Arizona State University



James TorlaDr. Naala BrewerMAT 267 Calculus III for Engineers28 April 2017The Problems with Conceptual CalculusIvars Peterson, the Director of Communications for the Mathematics Association of America once said, "To most outsiders, modern mathematics is unknown territory … its landscapes are a mass of indecipherable equations and incomprehensible concepts.” This incomprehensibility is often the result of the unpredictable results of mathematical theorems. While mathematical theorems -and mathematics itself- are simply the expressions of logic and deterministic applications of proofs to various problems, the average person tends to judge math and results two mathematical expressions with the same measure and round the possibility as everyday life. While the use of common everyday logic is appropriate for most types of early math and algebra, the more advanced styles of math often House seeming paradoxes that are at odds with the same logic that leads in the early stages of math. When math progresses beyond the point at which support system, common physical proofs and understanding, the understanding of those learning them becomes limited and anything learned seems at odds within itself. This causes many students of higher mass to unconsciously reject they’re learning even as they utilize them, thus causing in increasingly durable barrier against further understanding of mathematics.The first crack that seeming paradoxes exploit into distrust and distaste for the higher maths is one of the fundamentals that allow so many people to learn the basic math, common application and simple physical logic. It is through the method of observing the world, rather than simple rote, that teaches basic math. For example, 1+1=2 is one concept which – though the notation is learned in the classroom–the actual expression is used every day in a multitude of ways by the average person. This means that the logic behind this simple arithmetic statement –and all the ones that stem immediately from it– is entirely congruent with most people’s understanding of the world. The next way in which someone learn math is one that is not so natural; rather than simply learning the specific notation of the mathematical statement people must first be shown in example of the mathematical concept’s use. This method of “show and tell” is of the type used for such theorems as the Pythagorean theorem; a theorem which can be easily learned and understood by measuring the size of a right triangle, but one which requires a modicum of understanding and oversight from a more experienced mathematician to teach. While these types of theorems take years to learn (generally up to sixth grade) they are not intrinsically complex there are very few steps of logic required to grasp most of these concepts and very little trust is required in either teachers or in mathematics it’s self simply because the average person can see the concept in actions themselves.Generally, it is at the next of concept level that the dependency on concrete examples starts to damage student understanding. This level is filled with such concepts as systems of equations, concepts that are not easily observed in nature nor are they ones that I can easily be demonstrated – like the Pythagorean theorem. Rather, the fundamental understanding of these concepts rest on pure mathematical logic and a deep understanding of the system of proofs and syntax. This method for learning mathematics is entirely learned Indy is no great surprise that many students are lost or disheartened by this level of math. Yet even this slight barrier is nothing compared to the one that most students of mathematics encounter in early and mid-level calculus. As one explores the theorems and proofs of calculus one will sooner or later encounter paradoxes, or at least apparent paradoxes. These paradoxes are the final injury that causes the growing schism between concrete logic and mathematical proof logic to rupture, either permitting the student to continue on their mathematical journey or creating such distrust in the concept of math that any future attempts to learn higher math are stymied. One perfect example for this type of concrete-logic defying paradox is Gabriel’s Horn. Gabriel’s Horn, or Torricelli's trumpet, is a mathematical figure which is created by the service revolution of the function y=1/x around the X axis for the domain x≥1. When the surface is graphed the parametric equations are used:(1)(2) Thus when one wants to graph the volume of the Horn, one simply takes the integral the area of infinite circular slices (parallel to the x-axis, thus in terms of y) of the Horn, such that:(4)(5)(6) The resulting volume of Gabriel’s Horn is that the horn can be filled with π units (for simplicity’s sake her on referred in feet) of paint. This result Is –in the end of itself–a logical enough conclusion; you can fill a mathematical object with hypothetical paint, and in this case it happens to be π ft3. However, this statement taken in context with the next information about Gabriel’s Horn can cause doubts as to the validity of the math that lead to such information or mistrust about the seeming incongruity between math and the “real world”. When one wants to calculate the surface area of Gabriel’s Horn one does so in a manner much like before, however this time instead of using the equation for the area of a circle one uses the derivative for the area of the circle (simply because the circumference equation is the derivative of the circular area equation):(7)(8)(9)(10)(11)(12) The clear result of this calculation is that the amount of paint required to paint the inside of Gabriel’s Horn is infinite. And if this is where the incongruities arise. If one had a water bottle and one filled said bottle with paint logic would dictate that if one poured out the paint some would remain behind, having coated the sides of the bottle. However, this “concrete” lineup with the results of the math; since the surface area is infinite filling up the inside with paint cannot possibly coat the entire surface area of the shape, despite the fact that it is entirely possible to fill up the shape with paint. This incongruity and many others like it, while being only problems of simple logic that have solutions, are enough to discourage some people from attempting to garner further understanding of advanced mathematics. When complicated partially understood mathematical proofs produce results that seem to act against the common, everyday logic to which all are accustomed one of the two has to give. And more times than many it is the more advanced, but less intuitive, mathematics that lose favor. Now this is not a failure of calculus, rather this is a failure of the system of understanding on which Basic necessary mathematics are understood. Because the common, necessary arithmetic and algebra are so built off of the concrete logic that one can use without any mathematical understanding there is limited incentive to truly believe the Results that seem to violate the fundamental concepts. As long as the basic mass are taught with physical rationale students will be primed to reject further knowledge at more events mass once the foundations of their understanding are rejected. ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download