The purpose of economic models

I have an issue with the study of economics as a whole.

The emphasis of economists on elaborate mathematical models has been pilloried in a media disillusioned with economics' ability to predict recessions and such.

However, those articles have ripped on the math, and encouraged a more "common sense" view, rather than ripping on the way in which the math was used.

Common sense doesn't always serve well in economics. It's important, but many seemingly-logical things end up not making sense, or many seemingly-logical things conflict with one another. For example, any argument over protectionism fights against the "common sense" view that protecting American steelworkers helps Americans, or keeping home prices low via rent control helps Americans. Logical arguments can be made for taxation vs stimulus responses to recessions, and without strong theoretical and empirical work, it's impossible to determine what works.

However, the way in which math is USED is critical. In my opinion, because of the nature of necessary assumptions, math should be used to illustrate, not represent, in any possible case. Creating elaborate representations of consumer behavior under certain conditions is far less useful than illustrating a particular effect in a way that is robust to the assumptions made. This requires no less mathematical elegance, but it ensures that economic conclusions are painted in a realistic manner, instead of an elaborate, and internally consistent, but irrelevant manner (it's like the philosophy difference between validity and soundness - internally valid economic theory has its uses in certain situations where the assumptions fit, but externally sound economic theory is priceless).

I understand that in certain situations (especially macro), models need to be made, but assuming specific distributions or populations is far less useful than a Williamson-esque style of explication of the effects, neutral of population characteristics. Perhaps many, many models will have to be created to note consistencies, but the general idea is that amazing theories built like a house of cards on crappy assumptions (I'm looking at you, Fama, French, Markowitz, and the other proponents of efficient markets and efficient portfolios) are far less useful than the emphasis placed on them by economists. Macro models that are more agnostic to their own assumptions would also create less "Mulligan vs Krugman" sorts of arguments where both sides make stupid points based on models that are only partially relevant.