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Many students have a fairly good grasp of the topics they are learning but fall down because they miss fatal errors in their work. Some don't check for errors at all, while many simply can't find them. Indeed, many of them get very frustrated when they know they are wrong somewhere (perhaps because they checked at the back of the book) but can't find where.

The skill of checking your own work is a very useful skill, but as with any skill it is easier to get good at with some specific strategies.

I am specifically interested in problems involving some sort of calculation such as integration, solving equations, finding a basis for a subspace. (Proofs are an entirely different thing and have different sorts of "errors".)

What strategies can be taught to help students find and correct their own errors, and how can they be taught to students?

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  • $\begingroup$ I had trouble picking appropriate tags for this one. Feel free to make suggestions/edits if you think it would help. $\endgroup$ – DavidButlerUofA Jan 17 '15 at 12:56
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    $\begingroup$ It might be helpful if you clarify what subset of students to which you are referring. I would imagine that different strategies exist for, say, Algebra students versus students learning proofs, for example. $\endgroup$ – Andrew Sanfratello Jan 17 '15 at 13:53
  • $\begingroup$ If you are lucky enough to have a "nice" answer (e.g. $\frac{3}{17}$ or $\frac{5\pi}{128}$) then often times you can pinpoint the place where your answer starts to diverge from the known correct answer. For instance, where did $\pi$ come from? Did any of my steps generate a value of $\pi$? But this is hard to teach in general--it's basically equivalent to debugging--i.e. finding your mistake (given that you even know that your solution is wrong). $\endgroup$ – Jared Jan 18 '15 at 5:17
  • $\begingroup$ @Jared, are there no strategies that you can use for debugging? $\endgroup$ – DavidButlerUofA Jan 19 '15 at 4:04
  • $\begingroup$ Please add the "students-mistakes" tag. Stack Exchange does not let me suggest it via an edit. $\endgroup$ – Jasper Jan 19 '15 at 15:57
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Teach the students to perform a sanity check at the end of every problem, and a check-by-substitution when practical at the end the problem. If either check fails, they can use a technique for finding errors, such as:

  • Rory's binary search for the mistaken step.
  • Keeping a "top 10 list" of most common mistakes, such as some of the following:

    ** (not) distributing a -1
    ** multiplying by 2 instead of squaring
    ** adding instead of subtracting (or vice versa)
    ** adding instead of multiplying (or vice versa)
    ** slipping a decimal
    ** rounding errors
    ** mixing up 2s and 5s, or 2s and 7s, or 3s and 5s, or 3s and 8s, or 0s and 8s.
    ** forgetting to "carry the 1" in addition or subtraction problems.
    ** transposing digits.
    ** "off by one" errors when setting up a problem.
    ** forgetting to change the direction of an inequality when multiplying by a negative number.
    ** accidentally dividing by zero, or not splitting a problem into parts
    ** mistakes when cross-multiplying ratios
    ** mistakes when adding/rationalizing/simplifying fractions.
    ** omitting units, or mis-converting units.
    ** confusing sines and cosines
    ** using the wrong formula for the area or volume of a shape.

  • Look for spots where any of the "top 10 list" operations were performed, and double-check them.

  • Encourage the students to keep track of some of their more common mistakes. If they notice a pattern, provide help. For example, if they are mixing up similar-looking digits, first try teaching them how to make numbers more clearly. (Drafting classes teach how to make very legible numbers.) If that fails, they might have dyslexia; there are programs that can help.
  • When working the problem, make a note when a step is hard or confusing. If the check-by-substitution did not work, double-check those steps.
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  • $\begingroup$ Gerardo's post-exam after-action reports seem to be along the lines of "Encourage the students to notice their more common mistakes." $\endgroup$ – Jasper Jan 21 '15 at 2:17
  • $\begingroup$ This top 10 list has 17 items on it. :-/ I guess that's why it's a "top 10 list" and not a top 10 list. As a more constructive comment, might I suggest making the "perform a sanity check" part of this answer stand out a bit more, perhaps by converting it to a bullet point? It's easy to overlook by heading directly to the bulleted list. $\endgroup$ – shoover May 26 '17 at 18:28
  • $\begingroup$ @shoover -- This is not a one-size-fits-all list. Instead, it is a pick-and-choose list. Some students have lots of trouble with some items on that list; other students have trouble with other items. Each student will have their own top ten list, with "some of the" items in the list. $\endgroup$ – Jasper May 26 '17 at 20:27
  • $\begingroup$ Ah, I skimmed right past the "some of" in "such as some of the following". $\endgroup$ – shoover May 26 '17 at 21:22
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When I first started asking students if they "checked their answers" on a test, a number of them asked me what I meant. This was specific to algebra, and I told them they should take the answer, say, the X intercepts, and put those back into the equation to see if it resulted in Y being zero. Many had never heard of this, and thought that checking simply meant to go back through every calculation.

