How can I reduce the occurrence of algebra mistakes in my solutions?

Question

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?

Answer 1

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.

Meditate
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.

Answer 2

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?