Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving [Paperback]

Sanjoy Mahajan (Author), Carver A. Mead (Foreword)

Book Description

Publication Date: March 5, 2010 | ISBN-10: 026251429X | ISBN-13: 978-0262514293 | Edition: New

An antidote to mathematical rigor mortis, teaching how to guess answers without needing a proof or an exact calculation.

Editorial Reviews

Review

In problem solving, as in street fighting, rules are for fools: do whatever works--don't just stand there! Yet we often fear an unjustified leap even though it may land us on a correct result. Traditional mathematics teaching is largely about solving exactly stated problems exactly, yet life often hands us partly defined problems needing only moderately accurate solutions. This engaging book is an antidote to the rigor mortis brought on by too much mathematical rigor, teaching us how to guess answers without needing a proof or an exact calculation.In Street-Fighting Mathematics, Sanjoy Mahajan builds, sharpens, and demonstrates tools for educated guessing and down-and-dirty, opportunistic problem solving across diverse fields of knowledge--from mathematics to management. Mahajan describes six tools: dimensional analysis, easy cases, lumping, picture proofs, successive approximation, and reasoning by analogy. Illustrating each tool with numerous examples, he carefully separates the tool--the general principle--from the particular application so that the reader can most easily grasp the tool itself to use on problems of particular interest. Street-Fighting Mathematics grew out of a short course taught by the author at MIT for students ranging from first-year undergraduates to graduate students ready for careers in physics, mathematics, management, electrical engineering, computer science, and biology. They benefited from an approach that avoided rigor and taught them how to use mathematics to solve real problems.Street-Fighting Mathematics will appear in print and online under a Creative Commons Noncommercial Share Alike license.

"Many everyday problems require quick, approximate answers. Street-Fighting Mathematics teaches a crucial skill that the traditional science curriculum fails to develop: how to obtain order of magnitude estimates for a broad variety of problems. This book will be invaluable to anyone wishing to become a better informed professional."--Eric Mazur, Balkanski Professor of Physics and of Applied Physics, Harvard University

"All students and teachers of mathematics and science, whatever their level, will find a wealth of fun and practical tools in this fantastic book." David MacKay, Fellow of the Royal Society, Professor of Natural Philosophy, Cavendish Laboratory, University of Cambridge, Chief Scientific Advisor, UK Department of Energy and Climate Change

About the Author

Sanjoy Mahajan studied mathematics at the University of Oxford and received a PhD in theoretical physics at the California Institute of Technology. He is now Associate Director of the Teaching and Learning Laboratory and a Lecturer in the Department of Electrical Engineering and Computer Science at MIT. Before coming to MIT, he was a Fellow of Corpus Christi College, Cambridge, and a Lecturer in Physics in the University of Cambridge.

Product Details

• Paperback: 152 pages
• Publisher: The MIT Press; New edition (March 5, 2010)
• Language: English
• ISBN-10: 026251429X
• ISBN-13: 978-0262514293
• Product Dimensions: 7 x 0.2 x 9 inches
• Shipping Weight: 8.8 ounces (View shipping rates and policies)
• Average Customer Review: 4.1 out of 5 stars  See all reviews (15 customer reviews)
• Amazon Best Sellers Rank: #172,380 in Books (See Top 100 in Books)

More About the Author

Sanjoy Mahajan obtained his PhD in theoretical physics from the California Institute of Technology, and has undergraduate degrees in mathematics from Oxford University and in physics from Stanford University. Due to his many inspiring teachers, he became interested in science teaching, an interest he followed as faculty member in the Physics Department at the University of Cambridge and as a Fellow of Corpus Christi College, Cambridge.

While at Cambridge, he helped start the African Institute for Mathematical Sciences (AIMS) in Cape Town, South Africa, where he was the first Curriculum Director and taught the first courses in physics and computer science. There he wrote the free software to automate barcoding and cataloging the 5,000 donated books that started the Institute's library.

At MIT he has taught courses across the Institute in the mathematics, electrical engineering, and mechanical engineering departments. He is currently Visiting Associate Professor of Applied Science and Engineering at Olin College of Engineering.

