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THE CONCEPT OF INFINITY FROM K-12 TO UNDERGRADUATE COURSES

Abstract

Studies have shown that a solid background in mathematics and physics is important to the success of an engineering student. The concept of infinity is one of the most important, and yet difficult links in the mathematics sequence for undergraduate engineering students. The concept of infinity can be taught to K-12 student with hands-on exercises that use an intuitive approach for teaching the concept. However, engineering students require a more mathematically rigorous presentation. This paper presents a method for teaching the topic of infinity in freshman level mathematics course on discrete mathematics for engineering students, based on the ideas of bijection and equivalency within the topic of set theory. We also present some ideas of how the concept of infinity can be targeted in the K-12 environment.

I. Introduction

As part of long-standing efforts to enhance engineering education, the ASEE surveyed prevailing trends in K-12 education1. Aiming to determine teachers' attitudes towards engineering as an intellectual and career challenge for their students, the ASEE study reveals an interesting paradox. It discovers that an overwhelming majority of teachers are positive about exposing their students to the discipline of engineering. Agreeing with the statements, such as "Engineers are fun people", "Engineers love their job", "Engineers make people's lives better", the majority of teachers believe that engineering is a noble and challenging profession, gaining in social recognition and tangible rewards. However, when asked about the accessibility of the profession of engineering, the majority of teachers expressed a strong feeling that many of their students have no chance to succeed in the engineering world. The teachers feel that1 "majoring in engineering in college is harder than majoring in many other subjects... – a feeling they likely pass on to their students."

The dichotomy, revealed in this ASEE study, pinpoints a peculiar inconsistency in grasping the nature of the profession of engineering. Engineers are perceived as smart, wise, knowledgeable professionals who work with tangible objects to solve practical problems. In their work, engineers are engaged in a prolific intellectual activity that demands a great deal of self-imposed discipline and concentration. As a result, they are stereotyped as isolated abstract thinkers with profound insights, often single-minded, awkward, weird and socially inept. In other words, the abstract thinking engineer is often perceived as a "nerd" or "geek", logically contradicting the image of a practical engineer with "hands-on" ideas and the ultimate goal of designing, creating, and developing products and processes to solve practical problems.

Much of the gap between the sense of concrete and abstract in engineering lies in the poor scientific and mathematical background of engineering freshmen. The indispensable disciplines of mathematics and physics, based on non-intuitive models, are sometimes inadequately treated in the K-12 community.

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Klass-Tsirulnikov, B., & Katz, S. (2006, June), The Concept Of Infinity From K 12 To Undergraduate Courses Paper presented at 2006 Annual Conference & Exposition, Chicago, Illinois. 10.18260/1-2--293