The Art of Insight in Science and Engineering “is an antidote” to the issue that current physics courses provide “surprisingly little instruction and practice in that invaluable art … [of] back-of-the-envelope calculations,” writes reviewer Adam Lidz. “After working through the book,” he continues, “readers will feel equipped to come up with rapid-fire, approximate solutions to unfamiliar and complex problems, to pose their own questions, and to explore.”
On leave from the Franklin W. Olin College of Engineering in Massachusetts, Mahajan is currently MIT’s acting director of digital residential education and a visiting professor of electrical engineering and computer science. Before his post at Olin, he taught physics at the University of Cambridge and was a founding faculty member and the first curriculum director of the African Institute for Mathematical Sciences in South Africa.
Physics Today recently caught up with Mahajan to discuss The Art of Insight in Science and Engineering.
PT: What inspired you to write this book, and how is it different from Street-Fighting Mathematics?
MAHAJAN: I had wanted to write such a book since I was a graduate student at Caltech 20 years ago. While preparing for the physics qualifying exams and serving as a teaching assistant for the famous Order-of-Magnitude Physics course, I learned more physics than I had in the years of my undergraduate degree. Those three months showed me that physics teaching and learning had much room for improvement and how approximation and insight could fill the gap. I wanted to share the news! Inspired by Linux and the free-software movement, I wanted to make the news freely available—and am happy that MIT Press was willing.
In teaching what I saw as the art of approximation, I organized the course by topics in physics and engineering. However, this organization limited the material’s generality. Instead, reasoning tools, such as dimensional analysis or proportional reasoning, are useful far beyond any one example. So, I reorganized my teaching around reasoning tools.
My first foray into tool-based teaching was Street-Fighting Mathematics. Both books share the philosophy that rigor leads to rigor mortis. However, The Art of Insight has much less mathematics—just arithmetic, algebra, and a smidgen of calculus—and much more physics, chemistry, and engineering. The goal in Street-Fighting Mathematics was to integrate and understand a lot of applied mathematics. The goal in The Art of Insight is to understand a lot of the world with only a little mathematics.
PT: What type of feedback have you received?
MAHAJAN: So far, so good. One colleague, a physics department chair, told me that he would like to provide the book to all incoming graduate students. I have also been invited to teach the approach to biology graduate students and to physics teachers from around the world.
The reviews (including the one in Physics Today) have also been gratifying. Even the mild criticisms in other reviews—that “Mahajan clearly comes from a physics background” (in Science) or that “various parts of the book will make a mathematician cringe” (in MAA Reviews)—struck me as compliments!
A diagram, taken from the book, of the chapter organization in The Art of Insight in Science and Engineering.
PT: Is the ability to estimate a product of nature or nurture?
MAHAJAN: It is both. It is nature, in that we are all born with it. Our “number” sense, as Stanislas Dehaene shows in his wonderful book The Number Sense: How the Mind Creates Mathematics (revised and updated edition, Oxford University Press, 2011), is our evolutionary heritage. It consists of an exact system for one, two, and three and an approximate system, based on a logarithmic scale, for larger quantities.
But the ability to estimate is also nurture. We can limit the approximate-number system’s scope by neglecting ratio reasoning and by teaching the fear of being wrong—by teaching rigor mortis. Fortunately, we can enhance the scope by teaching ratio reasoning and by teaching courage, the courage to shoot first and ask questions later (not a policing recommendation!), and by providing reasoning tools to start shooting.
PT: What effect do you think modern technology has had on the ability of people to make educated guesses? As a lecturer, are you finding that science and engineering students are less capable than previous generations of performing back-of-the-envelope calculations?
MAHAJAN: The effect has been negative. Calculators in school mathematics teaching, for example, have been a disaster. Because calculators perform arithmetic so much faster than people do, we allegedly no longer should teach students arithmetic fluency. Similarly, because cars drive so much faster than walking, we should no longer need to walk! But driving everywhere rots our leg muscles—just as using calculators all the time rots our mental muscles.
Compounding the problem, graphing calculators have done for students’ understanding of functional relationships what regular calculators have done for students’ understanding of number and scale. How can you make back-of-the-envelope models and estimates without a sense for scale or functional relationships?
The damage is fortunately not permanent. But it needs active remediation. Thus, one skill that I always teach my students is estimating to one-half of a significant figure, by rounding every number to “one” or “few” times an integer power of 10.
PT: What books are you currently reading?
MAHAJAN: The Wealth of the Commons: A World Beyond Market and State (Levellers Press, 2012) edited by David Bollier and Silke Helfrich; Fluent Forever: How to Learn Any Language Fast and Never Forget It by Gabriel Wyner (Harmony, 2014); and Does Education Matter? by Brian Simon (Lawrence and Wishart, 1985).
Despite the tumultuous history of the near-Earth object’s parent body, water may have been preserved in the asteroid for about a billion years.
October 08, 2025 08:50 PM
Get PT in your inbox
Physics Today - The Week in Physics
The Week in Physics" is likely a reference to the regular updates or summaries of new physics research, such as those found in publications like Physics Today from AIP Publishing or on news aggregators like Phys.org.
