Apologies is this is the wrong forum; point me to the right one if that's the case.

In this particular case, I'm studying limits (in an intro Calc course, and on Khan Academy) but the question applies to any problem-solving that involves algebra. I seem to make an inordinate number of minor errors in evaluating algebraic expressions--somewhere on the order of 1 error : 2 problems. The most common is dropping a negative sign along the way, or incorrectly reducing a simple fraction.

Now, it seems obvious that the answer is "practice", but I am concerned that I have some kind of bad habit that I keep practicing, since I am assuredly practicing a great deal, and keep running into this issue.

What steps can I take to stop making these errors, and to catch them before I hand in the tests?

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    $\begingroup$ If I'm keen on the correctness of algebra, I generally do it forwards, then backwards; e.g. if I split $\frac{a+b}{ab}$ into $\frac{1}a+\frac{1}b$, I might check it by combining $\frac{1}a+\frac{1}b$ into $\frac{a+b}{ab}$. You can do plenty of sanity checks like that. I also sometimes plug in constant for variables (i.e. if I know of a solution, I see if it works. If it fails at a step, it means the step is wrong). It probably doesn't really build mathematical ability to do this, but it does catch errors. $\endgroup$ – Milo Brandt Apr 14 '15 at 3:39

I have the same problem. This barely qualifies as an answer but here's 3 things I've found helped:

Slow down
I find that many, if not most, of my errors surface when I'm working a problem too fast. When I deliberately slow myself down, to the point where it feels like I'm dragging, I make fewer mistakes.

Do the problem twice
I will attempt a problem and solve it. Then, pretending as if I'd never solved it before, try solving it a second time. When doing this I sometimes catch errors.

It seems I produce fewer errors when I've achieved some threshold of focus. Subjectively, it feels like I'm synchronous with the problem and there's a kind of mutual working out happening; I'm working the problem and the problem is working me. Most of the time it's incredibly satisfying. Meditation has helped me develop my ability to focus. If you've never considered meditation or have avoided it because of the "New Age" patina or because you thought it mandated some specific (and looney) belief system then I recommend Vipassana meditation. You don't have to adopt any dogma to practice Vipassana. I think of it like exercise: What dogma do you have to adopt to enjoy the benefits of jogging?

I hope that helps.

  • $\begingroup$ Do the problem twice is a good check. It is even better to do it twice with a space of time between the executions. Otherwise you are more prone to make the same mistake twice. $\endgroup$ – Ross Millikan Apr 14 '15 at 3:46
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    $\begingroup$ Thanks. The funny thing is that I read this answer after sitting down to do a problem set and found that I made far fewer algebra errors...and had been meditating in the minutes immediately prior. $\endgroup$ – Ben Apr 15 '15 at 18:27

Limiting cases are a good check. If there is a parameter in the problem, think about what your answer means if it goes to $0$ or $\infty$ or the limits of the range. Often that simplifies your answer. If you don't get the right one, you have an error. Dimensional analysis is a powerful tool. If you have an equation to solve, let $x$ have a unit. This will force the constants to have units, too. Then if you add terms with different units, something is wrong. Solving a quadratic, you have $ax^2+bx+c=0$, so $a$ has units of inverse $x^2$, $b$ has units of inverse $x$, and $c$ is unitless. The quadratic formula respects this. If your answer doesn't, it is wrong.

A big one is plug your answer into the original equation. Does it check?

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    $\begingroup$ Thanks Ross, this is good advice in general, and I follow it in general. In fact, these sorts of moves are how I am able to identify the fact that I make the mistakes. The question that I'm trying to solve, though, is how to reduce the occurrence of them in the first place. $\endgroup$ – Ben Apr 15 '15 at 18:25

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