Raising the Level of Reading Comprehension of Students at Community Colleges

Emphasis on Student Learning Objectives (SLOs) and grades should not divert us, the math faculty at community colleges, from our main goal: a meaningful and quality education for our students. SLOs and evaluations are necessary but we need to recognize that other factors are also important. One such is reading comprehension, the ability of students to understand what they are reading, particularly word problems. While students can answer straightforward questions like “Let A and B be events with P(A) = 0.8, P(B) = 0.1 and P(B|A) = 0.2, Find P(A and B)” or solve quadratic equations like x² – 7x + 11 = 0, they are sometimes unable to parse sentences in word problems to figure out what needs to be done, far less solve them. 

Yet it is word problems that help students connect with the real world, encourage them to think about relationships between numbers, and reveal interdisciplinary connections between mathematics and subjects such as English, physics, astronomy, chemistry, biology and environmental science.

Here is an example from statistics that illustrates how a lack of reading comprehension becomes a barrier for students to solve word problems.

Statistics (section 7.2, Elementary Statistics by Navidi and Monk): According to the National Health Statistics Reports, the heights of adult women in the United States are normally distributed with a mean of 64 inches and a standard deviation of 4 inches. If three women are selected at random, what is the probability that at least one of them is more than 68 inches tall?

The first difficulty students face is the phrase “At least 1”. The second is with the meaning and implication of the word “random.”

Students had learned one of the probability formulas in a previous section: “Probability (At Least 1) = 1 – Probability (None)”. They have no difficulty running “normalcdf” in their calculators to determine the probability when the parameters are explicitly given. However, connecting the formula and the idea of randomness and “normalcdf” to this problem seems beyond the capacity of most students. It comes down to a reading comprehension issue.

After carefully parsing the sentence “If three women are selected at random, what is the probability that at least one of them is more than 68 inches tall,” they slowly begin to make the connections. To ensure comprehension, I ask students to write complete sentences describing the steps they use to solve word problems such as this “if you want full credit.”

That last clause gets their attention.

This is a typical writing sample from approximately 60% of the students (the other 40% struggle to express themselves) who write complete sentences to describe the steps:

a) Find the probability that any one of the three randomly selected women is shorter than 68 inches by running (TI-84) normalcdf (0, 68, 64, 4) = 0.841. That is, the probability that a woman picked at random has a height between 0 and 68 inches is 0.841.
b) Since the three women are selected at random (no connection between them, that is, they are “independent” of each other), the probability that ALL three women are shorter than 68 inches is, by the multiplication law of probability,
P (A and B and C) = P (A) x P(B) x P(C) = (0.841)³ = 0.595.
c) Apply the “At Least 1” formula: Since the sum of all probabilities = 1, and since “At Least 1” includes all possibilities other than 0 or None, “At Least 1” and “None” include ALL possibilities between them. They are complements of each other. Therefore,
P(At Least 1) + P(None) = 1; P(At Least 1) = 1 – P(None)
Probability (At least One Woman taller than 68 inches) = 1 – 0.595 = 0.405

(Occasionally, a few students will go further and fill in more details. This is typical of what they write: To calculate P (At Least 1) directly requires the calculation of 7 different probabilities for this particular problem.
1. A is taller than 68 inches but not B and C  
OR

2. B is taller than 68 inches but not A and C  
OR
3. C is taller than 68 inches but not A and B  
OR

4. A and B are taller than 68 inches but not C  
OR

5. A and C are taller than 68 inches but not B  
OR
6. B and C are taller than 68 inches but not A  
OR

7. A, B and C are all taller than 68 inches

The only other option is
8. All of them (A, B, and C) are equal to or shorter than 68 inches, that is, NONE are taller than 68 inches.

The sum of all 8 probabilities = 1. So a) either I calculate the probabilities for options 1 through 7 individually and sum them, which is tedious and can lead to mistakes, or b) I do option 8 and subtract it from 1, which gives me the sum of probabilities for 1 through 7. It's easier to use option b, a neat trick!)

One or two students who take meticulous notes of what I emphasize in class will also add something like this:
“Even though entering actual heights between two boundaries gives the area under the bell curve, which is equivalent to the relevant probability, the calculator converts the heights into their corresponding z-scores ‘behind the scene.’ The area under the curve can be interpreted as probability only when the actual values, the heights in this case, are converted to their corresponding z-scores.”

I insist on complete sentences to explain the solutions to word problems because it becomes a test for students to see how well they understand the problems, that is, how good their reading comprehension is. Reading carefully clarifies their thinking, which, in turn, leads to clear writing. Reading and writing reinforce each other in a creative loop. Since language is the basis of thought, reading and writing well allow students to think well too. Students discover that this is true not just for English but also for math. 

I find it helpful to emphasize to students that they can often figure out solutions to hard problems as they go along. Many students, at least initially, have the mindset that they can only start when they have figured out the entire solution, so they never start!

(Other examples from statistics: Write complete sentences explaining the meaning of a confidence interval or the implications of rejecting or not rejecting the null hypothesis in a given context. Explain why switching events in conditional probability (“confusion of the inverse”) leads to different probability results. Describe a “black swan” event and whether or not you have experienced one that had a significant impact on your life. Should you buy that warranty or that lottery ticket? Why or why not?)

There is a lot of resentment in the beginning (typical reaction: this is not an English class!) but gradually students come around to appreciate the symbiotic relationship between reading comprehension and clear thinking and writing.

