We shared some thoughts about generalizing a couple of weeks ago, but that was aimed at understanding the process and using it as individuals. What we’re offering now is a generalizing hack to help students get the benefits of generalizing without them having to fully understand how it works. (And let’s be honest…when it comes to useful tools, being able to use them is usually more important than understanding how they work. Learning tools are no different; understanding how a graphing calculator app works, for example, has precious little to do with the learning that can be done through using the app.)
There is a simple progression of steps that leads to being able to generalize knowledge:
- Solving a problem in a familiar context
- Solving a problem in a similar context
- Solving a problem in a more difficult (less similar and less familiar) context
- Solving a problem in a very difficult (least similar and least familiar) context
This based on the Methodology for Generalizing Knowledge and gets right to the heart of generalizing by building the ability to generalize concepts, processes, and principles. We call it Generalization after Four Problems.
The G4P Hack (so named because it looks cool) challenges students to take what for teachers is a fifth step:
Create your own hard problem
This is a hack because it doesn’t require that students first be aware of having taken Steps 1 through 4. In fact, one of the common insights students have when they create their own hard problem is that they can already take the first four steps (though they may not think of it that way).
Here’s how it works:
In a basic math course, groups of students are working on adding and subtracting mixed numbers. One group just finished solving a problem and says it was “really hard”.
Their instructor responds,
“Hmm. Why don’t you create another hard problem, and we’ll share it with the other groups to see if they can solve it.”
The group gets to work. Here’s how it goes:
Student 1: We could change the whole number, but even if we use a 10-digit number, it doesn’t matter. We all know how to add and subtract, no matter how big the number is.
Student 2: Yeah. We could use uncommon denominators, like 43 and 279 and that would be harder but even if there isn’t a common divisor, we all know how to make common denominators. It would just take more time.
Student 3: We could use an improper fraction like 17/3 so they’d have to carry and borrow…but that’s stuff we already know too.
The students share a realization that these are the ONLY three things they can do to make this kind of problem hard. And more importantly, that they know how to do all three of them.
Without being aware of it, the students quickly discarded the contexts that were most familiar, most similar, and even less similar (Steps 1, 2, and 3), as being known and easy. The G4P Hack focused them on the least familiar/similar context for a problem (Step 4). In doing the work to create that problem, the students identified all the variables this kind of problem can have.
How much more practice do these students need, given not just what they’ve done but their realization about what they know and can do? We would say NONE. They have successfully generalized adding and subtracting mixed numbers. There is no problem of this kind that they can’t solve AND THEY KNOW IT.
That’s the beauty of The G4P Hack! It is a simple tool that is a kind of magic wand from a teacher’s perspective. Challenge students to create their own hardest problem in whatever subject and area they’re currently focused on, and they’ll work their way through the steps of generalizing and hone in on the variables specific to the problem type. And like hiding vegetables in dessert, they’ll have gotten something that’s really good for them even as they had fun trying to stump other students. Or their teacher.
