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A Big List of Interesting Mathematics Books Released in 2011

Antonio Cangiano — Fri, 16 Dec 2011 19:11:17 +0000

The following is a list of 46 interesting math books that were released in 2011.

As some of you know, I run a service called Any New Books?, which emails you a list of new books that are related to the categories of your choice each week. For the most part I pulled this list from the weekly staff picks there throughout this past year, just in time for your Christmas shopping.

The books are ordered by their current sale rank on Amazon (from the most popular to the least popular at the moment, with hardcovers first). I hope this page will help you discover a few titles you may have not have noticed yet.

[Programming book list]

	The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Sharon Bertsch McGrayne ISBN: 0300169698 Publisher: Yale University Press Publication date: May 17, 2011 Binding: Hardcover Estimated price: $17.60 Share this book on Twitter \|Facebook
	The Man of Numbers: Fibonacci’s Arithmetic Revolution Stores: USA \| Canada \| Italy \| Kindle \| UK Kindle By Keith Devlin ISBN: 0802778127 Publisher: Walker & Company Publication date: July 5, 2011 Binding: Hardcover Estimated price: $14.50 Share this book on Twitter \|Facebook
	The Mathematics of Life Stores: USA \| Canada \| Italy \| Kindle By Ian Stewart ISBN: 0465022383 Publisher: Basic Books Publication date: June 7, 2011 Binding: Hardcover Estimated price: $12.13 Share this book on Twitter \|Facebook
	The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1 Stores: USA \| UK \| Canada \| Italy By Donald E. Knuth ISBN: 0201038048 Publisher: Addison-Wesley Professional Publication date: January 22, 2011 Binding: Hardcover Estimated price: $53.15 Share this book on Twitter \|Facebook
	SuperCooperators: Altruism, Evolution, and Why We Need Each Other to Succeed Stores: USA \| Canada \| Italy \| Kindle By Martin Nowak, Roger Highfield ISBN: 1439100187 Publisher: Free Press Publication date: March 22, 2011 Binding: Hardcover Estimated price: $8.33 Share this book on Twitter \|Facebook
	Undocumented Secrets of MATLAB-Java Programming Stores: USA \| UK \| Canada By Yair M. Altman ISBN: 1439869030 Publisher: Chapman and Hall/CRC Publication date: December 12, 2011 Binding: Hardcover Estimated price: $60.57 Share this book on Twitter \|Facebook
	Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Paul J. Nahin ISBN: 0691144257 Publisher: Princeton University Press Publication date: August 28, 2011 Binding: Hardcover Estimated price: $18.74 Share this book on Twitter \|Facebook
	Stats: Data and Models (3rd Edition) Stores: USA \| UK \| Canada \| Italy \| Kindle By Richard D. De Veaux, Paul F. Velleman, David E. Bock ISBN: 0321692551 Publisher: Addison Wesley Publication date: January 8, 2011 Binding: Hardcover Estimated price: $98.99 Share this book on Twitter \|Facebook
	Functional Analysis: Introduction to Further Topics in Analysis Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Elias M. Stein, Rami Shakarchi ISBN: 0691113874 Publisher: Princeton University Press Publication date: September 11, 2011 Binding: Hardcover Estimated price: $60.00 Share this book on Twitter \|Facebook
	Cluster Analysis Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Brian S. Everitt, Dr Sabine Landau, Dr Morven Leese, Dr Daniel Stahl ISBN: 0470749911 Publisher: Wiley Publication date: March 8, 2011 Binding: Hardcover Estimated price: $59.99 Share this book on Twitter \|Facebook
	Viewpoints: Mathematical Perspective and Fractal Geometry in Art Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Marc Frantz, Annalisa Crannell ISBN: 0691125929 Publisher: Princeton University Press Publication date: July 25, 2011 Binding: Hardcover Estimated price: $35.31 Share this book on Twitter \|Facebook
	The Glorious Golden Ratio Stores: USA By Alfred S. Posamentier, Ingmar Lehmann ISBN: 1616144238 Publisher: Prometheus Books Publication date: November 22, 2011 Binding: Hardcover Estimated price: $14.46 Share this book on Twitter \|Facebook
	One, Two, Three: Absolutely Elementary Mathematics Stores: USA \| UK \| Canada \| Italy \| Kindle By David Berlinski ISBN: 0375423338 Publisher: Pantheon Publication date: May 10, 2011 Binding: Hardcover Estimated price: $9.89 Share this book on Twitter \|Facebook
	Game Theory and the Humanities: Bridging Two Worlds Stores: USA \| UK \| Canada \| Italy By Steven J. Brams ISBN: 0262015226 Publisher: The MIT Press Publication date: March 4, 2011 Binding: Hardcover Estimated price: $28.81 Share this book on Twitter \|Facebook
	Majority Judgment: Measuring, Ranking, and Electing Stores: USA \| UK \| Canada \| Italy By Michel Balinski, Rida Laraki ISBN: 0262015137 Publisher: The MIT Press Publication date: January 28, 2011 Binding: Hardcover Estimated price: $31.89 Share this book on Twitter \|Facebook
	The Oxford Handbook of Random Matrix Theory Stores: USA \| UK \| Canada \| Italy By Gernot Akemann, Jinho Baik, Philippe Di Francesco ISBN: 0199574006 Publisher: Oxford University Press, USA Publication date: September 25, 2011 Binding: Hardcover Estimated price: $173.05 Share this book on Twitter \|Facebook
	An Elementary Introduction to Statistical Learning Theory Stores: USA \| UK \| Canada \| Italy By Sanjeev Kulkarni, Gilbert Harman ISBN: 0470641835 Publisher: Wiley Publication date: August 2, 2011 Binding: Hardcover Estimated price: $73.99 Share this book on Twitter \|Facebook
	The Ultimate Challenge: The 3x+1 Problem Stores: USA \| UK By Jeffrey C. Lagarias ISBN: 0821849409 Publisher: American Mathematical Society Publication date: January 14, 2011 Binding: Hardcover Estimated price: $59.00 Share this book on Twitter \|Facebook
	Connections in Combinatorial Optimization Stores: USA \| UK \| Canada \| Italy By Andras Frank ISBN: 0199205272 Publisher: Oxford University Press, USA Publication date: June 1, 2011 Binding: Hardcover Estimated price: $94.04 Share this book on Twitter \|Facebook
	The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Burt C. Hopkins ISBN: 0253356717 Publisher: Indiana University Press Publication date: September 7, 2011 Binding: Hardcover Estimated price: $30.00 Share this book on Twitter \|Facebook
	The Evolution of Principia Mathematica: Bertrand Russell’s Manuscripts and Notes for the Second Edition Stores: USA \| UK \| Canada \| Italy By Bernard Linsky ISBN: 110700327X Publisher: Cambridge University Press Publication date: July 11, 2011 Binding: Hardcover Estimated price: $138.09 Share this book on Twitter \|Facebook
	Comparing Groups: Randomization and Bootstrap Methods Using R Stores: USA \| UK \| Canada \| Italy By Andrew S. Zieffler, Jeffrey R. Harring, Jeffrey D. Long ISBN: 0470621699 Publisher: Wiley Publication date: June 15, 2011 Binding: Hardcover Estimated price: $64.99 Share this book on Twitter \|Facebook
	Lectures in Game Theory for Computer Scientists Stores: USA \| UK \| Canada \| Italy By Editors at Cambridge University Press ISBN: 0521198666 Publisher: Cambridge University Press Publication date: February 14, 2011 Binding: Hardcover Estimated price: $50.00 Share this book on Twitter \|Facebook
	Probability Concepts and Theory for Engineers Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Harry Schwarzlander ISBN: 0470748559 Publisher: Wiley Publication date: March 1, 2011 Binding: Hardcover Estimated price: $67.99 Share this book on Twitter \|Facebook
	Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form Stores: USA \| UK \| Canada \| Italy By Kevin O’Meara, John Clark, Charles Vinsonhaler ISBN: 0199793735 Publisher: Oxford University Press, USA Publication date: September 16, 2011 Binding: Hardcover Estimated price: $71.55 Share this book on Twitter \|Facebook
	The Art of R Programming: A Tour of Statistical Software Design Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Norman Matloff ISBN: 1593273843 Publisher: No Starch Press Publication date: October 12, 2011 Binding: Paperback Estimated price: $21.99 Share this book on Twitter \|Facebook
	Here’s Looking at Euclid: From Counting Ants to Games of Chance – An Awe-Inspiring Journey Through the World of Numbers Stores: USA \| Canada \| Italy \| Kindle By Alex Bellos ISBN: 1416588280 Publisher: Free Press Publication date: April 19, 2011 Binding: Paperback Estimated price: $4.50 Share this book on Twitter \|Facebook
	Matlab, Second Edition: A Practical Introduction to Programming and Problem Solving Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Stormy Attaway ISBN: 0123850819 Publisher: Butterworth-Heinemann Publication date: August 11, 2011 Binding: Paperback Estimated price: $35.44 Share this book on Twitter \|Facebook
	The Best Writing on Mathematics 2011 Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Editors at Princeton University Press ISBN: 0691153159 Publisher: Princeton University Press Publication date: November 27, 2011 Binding: Paperback Estimated price: $12.14 Share this book on Twitter \|Facebook
	Algorithmic Puzzles Stores: USA \| UK \| Canada \| Kindle By Anany Levitin, Maria Levitin ISBN: 0199740445 Publisher: Oxford University Press, USA Publication date: October 14, 2011 Binding: Paperback Estimated price: $21.13 Share this book on Twitter \|Facebook
	Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 Stores: USA \| Canada By Jessica F. Shumway ISBN: 1571107908 Publisher: Stenhouse Pub Publication date: April 10, 2011 Binding: Paperback Estimated price: $17.88 Share this book on Twitter \|Facebook
	Agent-Based and Individual-Based Modeling: A Practical Introduction Stores: USA \| UK \| Canada \| Kindle \| UK Kindle By Steven F. Railsback, Volker Grimm ISBN: 0691136742 Publisher: Princeton University Press Publication date: November 6, 2011 Binding: Paperback Estimated price: $41.39 Share this book on Twitter \|Facebook
	Foundations of Geometry (2nd Edition) Stores: USA \| UK \| Canada By Gerard Venema ISBN: 0136020585 Publisher: Addison Wesley Publication date: July 16, 2011 Binding: Paperback Estimated price: $60.35 Share this book on Twitter \|Facebook
	Taming the Infinite: The Story of Mathematics from the First Numbers to Chaos Theory Stores: USA \| UK \| Canada \| Italy By Ian Stewart ISBN: 1847247687 Publisher: Quercus Publication date: November 1, 2011 Binding: Paperback Estimated price: $6.03 Share this book on Twitter \|Facebook
	The Number Mysteries: A Mathematical Odyssey through Everyday Life Stores: USA \| Canada \| Italy \| Kindle By Marcus du Sautoy ISBN: 0230113842 Publisher: Palgrave Macmillan Publication date: May 24, 2011 Binding: Paperback Estimated price: $10.66 Share this book on Twitter \|Facebook
	The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition Stores: USA \| UK \| Canada \| Italy \| Kindle By Stanislas Dehaene ISBN: 0199753873 Publisher: Oxford University Press, USA Publication date: April 29, 2011 Binding: Paperback Estimated price: $20.04 Share this book on Twitter \|Facebook
	Introduction to Proof in Abstract Mathematics Stores: USA \| UK \| Canada \| Italy By Andrew Wohlgemuth, Mathematics ISBN: 0486478548 Publisher: Dover Publications Publication date: February 17, 2011 Binding: Paperback Estimated price: $12.89 Share this book on Twitter \|Facebook
	Wizardry: Baseball’s All-Time Greatest Fielders Revealed Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Michael Humphreys ISBN: 0195397762 Publisher: Oxford University Press, USA Publication date: March 30, 2011 Binding: Paperback Estimated price: $11.00 Share this book on Twitter \|Facebook
	Differential Geometry: Bundles, Connections, Metrics and Curvature Stores: USA \| UK \| Canada \| Kindle By Clifford Henry Taubes ISBN: 0199605874 Publisher: Oxford University Press, USA Publication date: December 1, 2011 Binding: Paperback Estimated price: $36.20 Share this book on Twitter \|Facebook
	Fatal Numbers: Why Count on Chance Stores: USA \| UK \| Kindle \| UK Kindle By Hans Magnus Enzensberger ISBN: 1935830015 Publisher: Upper West Side Philosophers, Inc. Publication date: February 2, 2011 Binding: Paperback Estimated price: $10.05 Share this book on Twitter \|Facebook
	Matroid Theory Stores: USA \| UK \| Canada \| Italy By James Oxley ISBN: 0199603391 Publisher: Oxford University Press, USA Publication date: April 22, 2011 Binding: Paperback Estimated price: $53.16 Share this book on Twitter \|Facebook
	Visual Thinking in Mathematics Stores: USA \| UK \| Italy \| Kindle \| UK Kindle By Marcus Giaquinto ISBN: 0199575533 Publisher: Oxford University Press, USA Publication date: November 14, 2011 Binding: Paperback Estimated price: $30.57 Share this book on Twitter \|Facebook
	Mathematics, Ideas and the Physical Real Stores: USA \| UK \| Canada \| Italy \| Kindle \| UK Kindle By Albert Lautman, Simon Duffy ISBN: 144112344X Publisher: Continuum Publication date: August 4, 2011 Binding: Paperback Estimated price: $14.50 Share this book on Twitter \|Facebook
	The Algebra Solution to the Mathematics Reform: Completing the Equation Stores: USA \| UK \| Canada \| Italy By Frances R. Spielhagen ISBN: 0807752312 Publisher: Teachers College Press Publication date: June 17, 2011 Binding: Paperback Estimated price: $17.43 Share this book on Twitter \|Facebook
	A Career in Statistics: Beyond the Numbers Stores: USA \| UK \| Canada \| Italy By Gerald J. Hahn, Necip Doganaksoy ISBN: 0470404418 Publisher: Wiley Publication date: June 28, 2011 Binding: Paperback Estimated price: $48.55 Share this book on Twitter \|Facebook
	The Star and the Whole: Gian-Carlo Rota on Mathematics and Phenomenology Stores: USA \| UK \| Canada By Fabrizio Palombi ISBN: 1568815832 Publisher: A K Peters/CRC Press Publication date: August 24, 2011 Binding: Paperback Estimated price: $20.07 Share this book on Twitter \|Facebook

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Eberhardt Rechtin and the Barrier Course: A Case Study in Math Education

John F. McGowan, Ph.D. — Tue, 05 Jul 2011 10:55:12 +0000

At present, at any rate, very little evidence exists that great mathematicians and calculating prodigies have been endowed with an exceptional neurobiological structure. Like the rest of us, experts in arithmetic have to struggle with long calculations and abstruse mathematical concepts. If they succeed, it is only because they devote a considerable time to this topic and eventually invent well-tuned algorithms and clever shortcuts that any of us could learn if we tried and that are carefully devised to take advantage of our brain’s assets and get get round its limits.

