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.
Christopher Danielson said it so beautifully yesterday:
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.
This hits home. I have been looking over my daughter’s algebra homework with her (trying to help). She just wants to plug numbers into equations, and insists the answers are correct even if they make no sense in the context of the problem. I think more along the lines of tables, and/or making trial and error attempts, and then backing into an equation, but that’s not how they roll at school.
Since you have asked for perspective, let me offer some from language arts. What you have described is a symptomatic across the language arts discipline as well. Like math, it is important to build a procedural foundation, but all too often it ends there. In language arts, students are more concerned with the right way to write (because that is what the teacher teaches– spelling, grammar, punctuation, organization…and that is what is easy to correct…but there is no life in the writing). Students need to be able to experiment and play with language without the tyranny of the red pen. Sometimes a fragment is bad; sometimes it is good; there is no “Never begin a sentence with ‘and’, ‘but’, ‘it’ or ‘because rule”, and it is ok to take liberties with language to create a meaningful message. The obligation is to teach the effectiveness of language in communicating an idea. Years from now, students will not remember the difference between a dependent and an independent clause. So? Can they write clearly, concisely and coherently? If so, then they have probably developed the courage to put ideas to paper.
PS: Any errors in the comments above are subordinate to the ideas.
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