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 of overlooking 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 likely ("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 pheonmena), 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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    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? – Rusty Core Dec 3 at 21:26
  • It sounds like Paul isn't teaching them as part of an official class, and so might not be assigning grades on anything. – Traversal in the Forest Dec 3 at 21:32
  • 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. – guest Dec 3 at 21:34
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    @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. – Paul92 Dec 3 at 21:35
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    "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. – Rusty Core Dec 4 at 0:33
up vote 5 down vote accepted

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!

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It might help to give costly examples of such mistakes.

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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    With the same idea, I would just give minus full credit for any question with the wrong sign. – Pere Dec 6 at 12:22

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.

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  • Ah, just as with relative phases in quantum mechanics. – Vandermonde Dec 5 at 5:38

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