Stanislas Dehaene
The Number Sense, pages 7-8

Eberhardt Rechtin was a system engineer who helped develop the Deep Space Network at the Jet Propulsion Laboratory, served as Director of the Advanced Research Projects Agency (now DARPA), Assistant Secretary of Defense for Telecommunications, chief engineer for Hewlett-Packard, President and CEO of the Aerospace Corporation, and finally a Professor at the University of Southern California before his retirement. In 1995, he gave an interview to Frederik Nebeker of the IEEE Center for the History of Electrical Engineering on his career which is available on-line here. This is a far ranging interview covering a lengthy and distinguished career. In this interview, he discusses his experience with a “graduate barrier course” at Caltech (the California Institute of Technology) while getting his Ph.D. in Electrical Engineering (awarded in 1950). This was a highly mathematical course titled Electromagnetics taught by Professor William R. Smythe. According to Rechtin’s account, this course was designed to get rid of Ph.D. students who could not “cut it” at Caltech in the 1940s. This article explores what the barrier course may or may not have actually been doing.

By his own account, Rechtin had been a straight A student until he took this barrier course. He flunked the course to his surprise. Although Caltech allowed students to retake the course, the students who flunked usually failed on a second attempt. At least according to Rechtin’s account many years later, the odds were very much against him. He studied the book for the course over the summer, working through problem after problem, apparently without too much success. Then he realized that every problem in the book had two ways to work out the answer. One was apparently the standard, brute-force answer which took a long time, too long for the short tests and exams, and was tedious to perform. This was what he had been doing. But in every case, there was a quick way to solve the problem by reusing mathematical solutions to other problems that had been worked out by mathematicians or engineers previously. In his account, he mentions a problem that could be solved quickly using Bessel functions. He knew nothing about Bessel functions when he took the course the first time. There was always a “trick” solution to the problems that involved reusing known advanced mathematics. Rechtin took the course a second time and passed easily according to his account.

Rechtin seems to have interpreted the Electromagnetics barrier course as a sort of intelligence test in which the smarter, better students by Caltech standards would figure out as he did that the problems were solvable by reusing various known pieces of advanced mathematics such as Bessel functions. He also took it as a lesson for his career, to always look for quick ways to solve a problem by reusing known mathematics or previous work: don’t reinvent the wheel — certainly good advice. But is that actually what happened; was the effect of the barrier course on other students what Rechtin thought or even what the Caltech professors thought?

The Deliberate Practice Interpretation of the Barrier Course

Deliberate practice is the central concept of K. Anders Ericsson‘s theory of expert performance, which has recently been popularized by science writer Malcolm Gladwell in his book Outliers, previously reviewed in the article Debating Deliberate Practice. Deliberate practice is somewhat vaguely defined which is one of the major problems with the theory of expert performance. Ericsson uses the example of the backhand in tennis, which is a relatively rare move in the game. Tennis players who repeatedly practice rare moves such as the backhand will, in general, defeat players who do not engage in specific deliberate practice of the backhand or other rare moves. Someone who engages in deliberate practice of this type may well defeat players with many more years of experience playing the game, but relatively little practice of rare moves. This is sort of the concept of deliberate practice. In some contexts, Ericsson uses deliberate practice in a more general way to refer to a process of continuous self-improvement and conscious analysis of one’s performance and errors.

In intellectual activities such as mathematics, the notion is that, especially in a timed contest or exam, if the mathematician encounters a problem that is too complex, lengthy, and so forth to solve from first principles in the limited time available, a few hours for most exams in most college and university courses, the mathematician will fail. On the other hand a mathematician who has specifically studied and practiced this specific type of problem, such as an electromagnetics problem that is solved with Bessel functions, will solve the problem quickly and easily. There will be a dramatic difference between the two on many exams.

Ericsson’s theory emphasizes specific knowledge in a specific field or discipline. Ericsson largely rejects the notion of genius or general intelligence as well as an inborn aptitude for a specific subject. There are no born mathematicians. It is all study and practice, and a special kind of practice — deliberate practice. Deliberate practice is critical to Ericsson’s theory. There are clearly many examples of mathematicians or chess players or musicians who have many, many years of experience, but do not perform at the expert or “star” level. Why do some people with a few years of experience, often ten years, outperform people with decades of experience, especially in intellectual activities where physical aging is not as large a factor as in sports?

In fact, the barrier course that Rechtin encountered sounds like a good example of deliberate practice. The problems apparently required detailed specific knowledge such as a knowledge of Bessel functions. In the absence of this, the problems took too long to solve in the limited time available, a few hours usually. Once he figured out what was going on, Rechtin probably spent many hours studying Bessel function and other specific mathematical methods, although he does not explicitly say this in his interview.

What did the barrier course actually do?

It is far from clear what the barrier course actually did or what it was actually supposed to do. People, families, and cultures have different beliefs and attitudes toward study and practice. In the United States “rote memorization” or “studying to the test” is generally deprecated and “thinking things through from first principles” or “thinking for yourself” is often glorified, at least in theory. It is not unique to the United States. The author has heard parents from India, for example, express concern that their child was not being taught to think things through in school in the United States. The common stereotype is that Asian cultures such as China and Japan place a strong emphasis on heavy practice. Students from a background that emphasized practice, drilling, and were already studying technical minutia like Bessel functions would have been likely to easily pass the barrier course. On the other hand, students who were accustomed to “thinking things through,” and Eberhardt Rechtin sounds very much like this kind of student in his interview, would tend to fail. It often would not occur to the “think it through” students to engage in deprecated “rote memorization” unless someone told them. Rechtin is clear that no one, neither the other graduate students nor the faculty, would tell him how to pass the course; he had to figure it out on his own or already know what to do.

In his account, Eberhardt Rechtin interpreted the barrier course as an intellectual puzzle that he figured out. That is, he thought the problem of the course through and realized that he needed to reuse existing mathematical knowledge such as Bessel functions and this general reasoning insight was the whole point of the barrier course. Maybe it was. Maybe it wasn’t. He probably interpreted what he experienced from his personal and cultural background. The barrier course could just as easily have had the effect of selecting rather unimaginative students whose high performance was a consequence of heavy drilling and who had poor abilities to think things through. One can imagine professors eager for unimaginative drones to perform intellectual drudge work and not think things through and ask unwanted or unsettling questions: Professor Millikan, after reviewing your papers, I am pretty sure your theory that cosmic rays are caused by nuclear fusion in deep space is all wrong for reasons X, Y, and Z.

In fact, the effect and the selection of students could have been completely random. Some students would have figured out the trick immediately without flunking the course, unlike Eberhardt Rechtin. A few might have figured it out and passed on the second try as Rechtin did. Many might have simply glided through the course because they were already practicing or rapidly assimilating existing specialized knowledge (maybe they could learn existing knowledge through study — reading a textbook about Bessel functions, for example — with little practice or drilling). The barrier course could have selected several different types of students. By his own account, Rechtin’s experience was very unusual; most students who flunked did not pass on the second attempt.

Conclusion

Obviously, one should not draw firm conclusions from a single case, let alone a verbal account of something that happened over forty-five years before. Nonetheless, Eberhardt Rechtin’s account is similar to other selection procedures that the author has experienced or heard of in graduate programs in mathematical fields such as physics or electrical engineering. These procedures usually have the ostensible purpose, whether stated or not, of selecting the “best and brightest” as conventionally defined. They also often serve as a rite of passage, not perhaps unlike boot camp in the Marines or hazing in a fraternity, and this may be their true purpose and function.

For students, there are some probable lessons from this case study. Some tests and exams can be worked out in the time available from first principles. This often seems to be true of math and science problems in elementary, middle, and high school (K-12). An emphasis on first principles and general reasoning methods will likely succeed with these problems, tests, and exams. Some tests and exams have trick problems that require specific knowledge learned in advance of the test like Bessel functions in Rechtin’s account. These require specific study and possibly heavy practice to master and overcome. These problems appear to be more common in more advanced math, science, and engineering courses at colleges or universities. For parents and teachers, it is likely important to teach students to be aware of this and to identify the situation to the extent that this is possible.

This case study also illustrates the difficulty and perhaps impossibility of distinguishing between specific knowledge and hypothetical general intelligence or special aptitude (a born mathematician) using tests and exams. Is there a mental horsepower and, if so, what is it? If there is a mental horsepower, is it a single attribute or several? Did the barrier course select “geniuses” who figured out the trick as Rechtin did or did it select intellectual “drones” who had already memorized the answers or both? It may be that some exceptionally intelligent students were able to pass the barrier course without the specific knowledge of Bessel functions and other mathematical methods that Rechtin had to acquire through study and practice. It may be that some students passed due to heavy practice of special methods such as Bessel functions or rapid absorption of existing knowledge through study (whether due to some innate ability to learn existing knowledge easily or studying the right, unusually clear textbook, for example, greatly reducing the need to practice). The selection of students who could “cut it” may have been largely random.

About the Author

John F. McGowan, Ph.D. solves problems by developing complex algorithms that embody advanced mathematical and logical concepts, including video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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When Will I Use Math?

Antonio Cangiano — Sat, 21 May 2011 19:59:36 +0000

There is a new online initiative called We Use Math, which is meant to address and answer one of the most common questions that high school students have: when am I ever going to use math in the real world?

The idea behind this project is to motivate students to pursue mathematics-based careers, by helping them understand just how much math actually influences and shapes society. “We Use Math” is extremely well done, and I highly encourage you to check it out and pass it along – especially to any students you know.

Oh, and on a less serious note, there’s also the alternative take on this topic.

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Knowledge Is Power… Maybe

John F. McGowan, Ph.D. — Mon, 02 May 2011 20:39:40 +0000

At least since Sputnik in 1957, various tests and reports such as the Trends in International Mathematics and Science Study (TIMSS) have asserted that students in the United States perform surprisingly poorly in math and science compared to students in other nations. The poor state of education is a perennial topic in the United States. Most people would like to see better teaching and education. Particularly with respect to math and science, this goal is often phrased in terms of producing more Einsteins or Edisons, exceptional or expert performers who would presumably, so the theory goes, make more scientific and technological breakthroughs, spurring progress and economic growth. President Obama returned to this time worn political cliche in his State of the Union 2011 “Sputnik moment” speech:

Maintaining our leadership in research and technology is crucial to America’s success. But if we want to win the future — if we want innovation to produce jobs in America and not overseas — then we also have to win the race to educate our kids.

Think about it. Over the next 10 years, nearly half of all new jobs will require education that goes beyond a high school education. And yet, as many as a quarter of our students aren’t even finishing high school. The quality of our math and science education lags behind many other nations. America has fallen to ninth in the proportion of young people with a college degree. And so the question is whether all of us — as citizens, and as parents — are willing to do what’s necessary to give every child a chance to succeed.

This article is a follow up to Debating Deliberate Practice about Malcolm Gladwell‘s bestseller Outliers and deliberate practice, a theory of expert performance developed by psychologist K. Anders Ericsson. Outliers has a great deal to say about mathematics and mathematics education in the chapters titled “Rice Paddies and Math Tests” and “Marita’s Bargain.” Gladwell argues for a reduction or elimination of summer vacation and more work and practice as a solution to the perpetual purported problems with math and general education in the United States. In “Marita’s Bargain” he showcases the KIPP (“Knowledge Is Power Program“) Academy in the South Bronx in New York City as an exemplar for a new educational system. Most people would like to see better teaching and education. Many people would like to know whether deliberate practice is the proper way to achieve expert performance. This article takes a closer look at KIPP and the controversies surrounding it with a special focus on mathematics and also the use of mathematics in public policy debates.