Unfortunately, tests are often structured in a way that the average student barely has enough time to finish the exam, let alone time to check their work. Same for homework.

Still, I believe the practice of checking work is a good idea and should be encouraged.

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  • $\begingroup$ Agreed, as it can be part of the process. Often students are asked to solve say, 10, similar problems as a homework assignment. Perhaps 6-8 plus "show answer is correct" would be helpful. $\endgroup$ – JTP - Apologise to Monica Aug 30 '16 at 12:46
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I teach high-school calculus, and many questions can be checked by a graphing calculator. So my first strategy is to teach the use of the calculator and using it to check answers when possible.

However, often a student will see that his final answer disagrees with the calculator but he does not know which of his steps introduced an error.

I have found that I need two more strategies to avoid this. First, I encourage my students to check some of their steps if the final answer seems wrong. By using a "binary search" they can narrow the problem to just one step and sometimes even to a part of one step. They can then focus their attention to figure out the exact error.

Let me explain the search for errors more. The best steps to check are the key ones. (For example, in finding a normal line to a point given by its $x$ coordinate on the graph of a given function, the steps to check are the $y$ coordinate of the point, the derivative of the function, the slope of the graph at the point, the slope of the perpendicular line, and finally the equation of the normal line.) If there are no obvious key steps, the student should check a step near the middle of his calculations. If that step checks, ignore the steps in the first half and check the step halfway into the second half; if it does not check, ignore the second half and check the step halfway into the first half. At each stage decide which half of the remaining steps hold the error and cut that group into half. (For example, if there are 16 steps then one possible series of checks is 8 which doesn't check, 4 which does, 6 which doesn't, and 5 which doesn't. There then must be an error in step 5.) This halving strategy is similar to the "binary search" used in computer science. When a step is pinpointed, sometimes it can be broken down into smaller steps which can be checked.

Another strategy is to teach a mild distrust of the calculator. Sometimes the answer is correct and the calculator is wrong. I teach them several reasons the calculator can be wrong. The student may have mistyped the expression (the most common problem). The student may misunderstand the proper syntax ($xsin(x)$ is not interpreted as $x\cdot \sin(x)$ but as a new function $xsin$). And occasionally the grapher simply makes its own errors (smoothing out a discontinuous function, not extending a graph close enough to a singularity, etc.).

So, overall, I try to teach using the graphing calculator as a helpful but not entirely reliable tool whose main purpose is to check answers gotten in another way. I teach these concepts by doing them in practice problems at the board. I let students guide the solution, and sometimes they get it wrong. I often just proceed, and when we discover that we have gone wrong somewhere I demonstrate how to find the error. I deliberately choose some examples that lead to grapher error, often examples that came up in previous years. This way I do not even need to be embarrassed when I make a mistake: it leads to finding the bug, and I give extra credit to the person who discovered that there was a mistake.

Details: I use the TI-Nspire CX calculator, without a computer algebra system. So far this seems to have the best mix of power in graphing and in numerical calculations without allowing the student to avoid doing his own algebra. At least, it seems the best mix for high school precalculus and calculus. I use the Teacher Software to emulate the calculator on my school computer and project the results on the whiteboard. It took me years to get the funds for all this but I have a pretty decent setup now.

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  • $\begingroup$ What do you mean by "binary search"? I ask because this is the part I want - the part that is the actual strategy students can use to find their errors in their work - and I'd like a description of what this strategy involves and how you teach it to them. $\endgroup$ – DavidButlerUofA Jan 17 '15 at 21:18
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    $\begingroup$ @DavidButlerUofA: Good questions. I tried to be brief at first, but now see the additions to my answer. $\endgroup$ – Rory Daulton Jan 18 '15 at 0:50
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One possibility to encourage sanity checks is to practise sanity-check-type questions and include them in tests. An integration-related example could be:

A student has calculated the area bounded by $y=x^2$, the $x$-axis, $x = 0$ and $x = 4$ to be 128 square units. Without using an integral, explain why 128 square units is too large to be a correct answer.

Sketching the graph and calculating the area of the rectangle with base 4 and height 16 reveals the area to be at most 64 square units. (In fact, an improved upper bound thanks to convexity is 32, found by calculating the area of the triangle of base 4 and height 16.)

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    $\begingroup$ I really like these kinds of questions. They can engage high level reasoning and explaining skills. I have also used them as introduction questions for an exam to remind students of common errors and I think it helps them be mindful of their mistakes. $\endgroup$ – BBS Aug 24 '16 at 14:22
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Give your students a multi-step problem and 5 different "solutions" written by fictitious students (you, really).

Say:

Here are five different solutions to this problem. First, rank them from best to worst. Second, discuss your ranking with the student next to you and determine appropriate criteria for assessing solutions. Third, list what you believe the most common mistakes are and how one might prevent or catch them.

Then it's class discussion time.