Precalculus word problems are good examples of showing interdisciplinary connections. Example: Throwing an object upward to calculate the highest point reached and the time it takes to get there and fall back to earth under the influence of gravity shows the connection between math and physics. Exponential functions describing radioactive decay and carbon dating show the connection between math, physics, chemistry, archeology and paleontology. Extinction of species shows the connections between math, biology, environmental science and climate change. A mathematical model for how we forget what we learn over time shows the connection between time and memory. And so on. One writing exercise I assign students is to describe how the irrational number “e” harnesses the power of infinity in a limiting sense, in situations where things happen continuously, like birth and death in a population. (Unintended humor: A student wrote that “e” captures eternity rather than infinity!)

Some students ask for extra-credit projects because they are falling behind and want to bring their grades up. One project I often assign is to define the meaning of 10 words in both day-to-day context and mathematical contexts and to construct a sentence for each. Example: “irrational” usually means unreasonable or illogical but in mathematics, an irrational number, such as pi or e, is a number that cannot be expressed as a ratio of two integers. As a decimal, an irrational number neither terminates nor repeats.

Example: Define the following words in their mathematical and
non-mathematical contexts and write a sentence for each: Function, Eccentricity, Rational, Random, Sample, Population, Outlier, Probabilistic, Deterministic, and Complex.

We faculty are constrained by time. We have to teach courses, grade tests and quizzes, assess SLOs, maintain and monitor Learning Management Systems such as Canvas, track attendance, tutor students, maintain office hours and perform a host of other activities. Where is the time to raise the level of reading comprehension and encourage writing with clarity and precision? How can we instill the habit of paying deep attention and cultivating such skills as patience, curiosity, discipline and grit, necessary for academic and professional success, when we are constantly juggling time to complete so many basic faculty duties and responsibilities?

There is no easy or single answer to this. Perhaps the first step is to recognize that we need to look beyond SLOs, grades, performance and achievement by integrating some habits and practices in our teaching that can help students think clearly and independently and live courageously and confidently. One such practice, in my opinion, is to improve their reading comprehension by paying attention to what they read (difficult, given the continuous digital distractions) and writing the steps clearly and precisely as they slowly work their way toward solving word problems.

Good mathematics, like good reading and writing, requires an appreciation of structure, beauty, rhythm, and pattern. If we can make this idea an integral part of our teaching, as best fits our respective temperaments, we may consistently experience the joy that comes from shaping minds, semester after semester.

Connecting Reason and Faith on a Social Media Platform

Iconic Silicon Valley companies currently confront a credibility crisis of motive and trust. Helping spread fake news and propaganda, turning users into lab rats, mining personal data through addictive apps, stashing away billions of dollars in off-shore tax havens as disclosed by the Paradise Papers, suggest that under the veneer of connecting all and doing no harm, something more sinister is brewing in these Internet juggernauts.

Recent data reveal that on Facebook alone, as many as 126 million Americans were exposed to fake news stories during the 2016 U.S. presidential election, eroding our democracy. Russian operatives also created close to 3,000 fake Twitter accounts and over 1,100 videos on 18 Google channels. During a recent Congressional hearing, Senator Dianne Feinstein bluntly told representatives of Facebook, Google and Twitter: “You bear this responsibility. You’ve created these platforms.”

Americans are souring on tech titans who symbolize Silicon Valley and its increasingly questionable ethos. Not only are they accumulating wealth and power at an exponential rate, they are turning their companies into monopolies to silence discordant opinions and diminish, if not destroy, diversity.

Allow me to focus on Facebook. 

As the social media leader with over 2 billion active monthly users worldwide, can the tech giant remove our fears and restore our trust in the company, even if partially?

In his commencement address at Harvard this year, CEO Mark Zuckerberg expanded his company’s mission to include ‘three ways to create a world where everyone has a sense of purpose: by taking on meaningful projects together, by redefining equality so everyone has the freedom to pursue purpose, and by building communities across the world.” Zuckerberg defined purpose as “that sense that we are part of something bigger than ourselves, that we are needed, that we have something better ahead to work for.”

In other words, it’s not only about connecting all of humanity, it’s infusing these connections with a sense of purpose. Who can argue with that? The problem is that noble mission statements can often mask fiercer motives, especially when a company develops a global reach.

Yet it is plausible that the idea of purpose can transform the way we connect with one another in the virtual world.

Imagine that Facebook creates a platform that connects proponents of reason and faith, two ancient antagonists that have caused much sorrow in the world.

Let me clarify what I mean in a specific context: Silicon Valley.

According to a 2010 survey (the numbers could have only gone up since then) there are over 450 churches, synagogues, mosques and temples in the Santa Clara County alone, serving over 40% of the Valley’s 2 million population. The remaining 60% includes not only atheists but also those with complicated relationship to their faiths, particularly millennials who shun organized religion and pray in their own way but not in traditional places of worship.

If such a platform were created, could that be an example of the kind of meaningful project Zuckerberg envisions? Perhaps. After all, meaning, purpose and transcendence can flow from both the secular and the sacred. Any connection between the two can not only inspire fresh views on, say, stem-cell research and global warming but also deepen our understanding of how love, justice, suffering and forgiveness shape human affairs.

At the same time, we must also note that faith and reason can coexist within the same person. 

Perhaps no one exemplified this as persuasively as physicist Charles Townes (1915-2015). An article he wrote half-a-century ago in the IBM journal “Think” provides insights into the evolving nature of relationship between science and religion. After building the case that the two shared fundamental similarities - revelation in one is epiphany in another, for instance – Townes concluded that the two will eventually converge. “I believe,” he wrote in 1966 in The Convergence of Science and Religion, “this confluence is inevitable. For they both represent man’s efforts to understand his universe and must ultimately be dealing with the same substance.”

