Many real problems can be solved in a `one-off' way, tackling the particular case that is of immediate interest, then passing to the next task. However, if related problems are going to be posed in the future, then this `one-off' tactic is usually not the best way to proceed. We need to consider a general method--a strategy, or a methodology--for solving problems of a similar type. We endeavour to develop it to the point where it can be applied in a range of situations.
The specific, problem-focussed approach is sometimes referred to as `tactical research', while the more generalist approach is an aspect of `strategic research'. Here I discuss several examples of problems that are best solved through strategic research. In each, mathematical techniques are employed to obtain a broad understanding, and to suggest general methods of solution. The insights gained in this way are applied to produce problem-solving methodologies that have wide application.
The first example, on fractal methods for characterizing surface roughness, illustrates how mathematical theory has been used to develop methodology for solving a variety of curve and surface analysis problems, in financial, industrial and ecological settings.
An example, on image analysis, points out that problems in areas ranging from medical diagnosis to the quality of television reception can be solved using related methods.
My concluding example notes the substantial technological advances that have been achieved in the aerospace industry by applying mathematical theory.