Mental Math Gone Wrong?

Maybe this was a good idea, maybe not.

I was trying to figure out how to start class. My 8th Graders have been studying the Pythagorean Theorem. I knew I wanted to start with some mental math* but wasn’t sure how to start.

This desire to often begin class with some mental math is, at this point, sort of an instinct. On the one hand you need instincts when you’re planning class, because otherwise everything takes forever as you get sucked into a recursive vortex of decision-making. But is it a good instinct? I don’t know how to think about that.

The way I teach the Pythagorean Theorem, being able to mentally chunk a tilted square into triangles and squares (rather than trying to count each square or triangle) is an important part of the skill. It helps kids quickly see the area of squares, freeing up their attention to focus on the relationship between the squares built on the sides of triangles.

Yesterday, we explicitly talked about the Pythagorean Theorem in terms of the area of squares built on a right triangle’s sides. The plan for class was for kids to get better at using it in all sorts of different problems.

So, I decided to build a string of squares built on the hypotenuses of right triangles, and ask kids to find the square-areas in sequence, building up to a generalization. We start: What’s this square’s area? Put a thumb up (please don’t wave a hand in someone else’s face) when you’ve decided. What is the area? How do you know? OK here’s your next tilted square, etc.

image (6).jpg
Sloppy picture so you know it’s the real deal

Here’s where my teaching got sort of mushy. The really important skill isn’t finding the area of tilted squares. What kids really are going to want to know, later on, is the Pythagorean relationship between right triangle sides and areas.

So here’s the question: did this string of problems draw attention to the important math?

Turns out, it didn’t. Kids made the generalization in the last step (as far as I could tell from eavesdropping on their conversations) entirely on the basis of the earlier examples. And those areas were found by chunking up the area. In other words, this was arithmetic-generalization. They didn’t use the Pythagorean relationship.

What were my options, when I realized this? I was happy that kids were able to mentally dissect these tilted-squares, but was a bit disappointed that they didn’t start noticing Pythagorus here. I lost a chance to help them try out using that relationship. Since the rest of the class was designed to help them practice this theorem, it became important for me to prompt their memory of it at the start.

What can you do, right? Impossible to predict kids perfectly. Except that I could have prompted the Pythagorean relationship after the first example didn’t go the way I expected it to. I could have said — after I made sure that students were not going to — that this tilted square’s area could be found using Pythagorus, and then I’m sure I would have gotten more kids to play out this relationship in their minds for the rest of the string.

That’s not what happened, though, so I weakly finished the string with my own personal observation that, hey, we could’ve used PT here. The kids shrugged. OK. I pulled out a quick problem that did prompt kids to use the Pythagorean Theorem, but by then I’m not sure I had everybody on board. We finish, and kids are getting jittery. We’ve used up* whatever whole-group learning time we were going to get at the start of class, so I started problem-solving time.

That’s definitely how I see things right now, at least. Again, I don’t know if this instinct is a good one.

Class went OK after that. But I’m still trying to figure out whether I did this right. Should I have designed the initial string differently? Should I have reacted differently?

(And, the sort of meta-question I have is what exactly it would mean for me — or any teacher — to know how to do this better. Where does that knowledge come from? Can it be shared?)

5 thoughts on “Mental Math Gone Wrong?

  1. I’m curious about the goal for this string. In one spot it sounded like the goal was to get fast at finding the area of tilted squares by noticing patterns and decomposing into squares and triangles (maybe as a prerequisite to doing some later Pythagorean calculations?). In a later spot it sounded like the goal was to notice that this is an application of the Pythagorean theorem? Or maybe to notice how the final area of the square was related to the shape of the triangle drawn around the outside?

    Was the n^2 + 1 case the only case? Or did kids look at different cases but treat each one as its own number pattern?


    1. Max, yeah, maybe that’s my problem, being confused about my goals. Maybe I was trying to do two things at once, and I ended up doing one.

      Is it bad to try to do two things at once?


      1. Is it bad to try to do two things at once? Maybe only when they are in conflict? Like it’s not bad to try to walk and chew gum, but it is bad to try to play the trombone and chew gum.

        So were these two goals in conflict? Maybe? Like wouldn’t the early research of cognitive load theory suggest that the kids are going to struggle to both attend to how to easily get the numbers and see the Pythagorean pattern behind the numbers?

        What CLT suggests to me is to repeat for squares at different slants and when kids are seeing the pattern across the patterns, then expect the Pythagorean connection as they try to predict the area in terms of outer square size and slant for any outer square and slant.

        Also! I have always found old Pythagoras to be a sly and annoying guy. Where on earth did he get the idea to compare squares on the sides or right triangles? Why should I be expected to understand that — why can’t I just memorize the formula and move on. His was a rare stroke of insight and I’m happy to reap the fruit of his labors. But now! Thinking about slants and outer squares it makes sense that there would be a relationship there. A provable one, at that! And then that relationship turning out to unlock all of trigonometry… well it’s almost as cool as discovering the relationship between complex numbers and trying to scale and rotate on the coordinate plane…


      2. What CLT suggests to me is to repeat for squares at different slants and when kids are seeing the pattern across the patterns, then expect the Pythagorean connection as they try to predict the area in terms of outer square size and slant for any outer square and slant.

        This is a nice idea!

        (By the way, I realize that I wasn’t clear enough. [I’m going to edit the post to make this clearer.] I wasn’t asking kids to discover the Pythagorean relationship here. The relationship was stated explicitly during the previous class. I was hoping kids would remember it and use it. The idea is still a nice one, though.)

        By the way, I’m with you — the tilted square perspective on PT is really lovely. I first learned about it through Malcolm Swan (may his memory be a blessing)’s Shell Center. Then I learned about it through Henri Picciotto’s Geometry Labs, and then finally in CMP’s PT unit. I’ve become a true believer!


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