I am teaching entry calculus to a bunch of students outside class (more like complementary to their math classes, without making much connections) and I can teach on a much more individual level than in a class.

There seems to be a tendency to overlook mistakes such as a minus sign instead of a plus sign. If the answer is -0.954 and I point out that it should be 0.954, some of them are taking it more lightly ("oh, it was just the sign"), while if the answer doesn't look similar, they spend more time and think more about why is it wrong.

This goes also when analysing deeper concepts. For example, in doing some rather practical applications of the material (such as applications of the limit in modeling/estimating some phenomena), they are taking mistakes in modelling much lighter than mistakes in the computation.

How to convince them that those are actually real mistakes, and not to be taken more lightly than other mistakes?

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    $\begingroup$ What, grading does not help? Wrong sign means wrong answer means "no credit for you," the task is not done, period. No points for playing. If they don't care neither about correct answers nor about good grades, why you should care more? $\endgroup$ – Rusty Core Dec 3 '18 at 21:26
  • $\begingroup$ It sounds like Paul isn't teaching them as part of an official class, and so might not be assigning grades on anything. $\endgroup$ – Traversal in the Forest Dec 3 '18 at 21:32
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    $\begingroup$ Grading, for sure. But Escalante also had cute little names for common mistakes (facemask or the like, often sports analogies). Makes the kids laugh but also with a gentle little bit of derogatory sting so they don't want to mess up again. [Note that this method can be used if grading is not an option.] Maybe "backwards man" or figure out something cute. $\endgroup$ – guest Dec 3 '18 at 21:34
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    $\begingroup$ @RustyCore As I mentioned (not sure how clear it was), this is a much more informal setting, not a classroom setting. Therefore, there is no credit system. To be more specific, we're talking about a bunch of kids that approached me, looking to learn more. And they indeed, are eager to learn since they do everything out of their own will (they simply send me the "homework" or setup a meeting time, i discuss it with them - no pressure or repercussions). As mentioned, they do want to learn, but take lightly some mistakes. Also, it's one to one, so it's not in a group setting either. $\endgroup$ – Paul92 Dec 3 '18 at 21:35
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    $\begingroup$ "why what they feel like are small mistakes (e.g. wrong sign) are important" - small mistakes??? Would they prefer to receive +\$20 or -\$20? Would they prefer to drive West instead of East? Would they prefer to accelerate instead of braking? Imagine an error like this made in a car throttle ECU. Speechless. $\endgroup$ – Rusty Core Dec 4 '18 at 0:33

This is an interesting puzzle, for sure! A few thoughts come to my mind.

The Joy of Peer Review

I'm a huge fan of having students review each other's work. All the research shows that if done properly, it is one of the most effective ways to reinforce existing knowledge and deepen learning.

Since you aren't working with these students in a group setting, this becomes more challenging. However, I still think it is possible. Make copies of their work as they bring it to you or take pictures of what they write on a whiteboard. Then present one student's work to another and say "so and so was working on this problem and came up with this solution. Do you agree with it? Why?"

Since our educational system places a high emphasis on original, independent work, I would recommend telling your students why you are doing this beforehand. My favorite talking points include:

  • Most "real" mathematics is done in groups. Adult mathematicians who discover interesting, cool things don't work by themselves.
  • If you are researching a new topic and find a solution online, you need to be able to understand if it works and point out any flaws that it has. This helps everyone do better math.
  • Thinking about work from this angle provides different exercise for their brains. Athletes don't only do one kind of work out and expect to get stronger. Flexible thinking will make them see different things.

Present them with a wide variety of other work. Make sure to give them correct solutions as well as incomplete or incorrect ones. You can also give them a correct solution but where work was not clearly shown. They can learn from all of these examples.

When they spend 30 minutes trying to understand why someone else's function doesn't graph the way that it should only to find that the sign was flipped, they will be more diligent with their own work.

Let Them Find Their Own Mistake

In addition to letting them find each other's mistakes, you can let them find their own just like they would if they were editing a paper for an English class.

When they come to show you work, don't instantly tell them whether their answer is right or wrong. Instead, ask them to prove its correctness. Again, you can cite the idea that mathematicians who do interesting work aren't working on problems where the answers are known already.

Ask them how they can prove it to you/themselves. If they aren't sure, you can offer suggestions:

  • Graph it!
  • Explain how they reached the answer
  • Solve it "backwards"

If you have a chance to gather several students together, you can have them "present" on problems and encourage them to ask hard questions, point out problems in each other's solutions, or express confusion with the way something was explained or proven.

They will get into a habit of checking their own work, which will make them even more independent as learners and will eliminate this sort of problem.

Good luck! It sounds like you have an awesome group of students!


Can I answer as a learner? Three observations.

(i) When I was at school, I was impatient to get on with learning the next bit. So I rushed through a maths exercise partly because of enthusiasm. Presumably the enthusiasm is something to be encouraged, but also steered in the right direction. (Can it be harnessed into enthusiasm for finding mistakes?)

