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Unformatted text preview: MATH 74, SPRING 2009 DAN BERWICK EVANS 1. Introduction: Linguistic and Cultural Gaps in Mathematics Why dont middle and lower division classes adequately prepare students for upper division mathematics? This question has troubled teachers and students alike. From the students perspective, as they move from lower division to upper division they notice a bizarre shift in focus: before we devoted our time to getting the right answer (typically a number or a function) whereas now, suddenly, these questions are no longer important. Instead, we start asking why things are true and how they are truewe start proving things. So why arent students ready for proofs? From the teachers perspective (or at least this teachers perspective) this stems from two gaps in lower division education: a language gap and a cultural gap. Typically the former is addressed in a class on logic. The latter is not recognized as frequently and has only grown because of a strange reluctance of math teachers to admit math is anything more than a cold way of calculating various quantities. Perhaps this is not really surprising; it is very difficult to impart the cultural significance of certain theorems to students for whom math is but a tool they will use elsewhere. Indeed, it is unclear if such ramblings have any place in helping students learn calculus or algebra, and often these students simply dont care where the ideas came from. The focus is on getting the right answer, and any technique that achieves this end will do. We will spend many weeks learning logic and trying to bridge the language gap. However, to begin I want to keep things a bit less formal and discuss the cultural gap between lower and upper division math. It is very difficult to describe to non-mathematicians why math is exciting and fun. Comments to this effect tend to make a person look like a lunatic at cocktail parties. So, like any graduate student worth his salt, I will steal someone elses words to describe the situation: 1 A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas . . . . The mathematicians patters like the painters or the poets must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics. Clearly Hardy is excited by math, but what exactly is he talking about? Any student who has suffered through integration techniques, diagonalized a large matrix, or factored a complicated cubic knows that mathematical calculation can often be very ugly. The point is that these calculations are not the essence of mathematics; they are merely jazzed up arithmetic. The meat of what we do is in understanding the why . Why do certain integration technique work? Why can a given matrix be diagonalized? And why are we even able to factor cubics? The goal of the mathematician is to find anbe diagonalized?...
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This note was uploaded on 05/04/2011 for the course MATH 73 taught by Professor Danberwick during the Spring '09 term at University of California, Berkeley.
- Spring '09