What is KIPP?

The Knowledge Is Power Program, often referred to by the acronym KIPP, is a national non-profit network of charter schools serving disadvantaged inner city neighborhoods such as the South Bronx. The vast majority of students at KIPP schools, most of which are middle schools (5th through 8th grades), are black and Hispanic children from poor families. KIPP seemingly has had excellent results based on standardized tests of reading and math skills. In addition to its appearance in Outliers, KIPP has been the subject of a laudatory book Work Hard, Be Nice by Washington Post education writer Jay Mathews. KIPP is also the subject of intense political controversy.

Charter schools are a special kind of public school designed, in principle at least, to avoid the bureaucracy and other limitations of traditional public schools. Charter schools can be organized by various groups, which include for-profit school companies, which apply for and receive a charter from the state along with funding for each student. The charter schools frequently receive additional funding from donations and other sources in addition to the public funding. The large majority of charter schools are organized and operate in poor, disadvantaged, “inner city” areas like the KIPP charter schools. Most students at charter schools are black and Hispanic like the KIPP charter schools. In many cases the teachers at charter schools do not belong to the powerful teachers unions: the American Federation of Teachers (AFT) and its affiliates or the National Education Association (NEA). A substantial number of charter schools are run by for-profit school companies, known as Education Management Organizations (EMOs). There are suspicions that KIPP is a stealth school company that will convert from its current non-profit status to a for-profit school company to be listed on the stock market when the time is right.

The education market in the United States is about $500 billion per year. The US spends about $10,000 per year per student on about 50 million students. The “School Enrollment in the United States — Social and Economic Characteristics of Students” report from the US Census Bureau (dated October 1999) gives a total of 49,339,000 students in public school in the United States (Table A, page3). A total of 8,556,000 public school students are identified as “Black” and 8,081,000 as “Hispanic (of any race)”. The National Center for Education Statistics (NCES) reports that the US spent $9,683 per pupil in the 2006-2007 school year ($10,041 in inflation adjusted 2007-2008 dollars according to the report). The United States probably spends around $160 billion on public education for black and Hispanic students. Most of this money is channeled through the traditional public school system run and administered by school teachers and administrators who are essentially civil servants with a high degree of job security (a tenure system) and most of whom belong to a teachers union.

Many conservative, libertarian, and business groups have embraced charter schools as a vehicle for school reform which often appears synonymous with bringing the “miracle of the market” to education and eliminating the unions. As one might imagine, the education unions and their allies are not happy about this and generally have a dim view of the charter school movement. Charter schools have gained significant support in both the Republican and Democratic parties, and in liberal as well as conservative circles. The Obama Administration has made expanding support for charter schools a part of its education program. Charter schools have a mixed performance record. Some are better than the traditional public schools in their district, some are about the same, and some are worse as measured by standardized test results. The KIPP schools, as well as a few other charter schools, however, appear to have a consistently good record compared to comparable traditional public schools, making them the flagship for proponents of charter schools and associated reforms, and therein lies the rub.

The first KIPP school was founded in inner city Houston, Texas in 1994 by David Levin and Mike Feinberg, recent alumni of Teach for America (TFA), which is closely associated with KIPP. Teach for America was founded by Wendy Kopp based on her 1989 Princeton senior thesis proposing a sort of Peace Corps in which recent Ivy League graduates would become teachers in disadvantaged inner city schools. Wendy Kopp and Malcolm Gladwell have appeared together to discuss education reform at the New York Public Library. Wendy Kopp worked closely with Richard Barth who eventually became her husband. Richard Barth left TFA to become an executive at Edison Schools, the controversial for-profit school company founded by Chris Whittle. After Edison, Richard Barth became the CEO and President of the KIPP Foundation. KIPP has grown substantially since 1994. KIPP acquired the substantial backing of the late Donald Fisher, a co-founder of The Gap clothing stores. Fisher was also an investor in Edison Schools. His son John Fisher is the current Chairman of the Board of Directors of the KIPP Foundation.

At one point, Edison Schools was the darling of conservative, libertarian, and business school-choice/school privatization advocates, arguing that a private for-profit company listed on the NASDAQ could run schools more cost-effectively and with better results than the inefficient, bureaucratic traditional public schools dominated by the teachers unions. Edison has had many problems. For example, it apparently cost more for Edison to run a school than a traditional public school. Although Edison still exists as Edison Learning, its star has dimmed and similar hopes appear to be now pinned on KIPP and other charter school operators.

KIPP’s Numbers

KIPP posts an annual “report card” on its web site for each KIPP school as well as other information on the KIPP program. There are also a number of reports, both independent and funded by KIPP, on the performance of KIPP. Figure 1 shows the relative test scores for the KIPP Academy New York, the school showcased in Outliers.

Figure 1: KIPP Academy New York (South Bronx) Relative Test Scores

The “State Criterion-Referenced Test” seems clear to interpret. The “Norm-Referenced Test: Measures of Academic Progress (MAP)” is not shown (cut off). Although also positive, the author found the explanation of what was being measured incomprehensible.

The “report card” gives the following general figures about the KIPP Academy New York which may be helpful in setting the context.

KIPP ACADEMY NEW YORK

250 East 156th Street, 4th Floor
Bronx-New York, NY 10451
718-665-3555 | www.kippnyc.org

School leader: Blanca Ruiz
Year founded: 1995
Grades served: 5-8
Student enrollment: 268
Number of teachers (FTE): 21
Per pupil funding: $13,675
Facility type: Provided by district
Size of school: 25,650 sq. ft.

Note: FTE is an acronym for Full Time Equivalent as in equivalent to a full time employee. For example, two part-time employees could make 1 FTE.

Figure Two shows the student demographics from the “report card”:

Figure 2: KIPP Academy New York (South Bronx) Student Demographics

There are a number of KIPP schools in New York City. The main KIPP NYC web site provides many overall statistics including some statistics on longer term outcomes (retrieved April 29, 2011):

To and Through College

Over the last seven years, 93 percent of KIPP NYC alumni graduated from high school. By comparison, 46 percent of African American and Hispanic students across New York City graduated from high school during 2005-08.
87 percent of KIPP NYC alumni have matriculated to college. By comparison, 21 percent of African American and Hispanic male high school graduates attend college in New York State; 36 percent of all low income New Yorkers ages 18-24 matriculated to college during 2003-08.
To date, 31 percent of KIPP NYC alumni have earned bachelors’ degrees within six years of high school graduation. Our objective is to have 75 percent of our students graduate from college. We’re currently on track to hit a 50 percent 6-year graduation rate in the next two years; nationally, the college completion rate in low income communities is 10 percent.

NOTE: The comparisons in the three bullet points differ. The first bullet point compares KIPP NYC alumni favorably to African American and Hispanic students across New York City. The second bullet point compares KIPP NYC alumni favorably to African American and Hispanic male high school graduates in New York State and all low income New Yorkers aged 18-24. The third bullet point compares KIPP NYC alumni favorably to the national college completion rate in low income communities. This sort of slicing and dicing of data should make one uneasy and it is very easy to do today with widely available powerful computers and statistical tools.

Nonetheless, KIPP schools generally report positive results compared to comparable traditional public schools. Has KIPP found the magic formula for better education? Is it deliberate practice as Outliers implies?

Pump Up The Volume

In the United States, there is a genre of movie that features a new teacher, frequently young and idealistic, at a school, frequently in a poor crime-ridden inner-city neighborhood, who, after some problems and setbacks and usually a battle with the heartless or narrow-minded school administration, manages to inspire his or her students, generally a bunch of losers, to perform spectacularly well. Some are fairly serious like Dangerous Minds in which Michelle Pfeiffer stars as a former US Marine who inspires (yes) a bunch of losers. Others are over the top like the Arnold Schwarzenegger film Kindergarten Cop or The Substitute, which stars Tom Berenger as a mysterious special forces soldier who substitutes for his teacher girlfriend, who has barely survived an assassination attempt by a high school gang, in a tough inner city school. The public relations stories of Teach for America and KIPP very much fall within this tradition as Malcolm Gladwell notes in Outliers.

The 1990 movie Pump Up The Volume (not a very good movie) features a star school principal who turns her high school around, achieving winning SAT scores, and the adoration of the clueless parents. The secret to success? The principal illegally engineers the expulsion or removal of the school’s misfits and losers who score badly on the SAT test — using trumped up disciplinary charges and other tactics. This is what is known as sample bias. If you only test the good students, then you will get good scores.

Critics of KIPP and other high performing charter schools frequently allege a Pump Up The Volume scenario, whether intentional or not, to explain the test scores. There are two sources of bias that are frequently suggested. First, students and their parents choose to apply for a position at a KIPP or other charter school. By default, they would simply stay at their traditional public school. So there may be a selection effect in which more motivated or capable students and families apply to the KIPP schools. This does not require any skullduggery on the part of the KIPP schools. Second, students may leave KIPP after being accepted, known euphemistically as attrition or student mobility. The education business has more than a little jargon. It is here that the KIPP schools could consciously, unconsciously, or accidentally push out the “losers”. Critics often cite a study by SRI International (the think tank in Menlo Park, California formerly known as the Stanford Research Institute) funded by the William and Flora Hewlett (as in Hewlett-Packard) Foundation of the KIPP schools in the San Francisco Bay Area that showed a high rate of attrition in these schools.

KIPP has retained Mathematica Policy Research Inc. to conduct a comprehensive longitudinal study of the KIPP program and schools, probably in no small measure to counter the suggestions that KIPP gets rid of the losers to pump up its numbers like the nefarious school principal in Pump Up The Volume. Mathematica has presented a fifty-one page working paper “Student Selection, Attrition, and Replacement in KIPP Middle Schools” at the 2011 Annual Meeting of the American Educational Research Association (AERA). Mathematica has also produced an interim preliminary report which is highlighted on the main KIPP web site. This is a war waged with advanced statistics, regression models, and other sophisticated, ostensibly scientific, objective tools.

Infinite Regress

Modern public policy debates are frequently waged, at least in public, by think tanks such as the libertarian Cato Institute or the progressive Center for Economic and Policy Research (CEPR) (two heavily quoted and cited think tanks in the major media). These think tanks routinely produce scholarly studies, books, reports, and so forth that usually seem quite sophisticated, conform to most of the standards and forms of scholarship or science, and frequently incorporate sophisticated statistical and mathematical analyses. The think tankers usually seem extremely intelligent, highly educated, sincere, passionate about their beliefs and committed to the truth and the betterment of the world. The only problem is that they almost always produce studies and rigorous statistical and scientific analyses that confirm their beliefs. With a little practice, one can usually anticipate where a Cato Insitute or a CEPR will come out on an issue: often in the form of scholarly studies with data, seemingly rigorous analyses, and often sophisticated statistics. It is also extremely rare to find a think tank producing a report or study that concludes “we don’t know what is going on”. Think tanks live in an alternate universe where the think tankers have all the answers even to extremely complex and puzzling scientific and technical problems.

Now, one cannot claim that think tankers never come to contrary conclusions and reverse their positions because they believe that the accumulation of evidence fails to support their original position. For example, in the current controversies over education, Dianne Ravitch, a scholar and former education official in the George H.W. Bush (Bush I) administration, is well known for reversing many of her views. Nonetheless, this is quite rare and it is almost always an isolated individual rather than an institution or organized group. In fact, the apostate will often withdraw from the now offended institutions or groups. It is extremely rare to encounter anything like a retraction or explicit reversal of doctrine from a think tank or similar group. Sometimes these organizations will simply stop talking about something. For example, many liberal and progressive groups simply stopped talking about school busing after the Reagan Administration mostly killed it in the 1980′s.

How does this happen? Aren’t scientific and statistical methods supposed to provide definitive answers in a rigorous objective way? In most of these controversies where advanced statistical methods are used in an attempt to settle public policy issues “scientifically”, the issue of sample bias or some sort of measurement error frequently arises. In practice it is often extremely hard to design an experiment, study, or survey that could not be strongly affected by some sort of bias or measurement error. The commonly used statistical methods have limited ability to unambiguously detect or eliminate bias or measurement error. Thus the typical “scientific” public policy debate goes something like this:

Free Market Think Tank: Our new study of widgets shows conclusively that the free market widget producers in (insert allegedly free-market oriented organization, county, state, or nation here) outproduce the lazy socialist widget producers in (insert allegedly socialist organization, county, state or nation here) by (insert percent) percentage points. This is statistically significant at a (insert confidence level here) level.

Progressive Think Tank: That sounds fishy. We will have our staff of passionate young research associates examine the raw data. (Suspicious) You do have the raw data, don’t you?

Free Market Think Tank: Er, um, well, yes, you can look up the data on (fill in obscure source of technical data such as the Widget Producers Trade Association here).

