Direct Instruction of Problem Solving

Let’s say that you (1) believe that direct instruction (i.e. explicit worked examples with a procedure followed by repeated practice) is key to instruction and that (2) as a mathematical goal, students should be able to solve problems that they have never seen before and are likely to get stuck on, at some point.

What would that classroom look like?

Well, you’d probably provide some explicit instruction at the beginning of class in how to solve a new problem. Except that, of course, there is no guaranteed procedure for solving new problems, so you’d probably be explicit that you’re modeling good moves that your students would practice using later. A worked example.

And then you’d probably give them a chance to work on a non-routine problem that uses some of the moves or strategies that you’ve explicated. Maybe a quick one in semi-whole group, so that students don’t reinforce a mislearned move and get feedback before trying a problem on their own.

And then you’d probably give practice. Except that practice with problem solving means getting stuck and then getting unstuck with some move, strategy or technique.

You’d give students feedback about how well students used the moves that you’ve explicitly taught.

Maybe I’m wrong, but this doesn’t sound so different to me from the classroom imagined by opponents of direct instruction.

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13 thoughts on “Direct Instruction of Problem Solving

  1. It’s not clear to me that proponents of direct instruction believe that classroom instruction should or can prepare students to solve novel problems. That task is often passed on until students achieve a higher level of mastery (undergrad or grad school). It’s sort of like how most music programs don’t have any goals around preparing students to create (or often even interpret or appreciate) music, merely to competently play the music of others. The argument you are making might end up implying that instruction that has as a goal students applying heuristics in novel situations can’t look like direct instruction, at least in the sense of highly chunked, very carefully scaffolded, I-do-we-do-you-do practice preceded by explicit explanations. Also, the timing of explicit explanation in relation to struggle is a difference between some models that seems to make a difference, at least in some research studies.

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    1. It’s not clear to me that proponents of direct instruction believe that classroom instruction should or can prepare students to solve novel problems.

      Agreed! But isn’t it good to know what we’re really arguing about?

      Also, the timing of explicit explanation in relation to struggle is a difference between some models that seems to make a difference, at least in some research studies.

      Interesting. I wonder if this has to do with teaching the cultural expectation that we struggle on problems in math. The idea that this is an important element of transfer is an idea I associate with Greeno.

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      1. In terms of the struggle, I was thinking more of the work of Manu Kapur, and the “productive failure” research that indicates that struggling to make sense of an ill-defined problem before being shown a tool to efficiently think through the problem leads to better long-term retention of the tool. Example: being asked to rank a bunch of tennis players based on a mass of data, and making sense of that situation using informal methods, leads to better retention of standard deviation when it is taught after going through the messy struggle. (The data is about performance on one aspect of the game; some players are very consistent, some less so; students are already familiar with measures of center but have no tools to quantify spread before the lesson).

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      2. The discussion in this paper is short and says it better than I could gloss: http://www.manukapur.com/wp40/wp-content/uploads/2015/05/CogSci08_PF_Kapur_etal.pdf but I liked the key word *discern* — productive failure prior to lecture may support students to discern key ideas in a well-structured lecture. They do point out that there’s a lot going on in the productive failure condition including collaboration, unscaffolded solving of carefully crafted problems, and delay of structure, and it’s not clear what each contributes.

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      3. Got it. I’ll take a closer look soon.

        One of the recent bits of reading that hit me hard was in Lesh & Zawojewski (2007) was their note that in many studies that show strong effects for metacognitive questioning it was unclear whether it was newly developed metacognition that was helping learning. Maybe kids were just spending more time thinking about the content? they ask.

        This is a new way of being skeptical for me, and I find it troubling, in a good way.

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  2. This is very interesting – starting with the title! I think one core conception is that “learning [can? should? ideally does?] look tidy” (which to me connects to the role of right answers in the learning process). Also I think it relates to some conceptions about math as a body of knowledge, and even more as a discipline (that is, the doing of math, more than the corpus of mathematical objects, relationships, and truths). Your description evokes strong memories for me of my own experiences as a solver and learner at Ross/OSU (parallel to PROMYS — hi Justin!) and later (when I became a teacher), at PCMI (hi Bowen!)

    The discussion in the comments also reminds me of the research on demonstrations in science classes — apparently without some kind of “activation” stage before the demonstration, watching it can actually REINFORCE misconceptions rather than repairing them (to the degree that students will mis-remember what happened in the demo as being consistent with their prior misconception!) I was just looking for the source on this and I couldn’t find it, but I think I initially found it through the work of Derek Muller (“Veritasium” on YouTube), so if someone reading this doesn’t point us there, maybe I will just email him to ask…

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  3. My guess (from recent conversations) is that direct instruction proponents would suggest that novel problem solving in [domain x] requires that students already know [domain x]. Strong content knowledge precedes problem solving, in other words.

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