Most Helpful Customer Reviews

85 of 86 people found the following review helpful

5.0 out of 5 stars This book makes you smarter March 24, 2010

By N

Format:Paperback

Dr. Mahajan's has been teaching this sort of course for a while now and has been generous enough to make almost all his material available through his course web pages at MIT for The Art of Approximation and Street-Fighting Mathematics. If you want to preview this book I suggest you check those websites out.

The book is reminiscent of Consider a Spherical Cow by Harte, The Art and Craft of Problem-Solving by Zeitz, and How to Solve It by Polya. Although much of the book focuses on how to avoid doing integrals and taking derivatives, it presumes the reader is familiar with calculus. In this respect it's different from the books I just mentioned and other ones out there on approximation, e.g. Guesstimation. The example problems are diverse, most are borrowed from physics, geometry, and math, with a few are that are Fermi-type "real-world" scenarios.

My main complaint is that the book is so short. I wish the author had combined this book with material from his Order of Magnitude Physics and Art of Approximation courses, which are marvelous not only for the problem-solving on display but also for the physics content.

This type of book addresses a serious gap in American math-science education. Learning techniques for approximation allows one to tackle the sort of ill-posed problems one is most likely to encounter in the real-world. It is also intimately tied to recognizing the salient features of a problem, such as the physical principles involved in a physics problem or the most questionable assumption in an economic model. Street-Fighting Math deserves a wide readership and will hopefully influence other math-science teachers and authors.

55 of 56 people found the following review helpful

5.0 out of 5 stars important tools for scientists and engineers June 9, 2010

By John Regehr

Format:Paperback

The Trinity test occurred on a calm morning. Enrico Fermi, one of the observers, began dropping bits of paper about 40 seconds after the explosion; pieces in the air when the blast wave arrived were deflected by about 2.5 meters. From this crude measurement, Fermi estimated the bomb's yield to be ten kilotons; he was accurate within a factor of two. Although Street-Fighting Mathematics does not address the problem of estimating bomb yields, it gives us a reasonably generic toolbox for generating quantitative estimates from a few facts, a lot of intuition, and impressively little calculus. As one of the reviews on Amazon says, this book makes us smarter.

Street-Fighting Mathematics -- the title refers to the fact that in a street fight, it's better to have a quick and dirty answer than to stand there thinking about the right thing to do -- is based on the premise that we can and should use rapid estimation techniques to get rough answers to difficult problems. There are good reasons for preferring estimation over rigorous methods: the answer is arrived at quickly, the full set of input data may not be needed, and messy calculus-based or numerical techniques can often be avoided. Perhaps more important, by avoiding a descent into difficult symbol pushing, a greater understanding of the problem's essentials can sometimes be gained and a valuable independent check on rigorous -- and often more error prone -- methods is obtained.

Chapter 1 is about dimensional analysis: the idea that by attaching dimension units (kg, m/s2, etc.) to quantities in calculations about the physical world, we gain some error checking and also some insight into the solution. Dimensional analysis is simple and highly effective and it should be second nature for all of us. Too often it isn't; my guess is that it gets a poor treatment in secondary and higher education. Perhaps it is relevant that about ten years ago I went looking for books about dimensional analysis and found only one, which had been published in 1964 (Dimensional Analysis and Scale Factors by R.C. Pankhurst). If Mahajan had simply gone over basic dimensional analysis techniques, it would have been a useful refresher. However, he upps the ante and shows how to use it to guess solutions to differential and integral equations: a genuinely surprising technique that I hope to use in the future.

Chapter 2 is about easy cases: the technique of using degenerate cases of difficult problems to rapidly come up with answers that can be used as sanity checks and also as starting points for guessing the more general solution. Like dimensional analysis, this is an absolutely fundamental technique that we should all use. A fun example of easy cases is found not in Street-Fighting Mathematics, but in one of Martin Gardner's books: compute the remaining volume of a sphere which has had a cylindrical hole 6 inches long drilled through its center. The hard case deals with spheres of different radii. In contrast, if we guess that the problem has a unique solution, we're free to choose the easy case where the diameter of the cylinder is zero, trivially giving the volume as 36' cubic inches. Many applications of easy cases are simple enough, but again Mahajan takes it further, this time showing us how to use it to solve a difficult fluid flow problem.

