Have You Seen an Integral? Visual, intuitive and Relevant Explanations of Basic Engineering-related Mathematical Concepts

Abstract Today’s students are exposed to information presented in visual, intuitive and concise ways. They expect explanations for why a subject is important and relevant, as well as for its potential use. In order to adapt to students’ learning preferences and styles, efforts must be made to further modify teaching methods to include relevance of the material to daily life experiences. The material should also be presented in easy-to-comprehend, visual, and intuitive ways. This is most relevant in math courses that are usually taught with little or no connection to other disciplines, and in particular engineering. This paper focuses on introducing basic math concepts by linking them to daily experiences using relevant analogy-based examples, to be introduced prior to delving into purely mathematical explanations and proofs. The paper shows tangible physical explanations of concepts in calculus, specifically on topics such as: (a) Integration and differentiation. To explain these concepts, the paper uses several examples such as (1) relations between steering wheel angle of a car and the physical angle of the car in world coordinates, (2) relations between water flow and its accumulation in a container, (3) elevator directional motion, and (4) energy and its temporal rate-of-change during running, walking, sitting, and sleeping. It also shows some unexpected examples that relay to very basic daily observations such as the relation between moving shadows to differentiation and integration. (b) First order differential equation and time constant of first order system. Based on accumulated teaching experience, some helpful examples are: (1) battery charging a mobile phone at different initial charging values, and (2) cooling rate of coffee. There are of course many other examples, but not all of them are as impactful (e.g., radioactive decay and carbon dating). These ideas are shared so that instructors can use them to enhance understanding of engineering-related math concepts, and to show their relevance. We refer to this approach as “work in progress.” When using the above examples (and many others), students have demonstrated better, clearer understanding of difficult concepts. Even though this was not an official assessment, based on similar experience that was gained and assessed by the author multiple times in other engineering related subjects (Control Systems, Digital Signal Processing, Computer Algorithms, and Physics), it is believed that the approach has a great potential.