Students and teachers alike may become frustrated by traditional process, walk away from the process, and cause learning to cease. In the long run, that is a disaster; we need bridges that stand the test of time; we need skyscrapers that withstand earthquakes. All rely on a firm grounding in traditional mathematics, based on the Law of Excluded Middle. The Real World Unlike numerous approaches to the abstract world, the Law of Excluded Middle does not hold in much of the real world. Politicians

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... may argue from one point of view that their agenda, and only their agenda, is the correct one: that there is no middle ground-only black or white with no shades of gray. Take it or leave it. Again, such an approach can lead to frustration for both the decision-maker and the citizen; furthermore, as with the situation in the abstract world, such frustration can lead to disaster in the implementation and enforcement of decisions and consequent laws. Where else do we see this sort of approach, where the Law of Excluded Middle is invoked inappropriately? Perhaps almost anywhere that humans interact with each other or with the environment. Think about The Law of Excluded Middle the next time your opinion differs with that of another. Is someone trying to use this Law when it does not apply? Can you create a situation to move the discussion from one extreme, into the Middle, where perhaps it belongs-as a transition from one extreme to another? Mathematical Models Given the contrasting situations between the abstract and real worlds in relation to the Law of Excluded Middle, how then does it make sense to attempt to use mathematics, based on that Law, to create models of real-world phenomena, not based on that Law? Thoughtful models typically consider this dilemma. One approach is to make the model fit the real-world situations in or near the Middle and have a procedure for reducing the effect of the outliers: fit the 'gray' and deal with the 'black' or 'white' situations. If creators and users of such models understand these limitations, useful results may be obtained. For example, models simplify complicated situations and thus may offer a clear picture of how structure and function might work interactively to create a smoothly functioning 33 nd year