WIP: On Teaching and Learning the Concept of an Inverse Function: A Visual and Intuitive Approach

Daniel Raviv

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Conference: 2022 ASEE Annual Conference & Exposition
Location: Minneapolis, MN
Publication Date: August 23, 2022
Start Date: June 26, 2022
End Date: June 29, 2022
Conference Session: Incorporating Technology in the Classroom
Page Count: 29
DOI: 10.18260/1-2--40682
Permanent URL: https://peer.asee.org/40682
Download Count: 687

Paper Authors

biography

Daniel Raviv Florida Atlantic University

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Daniel Raviv received his B.Sc. and M.Sc. degrees from the Technion, and his Ph.D. from Case Western Reserve University in Cleveland, Ohio. He is a professor at Florida Atlantic University (FAU) where he is the Director of the Innovation and Entrepreneurship Lab. In the past he served as the assistant provost for innovation. Dr. Raviv taught at Johns Hopkins University, the Technion, and the University of Maryland, and was a visiting researcher at the National Institute of Standards and Technology (NIST) as part of a group that developed a vision-based driverless vehicle for the US Army (HUMVEE; 65 mph).
His related research work includes exploration of visual invariants that exist only during motion and can be used for real-time closed-loop control systems of cars and drones. He is also interested in teaching and learning innovative thinking, and how to teach innovatively. He is the author of five books: three on learning innovative thinking and two on teaching in visual, intuitive, and engaging ways.

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Abstract

Abstract

Students in today's generation learn differently: they rely less on textbooks and more on web-based explanations including short videos and animations. They repeatedly miss the “Aha! Moment” and ask for more experiential, visual, intuitive, tech-based, and easy-to-comprehend information. This is particularly true of math courses that are usually taught with little connection to other disciplines in science and engineering. As technology brings about a paradigm shift in the way people perceive and learn new information, adaptation to the students' new styles of learning should be explored.

This paper focuses on introducing a basic math concept, the inverse function, by linking it to daily experience using relevant analogies. It includes several examples of visualizations intended to aid comprehension of the concept prior to delving into purely mathematical formulas and proofs.

The paper starts with a general visual explanation of the concept of an inverse function, followed by visual, intuitive, and experience-based examples, including (1) non-mathematical examples, where the inverse exists, such as the idea of “negative” of a developed film, as well as where the inverse does not exist, such as getting a haircut; (2) visual diagrams of one-to-one relation, where the inverse exists, and one-to-many or many-to-one relations where the inverse does not exist; (3) Electrical Engineering examples such as using an op-amp input-output relationship where the output cannot be made the input (i.e., lack of inverse hardware); (4) Mechanical Engineering examples such as using a toothed wheel and pulley; (5) Computer Science examples, using the common “undo” operation as an inverse operation, and 3d reconstruction of an object using its shadow as a one-to-many case where inverse does not exist; (6) Physics examples, to show the inverse in temperature conversion; and (7) Mathematical transformations that have inverses, such as a mirror image of a kettle. The paper concludes with a set of fun related activities.

The contents of this work have been shared with students in a Calculus class using a remote (Zoom-based) classroom setting. The presentation was assessed following the lecture using an anonymous questionnaire. 29 students responded. Summary of the results to multiple questions show that the overall average response was very positive. This set of initial results indicates that the students preferred being introduced to the concept of inverse function visually and intuitively. Students found this approach to be very effective for learning and they highly praised the intuitive and engaging examples. This paper should be considered a work in progress. The presented illustrations, examples, and activities are meant to be a supplement to traditional presentations and by no means to replace existing textbooks or other pedagogical methodologies.

Citation
Format

Raviv, D. (2022, August), WIP: On Teaching and Learning the Concept of an Inverse Function: A Visual and Intuitive Approach Paper presented at 2022 ASEE Annual Conference & Exposition, Minneapolis, MN. 10.18260/1-2--40682

TY - CPAPER
AB - Abstract

Students in today's generation learn differently: they rely less on textbooks and more on web-based explanations including short videos and animations. They repeatedly miss the “Aha! Moment” and ask for more experiential, visual, intuitive, tech-based, and easy-to-comprehend information. This is particularly true of math courses that are usually taught with little connection to other disciplines in science and engineering. As technology brings about a paradigm shift in the way people perceive and learn new information, adaptation to the students' new styles of learning should be explored.

This paper focuses on introducing a basic math concept, the inverse function, by linking it to daily experience using relevant analogies. It includes several examples of visualizations intended to aid comprehension of the concept prior to delving into purely mathematical formulas and proofs.

The paper starts with a general visual explanation of the concept of an inverse function, followed by visual, intuitive, and experience-based examples, including (1) non-mathematical examples, where the inverse exists, such as the idea of “negative” of a developed film, as well as where the inverse does not exist, such as getting a haircut; (2) visual diagrams of one-to-one relation, where the inverse exists, and one-to-many or many-to-one relations where the inverse does not exist; (3) Electrical Engineering examples such as using an op-amp input-output relationship where the output cannot be made the input (i.e., lack of inverse hardware); (4) Mechanical Engineering examples such as using a toothed wheel and pulley; (5) Computer Science examples, using the common “undo” operation as an inverse operation, and 3d reconstruction of an object using its shadow as a one-to-many case where inverse does not exist; (6) Physics examples, to show the inverse in temperature conversion; and (7) Mathematical transformations that have inverses, such as a mirror image of a kettle. The paper concludes with a set of fun related activities.

The contents of this work have been shared with students in a Calculus class using a remote (Zoom-based) classroom setting. The presentation was assessed following the lecture using an anonymous questionnaire. 29 students responded. Summary of the results to multiple questions show that the overall average response was very positive. This set of initial results indicates that the students preferred being introduced to the concept of inverse function visually and intuitively. Students found this approach to be very effective for learning and they highly praised the intuitive and engaging examples.
This paper should be considered a work in progress. The presented illustrations, examples, and activities are meant to be a supplement to traditional presentations and by no means to replace existing textbooks or other pedagogical methodologies.

AU - Daniel Raviv
CY - Minneapolis, MN
DA - 2022/08/23
PB - ASEE Conferences
TI - WIP: On Teaching and Learning the Concept of an Inverse Function: A Visual and Intuitive Approach
UR - https://peer.asee.org/40682
DO - 10.18260/1-2--40682
ER -