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?
emergent math 
This is great and I think your answer is well thought out. It makes me think of the Cognitively Guided Instruction (CGI) research about the progressions students have as they’re learning. For example, when a student learns to add, they initially directly model all the objects in a story. So, 3 cookies and 5 cookies more often has pictures of 8 cookies. Eventually abstraction kicks in and they move to counting strategies where they begin by counting all of them (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13) then to counting on (5, 6, 7, 8, 9, 10, 11, 12, 13), then to counting from the larger (8, 9, 10, 11, 12, 13) and ultimately potentially to invented algorithms (8 = 3 + 5 so 5 + 5 is 10 and 10 + 3 is 13). In the case of a student not wanting to progress to a more advanced strategy, the best method I know of is to introduce a case that makes their strategy ridiculously inefficient and demands a new approach. For example, if you count on from the first number and not the largest, then 1 + 99 goes like 1, 2, 3, 4, …, 99. So, it begs a new strategy to go 99, 100.
In this particular case, maybe try something like “What # is 82.37 13.95% of?” I believe that their strategy will break down and they will realize that in some circumstances, another approach is needed. What do you think?