I've been doing math for a long, long, long time. But I still make "stupid mistakes" such as dropping a minus sign, or forgetting to follow through with the chain rule in differentiation. I'm not even very good at basic arithmetic.
I guess part of the problem is that I never had to work hard at getting the right answer in school, so I never practiced a lot. I've also been diagnosed with ADHD, dysgraphia and other learning-related disabilities.
In one of his lectures Dr. Leonardo(sic) Susskind stated that he was copying his results from his notes because if he tried to do the work on the fly, he would just make a bunch of algebra mistakes. I say this to point out that even people of august standing in mathematical fields have problems executing basic manipulations flawlessly.
One approach to improving my skills might be to start with a first grade math book and work through all the problems assigned to students from K to 12. It's something on my bucket list, but I haven't gotten around to it.
In lieu of repeating 12 years of pre-college math, are there any suggestions for improving symbol manipulation skills? I know, and believe the old adage practice makes perfect. I'm wondering if there are any kinds of focused exercises which efficiently address mechanical math skills.
This is a very common problem. The trouble is that routine mathematical manipulation is tedious for many people, and they just want to get through it as quickly as possible to reach the (hopefully) more interesting result at the end—typically by cutting corners and doing two or three steps at a time. This is where the mistakes creep in.
I don't think that doing lots of simple exercises is a good idea. Boredom is the origin of the problem, and adding boring tasks to your work will neither decrease your level of boredom nor help to overcome the effects of boredom.
Rather, I would take on interesting mathematical problems that are at about the right level to present a real challenge but not too hard either. Just accept for the present that you will make mistakes at the manipulative stages as you hurry through them.
When things do go wrong, there is the choice of checking back to find the error(s) or scrapping the erroneous work and starting again. Whether checking back or redoing, you will gain insight into the sorts of mistakes that are so easy to make. With these insights, you will learn to make immediate checks that will save laboriously generating mounds of garbage later. Part of what is learnt will probably be that doing certain steps in a pedestrian way is typically quicker in the end than taking them in multi-step strides.
This is of course a form of practice, but it is practice with a motivational thread running through it.