This is such a good question. I don’t have a great answer, and I’d like to try articulating why that is.

When people get in touch with me about this site, it’s often to talk about using the mistakes from this site in the classroom. As far as I can tell (and I can’t!), that’s how people who use this site tend to use the site. They take mistakes and ask their kids to analyze them. Why did this student make this mistake? Or, did this student make a mistake? What advice would you give them? What could they do better next time? And so on.

What’s the theory here? Why would this help learning?

Sometimes, when I’m talking to people, it sounds like people think that being aware of possible errors will safeguard students from future errors. Let’s call this type of instruction Teaching to Avoid Temptation. To teach this way is to ask students to reflect on errors, so that next time they won’t make them again. Will they be tempted to make those same mistakes? Maybe they will, but they’ll remember this conversation or their feedback on this last quiz and then they’ll know now to combine unlike terms or whatever.

As someone who spends all day working with children, I am skeptical that we can teach them to avoid temptations.

What we can do, though, is teach them some math that will help them think differently or more fluently about certain problems. Maybe analyzing and discussing math mistakes can do that?

I’m sure that some pieces of math mistakes can be great for teaching some new ways of thinking. But not all mistakes are fruitful for learning some math. What math could a kid learn from discussing how someone multiplied the base and power?

(Maybe I’m just not being imaginative enough?)

Anyway, as I was thinking about this I came up with two situations where a mistake can really liven up a whole-group conversation.

Situation 1: When there’s a wrong way of thinking that a lot of kids have, but you want an emotionally neutral setting to dispute it. So you invent a mistake (or you pull a mistake from this site) and discuss the wrongness of that mistake instead of one from your classroom.

Situation 2: When you want to isolate a strategy from the answer. Sometimes it’s hard to distinguish a strategy from a correct procedure. Drawing your students’ attention to a mistake that nonetheless tries something worthwhile might really help them focus on that worthwhile thing, maybe more than a correct attempt would.

The conversational work that kids will do would differ in those two situations. For Situation 1, kids are tasked with formulating justifications and reasons. (Is this right? If it’s wrong, why is this wrong? What would be right? Why would it be right?) For Situation 2, the work is articulating what was good about the solution attempt. That work might also involve using and practicing that helpful strategy. An easy move is to ask students to use that strategy to correctly complete the problem. Another is to ask students to use that strategy on a related problem, or a related set of problems.

That’s all I could come up with. You?

  • Anna Blinstein

    Those are the main ways that I use mistakes in my classroom, but one other way that I would add is in helping kids get better at the process of identifying errors. This is a little more general than your Situation 1 since it’s not necessarily focused on particular content, but rather on the ability to analyze a process, find logical flaws, and test the reasonableness of an answer, which are all skills I would want students developing and which analyzing possible mistakes helps them do. When I do this, I don’t necessarily tell them ahead of time that they’re looking at a mistake.

    • Bridget Dunbar

      I agree with Anna. Students tend to look at you funny when you ask them to review their work to find the error. The mentality is that “if I got it wrong, I must not know what I’m doing, so why are you asking me to go back and look at it.” In this case, explicitly teaching students how to go back through their work line by line in order to identify their own mistakes can possibly develop that attention to detail that so many students lack.

  • Jo

    I’ve never thought of this as Teaching to Avoid Temptation–I think I’m coming at it differently. I’ve really never thought of it as a way of preventing future mistakes. It has always been a way of coming at the deeper math in the problem for me. If you can analyze a mistake then I know you understand the math in a conceptual way and you’re not just following the procedure. If you can tell me that someone got that answer because they found the area instead of the perimeter then I know you understand the difference; if you tell me that the mistake is that someone added the denominators as well as the numerators then I know you have a pretty good understanding of what the denominator is. So, it’s never been about avoiding that particular mistake but about probing into why something works that way.

  • Depending on time constraints: One might consider “using errors as springboards for inquiry.” The quotation marks are because I am quoting from Borasi’s (1996) “Reconceiving Mathematics Instruction: A Focus on Errors” [Borasi having been a doctoral student under Stephen I. Brown of “The Art of Problem Posing” by Brown and Walter] in which 21 case studies of specific errors are explored. (E.g., 16/64 = 1/4 because you cancel the 6s from numerator and denominator!)

    Although I don’t think it’s included in Borasi’s book [on this matter, I may be mistaken!] an example would be the so-called “freshman’s dream” that:
    (a+b)^2 = a^2 + b^2.

    Is this ever true? (What if you replace 2 with n? Well, n=1 works; what about n=0? n=1/2?)

    There are many responses. One of them is to observe that the error implies:
    a^2 + 2ab + b^2 = a^2 + b^2, hence
    2ab = 0, i.e., ab = 0.

    And ab = 0 iff [a = 0 or b = 0].

    How would a student interpret this conclusion &/or its implications?

    Separately:
    You give the example of multiplying the base and exponent in an expression of the form a^b.

    Fine: You get ab.

    Is it ever true that a^b = ab? Yes, as a matter of fact it is.
    a = 0 is an example, as is b = 1. (Or combine them: a = 0 & b = 1.)

    a = 2 = b is, too.
    Are there others?