Progressive Think Tank: The…(long pause)…Widget Producers Trade Association? Couldn’t they be biased in favor of their greedy, self-serving corporate members?

Free Market Think Tank: Er, um, well, we’ll check on that.

(some time passes)

Progressive Think Tank: Our research associates have completed their analysis of your…flawed, biased data and found that the data from the (heavy sarcasm) Widget Producers Trade Association does not properly account for (fill in obscure technical issue that only widget manufacturing experts understand here).

Free Market Think Tank: Not so! We have reanalyzed the data from the Widget Producers Trade Association using the Blowhard Method (see path-breaking paper by Blowhard et al in the American Journal of Widget Manufacturing) and, well yes, a few numbers have changed slightly, but our study holds up and remains statistically significant at the (fill in confidence level here) level.

Progressive Think Tank: The Blowhard Method? Isn’t Blowhard…(long pause)…the Director of Research at Amalgamated Widgets, the notorious multinational anti-union widget manufacturer?

Free Market Think Tank: Well, yes, we go to the top experts and well, heh heh, they always seem to work for Amalgamated! They are the top widget company after all!

Progressive Think Tank: Well, we will have our staff of passionate young research associates take a further look at the data.

Free Market Think Tank: You do that. Meanwhile we will be putting out a new press release and appearing on CNBC, Fox, and the Colbert Report!

(some time passes)

Progressive Think Tank: Ahem, we have found that your reanalysis did not consider the well-known Sneezer Effect documented in Sneezer et al, in the International Journal of Widget Manufacturing. Sneezer demonstrated that when widgets are manufactured using the cold aluminum brazing method of Warner and Hancock, the Blowhard Method is inappropriate and biased. Once we account for the Sneezer Effect, your positive effect disappears and we find that the (insert euphemism for socialist here) widget producers outperform the corporate widget producers by (fill in percentage level here) percent at the (insert confidence level here) level. We are putting out a new press release and appearing on CNBC, Rachel Maddow, and the Daily Show!

This can and often does continue indefinitely as each side finds and/or rules out various scenarios under which some sort of bias or measurement error could have produced or invalidated the contested result (the free market failed, union workers are lazy etc.). These debates often get extremely technical and arcane as the specific potential sources of bias or measurement error frequently involve the technical details of the specific activity being debated. Frequently most third parties without detailed expertise in the subject matter (e.g. widget manufacturing) cannot follow the debate and lack the time and resources to come up to speed on the arcane issues of the topic. They frequently lose interest in the entire issue or simply revert to their ideological biases, preconceived notions and so forth.

The fancy statistical techniques such as regression analysis often have no or limited ability to deal with the potential sources of bias. Thus, the accuracy of the quoted statistical results depends on the original selection or measurement of the data. Even if one can rule out specific scenarios for bias or measurement error, one can usually (not always) come up with yet another scenario that requires a new experiment, study, or analysis to rule the new scenario out. And outright fraud or hoaxing is extremely difficult to rule out. One can almost always construct some sort of conspiracy theory to circumvent any seeming precautions against fraud or hoaxing. Of course, to maintain one’s serious scholarly reputation, one should never call it a “conspiracy theory.”

This infinite regress in which a new possible source of bias or error replaces each previously proposed source of bias or error as it is ruled out or demonstrated unlikely occurs in many scientific controversies, especially in fringe areas such as parapsychology or UFOs. Parapsychology is particularly relevant because laboratory parapsychology such as J.B. Rhine’s famous studies at Duke University involves extensive use of statistics. It actually proves extremely hard to design and carry out experiments in which there is no possible source of significant bias, measurement error, or fraud. Fraud or hoaxing. in particular, is often difficult to prove and even more difficult to conclusively rule out.

What to make of KIPP?

It is difficult to know what to make of KIPP. It is a small program with schools located exclusively in quite poor areas by US standards that are mostly out of the author and probably most readers’ experience. Both advocates and critics have potentially serious conflicts of interest; even a small fraction of $500 billion is a huge prize. In violent crime-ridden areas there are methods of encouraging poorly performing students to leave or otherwise rigging the results that probably could not be detected by conventional scholarly methods. This could happen without any actual intent on the part of the wealthy KIPP leaders. “Do whatever it takes” can have a very different meaning in the South Bronx than at Princeton or Yale.

It is reasonable to expect that the longer hours, Saturday classes, and longer school year, combined with more extensive drilling and so forth could produce substantial measurable benefits, especially on standardized tests. “Practice makes perfect.” and “Practice, practice, practice.” are ancient maxims.

If authentic, is KIPP’s success an exemplar of the theory of deliberate practice proposed by K. Anders Ericsson? In one sense of deliberate practice, it is clearly not. In some versions of deliberate practice, the need to practice relatively rare activities such as the tennis backhand is emphasized. The basic arithmetic and basic reading on standardized tests is generally not rare and does not involve rare trick questions. Ordinary practice or experience is probably all that is needed to improve performance on these tests. Indeed this may be one of the differences between math and science education at the K (kindergarten) through 12 (12th grade) level as opposed to college and university. Trick problems actually seem rare in K-12 whereas college and university level math and science exams often have many more of these and the preparation methods taught in K-12 school may be inadequate.

Another sense of deliberate practice is “continuous improvement,” emphasizing continuous self-examination of failures. In this sense, the KIPP programs may well be examples of deliberate practice judging from the descriptions of KIPP. Indeed, one might expect this to be helpful in improving performance even if deliberate practice is an incomplete or even mostly wrong theory of expert performance.

A more difficult question is what effect the KIPP programs have on general and abstract reasoning skills which can be difficult to measure on standardized tests, in part because even their nature is poorly understood. If this effect is negative, as I suspect it would be if the KIPP methods are largely equivalent to rote memorization, then the widespread adoption of KIPP methods could have a disastrous effect on public education and the economy. It is likely that these general and abstract reasoning skills are used heavily in major inventions and discoveries, commonly referred to as “breakthroughs”, an extremely rare phenomenon often cited as a justification for education reforms, especially math and science education reforms.

Conclusion

The vast majority of people would like to see excellent schools that enable a vibrant growing economy and productive scientific and technological progress. Unionized teachers and education entrepreneurs alike, for the most part, probably agree on the goal of well educated, happy, successful students. The question for most people of good will is how to achieve this goal. Seemingly sophisticated scientific, mathematical, and statistical methods often fail to provide the clear answers that most would like and may obfuscate the issues, substituting intimidating mathematical formulas, obscure statistical jargon, and arcane computer programming for genuine understanding.

Many debates like the KIPP debate would be improved by the full disclosure of the raw data (individual tests and test scores in this case) as well as the precise computer programs used to analyze the data. This question has arisen in the context of global warming for example. With the Internet and the World Wide Web it is certainly possible to do so. The analyses can be performed using freely available high quality statistical tools such as R or the MATLAB compatible tools Octave or Scilab or a number of others. Conventions could be established for the programs and data such as requiring human readable (plain English) variable and functions names that reflect the meaning of the function or variable: e.g. StudentAttritionRate for the student attrition rate and so forth. Many people are intimidated by cryptic symbols and computer code. A point-and-click, drag-and-drop graphical user interface (GUI) front end to the mathematical scripting code would be helpful in making the data analysis more accessible and credible to ordinary citizens as well as policy makers. The wxMaxima front end to the Maxima computer algebra system is a limited example of a graphical front-end for a mathematical scripting language. With student performance data and medical records there are clearly issues regarding protecting the privacy of the students or medical patients. With full disclosure, anyone could examine and reproduce the entire data analysis, step by step.

With current statistical methods, there does not seem to be a very good, or at least a very good well known way to deal with the persistent issues of bias and measurement errors or the infinite regress that often results from these issues. Rather, these central problems are dealt with in an ad hoc fashion that varies greatly from field to field, perhaps necessarily. Disclosing raw data alone cannot fully address these problems. It is, at minimum, necessary to have detailed information on how the data was selected and measured in the disclosure (metadata in computer jargon). Existing statistical methods are probably inadequate to deal with some of these bias and measurement issues. In part for these reasons, it is difficult to reach a definitive conclusion regarding KIPP either as an education reform or as an exemplar of the theory of deliberate practice.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

References/Suggested Reading

Spin Cycle: How Research Is Used in Policy Debates: The Case of Charter Schools
Jeffrey R. Henig
Russel Sage Foundation Publications
(February 2008)

The Death and Life of the Great American School System: How Testing and Choice Are Undermining Education
Dianne Ravitch
Basic Books
(March 2, 2010)

A Chance to Make History: What Works and What Doesn’t in Providing an Excellent Education for All
Wendy Kopp
Public Affairs
(January 25, 2011)

Crash Course: Imagining a Better Future For Public Education
Chris Whittle
Riverhead Hardcover
(September 8, 2005)

The Charter School Dust-up: Examining The Evidence On Enrollment And Achievement
Martin Carnoy (Author), Rebecca Jacobsen (Author), Lawrence Mishel (Author), Richard Rothstein (Author)
Teacher College Press & Economic Policy Institute
(April 30, 2005)

Work Hard. Be Nice.: How Two Inspired Teachers Created the Most Promising Schools in America
Jay Mathews
Algonquin Books; First Edition edition
(January 20, 2009)

No related posts.

Debating Deliberate Practice

John F. McGowan, Ph.D. — Mon, 18 Apr 2011 19:39:36 +0000

Outliers: The Story of Success (USA | UK | CANADA)
Malcolm Gladwell
Little, Brown, and Company, New York, 2008
309 pages

Outliers is the 2008 bestseller written by New Yorker magazine business and science writer Malcolm Gladwell, author of The Tipping Point, Blink, and What the Dog Saw. Gladwell is the son of mathematician Graham Gladwell and Outliers has a lot to say about mathematics, both indirectly and directly in the chapters “Rice Paddies and Math Tests” and “Marita’s Bargain”. Most importantly, Outliers has popularized the theory of deliberate practice espoused by psychologist K. Anders Ericsson.

Briefly, in the perennial “nature versus nurture” debate, Ericsson and, to a lesser extent Gladwell in his book, takes an extreme environmental position, attributing expertise and what is commonly referred to as “genius” or “talent” solely to many hours of “deliberate practice.” Ericsson and Gladwell argue that success is due to performing at least ten-thousand hours of this deliberate practice, typically over ten years. The title of chapter two in Outliers is The 10,000-Hour Rule. Gladwell, in particular, uses the term deliberate practice vaguely in the book and his public presentations, often using it interchangeably with practice, as will be critiqued further below. Ericsson is more precise in defining what he means by deliberate practice and distinguishing it from ordinary practice or experience. This is important because there are obviously many examples of scientists, mathematicians, athletes, chess players, and others with far more than 10,000 hours of ordinary experience or practice in their field who do not perform at the genius or star level that Ericsson and Gladwell are talking about.

In Ericsson’s theory, deliberate practice involves many hours of heavy practice of relatively rare tasks or activities. The example he has used on a number of occasions is the backhand in tennis. The backhand is relatively rare. Deliberate practice involves, amongst other things, practicing the backhand heavily to master this relatively rare move. Consequently, a tennis player who has engaged in this sort of practice can, on average, easily beat a player who has mere experience or even less intensive forms of practice. Now, in fact, Ericsson’s definition of deliberate practice is not quite this simple. Sometimes he refers to deliberate practice as practice in which there is a strong conscious focus on self-criticism and self-improvement, intensively examining one’s performance for errors and room for improvement. Deliberate practice is difficult to define across many different fields and activities. What is deliberate practice in tennis may not be deliberate practice in mathematics. What is deliberate practice in pure mathematics (theorem proving) may not be deliberate practice in applied mathematics (numerical computation, for example).

Outliers is largely composed of a series of stories that illustrate Gladwell’s main points. Many of these stories are open to alternative interpretations and some are highly questionable. This article will discuss a number of the stories in Outliers, pointing out their problems, with a special focus on deliberate practice and mathematics.

A Straw Man

Like many political works across the political spectrum, Outliers is guilty of setting up a straw man that is easy to debate and defeat. Gladwell treats the reader to some seemingly absurd quotes from Florida Governor Jeb Bush, President George W. Bush’s brother and President George H.W. Bush’s son, describing himself as a “self-made man”. In Outliers, Gladwell’s unnamed debate opponent, possibly Jeb Bush, is arguing that success is solely about individual talent and genius. There are no significant environmental, cultural, or other factors involved. Just because your Dad is the President has nothing to do with your relative success, say compared to that guy across town who’s Dad is a janitor. Certainly not. I did it all myself. Of course, this is a position taken by very few, probably not even Jeb Bush despite the quotes. Rather the actual debate is between those who view inherent talent, genius, hard work, and so forth as relatively more important on average than environmental factors, especially in “first world” nations such as the United States which ostensibly have democracy, free markets, and the rule of law.

Correlation is Not Causation

Outliers opens with the story of Roseto, an Italian-American community in Pennsylvania with a strikingly low rate of heart disease even though the residents of Roseto appear to share many of the allegedly unhealthy dietary habits of other Americans. Gladwell recounts how medical researchers have theorized that this healthy state of affairs is due to the traditional family and social culture maintained by the Italian-Americans in Roseto, in which respect Roseto apparently does differ from neighboring communities and the United States as a whole. This incidentally flies directly in the face of the medical orthodoxy blaming heart disease and heart attacks on diet, lack of exercise, fatty foods, and especially cholesterol levels which can be controlled by prescription drugs.

