Don’t forget Geometry when teaching Algebra

Right now, Michigan educators are trying to sort out the implications of the state switching to endorsing (and paying to provide) the SAT to all high school juniors statewide instead of the ACT as it had previously done. The SAT relies very heavily on measuring algebra and data analysis. This leaves plane and 3D geometry, Trig and transformational geometry under the label “additional topics in mathematics.” The new SAT includes 6 questions from this category. Compared to closed to 10 times that from more algebraic categories.

This comes up in every SAT Info session I lead. Should we just stop teaching geometry? All Algebra? Could geometry become a senior-level elective? If it’s not going to be on the SAT, then what?

These questions reflect a variety of misconceptions about the role of testing in curriculum decisions. In fact, these are the same misconceptions that are driving an awful lot of the decisions that are being made. And in this case, I think there’s more at risk than simply over-testing our students.

It would be a real shame to see Geometry become seen as an unnecessary math class. Because it’s not.

And to illustrate that, I’m going to tell you a story about a conversation I had with my daughter. She’s 6.

She wanted to try multiplication. She’d heard the older kids at school talking about it. So, I taught her about it and let her try some simple problems to see if she understood. And she did, for the most part.

So, I started to not only make the numbers bigger, but also reverse the numbers for some problems that she’d already written down. She had already computed 2 x 3, and 4 x 2, and 5 x 3, but what about 3 x 2, and 2 x 4, and 3 x 5. Were those going to get the same answers as their reversed counterparts?

She predicted no. So, I told her to figure them out to show me if her prediction was right or wrong. To her surprise, she found that switching the factors doesn’t change the product.

“Daddy… why does that happen? It should change, shouldn’t it?”

Translation: Daddy, how do you prove the Commutative Property of Multiplication?

How would you prove it?

The geometry teacher in me thinks about area when I see two numbers multiplied. We can model (and often do) multiplication as an array. It’s just a rectangle, right? A rectangle with an area that is calculated by it’s length and width being multiplied.

What happens if we rotate the rectangle 90 degrees about it’s center point? Now it’s length and width are switched, but it’s area isn’t. Because rotations are rigid motions. The preimage and image are congruent. Congruent figures have the same area.

If l x w = A, then w x l = A. 

Geometry helps prove this. Geometry also helps support a variety of other algebraic ideas like transformations of functions within the different function families. Connections between slope and parallel and perpendicular lines. There’s also the outstanding applications of algebraic concepts that geometric situations can provide. Right triangle trig, for example, is often a wonderful review of writing and solving three variable formulas involving division and multiplication. (A consistent sticking point for lots of math learners.)

As a teacher who spent years watching how Geometry presents such an environment for real, effective and powerful mathematical growth, eliminating it will leave a lot of holes that math departments are used to geometry content filling.

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4 thoughts on “Don’t forget Geometry when teaching Algebra

    • Twitter comment suggested perhaps easing back on Geometry in favor of statistics at the high school level. I would prefer to see some of the Algebra replaced with some good data analysis and statistics provided statistics is designed with moderate tech-filled computation and a ton of analysis and application.

      But I agree with you that Algebra (as it is typically taught) is often simply computation and algorithmic memorization. That’s a blog post for another day.

  1. I loved teaching geometry to students who didn’t feel confident in math because it was an opportunity to start something fresh that didn’t have nearly so many prerequisites. Students found success who had all but given up on math. And talking to my daughter tonight she was frustrated about her last test (word problems she just could not decipher) but feeling proud of her current work (rotation in the coordinate plane, which she can visualize without issue). Students have to see that math is so much more than manipulation of symbols in equations!

    • Tina, I’ve had the EXACT same experience you describe with students who were typically considered (by themselves or others) to be “not-math-students”. That’s why I used to pull as much of the computation out of my first semester of geometry as possible. Lots of drawing, folding, measuring, predicting, writing, talking, asking questions to each other. I wanted them to see that there’s more to math than crunching numbers and plotting points.

      And some of them found talent in places they didn’t know they had it… and wouldn’t have considered “math talent.” Geometry provides these opportunities better than anywhere else, at least at the moment.

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