A devout Christian, Townes was one of the greatest scientists of the twentieth-century, winning the Nobel Prize in physics in 1964 for inventing the maser and the laser. He tempered his idea of convergence: “Perhaps by the time this convergence occurs, science will have been through a number of revolutions as striking as those which have occurred in the last century, and taken on a character not readily recognizable by scientists of today. Perhaps our religious understanding will also have seen progress and change. But converge they must, and through this should come new strength for both.”

Can Facebook facilitate this convergence in some way? It's difficult to say but what is clear is that this dialogue in the age of social media will not happen through staged spectacles between well-known cerebral atheists like Richard Dawkins and prominent theologians. It can only happen when practitioners of reason and faith can explore the connection between the two in a spirit of humor, humility and curiosity.

Even if a social media platform makes the flow of ideas between proponents of reason and faith easier, including those who see no conflict between the two, it does not mean it will remove our suspicion of Facebook, or of Amazon, Google or Apple, companies that are also vying for global domination, particularly when the call for regulation and antitrust probes against the Big Four is gaining steam in the Valley and beyond.

But it’s a start. The unexplored region between technology and faith beckons people with open minds seeking rational and spiritual truths. Done right, it may even be that algorithms will someday lead to epiphanies and clicks to catharsis.

Richard Feynman and Bill Gates: An (Imaginary) Interview

(Updated on Sunday, 5 November 2017. Included reference to the excellent book
"The Quantum Labyrinth" by Paul Halpern, published in October, 2017. The book is about 'How Richard Feynman and John Wheeler Revolutionized Time and Reality.')

Bill Gates respects and admires "individuals who achieve something inspirational or who possess extraordinary character." Of these, one name comes up more often than others: the late great Nobel prize-winning physicist Richard Feynman.

Gates had planned on meeting with Feynman in 1988 but didn't get a chance. Feynman died of cancer in February of that year. "It’s an opportunity I’m sorry I missed," wrote Gates in the New York Times in 1995. "His book, Surely You’re Joking, Mr. Feynman, is a favorite of mine."

Feynman was a hero because, as Gates put it, "he was incredibly inspirational. He was an independent thinker and gifted teacher who pushed himself to understand new things.
I have enjoyed everything I’ve read about him and by him. I admired him deeply…"

In 1964, Feynman gave a series of lectures at Cornell University, the Messenger Lectures, under the title "The Character of Physical Law." Topics ranged from symmetry, probability and uncertainty in physical laws to techniques by which physicists seek new laws. The lectures were recognized for their extraordinary quality. "I have videotapes of physics lectures Feynman gave at Cornell decades ago," said Gates. "They are the best lectures I’ve seen on any subject. He shared his enthusiasm and clarity energetically and persuasively."

During an interview with CIO magazine in September 1997, Gates was asked: "Who would you invite to a dinner party?" Feynman was on the list, along with Einstein and Leonardo da Vinci. As recently as June, 2017, Gates said in an interview with TIME magazine that "One person I'm sorry I never got to meet is the physicist Richard Feynman. He had a brilliant mind and was a phenomenal teacher." (http://time.com/4786837/bill-gates-books-reading/)

On July 14, 2009, Microsoft Research, in collaboration with Gates, launched a Web site called Project Tuva that makes The Messenger Lectures freely available to the public for the first time. Gates purchased the rights to the seven lectures in the series to help kids get excited about physics and science. “I think someone who can make science interesting is magical. And the person who did that better than anybody was Richard Feynman. He took the mystery of science, the importance of science, the strangeness of science, and made it fun and interesting and approachable,” said Gates.

At the time of his death, Feynman had become everyone’s favorite physicist, thanks to the popularity of Surely You’re Joking, Mr. Feynman and What Do You Care What Other People Think. With these books, transcribed by his friend and drumming partner Ralph Leighton from taped conversations over a period of years, Feynman captured the public imagination as no other physicists had before him, with the possible exceptions of Albert Einstein and Enrico Fermi.

The Feynman Lectures on Physics, a set of lectures Feynman gave to undergraduates at Caltech in '62-63, is now a classic. (For a description of how the lectures came about, see the definitive article by Feynman's colleague Matthew Sands in Physics Today, April 2005. The lectures - Volume 1 as of this writing - are now available online for free on the Caltech website.

Feynman’s fame grew when he was appointed to the Rogers commission in 1986 to investigate the Challenger shuttle explosion. His dramatic demonstration on TV of the loss of resiliency in O-ring at freezing temperature as a principal cause of the Challenger accident made him a national celebrity. In applauding his performance, the physicist Freeman Dyson said: "The public saw with their own eyes how science is done, how a great scientist thinks with his hands, how nature gives a clear answer when a scientist asks a clear question."

Since he passed away in 1988, Feynman lore has continued to grow. Several books have been published, including Genius: The Life and Science of Richard Feynman (James Gleick, 1992), Most of the Good Stuff: Memories of Richard Feynman (American Institute of Physics, 1993), No Ordinary Genius (Christopher Sykes, 1994), The Beat of a Different Drum (Jagdish Mehra, 1994), Feynman’s Lost Lecture: The Motion of Planets Around the Sun (W.W. Norton, 1996), The Meaning of It All (Helix Books, 1998), The Pleasure of Finding Things Out (Perseus Books, 1999) Feynman's Rainbow (Leonard Mlodinow, 2003), Perfectly Reasonable Deviations from the Beaten Track: The Letters of Richard Feynman (Edited by Michelle Feynman, Basic Books, 2005), Quantum Man (Lawrence M. Krauss, 2011), and The Quantum Labyrinth (Paul Halpern, Basic Books, 2017).