Include in those solutions:

  • A range in the quality of solutions, from junk to A+ work.
  • The most common 2 or 3 mistakes.
  • A variety of checks, responses to the checks, and reasonableness judgments.
  • A variety of neatness, from practically LaTeX formatted to illegible scrawls. This can lead to a discussion on how to write down neat work and communicate mathematically on paper.
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First, let me endorse Jared's point that checking calculations and debugging computer programs have much in common. Good programmers build checks into their code before it evidences a bug.

Second, one general technique, which only works in circumstances where at least one variable is present, is: look at extreme values of the variables. An example from a program I was just writing this morning: I knew that as $h \to \infty$, a calculated angle $\theta \to 90^\circ$. So I placed that test into the code to halt with an error message if it was not (approximately) true.

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  • $\begingroup$ Do you think there is a decent way to "build checks" into your maths working? As someone with only a small amount of experience with computer programming, saying it's similar to debugging isn't particularly helpful to me for finding strategies for maths. $\endgroup$ – DavidButlerUofA Jan 19 '15 at 18:54
  • $\begingroup$ @DavidButlerUofA: I cannot speak so directly to the pedagogical issue, but certainly in my own math research I am constantly building in checks. Often they take this form: I know X to be true, so as I build a conjectural theory, when I reduce it back to what's known, it better show that X still holds. $\endgroup$ – Joseph O'Rourke Jan 19 '15 at 19:00
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This is something that I've been specifically grappling with in my college remedial algebra classes for the last few years. JoeTaxpayer's observations in his answer very much match my own (that many students have never heard of checking solutions to equations until I make a topic out of it -- in fact, I'm somewhat embarrassed by how many years I spent thinking it was "obvious" before I discovered this and made it a priority). The thing I want to add is:

One needs to get away from multiple-choice testing. If you already do this, that's great, but the institutional culture is such that many students see nothing but multiple-choice tests for the first 14+ years of math classes. The problem here is that it short-circuits the need for checking in the first place: if one's answer is in the multiple-choice options, then that itself serves as a kind of check. If it's close to an option then you're usually correct in picking that one. Otherwise, just picking an option at random is far less work than a proper check and justification. And yes, this is under a significant time constraint. So students never experience having something truly high-stakes depending on a single solution, with no outside safety net, and the value of really knowing that you're right.

Amazingly, I find that even if I do make this a priority in my elementary algebra classes, and promise (and follow through) that it's required on every single test all semester, and those tests are indeed not multiple-choice, many of my students simply give up on checking solutions, and leave that part on all the tests blank over and over again. "It's just too hard; I don't get it", they'll say; even after practicing the skill effectively every day for an entire semester. It's incredibly discouraging, and I wish there were some solid research on why that is.

Edit: I should say that there's an equivalent "block" that happens in our remedial college arithmetic classes when I emphasize checking answers by rounding/estimation on a daily basis. Students are agog, claim to have never heard of it before, cannot grok it all semester long, leave those sections blank on every test, etc.

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  • $\begingroup$ I wonder if students' struggles with error correction are partly caused by cognitive overload. They might need all their concentration just to deal with the required calculation, leaving nothing spare for picking up and using checking techniques. $\endgroup$ – J W Aug 20 '16 at 16:00
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One of the crucial points here is that verifying a result must be different than repeating the calculation, let me explain:

Verify that 221/12 = 18, rest 5: Instead of doing the calculation all over, calculate 12x18+5, which indeed equals 221.

Verify that -2 and 3 are solutions of x^2-x-6=0: Fill in the values: (-2)^2-(-2)-6 = 4+2-6 = 0, and 3^2-3-6 = 0

Verify that (x-1)^2+(y-2)^2=4 is a circle with center in (1,2) and radius 2 (bad example, I know, it's just for the sake of the argument). This means (for example) that the graph can't contain a point with a negative y-coordinate. Just try to put y=-1, then your equation becomes:

(x-1)^2+((-1)-2)^2=4
(x-1)^2+9=4
(x-1)^2=-5 => a square can't be negative, so no x-coordinate, hence no point with y=-1

...

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A pair of fresh eyes may see goodies and errors alike, that you cannot see yourself after writing the assignment.

Thus, I experiment with letting students peer-review (an anonmized fraction of) their upcoming assignment using the same rubrick as I intend to use when asessing their final submission. For this purpose, I have a class on Moodle (my highschool uses a local Danish LMS not supporting peer review).

Moodle, being open source and widely supported, has many innovative users worldwide, as for example the fine folks at UNSW Sydney, who have an excellent bunch of pages on teaching Students using Peer Review, supported with several video comments.

My experience tells that crediting students for their effort reviewing and assessing peers' work and inviting them to reflect on the benefits of actually doing some assessment themselves, increases retention and thus reduces the number of errors, as you were looking for.

Moodlecloud offers a commercial-sponsored 50-user licence for trying it out. You need to sysadmin a bit: Creating users and a course, enrolling users, before students have access to the course's ressources.

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