Chapter 3 is about lumping: replacing a continuous, possibly infinite function with a few chunks of finite size and simple shape. This is another great technique. The chapter starts off easily enough, but it ends up being the most technically demanding part of the book; I felt seriously out of my depth (it would probably help if I had used a differential or integral equation in anger more recently than 1995).

Chapter 4 is about pictorial proofs: using visual representations to create compelling mathematical explanations where the bare symbols are non-intuitive or confusing. This chapter is perhaps the oddball: pictorial proofs are entertaining and elegant, but they seldom give us the upper hand in a street fight. I love the example where it becomes basically trivial to derive the formula for the area of a circle when the circle is cut into many pie-pieces and its circumference is unraveled along a line.

Chapter 5 is "taking out the big part": the art of breaking a difficult problem into a first-order analysis and one or more corrective terms. The idea is that analyzing the big part gives us an understanding of the important terms, and also that in many cases we may end up computing few or none of the corrections since the first- or second-order answer may turn out to be good enough. Mahajan introduces the idea of low-entropy equations: an appealing way of explaining why we want and need simplicity in street-fighting mathematics.

Finally, Chapter 6 is about reasoning by analogy: attacking a difficult problem by solving a related, simpler one and then attempting to generalize the result. The example of how many parts an n-dimensional space is divided into by introducing some n-1 dimensional constructs is excellent, and ends up being quite a bit more intricate than I'd have guessed. This chapter isn't the most difficult one, but it is probably the deepest: analogies between different areas of mathematics can border on being spooky. One gets the feeling that the universe is speaking to us but, like children at a cocktail party, we're not quite putting all of the pieces together.

Back of the envelope estimation is one of the main elements of a scientist or engineer's mental toolkit and I've long believed that any useful engineer should be able to do it, at least in a basic way. Others seem to agree, and in fact quantitative estimation is a lively sub-genre of the Microsoft / Google / Wall Street interview question. Speaking frankly, as an educator of engineers in the US, our system fails somewhat miserably in teaching students the basics of street-fighting mathematics. The problem (or rather, part of it) is that mathematics education focuses on rigorous proofs and derivations, while engineering education relies heavily on pre-derived cookie-cutter methods that produce little understanding. In contrast, estimation-based methods require strong physical intuition and good judgment in order to discard irrelevant aspects of a problem while preserving its essence. The "rapidly discard irrelevant information" part no doubt explains the prevalence of these questions in job interviews: does anyone want employees who consistently miss the big picture in order to focus on stupid stuff?

In summary this is a great book that should be required reading for scientists and engineers. Note that there's no excuse for not reading it: the book is under a creative commons license and the entire contents can be downloaded as PDF. Also, the paper version is fairly inexpensive (currently $18 from Amazon). The modes of thinking in Street-Fighting Mathematics are valuable and so are the specific tricks. Be aware that it pulls no punches: to get maximum benefit, one would want to come to the table with roughly the equivalent of a college degree in some technical field, including a good working knowledge of multivariate calculus.

21 of 22 people found the following review helpful

5.0 out of 5 stars A treasure trove May 20, 2010

By Rahul Sarpeshkar

Format:Paperback

This book is a treasure trove of intuitive, practical, and brilliant mathematical techniques. Every person with an interest in mathematics, science, or engineering will enjoy this highly stimulating and fun book. For example, beautiful pictorial proofs of the inequality between the arithmetic mean and geometric mean illustrate the power, clarity, speed, and insight of intuitive visual methods versus traditional symbolic grungy methods. The power of working with operators to perform complex series summations is explained so clearly that both the beginner and the expert can experience the `aha' joy that comes with true understanding. Fluid-mechanics calculations via methods of dimensional analysis and the method of `easy cases', methods for quick integration by `lumping' that are remarkably accurate, techniques of successive approximation to solve physics problems with insight, rapidly converging series for pi, and pictorial proofs of series summations are just some of the many gems in this book. Sanjoy is one of MIT's best teachers and his book, therefore, reads like a novel. This book breaks new ground in the teaching of mathematics and will make powerful mathematical street fighters of all of its readers.