The problem is that what the researchers have demonstrated in Roseto is a correlation between an unusual lifestyle, for Americans, and better health. That is all. They have not proven this is the cause and the data would appear to directly contradict the causal theories of heart disease and heart attacks favored by medical researchers and pharmaceutical companies peddling anti-cholesterol drugs.

Indeed, this is a common problem in modern medicine and medical research. Many reigning views — paradigms in the overused language of the philosopher of science Thomas Kuhn — are based, on close examination, on statistical correlations between disease or health and various factors. Much of the modern theory of heart disease and heart attacks is based on the famous Framingham, Massachusetts heart disease study which showed correlations between various factors such as weight and diet and heart disease and heart attacks. However, there are other studies such as the Roseto study touted by Gladwell that appear, at least to the naive observer unschooled int he elaborate theories that modern researchers frequently produce to explain otherwise grossly contradictory data, to disagree.

Thus, the Roseto story may illustrate the primacy of culture over other individualistic causes as Gladwell proposes. Then again, it may not. It may simply be another data point indicating that we don not understand heart disease and heart attacks nearly as well as the medical experts believe (or, at least, claim).

It is Not What You Know; It is When You Were Born

Gladwell follows with a story about champion Canadian hockey players, who remarkably are nearly all born in the months of January, February, and March of each year. Shocking! Very few would disagree with the proposition that success in hockey or other competitive activities should be about talent, skill, determination, and so forth, and certainly not which month you were born in. This oddity is thought to be due to the annual age cutoff in hockey competitions in Canada which gives an advantage to kids born at the start of the year who are the oldest and thus largest and most mature members of their age cohort in each year.

Certainly this is unfair and a better system would compensate for this effect. But, how much does it tell us about the relative importance of innate talent, if it exists, and environment? Actually, it may tell us very little. After all, the vast majority of kids in Canada who try out for hockey don’t end up in the champion teams. What this oddity may tell us is only that, among those with the highest innate hockey talent, those born in the first few months of the year have an unfair advantage over their otherwise talented competitors.

If there is an effective cutoff in talent or size or some other variable innate to the person, then the very best will be sorted solely by external or random factors such as birthdates. This does not tell us that environment or culture is King, but rather that there is an upper bound to innate talent and there are more competitors at the upper bound than available positions. As the global population expands, we would expect more and more people to exist at the upper bounds of human potential.

As a specific example, suppose that innate hockey ability can be measured on a special scale from 0 to 100. The potential best hockey players in the world score 100. One-hundred thousand Canadian kids compete for 1000 positions on twenty national hockey teams. If there are more than one thousand Canadian would-be hockey players who score 100 on the innate hockey ability scale, purely environmental or chance factors will start to kick in and contribute to the players who end up on the teams. If there are 10,000 top hockey players competing for 1000 positions, the environmental effects are likely to be very strong. For example, only players born in early months in the year who have an unfair advantage in size and maturity may be selected. This does not mean innate ability is not very important relative to environment. In fact, 90,000 players fail because they lack the innate ability. Only 9,000 players fail to make the teams because of environmental effects.

Gladwell plays up the clustering of the birthdates of software entrepreneurs, corporate takeover lawyers, and others in several other places in the book. The same objection noted above applies to these examples as well.

A Hard Day’s Night

Outliers then introduces the concept of deliberate practice and the 10,000 hour rule in earnest. To illustrate Ericsson’s theory Gladwell tells the titillating tale of the Beatles sojourn in Hamburg, Germany where they ostensibly acquired ten-thousand hours of deliberate practice playing seven days a week, eight hours per day or more, at — yes — a strip club. Gladwell returns to this tale several times in Outliers and obviously enjoyed telling the story during his promotional appearances for the book; videos of many of his presentations are available on the Web.

With all due respect to the genuine musical talents of John Lennon, Paul McCartney, George Harrison, and Ringo Starr, popular music is clearly about a lot more than talent and indeed occasionally no genuine talent at all seems to be required. There have, for example, been scandals in which a performer could not even sing; the voice was provided by someone else, perhaps less photogenic and less sexy. There are very few fat or ugly people among popular music stars The manufacturing of pop stars and bands is so obvious that it is the subject of repeated satire in popular culture, movies, and so forth. The slimy music executive who manufactures a popular musician or band is a stock character in movies and television. More to the point, did playing songs at a strip club in Germany really qualify as the tedious deliberate practice in Ericsson’s theory? This is rather doubtful, but it is a catchy story with plenty of sex.

Is Bill Gates a Champion Software Developer?

Outliers also tells the tale of Bill Gates, the Microsoft founder and former CEO, whom Gladwell apparently interviewed. In particular, Gladwell tells the story of how Bill Gates supposedly acquired at least ten-thousand hours of deliberate practice on a time sharing computer as a kid in the 1960s. Outliers attributes Gates and Microsoft’s success in no small measure to this practice and Gates skills as a software developer.

As with the Beatles, it is difficult to evaluate this story. Did it really happen? Did Bill Gates practice on the time sharing computer qualify as Ericsson’s vaguely defined deliberate practice? Most importantly, is BIll Gates and Microsoft’s phenomenal success due to technical proficiency in software development?

The answer to this last question is probably not. Much of Microsoft’ success can be traced to a shrewd business deal in which Bill Gates sold an operating system, now known as DOS, that he did not have and did not create, to IBM for the new IBM PC. IBM then proceeded to make a series of colossal missteps over the next several years which culminated in sabotaging their OS/2 operating system and handing the PC software market to Microsoft, essentially an IBM supplier. Bill Gates is a shrewd businessman and negotiator; of this, there is no doubt. Was he a champion software developer: most probably not.

What is Deliberate Practice?

Both the story of the Beatles and the story of Bill Gates illustrate the problem of defining exactly what is deliberate practice. In his book and public presentations, Gladwell usually sidesteps this issue, often using the terms deliberate practice, experience, and practice interchangeably. Yet, the definition of deliberate practice is central to Ericsson’s theory. There were and are thousands of chess players with many more hours of ordinary experience or practice than Bobby Fischer when he became an International Grand Master in chess. The same with many other intellectual activities. Yet, many highly experienced individuals do not perform at an extremely high level like Bobby Fischer in chess or Tiger Woods in golf or Albert Einstein in physics. If just any experience or practice worked, most intellectual activities would be dominated by octogenarians. Yet, in many activities this is not the case. Why do some people with only 10,000 hours of experience grossly outperform others with 20 or 30,000 hours of experience? It is this problem that requires the definition of a special kind of practice — deliberate practice — to avoid invoking innate talent or genius.

Chris and Oppie

In arguing for nurture over nature, Outliers tells the stories of Chris Langan, allegedly the smartest man in the world, and the famous physicist J. Robert Oppenheimer, the scientific director of the Manhattan Project which developed the atomic bomb during World War II. Langan hailed from a poor, rather dysfunctional family and flamed out of a college allegedly due to administrative problems that led to the cancellation of his scholarship. He has had a succession of rather unimpressive jobs, is now a farmer, and won $250,000 on the One versus One Hundred game show. Oppenheimer was the brilliant son of a wealthy Jewish garment manufacturer in New York City and sailed through the attempted murder of his thesis adviser at Cambridge with a a slap on the wrist, to be later picked by General Leslie Groves as scientific director of the Manhattan Project — over many more highly qualified physicists such as Leo Szilard and Karl Compton, who do not seem to have had a track record of trying to murder someone.

Gladwell attributes Oppenheimer’s rather remarkable success, especially compared to Langan, to Oppenheimer’s sophisticated upper class social skills acquired during his posh upbringing — contrasted to Langan’s presumably poor working-class social skills. This is somewhat curious. By many accounts, Oppenheimer was quite arrogant and even admirers concede that his notorious arrogance contributed to his eventual downfall. And, ahem, he had tried to murder his thesis adviser.

In Gladwell’s account, Oppenheimer talks his way out of an attempted murder charge. There is not even a suggestion that Oppenheimer’s wealthy father pulled some strings (paid someone off) to make the potential attempted murder charges go away. A poor man’s felony is a rich man’s prank.

Why would General Groves have selected Oppenheimer with his history of attempted murder, mental problems, extensive connections to the Communist Party through his wife, mistresses, graduate students, and friends, abrasive, arrogant personality, and limited resume over Leo Szilard, Karl Compton, or many other more qualified physicists of the time? General Groves did not initiate the Manhattan Project. By most accounts, Leo Szilard, a friend and business partner of Albert Einstein and a rather murky character in his own right, was the original mastermind of the program. General Groves was brought in later as the project grew and quickly took a disliking to Szilard, forcing him out. Groves was also embroiled in feuds with other prominent physicists such as Karl Compton.

In selecting Oppenheimer to run the Manhattan Project, Groves was promoting a relatively young (38), second tier physicist who would be heavily dependent on Groves patronage for his position: certainly not the case with Szilard or Compton, for example. Oppenheimer’s dubious background probably ensured that Oppenheimer could be easily disposed of if he zigged when he was supposed to zag. Years later, in disputes over the Air Force’s atomic bomber program and the hydrogen bomb, Oppenheimer appears to have zigged when he was supposed to zag and, indeed, he was easily discredited and removed from influence, becoming a martyr for the American Left in the process.

This blindness to politics is one of the recurring problems with Outliers. In Gladwell’s account, the rich and powerful succeed because they have a better culture and because their parents take them to extra violin practice or pay for them to practice on time sharing computers. It is not genes but hard work, deliberate practice, that Gladwell uses to explain success. There is no politics, no string pulling, no skullduggery, no old boy networks or hidden agendas.

This real or feigned political naivete is particularly evident in Gladwell’s response to some of Chris Langan’s comments about Langan’s unhappy experiences with academia:

Even is his discussion of Harvard, it’s as if Langan has no conception of the culture and particulars of the institution that he is talking about. (Langan speaking) When you accept a paycheck from these people, it is going to come down to what you want to do and what you feel is right versus what the man says you can do to receive another paycheck. What? One of the main reasons college professors accept a lower paycheck than they could get in private industry is that university life gives them the freedom to do what they want to do and what they feel is right. Langan has Harvard backwards.

Oh, really? This argument that researchers and scholars in academia could make more money in industry and thus must be motivated by altruism and academic freedom is extremely common. It is repeated in both popular science and by academics themselves. It comes up in job interviews when academic researchers are recruiting people to work for their labs. It plays a role in the decision by many to go to graduate school, pursue a Ph.D., and try to become a professor, often a disappointing experience. It also gives the pronouncements of academics and researchers the added authority and credibility of the selfless truth seeker. But is it true?

Researchers and scholars may pursue a career in academia because they believe they have a special aptitude, whether inborn or acquired, for their specific field. I have talked with physicists who cited this as their reason for pursuing a career in physics. If one believes that one has a high general intelligence and could do whatever one wants, then one might pursue a career in physics or another academic field out of altruism. On the other hand if one believes one has a special advantage in a specific field, then one may pursue an academic career in preference to industry because one expects to do well specifically in that field, better than one could do in industry. Despite very high intelligence, Chris Langan did not do well in industry, at least until his game show appearance. Indeed, many of the stories that Gladwell recounts, which call into question the notion of general intelligence, undermine this argument.

The deliberate practice that Gladwell touts in his book is about practicing specific problems and techniques used in specific narrow fields whether these are chess or golf or the exotic techniques of particle physics. Deliberate practice is not the general intelligence of Lewis Terman that Gladwell ridicules. Someone who through happenstance or careful thought engages in huge amounts of deliberate practice to score well in tests in a specific academic field does not, in many cases, have easily transferable skills to industry. Gladwell’s own argument about the the true nature of success actually significantly undercuts the argument that he makes in discussing Chris Langan’s comments on Harvard.

Modern research is not a side activity of professors who are principally paid to teach students. It is often discussed in popular science and by academics themselves as if it were, but it is not. Rather it is largely funded by gigantic government bureaucracies such as the Department of Energy (DOE), National Institutes of Health (NIH), National Aeronautics and Space Administration (NASA), National Science Foundation (NSF), the Defense Advanced Research Projects Agency (DARPA), and several others. All told, the United States federal government spends about $100 billion on activities that are classified as research and development. Major universities such as Harvard receive billions of dollars in grants and contracts from these agencies. Faculty are recruited and rewarded most especially for bringing in grant money from the government. Modern researchers usually list the dollar amount of grants or contracts that they have received on their curriculum vitae The very top researchers in major fields such as molecular biology or high energy physics do fairly well financially; it is not clear they could have done better in industry. This does not mean that the vast majority of graduate students, post-doctoral researchers, and junior faculty, most of whom will not get a tenured faculty position and will eventually leave their field for industry, are not accepting low salaries and could do better in industry. In most cases, grad students and post-docs who are US Citizens, have a green card, or who otherwise can move freely from job to job could find a better paying job in industry.

