2025 ASEE Annual Conference & Exposition

With increasing reliance on technology, students are learning differently and often seek more interactive methods. They are exposed to numerous videos, websites, and applications that allow them to explore a topic without reading about it in textbooks. Students find these methods more engaging and informative than traditional presentations and reading materials. To meet students' new learning styles, instructors need to change the way they teach.
In mathematics, textbook-style lessons remain largely abstract. One topic that would benefit from a more intuitive approach is understanding the mathematical constant e and the function exp(x). This value and function frequently appear in both mathematics and real-world contexts, describing numerous STEM-related phenomena. However, many students struggle to develop an intuitive sense of the ex function, often overlooking its unique properties - such as the fact that the derivative of exp(x) is also exp(x) - and how this connects to real-life processes like growth and decay. Gaining a more practical and intuitive understanding of this function would help students apply their knowledge to solving new problems beyond the textbook.
To achieve this, we aim to present a new way of introducing e and exp(x). By focusing on visualization, we provide a more intuitive understanding of this function. In this paper, we discuss key features of ex, such as its derivative and integral, presenting them in a visual format that helps students gain a deeper comprehension of e and its functions. Highlighting these key features could also help students apply the function to new problems they encounter. Additionally, we discuss real-life applications of this function, showing how it has been used to solve problems related to growth and decay. This approach brings a clearer and less abstract understanding. Instructors may choose to incorporate this information into their presentations for the same purpose.
This paper should be considered a report on work in progress. The information presented is not intended to replace any existing curriculum or textbook chapters, but rather to supplement them by providing a deeper and more intuitive understanding. The content was shared with students in three classes, followed by a detailed questionnaire: 25 students in "Circuits 1," 41 students in "Stochastic Models for CS," and 8 students in "Modern Control," a graduate-level course. Students responded very positively to the presentation and the significance of e and exp(x). They overwhelmingly expressed that they valued visual, intuitive, and engaging methods of learning over traditional presentations or textbook readings.