One of the perpetually educating things about educating boys is what they struggle to learn. Abbey boys are generally selected for their ability to think abstractly, but that’s a road that everyone walks in their own time and in their own way. Not to get too Boethian on you, but the road from praxis to theoria is individualized.
When I teach Form V (11th grade) students the nature of voluntariness in Aquinas, we hit a simple test case early on. Since I’m about to wrap the Aquinas material in the next two weeks, I figured I’d comment on it a little.
In discussing how the will works (I abbreviate it “V” for voluntas), Aquinas considers how V acts without acting. One of his objections plays a cute little sophistry with the negative to illustrate the matter.
Consider how to apply a negative to the act of the will. Start with a positive:
I will to read a book.
There are two ways to apply the negative, since it is just an adverb. There are two verbs there, willing and reading, and either may support the negative.
I do not will to read a book.
I will not to read a book.
The first negation, on will, implies a simple lack of willing. The second negation, on read, implies a willing directly contrary to reading a book. It’s the difference between “it never came up” and “you can’t make me.” For Aquinas this is the difference between non-voluntariness and involuntariness. If you are willing to do a little violence to the Latin, you can express it by forcing a distinction between non volo and nolo (the objection, should you care, was collapsing the two).
Back to Abbey boys. It’s fascinating to see how the boys respond to this. They are all studying pre-calculus at this point, so it’s not like they have not been working abstractly in other areas. But the differences in how quickly the boys get this are fascinating. Some, the ones we would typically describe as simply brilliant, see the idea immediately. Others just need a little time, and some on-board scaffolding or re-wording to get it. But others hate this distinction because it’s just symbols dancing around on a page.
Another way to see how well they reason abstractly is when we return to this idea a month later. In I-II Q19 a2, Aquinas shows that circumstances do not make an act of the will good or bad using exactly the same reasoning.
Start with a positive act of the will:
I will to play video games.
Now add an essential circumstance which could make that act evil, “at the wrong time” (like when you should be at Mass, or when you should be studying, or something like that). Again, you can distribute that adverb to either verb, so that I’m either willing at the wrong time or playing at the wrong time:
At the wrong time, I will to play video games.
I will to play, at the wrong time, video games.
We needn’t worry about the direct aim of Aquinas here (I would need to pick a different object to illustrate his own point in Q19 a2). What I’m interested in, again, is my students. The same brilliant ones, plus a few who learn quickly, make the link back to Q6 a3 on their own. A great many others get it as soon as I remind them of the distributed negative. But a frustrating number groan and don’t get it. For some I think it’s the affront of having to remember old material from a month ago, but for quite a few others it’s the same problem as before: they really don’t get it.
I speak with one of my math colleagues often about the varying abilities with abstraction among our students. He’s our resident crusader for giving kids time to grow in this respect, because you can’t force abstract growth. As long as you don’t get a kid to give up by sending him through the meat grinder too many times and convince him he’s stupid, he will eventually be able to do the abstraction necessary for algebra. But there’s no forcing the growth; it’s a matter of steady repetition, correction, and patience. His classic algebra example, corresponding to mine above, is this chestnut:
(X squared + 4) is not equal to (X + 2) squared.
(Sorry, I’m not digging into math tools and taking a screenshot to paste in)
Anyway, he has the same experience I do with kids, although he’s working at the Form II-III level (8th and 9th grade). Even kids in Algebra II don’t all get this, and there’s nothing to do but keep sending them up and let them grow in their own time. But because of the dramatic difference in emphasis in our culture on the purpose and value of math and philosophy/theology, we have very, very different experiences in how this “problem” plays out. But that’s a post for another day.
Boy and abstraction. Always an adventure.