Abstracting the Concrete

Jose VilsonEducation

Last week in the classroom, I started dreading the idea of the two worst words for any regular teacher in this country: test prep. I hate it because it’s a contrived barometer of what they’ve truly learned, and en masse, becomes the data for metrics used to evaluate student progress, teacher competency, school preparedness, and demographic success rates. Unfortunately, only the people on the bottom of the totem pole ever address the malleability of these tests; they change at the behest of the emperor’s needs and not actually creating a standard for what certain grade levels should learn at any given part of their academic careers.

So instead, I decided to pull out the hardest questions from their predictive assessments (I’m a rebel) and help them understand how to do it. Let me show you. I took a problem like this:

Irregular Polygon

and actually made it more multi-dimensional.

Irregular Polygon 2

Nothing too shabby, but it’s interesting to see the kids’ faces of bewilderment. At first, only the more spatially inclined got how to do it. Then, as I started to show how the whole picture is nothing but a bunch of rectangles, the rest of the kids who were paying attention from the previous week’s lesson on the properties of a rectangle (it took a little reprogramming) were able to decipher the code on their own (for those not geometrically inclined, a hint: all parallelograms, including rectangles have 2 pairs of parallel and equidistant sides).

Of course, then I got really off-the-wall and gave them a similar figure and gave them completely off dimensions. Just to use the diagram above, the 10 was where the 7 is, and I mixed it up with some decimals, so the smart-asses in the class quipped, “But Mr. V, why didn’t you just put the measures where they belonged?”

I laughed, and like the quixotic teacher we’re used to seeing in the movies, I retorted, “It’s not about what you see. It’s about the idea behind what you see. Yes, the lengths are totally mixed up, and you’re already distracted by the mis-measurements. But if you understand that the longest length is the sum of the 2 shorter lengths parallel to it, then this should be no problem for you. Same goes with the the widths. Now, on the test, none of the lengths will be drawn to scale, so are you going to break out your ruler for every problem? I really hope not. I’m just making it obvious that you need to use your arithmetic skills to figure this out.”

Of course, one or two of them were still incredulous, but the rest understood. I was making them think for once. They couldn’t take anything for granted, and that’s important. Instantly, I found buy-in. I even differentiated by breaking the kids up in groups, and handing them cards with specially-made problems. Then they took it upon themselves to break out some chart paper and deliver how they did it to the rest of the class. I allowed for it only if I got to ask them critical questions to each and every student. Done and done.