soft question - How do you make less mistakes in math?

How do you make less mistakes in math? Do you try to be more alert, do you take your time more, or what? Usually I don't make that many mistakes, but sometimes (like now) I do math as I imagine I would do it if I was ever drunk. I just did a couple of problems and I'm confusing addition with multiplication, $\lt$ with $\le$ , and other stuff like this. I make most of my mistakes when I think about how to approach/solve a more open-ended or abstract problem, not when I actually write it down on paper.

Answer

I think that checking your work frequently is one of the best ways to deal with this. You can also check your work with varying degrees of formality: you can carefully go back through every computation, but you can also just look back and forth between your current line of work and the previous one to check that things seem to line up. This more informal form of checking is a good one to cultivate, because it doesn't take too much time, and can catch a lot of mistakes.

One example I use a lot: when multiplying out two expressions in parentheses, each with a lot of terms, I make sure that in the expanded expression I have the right number of terms; e.g multiplying $($two terms added together$)($three terms added together$)$ will give $($six terms added together$)$, and its pretty quick to check that you have six terms after multiplying out (quicker than computing the expansion all over again).

Related to this, another thing to try to practice is to read what you actually wrote (or what is actually written in the question), rather than what you think, or expect, to see there. (This is the basic problem with all proof-reading!) Concentration is important here, obviously, but simply being aware of the issue helps.

I think if you find that you really have trouble with concentration/alertness at a particular moment, taking a break and coming back to you work can save time in the long-term. Again, trying to cultivate a sense of what your current alertness and concentration level is can help. In general, just trying to be self-aware as you're working is helpful, and is something that you can get better at with practice.