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.
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.
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.
© 2011 John F. McGowan
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 [email protected].