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Does anyone have any experience giving students incorrectly "solved" math problems and asking them to identify this error? Being self-critical is one of the skills that I would like my students to take away from a remedial/intro college-level math class (you know, beyond numeracy, etc), but I am not sure how to support this kind of learning (besides using problems like "correct the following attempted solution.")

So two questions:

  1. Have you ever asked these kinds of problems? Did it help?

  2. Do you know of other strategies to support students learning how to check their own work?

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    $\begingroup$ I've done this sometimes. I have a website of math mistakes that might be helpful for you to draw on, thought it's mostly mistakes from younger kids. The site is mathmistakes.org $\endgroup$ – Michael Pershan Mar 16 '14 at 21:50
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    $\begingroup$ A similar question: matheducators.stackexchange.com/questions/2206/… $\endgroup$ – Aeryk May 24 '14 at 1:32
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    $\begingroup$ Math workbooks I used as a student had questions involving a math discussion between two people, and asked if they were right, or (if it was an argument) who was right....I think those questions are better, because students are not told "assume mistake, find it," but rather "is there a mistake? if so, what is it?" $\endgroup$ – Tutor Sep 2 '14 at 12:38
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Yes! I have used these a lot in an "intro to proofs" course. Typically, each weekly homework assignment has at least one problem of this variety, and I've written many like this for assignments and a text. Some thoughts about your posed questions and other ideas:

  1. Does this help? I think yes in several ways, but it's hard to tell. I don't put questions like these on exams, so the "becoming a better proof-reader" skill is something I expect students to pick up from the course, but it is not a major component on which they are assessed. I think that working on these exercises makes them better readers of their own work, but it seems nearly impossible to find a way to quantify or track this progress. I'm also not sure whether students are aware of whether these exercises are helpful. Many might see them as silly ("If we know it's wrong, why bother reading it?"), so I like to follow them up with a "Fix the proof, or give a counterexample to show why the claim is wrong".
  2. Other strategies? Not really, short of reminding and encouraging them constantly to do so. One idea, though: I've recently started devoting a bit of class time to discussing printouts that have actual student-written proofs from recent assignments. (I attached no names, but some students opted to share when it was their proof being discussed.) I noticed a very positive response from students after this, but I think this is because they're very curious about what the others are doing, and I'm not necessarily sure this will make them better readers of their own work. Still, I plan on repeating this kind of activity more and more often, and tweaking it to get a better understanding of what they get out of it.

Other thoughts:

  • Too many of these "find the flaw" problems are constructed by choosing a common error (dividing by zero, square-rooting an expression that is later evaluated to be negative, etc) and burying that error under symbolism and/or verbiage. This is unfortunate, and most students will then approach these problems from the viewpoint of, "Okay, I just need to know the right trick here". So, I've tried to come up with more interesting exercises that either (a) are based on actual student errors observed in the past, or (b) incorporate a flaw that's based on currently-learned material. For example, I have a "fun" problem that attempts to prove $\bigcup_{k\in\mathbb{N}} \mathcal{P}(\{1,2,\dots,k\}) = \mathcal{P}(\mathbb{N})$ (where $\mathcal{P}(S)$ is the power set of $S$). This one works out quite well because, usually, the students have trouble at first seeing why the claim is false; once I prod them into thinking about that (and disproving it with an example), they spot the error more easily.
  • Students have a strong tendency to merely disprove the claim, and not identify any logical/mathematical error in a proof. Sometimes, this can be obvious, even to them, but other times they get so twisted in their explanation of what went wrong that they end up writing a whole paragraph just to say, "This conclusion is wrong because ..." So if we go over these examples in class, I assume the role of the very stubborn yet erroneous proof-writer; I pretend that I am totally convinced my proof is right, and they have to work hard to show me why I'm wrong, and not just that I'm wrong. This role-playing activity really drives that point home, I've found.
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    $\begingroup$ I too have used that particular power set problem! Written up roughly as: P(Z+) = P{1} U P{1, 2} U P{1, 2, 3} U P{1, 2, 3, 4} U ... Since each of the sets on the right hand side is finite, we have written the power set of Z+ as a countable union of finite sets; this means it must be countable, i.e., P(Z+) is countable as desired. "Q.E.D." $\endgroup$ – Benjamin Dickman May 23 '14 at 23:13
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This will only address your first question.

I have a bit of experience with asking students to find mistakes. I have (only) done this in the classroom and have not asked students to find mistakes in anything written.

What I sometimes do is to tell the students that I will write down the solution and make three mistakes that they have to find. Most of the time the mistakes will be forgetting to close a parentheses or forgetting to write $\lim_{n\to \infty}$ when doing an improper integral. I might even make spelling mistakes. I haven't been able to evaluate how effective this all is, but I do believe that it helps. It targets the importance of correct notation. It also, in particular, works well as a way to get students to wake up during class and can be used as a tool to do just that.

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I use this regularly and have the impression it helps a lot in understanding arguments or getting missing points. Typical examples are

  • In induction proofs not checking the case $n=1$
  • In solving equations dividing by zero
  • Multiplying inequalities with negative numbers

What I try to do is to stress that this is a bad argument also by formulating a proof for a trivially false statement (like $1=0$). I do this because I always have a feeling that some students may not notice the mistake and try to memorize the proof.

Actually, this might be a slight problems: with this method you know how to recognize a mistake when you know there is one, but not necessarily become more self-critic.

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  • $\begingroup$ Thanks, András, I will have to keep that problem in mind. Perhaps give four proofs, one of which has an error, and have the students find the error? $\endgroup$ – David Steinberg Mar 18 '14 at 17:14
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I have done this when I have noticed a student blindly following a formula without checking the various limitations and assumptions involved. (Occasionally, I myself have been said student).

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This is a wonderful formative assessment (FA) tool that creates a greater amount of mathematical reasoning/communicating between students. The key to this technique of FA is creating a culture of self assessors and peer assessors while increasing their awareness of the factors of the concept you are covering. This will increase the amount feedback students can get in their own solutions. It also counters the culture of teaching mathematics as 'show 'n tell' teacher modelling "the way" to solve a problem which we are so want to do. I create 'student' work for the technique or pull the actual student work from their own approaches. Having a common set of solutions for students to discuss in their groups allow for a richer debrief if needed.

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FYI the (free, open source) textbook "Mathematical Reasoning: Writing and Proof" contains many exercises of this form: proofs with a variety of different kinds of errors in them.

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The other answers seem mostly to be about giving this sort of question in a proof-based course, but, in the spirit of your question (which seemed to be about more introductory courses), I have used this methodology with (I think) great success in Business Calculus, Calculus for Engineers, and Differential Equations (for Engineers). Basically, whenever I am preparing an exam question on which I find myself thinking "they'll all make the following important mistake", I convert it into a question of the form:

Your friend did the following ….
(30%) What mistake did your friend make?
(70%) What is the correct solution?

I haven't done any formal testing to see what impact it has on grades or, more importantly, understanding, but at the very least it forces them to recognise the mistake (rather than just making it, and getting no chance to remedy it).

Another, perhaps more pointed, tactic, which was suggested to me by a colleague, is to give students the chance to correct their own mistakes (submitted a week after an exam as part of that week's homework)—but to require them to explain what they did wrong and why, rather than just to give a correct answer. I like this in principle, but found it extremely hard to judge fairly whether something was a 'real' explanation.

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