Reminiscences by colleagues also appear from time to time in Physics journals, such as "Capturing the wisdom of Feynman" by Matthew Sands (Physics Today, April 2006) and "Memories of Feynman" by Theodore A. Welton (Physics Today, February 2007). A fascinating article on how Feynman approached the subject of piano tuning ("Stiff-string theory: Richard Feynman on piano tuning" by John C. Bryner) appeared in the December 2009 issue of Physics Today.

Feynman has even made it into the billboards! When Apple Computer began its "Think Different" series of ads featuring great scientists, artists, humanitarians and the like, the company chose Einstein and Gandhi among its first examples of the uncommon rewards awaiting those who dared to follow the beat of a different drum. "Can Feynman be far behind?" I wondered.

In November '98, I was driving in San Francisco's Mission District when I suddenly saw that familiar face with the knowing grin inviting commuters to ponder the mysteries of ... what? The photograph showed Feynman wearing the corporate T-shirt of Thinking Machine, a Boston-based company where he had briefly worked as a consultant in 1983. The shirt bore a schematic representation of Connection Machine - a cube of cubes - that he helped design for Thinking Machine. (It's the same photograph on the cover of What Do You Care What Other People Think?) Then, in April '99, Feynman "came" to Silicon Valley where I live. Anyone driving along Highway 101 in the South Bay could "see" Feynman teaching quantum mechanics at the California Institute of Technology, in front of a blackboard on which he had written matrices and differential equations. According to Caltech archives, the photograph was taken on May 2, 1963, during his "Lectures on Physics" period. (Feynman has a large number of fans in Silicon Valley, so there was disappointment when Apple "replaced" him with an image of an iMac five months later.)

Gates may have been most pleased, however, with the publications of The Feynman Lectures on Computation (edited by Anthony J. G. Hey and Robin W. Allen - 1996) and Feynman and Computation (edited by A. J. G. Hey - 1998.)

The first is a collection of lectures Feynman gave at Caltech from 1983 to 1986 as part of an interdisciplinary course called "Potentialities and Limitations of Computing Machines." The second contains contributions by distinguished computer scientists and physicists who were guest lecturers in Feynman's interdisciplinary course. It also contains reprints of Feynman's prescient articles on the physics of computing: "There's Plenty of Room at the Bottom" (1959!) and "Simulating Physics with Computers" (1982). Anyone reading these two books will agree that Feynman's insight and ingenuity make his lectures on computation almost as timeless as his physics lectures.

Feynman even has his own 37-cent first-class stamp! The US Postal Service has honored four American scientists - physicists Richard Feynman and Josiah Willard Gibbs, mathematician John von Neumann and geneticist Barbara McClintock. The stamps were issued on 4 May, 2005. The Feynman stamp shows the physicist in his 30s, framed by the unmistakable Feynman diagrams. The stamp came about thanks to the decade-long lobbying effort by Ralph Leighton, who organized a celebration on May 11, 2005, at the post office in Far Rockaway, the New York City neighborhood where Feynman grew up. On the same day, the street in Far Rockaway where Feynman lived - 2 blocks from the post office - was renamed in his honor, from Cornaga Ave to "Richard Feynman Way." The date was appropriately chosen: May 11 is Feynman's birthday.

Gates never met Feynman but it is fascinating to imagine a meeting between the two. Here is the whiz kid transformed into a wide-eyed pupil, marveling at the master’s facility with ideas and insights, wondering at the source of that magical genius that was uniquely Feynman’s. What does Feynman think of the current state of computing? How does he envision its future? Are any architectural breakthroughs in software imminent? Where is the limit and why?

---

An autumn afternoon. Gates is at the Feynman house in Altadena, Southern California. Feynman introduces Gates to his menagerie - one horse, two dogs, one cat, and five rabbits. Gates smiles as Feynman addresses each animal by name and inquires of its health.

Afterwards, they settle down in the book-lined living room to talk.

Gates: When did you first take an interest in computers?

Feynman: My interest in computers really grew with my interest in physics, which is to say, very early. I recall reading in high school about mathematical machines, tide predictors, area measuring devices, and all kinds of wonderful things about computing in the Encyclopedia Britannica.

Gates: How have computers helped you in physics?  

Feynman: When I try to solve a difficult problem, I ask myself: What is it that I can compute that will explain how this particular physical system behaves? This approach has served me well. I am interested in useful results, not abstractions, whether it’s in understanding how light interacts with matter or why helium behaves so strangely at low temperatures. I get useful results by coming up with numbers that can be experimentally verified. Computers play an important role in this verification process.

Gates: But isn’t your kind of computing different from the computing most of us are used to?

Feynman: It’s true the kind of computing I am interested in is based mostly on physical insights and mathematics - mathematics always in the service of physics, I must add - but computers are a big help. They can perform millions of important calculations a scientist may never have the time for. Sometimes computers can even suggest ideas one hasn’t thought of before. I find this exciting. For anyone curious about how nature works at the deepest level, a computer can be a valuable tool. I certainly find them so.

Gates: Back in 1942, when you and other scientists were working on the Manhattan Project, digital computers were several years away. What was computing like then?

