“Problems are often open-ended, paradoxical, and sometimes unsolvable, and require investigation before one can come close to a solution” (Zeitz x). When problems in a field can be defined as “open-ended, paradoxical, and sometimes unsolvable,” (Zeitz x) one might be able to reasonably assume that the field is difficult to conceptualize. Problem solving is a field of mathematics specifically designed around solving mathematics-related problems in competitions. Unlike the academic competitions seen in movies like “Mean Girls” or on trivia shows like “Jeopardy,” math competitions like The William Lowell Putnam Mathematical Competition are more akin to taking a test, where contestants write out proofs to complex mathematical concepts within a time limit of a few hours or so. These questions can range from trivial mind benders to full-length proofs, so preparation becomes a key element for success in the field. One useful text to help students new to the field prepare is “Art and Craft of Problem Solving” by Paul Zeitz, and this text will be the primary source for metaphors to be referenced in this essay. But why would there be any need to find metaphors in a mathematics textbook? Well, similar to other complicated fields, problem solving relies heavily on metaphor to explain certain concepts because, as Zeitz states in the prologue of his textbook, “Some branches of mathematics have very long histories, with many standard symbols and words. Problem solving is not one of them” (Zeitz 3). In fact, at least in the case of Zeitz, metaphor is a necessity for imparting an understanding of problem solving concepts on readers because some of these problem solving concepts are so abstract. As such, an understan...
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...epts they don’t understand to elements of real life, elements ranging from the conceptual metaphor Arithmetic is Object Collection to chess boards or maps. Without metaphor it would be extremely hard to introduce new students to even the idea of problem solving, where even the shorter problems actually take more than three or four steps as opposed to the common single step problem that frequents calculus and pre-calculus classes. As can be seen through the heavy involvement of metaphor in Zeitz’s Art and Craft of Problem Solving, metaphor is an integral part of the education of the problem solving process, particularly for understanding the heuristics of problem solving, analyzing the metaphors involved with word problems to find the intended direction for solving the problem, and certainly in understanding the common theorems that are useful tools for competition.
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