strong and experienced readers can learn things by reading that are completely outside their experience to date. A good reader can sometimes infer the meaning of a word just from context, even if they’ve never seen it before.

But it would be a mistake to jump from this to the idea that people don’t need to read about things that are within their experience, and for which they have examples ready to hand. On the contrary, the ability to read outside ones accustomed context is obtained by having read within a wide variety of well understood contexts.

That is, just as abstraction depends on examples, so too deep reading skill depends on knowledge about a wide variety of types of content.

comments

gemcode

Abstraction in Education