How to avoid careless mathematical errors?

I found this discussion on reddit “How to avoid careless mathematical errors?“:

Hi //math.

I am a high school student who happens to be VERY good at math, but who consistently fails to get As on tests due to careless errors. Most of the time, they come from forgetting a 0 after a decimal place, multiplying instead of dividing, putting a decimal point in the wrong place, or just factoring wrong. I actually had to drop a Precalc Honors class because I got Ds on tests from the sheer number of stupid mistakes I made, despite understanding the material very well.

I assume that this occurs because I work quickly, but if I work slowly, I run out of time on the test. Additionally, my handwriting is horrible, but there’s really nothing I can do about that. And even when I check my answers after finishing, I still miss my mistakes.

I know that this issue is going kill my college GPA, so I want to fix it as soon as I possibly can. What do you guys do to avoid stupid mistakes?

I read through the comments and found some really good ones:

  • First of all, everybody makes careless mistakes. All the time. Heck, a famous theorem in the field of partial differential equations had a simple sign error that nobody found for ten years! (The theorem turned out to be true anyways, thankfully.)
  • Hate to break it to you, but handwriting is a major cause of mental mistakes. Put in an hour of practice writing mathematics every day – forming numbers and letters slowly and accurately, and speeding up as the days go by. We all used to have shitty handwriting – look at any first-grader. Some of us still do; some of us have worked at it enough to fix the problem. When you say “there’s really nothing I can do about that” you’re defeating yourself before the battle’s even begun. Sun Tzu would be rolling in his grave right now. (As for handwriting: I’ve found my handwriting has improved somewhat by picking up a fountain pen, since it kinda forces you to be a little more careful – you should be able to get a disposable pilot vpen or similar at most newsagents, maybe try that.)
  • Develop in-place sanity checks. This is a hard one to do, which is why I saved it for last. Remember how when you learned how to solve problems with absolute values and square roots and such, sometimes you had to check all the possible solutions and throw out a few “trivial” solutions? Same basic principle here. You need to develop the habit of thinking to yourself, “Does that sound right?” and knowing how to judge if an expression sounds right. This is more subjective than I’d like, so I’ll try and give an example. Let’s say I ask you to find the inverse of the matrix [1,4;2,3]. You can work through all the row manipulations to get it to the identity and then work from there. But you know that your answer should consist of rational numbers whose denominators are +/- 1 or +/- 5, because the determinant of the matrix is -5. So immediately you have an in-place “sanity check”.
  • you should check your answers as often as possible. I would never find an inverse of a matrix without multiplying it by the original to check. In many cases you can check your work by a different method than the one you used to solve the problem. This is easy to find when there’s an obvious inverse to the problem: if you got A – B = C, check that C + B = A; if you got that the integral of f(x) is g(x), check that g'(x) = f(x). These are simple things you can do to ensure that your answer is right, that most people don’t do. If there’s no straightforward “inverse” operation, you may still be able to find an alternate solution method. Do some steps in a different order if that’s possible, plug in simple values at various points to verify your algebra, etc. You’ll have to be a little creative, but if you’re VERY good at math it might not be so hard for you.
  • Get used to your common mistakes. That way, you can think “In this situation, I usually do that mistake, so I’d better be extra careful on that point.”
  • Based on my experiences throughout high school and university, the majority of my errors were due to trying to do too many steps in my head. Write your logic down – it will be easier to follow-through and also to check later (if needed).

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