Feynman: We had these Marchant and Monroe computers - hand calculators with numbers, really - that were good for adding, multiplying, dividing, and so on. They were about a foot across and several inches high, with all kinds of levers on them that you pushed to get results. Unfortunately they broke down often. Metal parts wore thin and came out of alignment because of the pounding they took, and had to be sent back to the factory for repair. We just couldn’t afford the downtime - this was a wartime effort after all - and so some of us began to tinker. We would take the covers off and try to figure out ways to fix them. Pretty soon we got good at it and kept things going.

Then the calculations became complicated, way beyond the capacity of the Marchants. We had to get IBM machines - multipliers, tabulator, verifier, keypunch, sorter, collator and what have you. They were the best machines at the time. We managed to assemble them ourselves and came up with results that turned out to be very important.

Gates: I am curious about some of those calculations …

Feynman: The biggest challenge was to figure out how much energy would be released from various designs of the bomb. Then we had to narrow it down to how much energy would be released from specific designs that would be used in the actual bomb, and how much fissile material would be needed in each case. This was very complicated, nonlinear equations and all, and we had so little time! The experimentalists couldn’t help; they needed our results to carry on their work. We computed by simulation, using a primitive form of what you would now call parallel processing. But we rose to the challenge. Our calculations told us what we could and couldn’t do. Lots of important results, very accurate.

Gates: Fifty years later, how do you view your wartime efforts?

Feynman becomes pensive. "At Los Alamos, we were doing what we had to. We started for a good reason. We worked hard. It was exciting. We discovered, invented, and pushed the limits of science and technology to create a bomb to help us win the war. We won the war but afterwards many of us weren’t so sure about the bomb itself. We had second thoughts. The bomb took on a life all its own. I remember sitting in a restaurant in New York shortly after returning from Los Alamos to teach at Cornell, wondering: what if New York City were to become another Hiroshima! Everything around me would be smashed. What was the point of life, of all this creativity? It didn’t make any sense at all. It took me a while to shake off this feeling. Eventually I got busy with physics and moved to Caltech, but that’s another story.

Gates: What about your experience with computers, though?

Feynman brightens. "There’s no second thought about that," he replied. "The main idea I came away with from Los Alamos was that even simple, primitive machines could be used to calculate important results. And it keeps getting better!"

Gates: I just finished reading Feynman Lectures on Computation. I am intrigued by a remark you made at the end of one of the chapters: "In 2050, or before, we may have computers that we can’t even see!"

Feynman: What I was investigating in those lectures was the answer to a fundamental question: What

is it that we can and cannot do with computers today, and why? One issue was, how small could you make a computer? Was there any physical limitation to its size due to laws of physics? That led me to investigate the characteristics of a computer operating according to the laws of quantum mechanics. If we want to make extremely small computers, no more than the size of a few atoms, we would have to use the laws of quantum mechanics, not classical mechanics. So I began to analyze what you would call quantum computers. People thought that the uncertainty principle would be a limitation: that is, you wouldn’t be able to make a computer as small as you wanted because of the way time and energy, for instance, was related. I found to my surprise that quantum mechanics didn’t impose any limitations on the size of a computer, over and above those due to statistical and classical mechanics.

Gates: So you don’t have to worry about any unavoidable limitation arising from natural laws when you are trying to build the smallest possible computer?

Feynman: Exactly. Nature is quantum, not classical, so if you are trying to simulate nature, it had better be built on quantum laws. And if there’s no restriction there, you’re on solid ground! Of course, you have to consider the second law of thermodynamics, reversible computing and so on.

Gates: How would you write software for such a computer?

Feynman: That’s for you to figure out! I did my part!

Laughter.

Gates: I noticed in your lectures that you derived Shannon’s Theorem in three different ways, using concepts from statistics, geometry and physics! Why?

Feynman: I am an explorer. I like to find things out for myself. That’s how I understand anything. What I cannot create, I do not understand. If I can derive a theorem or a result independently, even when I know it has been discovered before, it means I understand it. That’s the only way I know how to learn. In the case of Shannon’s theorem, each method I used to derive it taught me something new.

"Besides," Feynman lowers his voice conspiratorially as Gates instinctively draws nearer, "what one fool can do, so can another!"

More laughter.

Gates: The topics of your lectures - coding and communication theory, Shannon’s theorem, quantum computers and such - are fundamental research topics. We have a growing research department at Microsoft but our focus is somewhat different. We are primarily interested in such things as: How do you increase people’s creativity through software? What will make computers easier to use, more responsive to the needs of the user, more natural? Can computers extend human cognition by assimilating speech and linguistics? These are the issues that interest us. Do you have any interest in these aspects of computing?

Feynman: Of course I do. Any tool that can make computers easier to use and more natural, as you say, is important. My own work in physics reflects this. I invented something called Feynman diagrams that allowed me to make complicated quantum calculations in one evening that used to take physicists six months! It was my moment of triumph, to realize that I had succeeded in working out something worthwhile.

So if you are inventing tools and products that simplify computing and at the same time open up complex problems for intelligent analysis, you are doing right by me.

Gates: Is there anything we should look out for?

Feynman: You need to make sure that the tools you create do not become more complex than the problems they are designed to solve.

Gates: One thing that concerns me very much is trying to anticipate the nature of computing ten or twenty years from now. We want to be as intelligent about it as we can, so that we can begin laying the foundation for it now, if that’s possible. Personally, I am looking for some good ideas to take us there …

Feynman: Well, it seems to me you need a new model for writing software, considering how important software has become in everything we do. I think that the next generation of software ought to be modeled after natural objects defined by natural laws. If you can model software objects after objects of nature, I think you will have moved on to something new and significant.

