I’m vowing to spend the precious time I have left teaching observing other teacher’s classrooms. When I’m observing I have the luxury of really focusing on how students are approaching a problem and asking them about their thinking without worrying about the other students who are waiting to talk to me.
Today I sat with a group of students who were trying to solve the sides of a 30-60-90 triangle. Someone had told them the “rule” for figuring out the sides so they didn’t seem inclined to follow the teacher’s suggestion of figuring it out with the Pythagorean Theorem. When the teacher asked how they knew that the long leg should be the short leg times the square root of three the kid said “mind reading.”
The students I sat by knew that the long leg was 6, called the short leg x, and knew that x*3^(1/2)=6 was the equation they needed to solve. Up to this point I think that the main reasoning they used was asking someone else how to do it and attempting to memorize the formula.
Then they asked me “Should I cube both sides to solve for x?” I’m assuming that they remembered someone saying that to get rid of square roots you square both sides and didn’t realize that in this case all they needed to do was divide 6 by the square root of three. They are not able to see it as just another number that follows the same rules as all constants because of the square root. Something about the square root triggered the silly notion to cube both sides. And when it is written out like it sounds absurd but the student thought that because the three was under the radical they should cube to address the issue.
I’m not sure how to start. I cautioned the kids to think carefully about what they were doing and not applying a random rule that they remember about square roots in all situations. I haven’t brought this up with the teacher yet but it was incredibly interesting to see that when he explained the solution he assumed that the only hard part was rationalizing the denominator and totally missed that these kids didn’t even know what a square root sign looked like or what it meant. I don’t think that the 10th grade geometry teacher is at fault for this algebraic understanding, and I’m not even sure if our school is at fault.
We have a bunch of teachers with small classes who care about their students and know math and teaching. We have a smart and dedicated student body who have dedicated parents. And yet some are completely lost when it comes to basic algebraic operations. Is there something we can do to fix this? How do we even start? Or is this normal? I don’t even know how to begin to figure this out.