In many respects, Gladwell’s account is not that different from the conservative Republicans like Jeb Bush that he is presumably debating. Instead of attributing the success of the upper class to some mysterious genetic advantage, Gladwell repeatedly invokes virtuous hard work, deliberate practice, and special cultural advantages such as superior social skills.

In comparing Chris Langan and J. Robert Oppenheimer, Gladwell is tearing down a straw man. Coming from a very wealthy family, Oppenheimer had enormous advantages in addition to genius. On the other hand, although a genius, Langan came from an extremely poor and dysfunctional family environment. If one argues that innate talent always wins out, then this comparison easily settles the argument: no it does not. Gladwell compares two performers for whom the environmental differences are extreme. Even those who believe innate ability is a very strong determinant, say contributing ninety percent of outcomes on average, might still expect to see dramatic differences in outcomes and performance between an Oppenheimer and a Chris Langan. What happens when we compare a genius like Chris Langan from a stable, middle class family with someone of average intelligence also from a stable, middle class family?

The Problem with Deliberate Practice

In the United States (at least since Sputnik in 1957), math and science education are always in trouble. The schools are always failing to teach math and science. There is always a desperate shortage of mathematicians and scientists. Ten Ph.D.’s for every tenure track position is way too few. The United States desperately needs more Ph.D.’s. Some other nation — Russia, Japan, and now China — is always running circles around the spoiled, wimpy, not to mention dumb American school kids. The sky is falling! Send more money! In the chapters “Rice Paddies and Math Tests” and “Marita’s Bargain” Gladwell discusses the perennial issue of math education in the United States and makes specific suggestions to improve math education based on the theory of deliberate practice as he sees it. Perhaps not surprisingly, the bottom line is that students should work, study and practice harder. Gladwell proposes to abolish summer vacation and replace it with many more weeks of drilling and drudgery in hopes of producing a nation of virtuous Asian-style math whizzes (China being the current foreign nation running circles around America’s mediocre math and science students).

There is another, less flattering name for deliberate practice: rote memorization. Anders Ericsson’s research focuses heavily on activities that involve short, timed contests or exams: competitive games like chess, sports, music performance, and, of course, academic exams and tests. If a test or competition involves activities that are too complex to figure out from first principles in the time allotted — even for a genius — then one must practice and memorize the activity to succeed in the test or competition. A tennis player does not have time to figure out how to hit a backhand during a tennis match. He or she had better know how to do it to defeat a player who does. Only the most extreme innate intelligence, like the fictional hero of the Pretender TV show, could adapt and perform in real time.

There is no doubt that there are many real world, relatively repetitive activities that require practicing and memorization. We rightly expect air traffic controllers and pilots to know what to do, how to fly an airplane, and so forth. We do not want them to be figuring out what to do while flying a plane with hundreds of passengers. But does memorizing the answer to a problem that we (humankind) already know how to answer teach the student how to solve a problem that we don’t know the answer to: curing cancer, free energy, an end to war?

Invention and discovery, research and development, software development, and many other activities, including, for example, responding to a new kind of emergency, depend on the ability of scientists, engineers, and ordinary citizens to figure out new things and improvise. Deliberate practice will not necessarily teach these important skills; in many cases, we do not even know what these skills are.

Gladwell quotes several famous education reformers from the nineteenth century with evident disdain for their enthusiasm for summer vacations and (OH MY GOD) play:

It is when thus relieved from the state of tension belonging to actual study that boys and girls, as well as men and women, acquire the habit of thought and reflection, and of forming their own conclusions independently of what they are taught and the authority of others.

Gladwell expresses astonishment at the nineteenth century notion that one could study too much or much or that it could have adverse consequences such as insanity, attributing such mushy thinking to the wimpy Western farming tradition as opposed to the virtuous work-until-you-drop tradition of the Asian rice paddy.

Yet, there is a striking pattern in many major invention and discoveries that the nineteenth century education reformers may have been well aware of. A very high proportion of major inventions and discoveries have been made on a break, a vacation for example, when not actively thinking about the problem. According to Greek historical accounts, Archimedes solved the problem of determining the gold content of the King’s crown without destroying the crown (a major breakthrough) while taking a bath. Kepler realized that Mars had an elliptical orbit over the Easter Holiday in 1605. The mathematician William Rowan Hamilton conceived of quaternions during a recreational walk. In his autobiography, Nikola Tesla described suddenly seeing the design of the alternating current motor in his head while watching the sunset and discussing Goethe. James Watt claimed he conceived of the separate condenser steam engine during a walk in the park. Erwin Schrodinger came up with the Schrodinger Equation on a ski vacation in the Alps. In his book on mathematical invention, the great French mathematician Jacques Hadamard concluded that this was part of the general pattern of inventions; they frequently occurred during a break.

There is also evidence in support of the nineteenth century notion that one can overdo study. The famous mathematician Felix Klein burned himself out in his scholarly duel with Poincare and lost his ability to perform many mathematical research activities, devoting himself to administration and mentoring other mathematicians. Several famous mathematicians including Georg Cantor and Kurt Godel developed serious psychological problems. Godel may have starved himself to death. Many inventors and discoverers have described a period of mental exhaustion after making their “breakthrough”. Modern software developers often experience “burnout” after prolonged programming projects. Software development has very high turnover rates and many people enter and, more to the point here, leave the field every year.

Deliberate practice presents the hazard of substituting rote memorization for deeper understanding. In many respects a form of studying for the exam, it likely can deliver high levels of performance on tests and exams, substituting easily measurable technical skills (such as basic arithmetic) for harder to measure abstract reasoning skills and “intuition” that are essential to solve many problems that we do not yet know how to solve.

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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Teaching Prime Numbers to Children

Antonio Cangiano — Wed, 16 Mar 2011 18:00:16 +0000

The other day I received a math book that’s geared towards children. It’s called “You Can Count On Monsters” (USA | UK | Canada) by Professor Richard Evan Schwartz. This mathematics book is unconventional in the sense that it’s mostly comprised of drawings, with relatively few words or formulas. It’s square shaped, enjoyably colorful, and printed on stark black paper which really makes the illustrations pop.

Though it might surprise some to know, this is one of the most popular math books in print today – it’s even an Amazon best seller (when looking at overall Amazon sales, not just in the math category). Even more surprising, the book isn’t brand new and therefore just riding a wave of immediate (but ultimately short-lived) success. It was released over a year ago and has brought in some incredible sales figures over that time. On Amazon, You Can Count On Monsters mostly has five star reviews, and it’s been well reviewed by several outlets, including NPR.

So what is this book all about? I like Publishers Weekly’s factual and succinct description of the book:

This compact, innovative book counts to 100 using prime numbers represented as monsters, each with identifying characteristics (two resembles a bee with two buggy eyes, and three is an angry-looking triangular creature). The book opens with explanations of multiplication, prime and composite numbers, and factor trees, then moves on to a list of numbers. Each prime number looks unique, while composite numbers are represented by scenes involving their prime monsters (eight is illustrated as three of the beelike twos, i.e., two times two times two. Readers may have difficulty deciphering the pictures, which come to resemble little works of abstract geometric art. But especially for creative learners, visualizing the roles each monster plays may lead to deeper number sense. Ages 4 to 8.

This book is charming, interesting, fun, and clearly comes at the topic in a child-friendly manner, yet it’s still fascinating enough to engage older readers, too. It builds on the basic concept of integer multiplications and moves on to intuitively (and visually) illustrate the concept of prime numbers, composite numbers, factorization trees, and even more advanced topics such as the Sieve of Eratosthenes, as well as an illustration of Euclid’s proof for the existence of an infinite number of prime numbers.

While the illustrations may be targeted at children, the math in this book is rigorous and non-trivial considering the age group it’s geared towards. The approach taken is remarkable as it doesn’t fall into the trap of presenting mathematics to children as a dry, mechanical series of steps in an algorithm that will eventually solve a given problem. It takes an exploratory approach where experimentation and imagination are associated with the general idea of doing math.

The most important lesson this book presents is how to reason when it comes to viewing numbers like mathematician (and by the same token, how fun it can be to think in terms of numerical patterns).

2 x 7 = 14

You Can Count On Monsters may help children overcome their trepidations about mathematics, and even aid in the development of a real penchant for math from an early age, which will obviously be of great benefit to any child as they go through their school years and on to adulthood.

In short, we highly recommend picking up this excellent (and delightfully illustrated) book for your children, and spending some quality time together with them as you explore the colourful pages of You Can Count On Monsters. Your youngsters will enjoy the experience – and it may even help you take a new approach to how you help your child learn math from here on out.

Sponsor’s message: Math Better Explained is an insightful ebook and screencast series that will help you deeply understand fundamental mathematical concepts, and see math in a new light. Get it here.

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Drexel University launches Financial Education in the Math Classroom site

Antonio Cangiano — Mon, 07 Mar 2011 18:00:29 +0000

The following is a guest post by Stephen Weimar (Director of the Math Forum) and Valerie Klein (Professional Collaboration Leader) from Drexel University.

Recent economic hardships have demonstrated that the understanding of financial and mathematical basics is critical in keeping this country afloat. Too many people learn their financial lessons the hard way as adults, which raises the question “How can we effectively prepare our children to become fiscally responsible adults?”

The answer lies in education. Along with parents, teachers play a vital role in equipping students with the skills they’ll need to become financially responsible. There is a growing need to provide teachers with fun, engaging, practical, and shareable resources on the topic of financial mathematics. In response to this ongoing need, Drexel University recently launched the Math Forum – Financial Education in the Math Classroom, a website dedicated to providing resources and realistic conversations on teaching financial concepts in the classroom. In learning to apply mathematics in a financial context, students build on prior personal knowledge to deepen their understanding of core mathematics concepts and their applications, and to address some of the fundamental competencies inhibiting financial literacy.

The Financial Education in the Math Classroom website features an online professional development course for teachers that focuses on exploring the relationship between problem solving and financial education in the math classroom. As part of this offering, teachers can solve financially situated math problems, and share resources, ideas and strategies for incorporating new techniques into their classrooms. Class discussions investigate the benefit of financial topics that are of interest to students. Developing a problem solving culture takes time, and time is a scarce resource in the classroom. Financial contexts presented in the classroom work better when they are authentic – using real ads for cell phones, TVs, or realistic savings rates – and more importantly, authenticity might vary within and across schools. Mathematical understanding enhances financial understanding and financial contexts create access to mathematics with which students might otherwise not engage.

What makes a good problem solver? Excellent problem solvers are able to look at a problem, think about the quantities, relationships and constraints, and make a model for the situation. When making sound financial decisions, adults do exactly this. For example, when adults are choosing a cell phone plan, thinking about mortgage terms or creating a budget, problem solving comes into play. Our goal is to provide problems for teachers that allow students to practice this kind of modeling and problem solving. Teachers continuously report problem solving exercises provide valuable learning experiences for their students.

The Financial Education in the Math Classroom section of The Math Forum is focused on providing rich problems and supporting the development and practice of problem solving strategies and communication using real world applications. For example, one of the exercises on the site, Texting, 1, 2, 3, asks students to explore the cost of a cell phone plan and decide under what circumstances an unlimited texting plan would be less expensive. The Math Tools on the site allow students to explore how to pay off credit card balances, how to save for college, and how to comparison shop for car rentals, among other concepts and ideas. Explanations from Dr. Math address contexts like comparison shopping and step functions in the real world.

Since its inception, the Math Forum has been dedicated to creating online spaces for teachers to connect, share resources and develop their practice. We have discussion boards, blogs, and multiple spaces for teachers to discuss educational topics, teaching strategies and to simply ask any questions they might have about teaching or mathematics. These interactions are organized, selected and published so that educators from around the world have constant access to community wisdom and best practices. Moving financial sense forward makes sense and cents for all of us.

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Watch Khan Academy on your mobile device

Antonio Cangiano — Mon, 03 Jan 2011 14:00:00 +0000

Khan Academy is an amazing non-for-profit project that’s chock full of mathematics, science, and other educational videos. Wide coverage, excellent explanations, and the fact that’s 100% free to use, have made this site a huge success and helped attract donations and praise from many, including philanthropist Bill Gates.

But I’m not here to talk about this terrific and very well known project per se. Instead I want to tell you about the fact that a major fan of the project released a nice looking application for Android, iPhone, and iPad, that enables you to watch and study Khan Academy videos while you’re on the go.

To install it, all you have to do is visit http://khanapp.com/ using the device you want to install this app on.

Khan App is brand new and while it’s still a bit rough around the edges, it shows definite promise. Even the folks at the Khan Academy team took notice and suggested (in a technical forum) that the author should get in touch with them to collaborate and further improve the application.

This useful application, much like the videos themselves, is absolutely free and you can pick it up here.

Sponsor’s message: Check out Math Better Explained, an elegant and insightful ebook that will help you see math in a new light and experience more of those awesome “aha!” moments when ideas suddenly click.

