The Meaning Behind the Symbols

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Categories: GSI Online LibraryTeaching Effectiveness Award Essays

by Aubrey Clayton, Mathematics

Teaching Effectiveness Award Essay, 2005

Mathematics is ultimately nothing more than a language of human ideas, but these ideas are expressed in an extremely concise and oftentimes apparently indecipherable code. As a result, many Calculus 1B students are frustrated by the amount of rote memorization they think is expected of them. Indeed, many textbooks reinforce this misconception by presenting much of the material without any motivation. So, for example, it’s natural for a student to feel overwhelmed by the definition of the limit of a convergent sequence of numbers: “A sequence s(n) is said to converge to a limit L if for all ε > 0 there exists N such that n > N implies | s(n) – L | < ε.” When they first see such a statement, students often give up and just do their best to survive the rest of the course. Therefore, my goal was to show the students that this definition is actually very natural and intuitive, and thus shouldn’t be dismissed as another useless fact to memorize.

My strategy to motivate this idea was to start with a problem and get the students to arrive at the definition without having seen it before. So, I challenged them to convince me that a concrete example of a sequence, the sequence 1/2, 3/4, 7/8, etc., was “tending” towards the number 1. Everyone agreed that it was, but no one could construct a persuasive argument. “What exactly does it mean to ‘converge’ to 1?” I asked. The conversation quickly focused on the differences between the sequence terms and the limiting value, and the class pointed out that these values were getting progressively smaller as the sequence continued. The difficulty, however, is that this value never reaches zero, and so the putative limit never actually appears in the sequence. The students struggled to find a convincing argument that it should still be the right answer, but without a precise definition, it became clear that there was no way to be sure of their claim, or even make sense of what their claim meant, and this was an important moment in their understanding of the material.

To get them started, I asked a student with a calculator to compute some more terms in the sequence as I copied them onto the board. The critical event occurred when after a few terms the calculator began simply outputting the number 1, which was obviously incorrect, as we had previously agreed. When I asked what had happened, a few clever students pointed out that the calculator had made a “round-off error,” since it wasn’t computing the exact values of the sequence, but rather rounding them to the nearest 8 or so decimal places. In other words, the calculator eventually couldn’t distinguish between the sequence terms and the number 1. Many students were already familiar with round-off error from their science classes, and so this also helped guide their intuition. After some more discussion, we agreed that this phenomenon would happen no matter how precise a calculator we bought, since they would all have some round-off error, however small. The students finally all agreed that this was the convincing argument we had been searching for. At last, the test we had come up with was that given a fixed round-off error, no matter how small, it was the case that eventually all the sequence terms were within that error of the limit. And this is exactly the definition of limit, just written out in words rather than symbols. After I showed them the book’s definition, most students were amazed at the similarity.

Many important lessons came out of this exercise, and its effects were seen in several other aspects of the course. In addition to learning about the need for a precise definition when making an argument, the class learned to make sense of the symbols in the book by rephrasing them in their own words. Also, they learned that mathematics actually uses a lot of common sense, even if it is sometimes apparently obscured by notation. As a result, I saw a dramatic improvement in their ability to read and understand proofs, and the students were confident and excited to learn, rather than frustrated at the prospect of a whole semester of memorization.