Gates: How so?

Feynman: For one thing, you can be sure that the software will do what it is meant to. If you push this idea further, the same software should be able to transform itself appropriately if the boundary conditions were to change. It seems to me no matter how well thought-out a computer program is, there’s always some unforeseen error in it. It’s not the fault of the designer or the developer, it’s the model on which the program is written. A large software system now seems to me to be like an elaborate sandcastle one builds at the shore. Suddenly a big wave hits and it’s gone!

Gates: So we have to change the foundation?

Feynman: I think so. You need to bind software development to criteria higher than standards and protocols. Numerical data modified by computers should be treated much as laws of nature govern the way objects behave in the real world.

Gates: But doesn’t that imply that people who write software have to be physicists as well?

Feynman: Not really. If you are talking about physical laws, the fundamental laws of nature are simple. That’s where their power and beauty come from. You go through all these complicated calculations and what comes out in the end is unbelievably simple. That’s what I talked about in those videos you have of mine. Besides, it can’t hurt to know a little physics. It’s a part of our culture!

Gates: That is true. Perhaps the new model could help in the area of testing too. Software testing has become very critical. We are always fighting product release deadlines against testing!

Feynman: Exactly. How do you test software against all possible failures? I don’t think you can, using current methods. But if software objects can be modeled after natural objects, testing becomes more straightforward. You have more confidence in the result. If the test fails, you may end up discovering something new and unexpected. That’s how it is in physics. There’s no fooling natural laws!

Gates: I remember the last sentence in your personal report on the Challenger accident: "For a successful technology, reality must take precedence over public relations, for nature cannot be fooled."

Feynman (obviously pleased): You get the idea!

Gates: Any example modeling software after natural objects?

Feynman: Some years ago, a bunch of guys were trying to build superfast computers using parallel processors. The company was called "Thinking Machines." My son Carl, who is interested in such things, joined them. Since the kids running the company didn’t know any better, they ended up hiring me too!

Anyway, the problem was to design a router that delivered messages from one processor to another. There were a million processors in that machine and it wasn’t practical to connect every pair of them. We chose a model where each processor needed to talk directly to only a few of its neighbors. The problem came down to figuring out the minimum number of buffers to hold messages for the router to operate efficiently.

I analyzed the problem by treating the router circuit diagrams as if they were objects of nature, the kind of stuff I do all the time in physics, and came up with a set of partial differential equations. The equations said five buffers per chip would do. Others predicted seven but in the end, it had to be five from a practical standpoint. My equations apparently saved the day!

Gates: We use certain types of object models for writing software. Coming to think of it, you can say these objects are quantized software.

Feynman: Yes! Small, self-contained software that solves just one specific problem that makes life easier for people can be called quantized software. See, you are already using concepts from physics, only you don't know!

Gates: What you are suggesting, then, is that future software should include ideas and concepts from physics as well as from computer science.

Feynman: Yes, but I think ideas should also come from biology. In fact, I think the intersection of biology and computer science could prove even more fruitful for developing software than physics.

Gates: It’s already happening. Bioinformatics is an emerging field that brings together ideas from biotechnology and computers. You studied biology for a while, didn’t you?

Feynman: Yes. I actually worked in the field during my sabbatical year at Caltech. This was after Watson and Crick’s discovery of the DNA spiral. My big moment came when Watson himself invited me to give a seminar on my work at Harvard. I am convinced biology has a lot to offer to computer science, especially in writing software.

Gates: In what specific ways?

Feynman: Well, insights can come from understanding how living organism function, from their adaptive, fault-tolerant, error-handling traits. It can come from studying how human genes are laid out, how the 'book of life' actually reads … There are thousands of possibilities, really.

Gates: That’s exciting! If I were not in computers, I would most certainly be working in biotechnology. I think we are only scratching the surface here. Have you been following the Genome project?

Feynman: Yes, and I think what scientists have accomplished is remarkable. It's a historic milestone to have mapped the genetic blueprint for human life and make it available to researchers!

Gates: The way we treat diseases and prevent them will be revolutionized!

Feynman: I certainly hope so. We can end suffering, at least to some extent, only if we know what disease really is. And when that happens, perhaps we can start talking more about health and less about disease.

Gates: One of my goals in life is to help eradicate diseases like smallpox, malaria, cholera and polio from the world. It's terrible that so many people still die from these diseases in this day and age. The success of the Genome project should certainly help.

Feynman: That will make it all worthwhile, won't it? In many ways, the Genome project reminds me of the Manhattan Project. I feel the same sense of excitement, the same anticipation. I wish I could start over again!

Gates: I am sure there's much you can contribute still.

Feynman: Is there any database that can store the Genome and sift through its data quickly?

Gates (momentarily taken aback): We are working on it. It is extremely important to create a database system that can meet these types of challenges.

Feynman: That should be a milestone for Microsoft. If biological data managers find widespread use, database research will pick up speed too! I want to see this applied to neuroscience as well. If we can track and analyze the activities of billions of neurons simultaneously, we will have made inroads into the working of the brain, perhaps our ultimate frontier.

They ponder the implications of the coming revolution in genetics and neuroscience. Both see beneficial possibilities but recognize that there are important moral and ethical issues to consider too.

Gates: It seems to me that a really good software engineer should be able to derive inspiration from different disciplines.

