I’m fired up today. I’m going to quasi-vent on my blog, and hopefully I will not offend too many people in the process. You’ve been warned.
When I walk into another teacher’s room and all of the students are silently working individually, I get sad. Forget the monotonous drudgery of textbook procedural homework. When students don’t talk to one another about mathematics, they most likely are not experiencing math as richly as they could if they were working together.
I had two separate and seemingly unrelated incidents today that frustrated me extensively so I’m hoping that the online community can give me some perspective in these areas.
First, Algebra 2. Solving equations. We were doing some practice on the whiteboards. We haven’t talked explicitly about solving quadratic equations but I wanted to do a little experiment.
Me: Solve x^2 + 1 = 37. [writing on the board]
A few were able to work to a solution, but it was the same students that I would have expected to do so regardless of how the question was presented.
Me: Ok, let’s try this a different way. I think of a number. I square it, then I add 2. My result is 27. What number was I thinking of?
Every single kid in the room was able to arrive and understand that the answer was 5. Some did this very quickly. Maybe you’re thinking “well, they didn’t know that -5 was also an answer.” News Flash! None of the kids remember that. None. They know the procedures they used before: quadratic formula, factoring, taking the square root, completing the square. But they have no idea why, they have no connection for what x really means and they have no conceptual understanding of a quadratic equation.
THE STEPS WIN, PEOPLE! The steps trump thinking. The steps trump number sense. The steps triumph over all.
Here’s a second example, equally as frustrating. I’m helping a student get caught up on his algebra assignments for another teacher’s class. I don’t teach this class, and I like this kid, so I don’t mind helping him at all.
So systems of linear equations. 8x + 9y = 15 and 5x – 2y = 17 (or some bolonga like that.)
Kid: I don’t get this, I mean, what is x? II know I can substitute numbers for it and get y, but what does it connect to?
Me: Read this word problem: You work two jobs. One you make $6/hr and the other $8/her. Last week you worked 14 hours and made $96. How long did you work at each job?
Kid: [3 minutes and an ounce of brain-sweat later] 8 at the first job, 6 at the second.
Seriously, to see this kid mentally crunch these numbers was magical. To him, that was common sense. To another kid, it might be a table. To a third, trial and improvement. Why can’t most kids do that? Because we (you) insist that they set up a system of linear equations every single time. And because the title of the section in the book is 3.1 Solving Systems of Linear Equations. And then we focus on the steps and the methods. Substitution. Linear combination. Elimination. Graphing. And then a year later, those are just fancy terms that math teachers use to make easy things difficult.
Procedural fluency is important, but it must be built on a foundation of conceptual understanding. The procedure should never lead the discussion, and in most high school math classrooms, it unfortunately is.