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Free Kindle Math Books

Antonio Cangiano — Fri, 31 Dec 2010 00:03:57 +0000

The CK-12 Foundation is a wonderful non-profit organization. Recently they released several free Kindle ebooks about Mathematics. Currently in the Kindle store you can find the following CK-12 Foundation math titles, all priced at $0.00:

Advanced Probability and Statistics

From a cursory look it would seem that the content of these ebooks is pretty solid and certainly worth checking out. While providing free math books isn’t a new concept, having said books formatted for your Kindle is a major plus (though there are a few concerns about their legibility in the reviews; these points are being addressed by the foundation).

More textbooks are available on the CK-12 Foundation site, which enables you to customize the content of a textbook through their intriguing “FlexBook” system. In their own words:

Traditional textbooks are both expensive and rigid. FlexBooks conform to national and state textbook standards. They are free, easy to update and easy to customize. With FlexBooks, you can customize your textbooks to support your innovative work in the classroom. The CK-12 Foundation provides FlexBooks free to anyone who wants to use them.

While it’s safe to assume that many of our readers already own a Kindle device at this point, it warrants mentioning that you don’t actually need one to read Kindle ebooks (there is Kindle software that’s available for Windows, Mac, iPad, and other mobile platforms that enables you to read Kindle ebooks sans owning an actual ebook reader).

However, on a related side note, if you’re buying a Kindle with the aim of reading technical material and textbook, I highly suggest that you opt for the DX version, because the screen on the regular sized one is too small to comfortably read such detailed material (particularly if you have free math PDFs you’d like to read).

Happy reading and all the very best to everyone in 2011, from the Math Blog team.

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The Quantum Mystics

John F. McGowan, Ph.D. — Mon, 04 Oct 2010 05:20:44 +0000

The 2004 movie/documentary What the Bleep Do We Know? has a simple answer to life’s problems: Quantum Mechanics! Can’t get a date? The answer: Quantum Mechanics. Hate your job? The answer: Quantum Mechanics. Worried about the long term health effects of your prescription anxiety pills? Have no fear. Quantum Mechanics is the answer. The movie intermixes a maelstrom of flashy computer generated special effects, the ultimately uplifting story of Amanda, a divorced photographer hooked on anxiety pills, who finds enlightenment and a good man through the power of quantum mechanics, and interviews with fourteen experts including physicist Amit Goswami, physicist Fred Alan Wolf, physicist John Hagelin, physicist David Albert, neuroscientist and pharmacologist Candace Pert, anesthesiologist Stuart Hameroff, a number of others, and, oh yes, New Age spiritual guru JZ Knight channeling Ramtha, the 35,000 year old spirit of a warrior from the lost continent of Lemuria. As it happens, the filmmakers and some of the experts are adherents of JZ Knight/Ramtha.

What the Bleep Do We Know? implicitly advances the theory that not only does human consciousness cause the mysterious collapse of the wave function in quantum mechanics but in fact can control or influence the outcome of the collapse, thus allowing human beings to influence or control events through the power of their mind and quantum mechanics. What the Bleep Do We Know? is vague about the actual textbook theory of quantum mechanics, not surprisingly since this theory gives no clear answer as to why, when, or how the quantum wave function collapses. The movie/documentary is one of the more prominent recent examples of the linking of quantum mechanics and mysticism, sometimes by experts with impressive resumes.

Like most popular science, the movie makes very limited use of mathematics: rendering the movie title in mathematical symbols on the DVD case and web site and a brief sequence of computer generated flying equations that are never explained. This is a very common technique in video, both science “fact” and science fiction where the audience is briefly shown a mathematical formula or equation in cryptic, often Greek, symbols with either no explanation or a vague explanation. Brian Greene’s The Elegant Universe PBS/Nova special, which manages to promote the highly mathematical field of superstrings with almost no mathematics, features a couple of scenes with cryptic equations on a blackboard or notebook page, for example.

What is Quantum Mechanics Actually?

The textbook theory of non-relativistic quantum mechanics is actually quite simple. Quantum Field Theory, sometimes abbreviated QFT, which purports to reconcile quantum mechanics and the theory of special relativity is extremely complex, abstract, and difficult to learn. The author once attended a lecture by the physicist Leonard Susskind in which he stated that it took ten years to learn Quantum Field Theory. This seems to be true for most people. In contrast, basic quantum mechanics is straightforward and could probably be explained in a movie/documentary like What the Bleep Do We Know?

The basic quantum mechanics formulated by Niels Bohr, Max Born, Werner Heisenberg, and their colleagues over the objections of Albert Einstein, Erwin Schrodinger, and Prince Louis deBroglie asserts that sub-atomic particles (and presumably macroscopic objects like tables and chairs) are characterized by a quantum state or quantum wave function usually represented by the Greek letter . This wave function is a complex number representing a wave with an amplitude and phase. For example:

The quantum wave function is governed by a wave equation, a partial differential equation. There are several different wave equations depending on the type of particle: the original Schrodinger Equation, the Dirac Equation, and the Klein-Gordon equation. As far as the author can determine, these equations were found empirically — that is they were educated guesses that turned out to match experimental data on certain types of particles. Subatomic particles are thought to have a mysterious instrinsic “spin” that is somewhat analogous to the spin/angular momentum of a macroscopic body. Different wave equations govern the quantum wave functions of particles of different spin. The basic Schrodinger Equation which describes the electron in the hydrogen atom approximately (the electron is a spin 1/2 particle which is exactly described by the Dirac Equation) is:

where is Planck’s constant divided by , is the mysterious “wave function”, is the potential such as the electrical potential of the hydrogen atom, and is the position.

Qualitatively, one can think of the quantum wave function as a physical wave like a water wave, a sound wave, or a radio wave. These familiar phenomema are governed by wave equations as well, similar to the Schrodinger equation in many ways. This similarity to known phenomena and mathematics made Schrodinger’s wave picture and his wave equation appealing to the physicists of his time (the 1920′s) and easy to use.

So far so good. The problem is that when one conducts experiments with electrons, one sees something consistent with point particles, not waves. Electrons are detected as points on photographic film or a modern electronic imaging device, physical tracks in a cloud chamber or bubble chamber. Waves interfere. Interference effects appear with electrons in accumulated patterns of many electrons. If one fires properly prepared electrons through a crystal, dots will appear on a sheet of photographic film. Each electron appears at a specific point on the film, unlike a conventional wave which would be spread out over the film. As many dots appear on the film, unexpectedly an interference pattern will appear as if the electrons are somehow also a wave diffracting through the crystal. Similar effects are observed with other subatomic particles such as neutrons.

At this point, the formal mathematical theory of quantum mechanics expressed in symbols gets fuzzy. Bohr, Born, Heisenberg, and their colleagues argued that there was something called a “measurement” or “observation,” which they did not define clearly. In most practical experiments, the measurement is the determination of the position of a subatomic particle such as an electron (a dot on a sheet of photographic film for example). The square of the amplitude of the quantum wave function was the probability of observing a particle at the position .

In general, the “measurement” was represented in symbolic mathematics as an operator acting on the quantum wave function . The measurement supposedly caused the quantum wave function to instantaneously jump to an “eigenfuction” or “eigenstate” of the operator. An “eigenfunction” is a simple concept with a fancy German name. An operator is something, such as taking the derivative or simply multiplying by two, that transforms one function into another function.

An eigenfunction of an operator is a function such that the operator transforms the function into a factor, known as an “eigenvalue,” times the original untransformed function:

With a position measurement, this meant the quantum wave function collapsed to a point, often represented by a so-called Dirac function where the Dirac function is zero everywhere except at and infinite at . This infinity is such that the Dirac function integrated to unity (1):

A measurement could also be a measurement of the momentum of the particle. In this case, the appropriate operator is the momentum operator:

In this case, the so-called eigenfunction or eigenstate of momentum is:

where the eigenvalue is the measured momentum . In this case, the probability of measuring a momentum is the so-called inner product of the original quantum wave function with the complex conjugate of the so-called eigenstate:

There is something remarkable and unexplained here. The quantum wave function jumps discontinuously and instantaneously from one form to another. If the measurement is a position measurement, the quantum wave function, which may be spread out over quite a distance, collapses instantaneously (faster than light) to a single point. What causes the collapse? What is a measurement? Schrodinger’s Equation and the other equations of basic quantum mechanics do not tell us. Representing the measurement as a symbolic operator does not tell us; it obscures the problem. Textbook quantum theory does not tell us what physical process actually causes the collapse, how the collapse occurs (it is presumably instantaneous), or why. In the textbook theory of quantum mechanics, the collapse of the wave function to a point is completely random . A conscious observer has no power to influence or control the outcome of the collapse (the measurement or observation).

The Quantum Measurement Problem

From the very beginning of modern quantum mechanics in 1927, some physicists, notably Albert Einstein, Erwin Schrodinger, and Prince Louis deBroglie, were dissatisfied with this hazy, hand waving explanation of what was going on. This has led to a number of attemps to clarify quantum mechanics including the Many World Interpretation of Hugh Everett, the pilot wave theory of Louis deBroglie and David Bohm, and a number of less well known theories and interpretations. Significantly, a few physicists including Eugene Wigner and John Wheeler speculated that consciousness might be directly involved in the collapse. Perhaps a conscious observer was needed to collapse the wave function. This in turn opened the door to a number of mystical possibilities.

The quantum measurement problem is a significant problem. It is often said that quantum mechanics makes accurate predictions for a wide range of physical phenomena and that it has never been wrong. There is supposedly no experiment that contradicts quantum mechanics in over ninety years of experiments in atomic, nuclear, and particle physics. Yet, quantum mechanics is a plastic theory. Because the theory does not provide a clear rigorous theory of when the measurement occurs, the predictions are ambiguous. If an experiment unexpectedly shows that a particle behaves as a wave, the measurement process is sufficiently ambiguous that one can probably argue that this is in fact what should have been seen. One simply didn’t understand quantum mechanics prior to the experiment. Conversely, if one observes particle-like behavior where one naively expected wave-like behavior, once again one only has to redefine what constitutes a “measurement” that collapses the wave function.

One of the problems that quantum mechanics was developed to solve is the stability of atoms. If matter is comprised of atoms and if an atom consists of a positively charged nucleus with negatively charged electrons somehow “orbiting” the nucleus, there is a serious problem. In the classical electromagentic theory of Maxwell, a charged particle that is accelerating, which can mean either changing speed or direction, will radiate electromagnetic waves. The atom should quickly decay. Yet, this was not observed. There was clear evidence that atoms contained a positively charged nucleus and seemingly electrons in something like an orbit around the nucleus. Nonetheless, the atoms did not decay.

In orthodox quantum mechanics, the electrons in an atom are uncollapsed wave functions. The phase of the wave function oscillates in time but the amplitude of the wave function is constant. This means somehow that the spatial distribution of the electric charge around the nucleus is constant — unchanging. Thus, there is no acceleration and the atom is stable. Amongst other things, the observed stability of atoms means that the wave function must refer to a physical reality, not to the observer’s state of knowledge about the electrons in the atom. If the electrons had a definite position unknown to the observer, e.g. a scientist performing an experiment, the atom would radiate electromagnetic waves and decay. The wave function cannot collapse for the atom to be stable.

These electrons in uncollapsed wave functions nonetheless appear to interact with other particles and objects, apparently without collapsing. For example, a mirror reflects light. This is thought to be due to the electric charge in the electrons in the atoms. Yet, seemingly, the electron wave functions in the atoms of the mirror do not collapse. The atoms in the surface of the mirror remain stable and do not decay as one might naively expect. So, too, if one pushes on the mirror, the firm resistance of the mirror to one’s hand is attributed to electrostatic repulsion between the electrons in the mirror and the electrons in the atoms of one’s hand. Yet, again, it does not seem that pushing on the surface of a mirror causes the electron wave functions to collapse and the atoms on the surface of the mirror to decay as one might naively expect. However, if one blasts the mirror with energetic x-rays, the x-rays will knock loose electrons which can be observed as tracks in detectors — seemingly point particles. The electron wave functions collapse and the mirror will physically degrade when bombarded with x-rays.

Why does one interaction or “observation” appear to collapse the wave function and another does not? To be sure, there is probably some hand-waving explanation of this seeming problem with quantum mechanics in the advanced physics research literature. But really, is it understood? Probably not.

These observations with mirrors and other macroscopic objects arguably bracket the conditions under which the wave function collapses. There are many practical and mathematical difficulties in determining what quantum mechanics predicts for a macroscopic object comprised of trilions of trilions of trillions of atoms and molecules. Presumably, scientists can study this behavior at the level of atoms, molecules, or small numbers of atoms with atomic force microscopes and other modern instruments and determine experimentally when and under what conditions the wave function is collapsing (if this is even a correct interpretation of the physical phenomenon). It is probably more accurate to say that experiments may be able to determine when and under what conditions the wave function collapses assuming that the wave function collapse really exists as a meaningful physical concept (for example, the pilot wave theory of quantum mechanics does not require the wave function to collapse to explain the puzzling observations).

Symbols and Concepts

The expression of quantum mechanics as a formal mathematical system of symbols (that could be programmed on a modern computer for example) does not contain a predictive mathematical formula for when and under what conditions the wave function will or will not collapse. The expression of the “measurement” or “observation” as a symbolic operation such as the operator does not answer this question. It actually sweeps the problem under a symbolic rug as Einstein, Schrodinger, and deBroglie realized. Rather physicists are using fuzzy verbal concepts such as Niels Bohr’s “complementarity” or unstated theories or speculations to specify when the wave function collapses and when it does not.