Feynman: Yes! I made the point in my Nobel lecture that a good physicist might find it useful to have a wide range physical viewpoints and mathematical expressions available to him. If everyone follows the current fashion in expressing and thinking about the generally understood areas, then understanding the open problems is limited. It’s possible that the truth lies in the fashionable direction. But if it is in another direction, who will find it?

I would make the same point to the new generation of software developers. Don’t limit yourself to what you know or what already exists. Be an explorer, not a tourist. Look across disciplines. Dare to follow the beat of a different drum. Your inspiration may come from the dance of molecules on a wave in the sea, the complexity of a beehive or an ant colony, the march of stars across the heavens, the nature of memory and language, the symmetry of a snowflake, and so on. There’s no end to it! After all, nature’s imagination is richer than ours, so why not use what we know of it to our advantage?

Gates: It all goes back to childhood curiosity, doesn’t it?

Feynman: Right! And it’s a tragedy we can’t hold onto some of that curiosity as we grow older!

Gates: Who are your scientific heroes?

Feynman (after a pause): There are three, really. Sadi Carnot, James Maxwell, and Paul Dirac.

He explains: "Carnot obtained a general principle of nature from the nuts and bolts of the thermal efficiency of steam engines. In one stroke, Maxwell unified electric, magnetic, and optical phenomena. And Dirac, a hero of mine ever since I read his book The Principles of Quantum Mechanics, discovered the relativistic equation for the electron."

Gates: What do you consider your most interesting discovery since winning the Nobel Prize?

Feynman: I liked my work on the theory of liquid helium. Another was discovering the laws of weak interaction with Gell-Mann. I also worked out something called the theory of partons to explain some of the properties of protons. Right now I am standing back. I am playing around with some ideas in my mind and I don’t have a clue where I'll come out. I guess that’s what makes it exciting, not knowing what strange territory one will end up in!

Gates: Any disappointments?

Feynman: Certainly. I’ve spent years trying to solve some difficult problems without success. The theory of turbulence is one. In fact, it is still unsolved. Another was my inability to understand superconductivity in which I worked for a couple of years. I should have grasped that one after my success with liquid helium but I didn’t.

Gates: Do the disappointments linger?

Feynman: Not at all! Even where I failed, I worked very hard and had a terrific time. People only hear of successes but not of failures. The important thing is to decide which problems are important and give them your best shots. If you succeed, fine. If not, you almost always end up learning something new!

Gates: You mentioned turbulence. Is that a part of the study of complexity?

Feynman: Yes. There are many phenomena we do not understand yet, from the flight of a swarm of bees to the self-organizing properties of neural networks in the human brain. How is it that a few basic rules can lead to such extraordinarily complex behavior? This is the fundamental question any theory of complexity must answer. And it’s proving to be a real challenge!

Gates: Does that reflect a failure of conventional methods for understanding complex systems?

Feynman: Probably, or else we would have solved these problems long ago! It’s the same idea in software: How do you make sure a software system consisting of 20 or 30 million lines of code works coherently? You have to retain a certain amount of skepticism about accepted ideas and keep an open mind about ideas that appear flaky. It also means any serious study of complexity will require us to explore the fundamental relationship between physics and biology to computation.

Gates: Meaning?

Feynman: Meaning whether or not there is a computational model of the universe and of biological systems that we need to consider and understand. The field is wide open and it’s undoubtedly going to require interdisciplinary research.

Gates: During the Manhattan Project you came in contact with some of the most powerful minds of the twentieth century - Enrico Fermi, Niels Bohr, Hans Bethe, John Von Neumann, Stanislaw Ulam, Robert Oppenheimer and others. Do you think there will be such a gathering of "monster minds" again?

Feynman: That was a different situation at Los Alamos. The project came about because we had to win a war. Besides, the science was also good, very good. We were discovering and inventing as went along. I don’t know that such a situation will reappear. Times have changed. It’s more structured now, more what you call "market-driven." But it really doesn’t matter, because there are many great minds about, brilliant people working in your field who are pushing the limits of technology, worrying about how people interact with computers, how software is written, and so on.

This was the moment Gates was waiting for, the real reason he had come to Altadena. Everything else was a prelude to this moment.

Gates: Will you consult for Microsoft?

Feynman’s response is instantaneous and unequivocal. "That’s the wackiest idea I ever heard!"

Gates is relieved. He has read enough about Feynman to know that the response really means Feynman is interested.

Gates smiles and suggests that his son Carl can perhaps join him too.

Feynman’s face glows. "Ain’t a bad idea at all," he says in his best Brooklyn accent.

Gates doesn’t want to leave any loose ends behind. "How about next month?" he asks. "We will give you a tour of the Microsoft campus and you can meet the people working on new ideas. I think you will like what they are doing."

But Feynman stops him. "Not possible," he says. "I am going on a trip to," he pauses dramatically and then says with a flourish, "Tannu Tuva!"

Gates is vaguely familiar with the planned trip to this place deep in Central Asia, outside of Outer Mongolia, with a capital named Kyzyl. One reason Feynman is interested in going to Tuva is because, in his own words, "any place that’s got a capital named K-Y-Z-Y-L has just got to be interesting!"

For a moment Gates thinks of asking Feynman if he can come along too. Then he sighs. Too many commitments! With a shock he realizes he too is a prisoner of schedules and deadlines, just like other worker bees!

"After you return from Tuva, then?" he asks, almost plaintively.

