Had a nice, quick twitter conversation with Anna (@borschtwithanna) yesterday morning. Anna reached out with a question about providing methods in an inquiry-based classroom.

If you teach PBL or PrBL, do you teach students standard algorithms too? @emergentmath @SchettinoPBL @NatBanting

— Anna Blinstein (@Borschtwithanna) October 23, 2014

Anna was conflicted due to her students’ unwillingness to deviate from their inefficient problem-solving strategy. Rather than setting up an equation…

@emergentmath “What # is 82 14% of?” They will make elaborate drawings and many lines of work to figure it out.

— Anna Blinstein (@Borschtwithanna) October 23, 2014

Setting aside for the moment that this is actually a pretty good problem to have (students willing to draw diagrams to solve a problem, even at the cost of “efficiency”), it does circle back to the age-old question when it comes to a classroom steeped in problem-solving: “Yeah, but when do I actually teach?”

The answer to that particular question is “um, kinda whenever you feel like you need to or want to?” The answer to Anna’s question is pretty interesting though, and I’d be curious what you think about it. Personally, I never had students that were so tied to drawing diagrams to solve a problem, that they weren’t willing to utilize my admittedly more prescriptive method. I do have a potential ideas though.

Consider **Systems of Equations**. This is a topic that is particularly subject to the “efficient” method vs. “leave me alone I know how to solve it” method spectrum. Substitution, elimination, and graphing were all methods that students “had” to know (I’ll let you use matrices if you’d like, I’m good with just these three for now).

Anyway, so I’m supposed to teach these three different methods for solving the same genus of problems. I want kids to know all three methods (generally), but also want to give them the agency to solve a problem according to their preferred method. Here are a few possibilities to tackle this after all three methods are demonstrated:

1) Matching: Which method is most efficient?

OK so matching is kind of my go-to for any and all things scaffolding. It’s my default mode of building conceptual understanding and sneaking in old material (and sometimes new material!).

In this activity students cut out and post which method they think would be the most “efficient.”

Students could probably define “efficient” in several ways, which is ok in my book. Also, it’ll necessitate they know the ins and outs of all three methods.

2) Error finding and samples of work

This is another go-to of mine. Either find or fabricate a sample of work and simply have students interpret. If you’re looking to pump up particular methods, consider a gallery walk of sorts featuring multiple different methods to solve a particular problem. The good folks at MARS utilize this in several of their formative assessment lessons. These are from their lesson on systems.

Students are asked to discuss samples of student work and synthesize the thinking demonstrated, potentially even to the point of criticism.

That’s a couple different ways to address methodology and processes that may turn out to be more efficient, while still allowing for some agency and inquiry on the part of the student.

What do you have?