House of Eratosthenes

As the definition of “learning disability” has noticeably broadened in recent years, I’ve harbored the suspicion that, although I wasn’t diagnosed with one in my childhood, if I had it to do over again today I might be. Every now and then some evidence comes along that elevates this from a suspicion to a near certainty.

I saw a web ad about some kind of learning program, don’t remember what it was. Girl calls woman. The girl is doing her homework, and can’t remember how to find the area of a triangle. So the woman reminds the girl that the area of a triangle is one half the length of the base times the height; so, what you do is multiply the base times the height, and then divide by two. Then they grin at each other. This bugs me, although I imagine others would wonder why.

I’m looking at it from the point of view of the instruction service provided: The formula was translated into a sequence of steps. It has me wondering about supply and demand, because when I needed help with homework, this particular instruction was in great supply even though the demand wasn’t there. And that’s mostly a ditto in the other half of my world experience, helping my son — translate a formula into a sequence of steps? Negatori. Choose the formula that applies, maybe. Figure out which bit of information in the “too much information” section is ripe for disposal before the real work starts.

And I remember the area-of-a-triangle thing pretty clearly: I was obsessed, like a dog going after a bone just out of reach, with the why. How come it is that this always works? For those who are a few years past this level of education, you can do this with right triangles, acute triangles, isosceles triangles, scalene triangles.

What makes it so? What if the width of the triangle is much greater than the base; do you use the base, or the width? And why? How is it to be demonstrated?

A chirpy math tutor just walking me through the steps of the formula, wouldn’t have done anything to resolve this. And in a way, it’s not a good demonstration of the problem because if I was off in the weeds meandering through all this, and I was interrupted with a task to just work out the area of a triangle as expected, I would’ve been able to do it fine. But it certainly was, and is, a distraction. And the other kids weren’t even wondering about it. But, to my way of learning, this kind of thing is a vital prerequisite to just getting the concepts down cold.

Hey look, even Wikipedia has the same kind of diagram (in the public domain) that got me going on all that. They’ve got a triangle with a base that’s narrower than its width. But if you follow the area-of-triangle link I embedded above, they don’t say anything about this scenario, they just give you instructions.

And, their instructions are equally lacking. Base times height times one-half? Or width times height times one-half? As it happens, if you simply ignore ramifications and consequences and hypothetical scenarios and simply stick to the script you’ll get it right, because the correct decision to make is: base. But can you come up with a proof?

See, there are those among us who can’t consider the lesson learned, with any confidence, until we construct some kind of a proof. After all, the question might be on the test: Base is a foot, height is two feet, width is a whole mile or more. Base times height over two, or with times height over two? And if it’s base times height over two, giving us a final area of one square foot, then how can that be, how does it work?

This step-by-step procedure-driven learning is like making your mind into a rake, nimbly navigating across the surface of such problems, while there are those of us who are more inclined in aptitude to work like pitchforks. We’re better suited for probing the conceptual material all the way down to the bottom, breaking up the clods, getting it all sorted out. Not so agile with the surface-spanning. It’s not that we’re slower with the work, the problem is that we’ve identified more of it to be done. Once we’ve identified it, we lack the ability to skip past it on command. It’s got to do with how the learning is done; it’s got to do with sequencing.

So yeah, the learning-disability mania lately really upsets me. It’s difficult to exhaustively identify all that’s going on here, but one of the things that seems to me to be undeniable is that LDs are being identified first as anomalies — this kid over here, isn’t learning & behaving quite the same way as all those other kids. We’ll work out why that is at some later time, but for now the important thing to do is to treat him differently. Well, that’s not going in the right direction. In my day I was held to the same standards as the other kids, and I had to figure out on my own how to make it work. Yes, there was some stress involved, there was suffering, some of it on me and some of it on my parents, and my teachers, and ultimately my grades. And, my adaptation to the world around me will always be incomplete. But then again, that’s true of any of us, isn’t it? Don’t we all have our little crosses to bear? Aren’t we all just a bunch of strangers in a strange land, in some way? Individual experiences are unique by their very nature, aren’t they?

And how does it improve the situation, to treat kids differently? All you manage to accomplish then, is to remove the incentive for getting the work done that really has to get done. They need to learn to map things out in their own way, so it can be programmed into their uniquely-laid-out brain circuits; they have to take responsibility for whatever translation tasks have to get done. It’s their job.

Somehow, somewhere, at some time, we’ve been sold this bill of goods that it’s the school district’s job to catalog all the students according to this iconoclastic brain-circuitry-layout, and start up as many special education programs as have to be started up — very much like instructing in as many different languages as are manifested in the native tongues of all the student body. See, we skipped past a dialogue we needed to have there. I don’t think that’s the way it should work. And on that particular note, I don’t think I’m one voice in the wilderness, I think there are others who see it the same way. But there, as in many other things, it seems we’ve settled on the answer that the system has to be all-understanding and all-knowing, while the individuals just sort of bumble along in whatever way they think makes sense.

This does not make for a graduating class of capable, productive and society-ready adults, which is what we all say we really want. It tends to produce, in my opinion, the opposite of this. Script-kitties. Experts in following sequences of steps, and when they’re done, the result should be like such-and-such…but what if it doesn’t work someday, what then? They won’t have the skills to sort that out, but that will be “okay” because it won’t be their job. The system we’ve managed to put together, from all I’ve learned about it, is a system that’s pretty sure it will all somehow work out fine. But, that’s the thing about bureaucracies: If & when it doesn’t work out fine, it isn’t anybody’s fault.

I can remember having that proof demonstrated to me when I was a kid. Geometry was fun. It was one of my favorite things to do with my dad, who knew some funny proofs with embedded errors that made it seem as though you could prove crazy things like “all angles are right angles.”
- Texan99 | 03/07/2013 @ 17:26
Wow, here it is. You can find anything these days! Turns out it was from Lewis Carroll. http://www.jimloy.com/geometry/obtuse.htm
- Texan99 | 03/07/2013 @ 17:29
The trick is that the height is based on a 90-degree angle from the base, not the length of that left-side. If you brought that bottom-left angle to 90, while leaving the “longer” width (the hypotenuse) attached, you’d easily see that the hypotenuse simply cuts the rectangle in half, which is why the formula could be expressed as “half the area of a rectangle with these dimensions”. The angles adding up is “why” it works; any change in one necessitates a change in the other, so the area stays the same.

(Haven’t looked at the link. Geometry was my worst math-related subject, so I had to try harder to understand them. The above is how I remembered it.)
- Daniel | 03/11/2013 @ 17:35

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Area of a TriangleThursday, March 7th, 2013

Area of a Triangle
Thursday, March 7th, 2013