In my intermediate algebra class, we spent almost 2 weeks studying lines, slopes, and intercepts. Students were instructed on several techniques for graphing lines: plotting random points, using the x- and y-intercepts, and finally using the y-intercept and two other points. So, students came away thinking that these are three different methods of graphing a line when actually they are the same thing – using points to draw lines. And they end up doing a lot of calculations that only encourage arithmetic errors of many types.

They were then asked to solve a system of equations graphically. One student very painstakingly expressed each of the given equations in the slope/intercept form and THEN substituted values for x to calculate y. She didn’t get two things, at least, that I see.

First, expressing each equation in slope-intercept form makes it unnecessary to then plot points and get the graph. Once you know the slope and the y-interept it is easy to draw the line.

Second, if your plan is to choose a value for x and then calculate y, there is no reason to use the slope-intercept form to begin with. Just plug and chug in the given equation.

So, what could I have done differently to make sure that my students made that connection? Because I’m pretty lazy and I’m always looking for ways to make problems easier – I assume my students are the same way. But the fact is they have a lot of energy but expend it most inefficiently in their problem-solving.

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March 24, 2011 at 10:05 pm |

I’ll hit you back with a question: Why are we, as math teachers, so invested in what solution methods our students use? We worry about whether their methods are the most efficient or insightful or standard, etc.

Yet at the same time, we talk about the importance of problem solving. Isn’t your student solving problems?

I agree that the two insights you offer are useful ones. But are they

necessaryfor the task at hand?And here’s another question. How do you think she is thinking about this mathematical situation? Not the way you and I do, we agree on that. But maybe she sees the y=mx+b form as comfortable and familiar. So the first thing she wants to do is get her equation into that form. And then maybe she sees this as a function-put in x, get out y. And furthermore, maybe that input/output idea is less salient to her when the equation is in standard form.

Could it be that she’s not just inefficiently spinning her wheels, but is navigating her way through a mathematical landscape by using her own familiar landmarks?

And if so, what does that mean for you as her teacher? What is/could be your role in helping find other, shorter routes through that landscape?