There is much talk about “productive struggle” lately. The arguments tend to be in the “we’re all saying the same thing” polemic. Tacking on explicit instruction at the end of a “productive struggle” session in which students have not been successful is not what we’re saying. Nor are we saying that “just in time” learning is productive. This is where students are expected to find the information needed to solve the problem as they are working it.
Learning anything new involves some struggle. The initial stages of learning a new procedure or problem-solving technique usually involves imitation. And as anyone knows who has learned a skill through imitation—e.g., learning a dance step, bowling, golfing, playing an instrument—what looks easy is often more complicated than it looks. So too with math. Students will struggle to get the procedure down correctly at first.
Let me give you an example, again from algebra. I will provide basic and explicit instruction on a type of distance/rate problem, using diagrams and other techniques. At first I might ask “Two cars go in opposite directions from the same spot; one going 60 mph and the other 70 mph. How far apart will they be in 3 hours?” Students have been given the distance = rate x time equation, and they learn how to solve such problems—it is relatively straightforward.
After a few like this I will give a similarly structured problem, but this time a different part of the problem is missing. Specifically: “Two cars go in opposite directions from the same spot; one going 60 mph and the other 80 mph. How long will it take for them to be 420 miles apart?” Students will definitely struggle with this but with some prompts from the teacher, they are able to build on the previous problem.
Amanda Vanderheyden talks about this in her interview with Anna Stokke. In that interview Amanda talks about “acquisition instruction” which is that stage at which students are learning and understanding new things and building fluency with what they have just learned. In the beginning stages the acquisition tends to be difficult so it is especially essential to provide appropriate guidance. Leaving them entirely on their own might work for some, but for most, too much struggle will result in labored responses. It is not “teaching them to swim”. Ultimately, students set up the equation as 60x + 80x = 420, and they will get x = 3 hours.
Once students have reached mastery in a particular aspect of math, and are in what Vanderheyden refers to as the generalization and adaptation stage of learning, it is then appropriate to have them work on variations of these problems that require more work. Thus, problems in distance and rate increase with complexity as students increase their knowledge of tactics and can generalize and adopt them to other similar problems.
What I see happening in the arguments about “productive struggle” is a nod to Vanderheyden’s “acquisition instruction”, but which has traces of the late Grant Wiggins. Specifically, Wiggins held that problems that employ scaffolding are not “authentic” and do not teach students to solve problems in novel settings.
The arguments I’m seeing strike me more as “fallacy of the heap” reasoning that the progressive camp uses to say that the difference between novices and experts is not as black and white as it would appear. Problems that escalate in difficulty represent a continuum, and the fallacy of the heap then comes in to confuse the issue. With no clear distinctions being made between problems that are within the acquisition phase, and problems in the generalization and adaption stage of learning, the practice of giving inappropriate problems in the name of “productive struggle” will likely continue.
A test I’ve used to determine the appropriateness of a problem is the reaction students may have when seeing the answer. If the reaction is generally an “OHHH” or “I should have got that”, it is likely an appropriate problem. If, however, the reaction is “There’s no way I could have got that,” then the problem lies beyond the acquisition stage of instruction, and the struggle is not productive—i.e., there is no generalization that occurs, and no transfer of knowledge to other problems.
In short, struggling to learn the breaststroke is not the same as struggling to keep from drowning. The latter doesn’t teach you how to swim.