Feynman looks out the window. The day has about an hour of sunlight left. Shadows are lengthening and a small wind is stirring the sycamore leaves.

"Why not?" he says, grinning, and extends his hand. Gates gratefully shakes it.

 "There’s a revolution coming in software and I think you will enjoy being there when it arrives," says Gates.

 "I’m sure I will."

Gates presses on. "Information is our most important asset. We plan to create great stuff out of it."

 "No," replies Feynman softly, "imagination is."

---

On the flight to Seattle, Gates is restless. He picks up an airline magazine and is immediately repulsed by its banal content. He wishes he can run up and down the aisle to work off the intellectual energy that has gripped him. A mechanical problem has grounded his private jet, so that's not a realistic option.

He reclines as far back as the seat allows and closes his eyes. "Must quickly put a team of good people together," he thinks. "Feynman is coming to Microsoft. Seek inspiration from across disciplines. Learn to tap into nature’s imagination. It’s time to take the company to another dimension."

(c) Hasan Zillur Rahim

Faith, Reason and the Templeton Prize

The 2007 Templeton Prize “For Progress toward Research or Discoveries about Spiritual Realities” was recently awarded to the Canadian philosopher Charles Taylor for his insights into the nature of the secular and the sacred and how one without the other can be perilous for mankind. “The divorce of natural science and religion has been damaging to both,” he said, “but it is equally true that the culture of the humanities and social sciences has often been surprisingly blind and deaf to the spiritual.” The 75-year old McGill University emeritus professor has called for new insights into the human propensity for violence, one that also takes “full account of the human striving for meaning and spiritual direction, of which the appeals to violence are a perversion.”

The American philanthropist John Templeton created the annual prize in 1973 to recognize research in spirituality and its possible confluence with science. He made it the most lucrative prize in the world – at more than $1.5 million, it is larger than the Nobel Prize – to emphasize that we are shaped more by our spiritual longings than by any other factor, and therefore advances in the understanding of spirituality should also begat more attention and recognition. (Given the 72 years headstart the Nobel had over the Templeton, this may take a while!)

Of late, religion, spirituality and God have been under assault by militant secularists whose ranks include prominent scientists. Leading the charge is Richard Dawkins, professor of public understanding of science at Oxford University. His book, The God Delusion, has been on the best-seller list for several months now. Dawkins suffers from no shades of gray. God is unnecessary, he says, because science - evolution, randomness, physical laws and such - can explain everything. If anything does lie beyond the scope of science, it has no meaning and is therefore irrelevant. His fellow-travelers include the neuroscientist Sam Harris (The End of Faith: Religion, Terror and the Future of Reason) and Tufts University philosopher Daniel Dennett (Breaking the Spell: Religion as a Natural Phenomenon) among others.

For every atheist or agnostic scientist or philosopher, however, there are at least a hundred who are passionate about their faith or at least open to the possibility of a Supreme Being. One such is the geneticist Francis Collins who led the successful effort to complete the Human Genome Project, a multidisciplinary enterprise to map and sequence the human DNA. Collins refutes Dawkins by asserting that God lies beyond the reach of science, beyond space and time, and so cannot be explained by science. God used His creative power to bring all creation into being. If we keep an open mind, we can detect God’s handiwork in the signs He has strewn about us, from the large-scale drama of the universe to the intricate world of sub-atomic particles. Author of The Language of God, he bemoans the fact that many of the current battles between atheists and fundamentalists have really been started by the scientific community, which he feels is an enormous tragedy.

Collins summarizes the beliefs of many scientists such as that of the astronomer Owen Gingerich who makes the point in God’s Universe of the existence of a Creator. The Muslim astrophysicist Bruno Guiderdoni draws inspiration from his faith in his research on galaxy formation. The fundamental mystery that animates physics and cosmology, he believes, is that the world is intelligible. The Nobel physicist Abdus Salam (1979) found in his faith the inspiration to delve into the mysteries and symmetries of fundamental particles. A list of recent Templeton Prize winners also illustrates the point: physicist Freeman Dyson (2000), chemist Arthur Peacocke (2001), mathematical physicist John Polkinghorne (2002), applied mathematician George Ellis (2004), Nobel physicist Charles Townes (2005) and mathematician John Barrow (2006). They were cited not for their scientific or mathematical discoveries but for their efforts to show in their distinctive ways that science and religion are two windows looking out on the same universe.

If scientists can be inspired by their faiths, can theologians and religious leaders be inspired by science? Certainly, and one example will suffice. In his book The Universe in a Single Atom: The Convergence of Science and Spirituality, the Dalai Lama writes eloquently about his fascination with science from an early age “It was not very long before the colossal significance of science for humanity dawned on me - especially after I came into exile in 1959. There is almost no area of human life today that is not touched by the effects of science and technology.” Yet he warns of the danger of trying to find within a purely scientific context answers to questions such as the meaning of life or good and evil. “The problem is not with the empirical data of science but with the contention that these data alone constitute the legitimate ground for developing a comprehensive worldview or an adequate means for responding to the world’s problems … By the same token, spirituality must be tempered by the insights and discoveries of science. If as spiritual practitioners we ignore the discoveries of science, our practice is also impoverished, as this mindset can lead to fundamentalism.”

The Templeton prize celebrates those who seek to reconcile the ancient adversaries of science and religion by confronting difficult questions head-on, such as those raised by Darwinian atheists and religious fundamentalists. It celebrates the middle ground between the dispassionate observer and the devout believer, suggesting that the two can be fused into one for a full and creative life.