(ii) I honestly thought that by doing as many steps as I could at once in my head, I was being good at maths. What I was actually doing was making sign errors and the like inevitable.

(iii) Finding misplaced minus signs and suchlike was hard. Most likely made worse by (ii). That was part of my reluctance to do it. I didn't actually believe that I could avoid the mistakes, however careful I was. So why try?

So, based on this historical sample of one person, what might work? What I needed to be taught was:

  • Practice thinking clearly, not thinking fast.
  • Don't do lots of steps at once. Do small enough steps for each one to be easy, and for there to be no doubt whether it's right or not. You'll also have more idea what you did.
  • If you do make the steps simple ones, it's much easier to check them afterwards and since you can find the errors, it's worth trying to after all.
  • If you go slowly enough to avoid the errors, maths is much more fun because you spend more time finding things out and less time wondering why it's not working.
  • $\begingroup$ "Can it be harnessed into enthusiasm for finding misrakes?" – Do you re-read your posts? $\endgroup$ – Rusty Core Dec 12 '18 at 5:59
  • $\begingroup$ @RustyCore Ha! Yes, I re-read obsessively in fact. But I made the mistake of posting from the Android app while overdue for an eye test. It doesn't allow zooming text and I can't see what I'm doing without a magnifier,which I then have to put down to make the correction which is now too tiny to see . . . what with that, and autogarble learning the typos as new words and inserting new horrors, it's quite easy for a word like misrakes to stay in. Which adds a new point to my post I suppose: do your maths in writing more than 2 mm high! (Typo now corrected.) $\endgroup$ – timtfj Dec 12 '18 at 12:12
  • $\begingroup$ @RustyCore The line pitch in my previous comment when unzoomed is almost exactly 2mm, making the actual letters about 1½ mm high. $\endgroup$ – timtfj Dec 12 '18 at 12:30
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    $\begingroup$ This is an interesting point. I don't know if it's due to a growing interest in the material itself and not in doing a pile of work or age or something else, but I noticed myself that, while I had the same tendency when I was younger, now I have a lot more patience for appreciating the results, thinking about them and so on. However, it might also be the case that right now I do more variate work, not 1000 exercises on the same topic. $\endgroup$ – Paul92 Dec 12 '18 at 13:04
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    $\begingroup$ I appreciate your personal insights. One interesting thing I heard recently was about Excel model creation (financial modeling). That basically every "step" should have a new column. IOW, not the super formula in one cell. but show the intermediate steps. I think it is good advice across many fields. $\endgroup$ – guest Dec 13 '18 at 21:31

It might help to give costly examples of such mistakes.

  • $\begingroup$ I believe that the Hubble telescope example was due to a sign error in the expression for its shape—and luckily, knowing the precise error made it possible to design corrective ootics (though I remember some initial worry that this might be impossible). $\endgroup$ – timtfj Dec 19 '18 at 0:36

Perhaps one approach would be to emphasize to them some general number sense and estimation. That way, they can get in the habit of stopping and thinking "Hmmm....does this answer seem reasonable?" before they circle it and move on. For example, an exponential growth problem could go wrong by this "simple mistake:" the student taking the rate of 4% and adding it to 1 to get 5 and then multiply the starting amount by 5 and moved on. So a population of 250 that grew by 4% ended up at $250(1+4\%)=1250$. Hopefully with some number sense, they will pause and re-evaluate their steps ("That seems like a really big jump for just a small 4%...it shouldn't be that big...Oh woops, I forgot to change 4% into 0.04 before adding the 1, it should really be $250(1+0.04)=260$, that looks better.")


Let me elaborate Rusty's last comment: you go to one of the students, and you tell him/her you'll give him/her -20 euros (or dollars, whatever the currency in your country), and you just open your hand.

The student will be waiting for you for the money but then you explain that, giving -20 euros means, (s)he has to give you 20 euros, and then you say:

"That's the importance of a sign!"

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    $\begingroup$ With the same idea, I would just give minus full credit for any question with the wrong sign. $\endgroup$ – Pere Dec 6 '18 at 12:22
  • $\begingroup$ If they're learning complex algebra you could give imaginary credits as well . . . $\endgroup$ – timtfj Dec 13 '18 at 21:59

I don't know "how deep" is your "course" and their knowledge, but a good demonstration of the importance of the sign might be an example with multiple parts, when you need to rely on your previous answers. Based on my failures, the complex analysis is a good example. There are contour integrals, where you need to break them apart into smaller ones. And once or twice I did miss the sign of an integral. And it did really mess up the final answer, for example, it could make the $2\pi + 2\pi =4 \pi$ into $2\pi -2\pi=0$, which is not just a sign error.

  • $\begingroup$ Ah, just as with relative phases in quantum mechanics. $\endgroup$ – Vandermonde Dec 5 '18 at 5:38

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