    Also, if it were generally true that a^b = ab, then we’d also have b^a = ba.
    And then: a^b = ab = ba = b^a would be true.
    Now, we know the general fact used here is wrong.
    But what about the conclusion? i.e., that a^b = b^a. Does that happen?

    Again, there are a couple of simple cases.
    But there’s also 2^4 = 4^2. (Both are 16.)
    That’s a pretty interesting example!

    Are there any others? [etc]

    • Michael Pershan

      These are interesting questions! To what extent, though, does their interest depend on their being common mistakes? What seems cool about these to me is the substantive math, not the mistakeness of the generalization.

      • Sorry, I don’t think that I am grasping your question [in the above comment]. You had written [in your original post]:

        “But not all mistakes are fruitful for learning some math. What math could a kid learn from discussing how someone multiplied the base and power? (Maybe I’m just not being imaginative enough?)”

        And so I tried to give a couple of examples in which mistakes could be fruitful for learning math (the freshman’s dream is a common mistake; the base & power multiplication one is literally from you) by indicating how follow-up questions could lead to “substantive math.”

        Maybe it will be of use if I respond instead to another item:
        “it sounds like people think that being aware of possible errors will safeguard students from future errors.”

        My thinking is that being aware of common errors can help safeguard students against, well, those common errors! And if you do this successfully (students aren’t misunderstanding how to use the equal sign; students aren’t distinguishing a triangle from “an upside-down triangle” [or an ice cream cone…], etc) then it’s not that students will stop making errors. We all make errors. (And if we don’t, then we aren’t challenging ourselves enough!)

        But if we know, in advance, what the common errors are, then let us try and prepare for them. Let us teach & scaffold & discuss in a way that might prevent common misconceptions, but let us also recognize that if we are successful [a big if!] then we will end up with un-common errors. And then let us use our collective imaginations to dream up how errors can lead to interesting mathematics.

        Common errors can provide a road map for what is ahead in our class discussions. And when we keep crashing into the same obstacle, then we need to update our maps. But when we get past the classic obstacles [whatever “classic” means for us at this time] then, okay, perhaps we arrive at our destination. Or perhaps there is some other strange obstacle, previously unrecorded, and we can use it as an opportunity to veer off into an interesting [mathematical] side-path.

        [Even if totally unrelated to your follow-up query, I hope the above has some use!]

        • Michael Pershan

          “But if we know, in advance, what the common errors are, then let us try and prepare for them.”

          Ah, I agree that it is helpful to know what common errors are and to plan with them in mind.

          But how do we teach students not to fall into these traps?

          One technique I see tossed around is the “Dead Puppy” technique. There are these posters that teachers post in their classrooms: “Don’t make these mistakes or this puppy will die!”

          The theory seems to be that kids will, though tempted to distribute an exponent, remember the warning issued by the poster and resist the temptation.

          Maybe that helps, maybe I’m just grumpy. But I think we should be in the business of teaching kids ways of thinking about stuff and giving them time to practice, not training them to ignore their temptations and intuitions.

          • …no, not the DPT. Or anything like it.

            I don’t think students are pre-programmed to “distribute an exponent.” That suggests to me a lack of detail in initial presentation (e.g., the notational significance, the importance of writing neatly — so that the numbers aren’t misaligned, motivating examples that don’t mislead in the way that 1^1 or 2^2 or 0^3 might, etc) or, perhaps, a lack of general sense-making (e.g., “when I’m faced with a math expression and don’t know what to do, I just try whatever operation I can”).

            I’m not sure if we are speaking [typing] past one another, but I’m enjoying the sample mistakes! (I looked over the volume ones just now; I’d probably sit down with that student to ask about their thinking process…)

  • Dan Meyer

    Let’s stipulate that problem posing can be generative and productive (Brown & Walter) and that interrogating contrasting cases help students attend to the deeper structure of problems (Schwartz). Then one approach we could take to #mathmistakes in class is to select a common wrong answer for a particular exercise and ask students to pose a problem for which that wrong answer was correct the correct answer.

    This is more valuable than problem posing w/o errors — eg. asking students to come up with a system of linear equations where (2,-5) is the intersection — because students wouldn’t have seen the contrasting case and experienced the operations already.

    • Michael Pershan

      I’m in! This could be a valuable activity, and I buy your argument that preserving the initial problem in the new problem can really draw out the contrast.

      The thing I don’t get is why it’s important that this be an error instead of just another case?

      If the problem is “What is the solution to this system of equations?” and kids mostly get it right — they answer (2, -5) and the answer really is (2, -5) — then I think it would still be valuable to ask them to pose a systems o’ equations problem that has (-1, -10) as a solution. And if a group of kids mostly get the question wrong — they answer (2, -5) and the answer really is (7, 409) — then what’s gained from asking them to pose a systems problem that has (2, -5) as a solution rather than (3, 9)?

      • Dan Meyer

        >> what’s gained from asking them to pose a systems problem that has (2, -5) as a solution rather than (3, 9)?

        If for no other reason, for the message it sends that errors are sources of interesting mathematical thought rather than sources of shame?

        • Michael Pershan

          I buy this, and for all my mathmistakiness I’m not thoughtful enough about these sort of things in my teaching.