How I do math that starts out as a mathematical expression
I learned how to do math on paper or blackboard, except for an interlude at a Montessori school, where we used physical media. After a while, any math problem that took the form of a listing of mathematical expressions was one to solve with successive string manipulations. The initial form implied a set of transformations and a write-up that I had to perform. By looking at what I just wrote, and plotting how to create a transformation closer to my final answer, but with just a few manipulations of the earlier expression, I would record successive approaches to the final answer.
"Move the expression there, put that number there, that symbol there, combine those numbers into a new number using that operator, write that new thing underneath that old thing, line up the equals signs, line up those numbers,". Repeat down the page as I write.
How I do math that starts out with a verbal description(for example, an algebra word problem)
Starting from the verbal description, get an idea of the mathematical expressions that I need to write to express what's given in the problem. Write those down. From there, go on to solve the mathematical expression with a successive transformation approach.
How I do math that starts out with a graphical description (for example, a graph of a function)
Similar to starting with a verbal description, get an idea of the mathematical expressions that I need to write to express what's given in the problem. From there, go on to solve the mathematical expression with a successive transformation approach.
NOTE: sometimes a verbal description could benefit from a picture, for example, in a physics problem. In that case, I would draw a picture of the physical system to help me identify the corresponding mathematical expressions to start with but also to help me feel like I "understand" what the verbal description depicts.
About internal visualizing vs using cognitive aids to do math
Cognitive aids reduce cognitive load for representing information. If you can choose between having an external picture that you can look at anytime of a graph, versus an internal picture of the same graph, then for most purposes, and to allow you the most freedom of operation in approaching a mathematics problem, I would suggest that you use the cognitive aid.
That's how I've always preferred to do math:
- write out a math expression or draw out the graph or diagram of the problem
- don't reduce write-outs of successive steps if that could lead to calculation errors
- keep it all neat on the page
For higher-level math, I think it makes sense to use a computer as much as you can to handle details and visualization, relying on its precision and memory while you concentrate on the identification and use of an algorithm that produces a useful solution to the problem.
Conclusions about how I do math
Basically, I do it the same way I have since I was little. I remember learning my times tables by memorization and writing them out, then paper and pencil work in class, then a bit of calculator work much later on with a TI-83, but mainly for large-number multiplication or division, or to check my work.
I took a graph theory class for my math minor, and I wish I still had my notes. I suspect that some of the answers were actually pictures, but I don't remember much of what I did in the class, it was 30 years ago.
Mathematics involving a computer is more or less the same. You write out equations, but you might be working with cell references or variable names or varying data sets, so you are basically stopping at the point that you turn a word or graphical problem description into a mathematical expression. Then you let the calculator or computer do the work.