Every few years I try this. It’s gotten to the point where I can no longer tell if this is actually helpful or illuminating, but below you’ll see the categories that I created when I tried to sort a bunch of mistakes that I’d logged on this site.
Enjoy, and please share any disagreements or alternate sortings that you see in the student work.
Mistakes Due To Limited Applicability of Models
Recursive rather than Relational Thinking
Circular rather than Rectangular Models of Fractions
Non-commutative rather than Commutative Model of Multiplication
Acting Out the Problem rather than Using a More Efficient Strategy
Mistakes Due To Applying Properties of a Familiar Model in an Less Familiar Situation
Linear properties applied in Non-Linear situation
One-Dimensional Distance applied in a Two-Dimensional Situation
Additive properties applied in Multiplicative situation
Side-times-Side Formula for Finding Area Applied in non-Rectangles
Area Properties Applied to Perimeter
Properties of some paradigmatic example of a shape applied globally 
Properties of a Fractional Parts of a Rectangle Applied To Other Shapes
Mistakes Due to Quickly Associating Something In Place Of Another
Squares Instead of Square Roots
Multiplying In Place of Exponentiation
Addition In Place of Multiplication
Changing the Numbers of the Problem
Operating on the “Answer” in an Open Sentence Problem
 This is a very mushed-together category. I’ve fallen into the trap of giving geometry short-shrift in the face of arithmetic and algebra. In general, I understand geometry thinking less well than I understand arithmetic/algebraic thinking. That category of “Properties of Shapes Overextended…” needs some serious breaking-down.
6 replies on “Classifying Math Mistakes”
Great work. This is going to make for amazing discussion in our teacher prep classes.
Yeah? I really wasn’t sure whether this sort of scheme is true/useful.
I suppose my biggest concern is about the distinction between “you’re using too weak of a model” and “you’re applying the properties of one model to a situation where they don’t apply.” I’m a bit worried that this distinction is hard to make precise, and that makes me worry that it’s fake.
I’m going to keep on worrying about that.
I’m also going to worry that I short-shrifted geometry by collapsing a lot of different things into “apply properties of some paradigm shape to a larger class of shapes.” I take it as a lesson of the van Hieles that this is how geometric conceptions develop, but we can get into a lot more detail, I think, e.g. “apply properties of a regular n-gon to all gons of same n” or something.
It’s not so much that your framework is true, but I like that it suggests teaching responses, and that even the act of classification is a good generalization for teaching. We know those math connections help our students and it just has to be (=”untested theory”) that the same kind of thinking will make us better teachers.
Maybe an additional characteristic could be progressing mistakes (like applying model in a new context) vs. dead-end mistakes (applying area ideas to perimeter or squares instead of square roots). Hmm. Maybe those are conceptual blind spots.
The more I think about (this tab has been open since you posted) the more I want to see these in a Fosnot&Dolk type landscape: concepts, models, strategies.
I’ve got the Fosnot books on my shelf, but I’ve never dived deeply into them. I’d be interested in seeing your take on applying either your distinction or the Fosnot framework to this analysis.
I think there is a distinction in the first two categories — I view one as a mistake of invalid reasoning, and the other is not.
I would not call using a weak model when a better one would do a mistake of reasoning. Sound reasoning can be applied with a weak model — it just leads to limited progress, or maybe even a complete solution that uses more effort than was truly needed.
In contrast, I think the most common reasoning mistake is a misunderstanding of quantifiers. The most common type of this seems to be:
Giving a few examples and expecting that proves a general rule, or what amounts to the same thing, expecting a rule applies generally from experience with limited examples.
I would say applying a familiar model in a less-familiar situation is usually an example of this.
Another type of reasoning mistake, which I don’t think falls in your above categories, is using a false converse of a theorem. An example: solving a radical equation and not checking the “solutions” to see if they are actually solutions. The source of this problem is that the usual algebra work verifies “if this equation has a solution, then that solution must be x=8 or x=3” or something like that. I think everyone has, at some point, instinctively understood the work instead to say “if x=8 or x=3, then x is a solution of the equation”. But the truth of this converse doesn’t come for free, which is why we check the possible solutions.
[…] Here Michael Pershan looks at posts over the last few years and attempts to classify the student mathematical mistakes into categories. […]