A long time ago I started thinking in general terms about how we go about solving problems. Here's the way I still see it.
Each level builds upon the previous one. We usually don't apply the same method to solve every problem. It should almost go without saying that we attain our highest level in our areas of expertise.
0. I've tried nothin', and I'm all out of ideas. This level comes by way of a Simpson's episode. Ned Flanders' beatnik mother said it to a psychologist. Unfortunately, I do think some people use this as a strategy.
1. Monkey See, Monkey Do. There really is no “understanding” of the solution here at all. It simply entails copying what someone else has done.
2. Trial and Error. Here you make a guess — sometimes educated, sometimes not. The result is then compared against the outcome to determine if a given solution is better or worse than the standard.
3. Intuition. Sometimes we do things without really being able to explain why. Often the results are quite good. I think of this as problem-solving on an almost subconscious level. The result is a synthesis of all of our experiences and everything we know. Personally, I find myself doing this quite a bit. It usually happens when I am too lazy (or very certain of the results) to actually do the math.
4. Engineering. At this level, calculation is required. The formulas are typically well known — or at least available in books.
5. Original Research. This is engineering where the formulas are not known (or published). You end up making theoretical models of behaviors and testing them against practical results.