I just remembered today that I wrote a post about how the constant velocity lab in physics matched up nicely with the linear equations stuff in algebra, but then I never followed up on the results. Here goes:

I started the unit on systems of equations with this problem. It was awesome that they took to it so quickly. With very little help from me, each group came up with a testable solution to the problem. Some used the chart, some took it a step further and graphed the lines. “Interesting that they were able to come up with such specific solutions then” you might say. Well, what they did was approximate the solution with the graph or chart. Then some groups used guess and check to find the number, and some groups used technology to find the intersection (Desmos or TI-84).

Next time, I need to make sure to let them test their solution with the actual cars. That was the plan, but I never went back and did it, d’oh!

Anyways, in the past** the sequence of learning happened this way:**

1. Learn different techniques to solve a system of equations. Graphing first, then substitution, then elimination.

2. Practice those techniques on a bunch of problems.

3. Apply the techniques to a “real world” problem.

Even as I type this, it seems like a logical way to do it. You can’t *apply* a skill until you *master* the skill right? I think that’s why this continues to be the way that so many math teachers teach this unit. But my experience starting with the real world context showed me that they can and will find solutions to problems even if they didn’t start out with all the right skills. There seemed to be two types of students in general:

1. Students who understood the problem already and came up with a solution fairly quickly. These students should be allowed to use their class time for extending the problem into different contexts and doing whatever assessment I’ve planned for this skill so that they can move on or tutor other students.

2. Students who don’t really understand the problem or how to solve it. I think that the standard way of teaching this unit tries to cater to these kids. **But this way is better for them too.** First, they benefit from just watching a student (#1) solve the problem (as they work in groups to create the whiteboards). They also are (hopefully) intrinsically motivated to learn the skills because they see how they can be used to solve the problem. These students can use their class time learning the basic skills of using an equation to generate an input/output table, and then plotting those points to make a graph.

**Note:** *They* were the ones that wrote those equations on the boards we started with. In physics class (also with me), they started with just the cars, some plastic markers, and some questions. The ladder of abstraction was climbed from there.

This is terrific! I think your breaking down students into two groups and observing that figuring things out for themselves is better _even for those that don’t understand the problem or how to solve it_ is incredibly valuable. Almost as important is your comment that the traditional way (learn different techniques to solve a problem; practice those techniques on a bunch of problems; apply the techniques to a “real world” problem) _seems_ logical. I’m essentially teaching a class of one student in Algebra II right now — a real opportunity to do things differently, but I keep find myself following the textbook’s traditional and “logical” approach. Sigh. Polya says the student should have a reasonable share of the work; yes, and a reasonable share involves their doing some serious thinking, not just blindly following rules we’ve told them. As Dan Meyer says, be less helpful 🙂 .