This is not unusual. When physical theories are expressed in purely symbolic forms such as Maxwell’s Equations or Schrodinger’s Equation, a great deal is often hidden in the definition of the symbols. The quantum mechanical is a good example. But, even in Maxwell’s Equations, exactly what is the definition of the electric field and the magnetic field . How are these symbols in the formal mathematics connected to real physical measurements in the laboratory or in the field? In physics and engineering, ultimately the abstract symbols must correspond to physical macroscopic objects and systems that form sense perceptions in the human mind, e.g. dots on a photograph, the deflection of a meter, and so forth. Somewhere in this process shapes and more generally concepts that the human mind can handle, but which we cannot (yet) express in purely symbolic terms usually appear.

Quantum Mysticism

There is a long history of physicists and other scientists wandering off into mystical or spiritual areas. During the Renaissance, the sharp separation and alleged conflict between science, religion, and mysticism largely did not exist. Johannes Kepler was an astrologer and mystic. Tycho Brahe was an astrologer and alchemist. Isaac Newton practiced alchemy and had a range of religous and mystical views. Even in the early nineteenth century, Michael Faraday was deeply religious although he tried to keep his personal religious views out of his scientific publications. Faraday and many of the early electrical experimenters and theorists were deeply influenced by mystical ideas about the unity of nature which led them to actively seek the evidence of a strong connection between electricity, magnetism, light, and chemistry which they found. What the Bleep Do We Know? features an entertaining and probably false story in which the Indians (Native Americans) were supposedly unable to see Christopher Columbus’s ships because they were not mentally prepared to see the ships — until a wise shaman somehow opened their eyes. Astonishingly, early electrical experimenters had been studying electricity extensively for at least twenty years without noticing that there were magnetic effects (fields in modern parlance) around wires carrying electrical currents. The invention of the electric battery in about 1800 made it possible to conduct extensive, detailed, repeatable experiments with electricity. The mystically inclined Hans Christian Oersted easily found the magnetic field around a current-carrying wire with a magnetic compass, something that anyone could have done for…twenty years. He saw what he was mentally prepared to see; it does happen (all the time).

The evolution and growth of modern science has been accompanied by the adoption of a militant, materialistic, reductionist, and atheistic creed by many leading scientists and popularizers of science (see, for example, works by Steven Weinberg and Carl Sagan). Nonetheless, there have been a number of modern physicists, including some with impressive resumes, who have bucked the trend and become enamored of various mystical, parapsychological, and spiritual ideas. Arthur Eddington developed a complex mathematical theory of everything with strong mystical elements with which he apparently became obsessed. This may have contributed to the notorious bitter feud between Eddington and Subrahmanyan Chandrasekhar in which Chandrasekhar’s discoveries threatened to shatter Eddington’s theory. The Nobel-prize winning physicist Brian Josephson is probably one of the best known, most prominent recent examples. Josephson became fascinated with extrasensory perception (ESP) and other exotic ideas. Although many of his ideas are rather materialistic, the physicist David Bohm became associated with various mystical and spiritual individuals and groups such as the spiritual guru Krishnamurti. Bohm’s ideas, in particular, have become intertwined wih many mystical ideas about quantum mechanics. John Hagelin, who appears in What the Bleep Do We Know?, features an impressive resume before leaving mainstream (government funded) theoretical particle physics to join the Maharishi Mahesh Yogi‘s transcendental meditation movement. Relevant to the theme of What the Bleep Do We Know?, the physicist and engineer Robert Jahn conducted experiments for years at Princeton claiming to detect small effects where people could slightly alter the outcome of random, presumably quantum mechanical processes such as radioactive decays or electronic noise.

The organized, mostly government-funded physics “community” has taken various measures to distance itself from these mystical, spiritual, and parapsychological theories and experiments. It is almost impossible to publish research papers on these sorts of things in major physics publications such as Physical Review. A few years ago, the prominent and influential arxiv.org preprint server shifted from a policy of being open to all to requiring an institutional affiliation, a sponsor to endorse the preprint, and some other “precautions,” generally to block publications by real or alleged “crackpots.” The mainstream physicist David Albert who appeared in What the Bleep Do We Know? rapidly distanced himself from the movie, claiming he was quoted out of context and did not realize the subject matter of the film; this may well be true.

It is difficult to know what to make of this sort of thing. Some important aspects of quantum mechanics are almost certainly not understood. Thus, it is not possible to rule out a direct role for consciousness or other mystical extensions to quantum mechanics. It is also clear the textbook theory does not require a role for consciousness as What the Bleep Do We Know? and similar materials claim — quite the opposite. Most physicists appear to be attracted to the modern field by the discovery that they are good at physics in school, which often means good at specific calculations or derivations. Modern physics is little more than a glory-driven professional sport like football or basketball — with plenty of egos and mindless competition and precious little “enlightenment”. Intrinsically, fundamental physics such as theoretical particle physics, cosmology, and so forth deals with fundamental questions about the nature and origin of everything. Some people come into physics primarily driven by curiosity, spiritual, or philosophical feelings about these basic questions. There is obviously an appeal to finding evidence of God or some comforting mystical reality in science and in empirically measurable phenomena such as quantum mechanical phenomena. More cynically, career prospects and salaries in physics are relatively poor; becoming a spiritual guru with a Ph.D. in Physics may be a career option, even a lucrative one in rare cases. Also, despite the general lack of mainstream government funding, over the years, a number of wealthy philanthropists such as the late Laurance Rockefeller and Robert Bigelow have funded a plethora of far out research topics such as parapsychology, crop circles, and UFOs.

Crypto-Mysticism in Mainstream Physics

Probably most readers of this article will find the story of a New Age guru allegedly channelling the 35,000 year old spirit of a warrior from the lost continent of Lemuria who promotes a questionable extension of quantum mechanics amusing, if not ridiculous, and may enjoy a smug sense of superiority over the poor benighted folks who believe this sort of thing. Yet, labels like cult, pseudoscience, bad science, junk science, and pseudomathematics are generally applied only to relatively small fringe groups with little money, power, or prestige. In fact, it is common to encounter religious and mystical language and imagery in popular physics articles. Sometimes it is deadly serious. Often the words are used in a way that lends itself to a double meaning. That is, a statement has one meaning if the words are given their common meaning in English, yet a different meaning if the “special meaning” used by physicists is used. In many cases, the religious or mystical language or imagery is ostensibly tongue in cheek, not really serious.

A fairly extreme example of this is the title of Leon Lederman’s book The God Particle, about the Higgs particle, the “Holy Grail” (another religious image) of particle physics. The title says it all. Is the Higgs particle really equivalent to God? Albert Einstein referred to God frequently in various public appearances. There is a famous statement by Einstein “Raffiniert ist der Herrgott aber boshaft ist Er nicht.” which is commonly translated as “The Lord is subtle, but not malicious.” Abraham Pais used this translation as the title of his biography of Einstein. What Einstein meant by “God” is not clear and the subject of some controversy. The famous American rabbi and Jewish leader Herbert Goldstein specifically asked Einstein in a telegram if he believed in God. Einstein allegedly responded with:

I believe in Spinoza’s God who reveals Himself in the orderly harmony of what exists, not in a God who concerns himself with fates and actions of human beings.

It is worth understanding that the philosopher Spinoza’s views were very far from either traditional rabbinical Judaism or Christianity. He was, in fact, excommunicated by his Jewish religious community. This custom of using the term “God” in a way that differs dramatically from common usage is frequent in the popular physics literature. Here is a fairly clear statement of this practice from Leonard Susskind’s popular physics book The Cosmic Landscape: String Theory and the Illusion of Intelligent Design (page 8):

I don’t know the religous beliefs of Davies or Greenstein, but I would be wary of too literal an interpretation. Physicists often use terms like design, agency, and even God as metaphors for what is not known — period. I have used the term agent in print and have been sorry every since. Einstein often spoke of God. “God is cunning but He is not malicious.” “God does not play dice.” “I want to know how God created the world.” Most commentators believe Einstein was using the term God as a metaphor for an orderly set of laws of nature.

Well, maybe. Why not use “what is not known” instead of “God,” which in common usage means something totally different?

It is also common to encounter language about “knowing God,” “glimpsing God,” “knowing the mind of God,”, or “glimpsing the mind of God” in the popular physics literature. There is an example of this in Brian Greene’s PBS/Nova special The Elegant Universe where a physicist talks about knowing the mind of God.

It is also common to encounter mystical or religious references that are supposedly tongue in cheek. For example, the PBS/Nova special The Elegant Universe contains a seemingly tongue in cheek story about the original supposed breakthrough in superstrings. The two physicists are struggling to perform a complex symbolic derivation. There is however a raging storm outside which threatens to distract them from sucess. Are the Gods themselves seeking to prevent the discovery? The physicists, of course, succeed despite the raging storm. It is all reminiscent of the jealous Creator of some Gnostic Christian sects who seeks to keep men ignorant, but knowledge gnosis of the true supreme God sets men free. Murray Gell-Mann used the mystical term “The Eightfold Way” to humorously label his theory of quarks. Particle physicists initially named two of the quarks “truth” and “beauty,” but later renamed them “top” and “bottom.”

The central idea of most mystical systems is the absolute unity of all things, an absolute One identified with God and often the universe as well. In the western Neoplatonic tradition, this is closely associated with the mathematical mysticism of Pythagoras and Plato, in which mathematics plays a central role in the nature and architecture of the universe. The quest for a unified field theory or theory of everything (TOE) in the popular physics literature bears a striking resemblance to this ancient mystical belief. Even the “landscape” of a myriad alternate universes, the subject of Susskind’s book, bears a suspicious resemblance to the myriad failed worlds that God creates before our universe in some versions of the Jewish Kabbalah, the Jewish mystical tradition. In the Kabbalah, one of the supreme goals is to learn and master the most secret name of God, which confers infinite power on whoever speaks it; this bears a marked resemblance to the quest for a secret equation, a theory of everything, that explains the entire of creation and which might confer great power as well (antigravity, warp drives, quantum teleportation, time travel, who knows?).

Scientism is the idea that natural science is the most authoritative worldview or aspect of human education, and that it is superior to all other interpretations of life. Its’ appeal to many scientists is easy to see. Science can take the place of traditional religion. With quantum mysticism, the mysticism and spirituality is out in the open, clearly recognizable — proudly touted, in fact, whatever its’ actual merits. The disguised (crypto) mysticism of the popular physics, and other popular science, literature is less clear, easy to deny as just words used in an odd way or as light hearted humor. No one is really claiming the Higgs particle is God…or are they? But the dangers from heavily funded and prestigious mainstream science gone off the rails would dwarf the dangers from even the largest New Age cult to insignificance.

Conclusion

Returning to What the Bleep Do We Know? and similar materials, caution is well advised. Consciousness is not a part of regular textbook quantum mechanics nor is it clearly necessary to explain the logical and philosophical problems with quantum mechanics. It may be that we will one day find that mysticism, parapsychology, or some similar exotic idea is involved in quantum mechanics, but we do not know this today nor are we close to an answer.

Speaking mathematically, it may be possible to find the explanation for the quantum measurement problem by simply adding a non-linear term to the Schrodinger Equation that causes the collapse of the quantum wave function under certain physical conditions, most probably some sort of interaction between the wave functions of different particles. It is very hard to know what the precise form of this additional term might be. There have been a few attempts in the physics literature. Most likely it is necessary to think carefully about the conceptual issues swept under the rug in the quantum measurement operators to make educated guesses about the nature of this term if it exists. It remains quite possible that solving these problems requires a radical change, discarding the Schrodinger Equation and other known mathematics almost entirely.

Suggested Reading/References

	The Quantum Ten: A Story of Passion, Tragedy, Ambition, and Science Sheila Jones Oxford University Press, USA 2008 336 pages
	Quantum Dialogue: The Making of a Revolution Mara Beller University Of Chicago Press 2001 365 pages
	Infinite Potential: The Life and Times of David Bohm F. David Peat Basic Books 1997 380 pages
	The Undivided Universe: An Ontological Interpretation of Quantum Theory David Bohm, Basil J. Hiley Routledge 1993 397 pages
	Opticks: Or a Treatise of the Reflections, Refractions, Inflections & Colours of Light-Based on the Fourth Edition London, 1730 Isaac Newton, Albert Einstein (Foreword) Dover Publications 1952
	Quantum Theory At The Crossroads – Reconsidering The 1927 Solvay Conference Guido Bacciagaluppi, Antony Valentini Cambridge University Press 2009 556 pages
	Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy J. S. Bell Cambridge University Press 2004 288 pages
	The Cosmic Landscape: String Theory and the Illusion of Intelligent Design Leonard Susskind Little Brown and Company New York 2005

About the Author

John F. McGowan, Ph.D. is a software developer, research scientist, and consultant. He works primarily in the area of complex algorithms that embody advanced mathematical and logical concepts, including speech recognition and video compression technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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