Monday, July 27, 2009


Here's a fun game that the NYC Teachers Math Circle members enjoyed playing in the outdoor classroom at Union College this evening. Albany Area Math Circle members who will be mentoring middle school students this year may also want to use this game with them.

"Hangmath" requires no equipment except for pencil and paper, or a chalkboard and chalk. It works like Hangman only with numbers.

Here's an example: one player privately writes out a 2 digit by 3 digit multiplication problem and keeps it hidden from the other players.

The first player creates an empty grid corresponding to the digits in the problem, for example:

_ _ _
x _ _
_ _ _
_ _ _ _
_ _ _ _ _

(EDIT: thanks to Prof. Moorthy for help with the html to line that up correctly!)

The other players then have to make guesses of the form "Is there a four in the tens column?" If yes, then the first player fills in all the fours in the tens column. Otherwise draw a part of the hangman.

Obviously, you can always get a guaranteed fill-in if you ask "Is there a zero in the ones column?" (Why?)

Initial guesses after that tend to be random (unless there was more than one zero in the ones column), but once one of your random guesses turns out to be a hit, you can use that information to make more intelligent guesses about the remaining

What I like about this is that there is a lot of logic and backwards reasoning and understanding that can go on in solving the problem. Playing this game really gives students a solid understanding of what they're doing when they use a multi-digit multiplication algorithm. They understand it inside out, upside down, and backwards, rather than as just a mindlessly memorized rote recipe.

The player who makes up the problem also can use some good logic and backwards reasoning in trying to design a problem that will be hard to crack.

Variation: You could also use this game with a long-division problem grid as well.

I like this game as an icebreaker with a group of students that may include some very shy and tentative students who may be unsure of themselves. Why? Because a certain amount of guessing/trial and error is necessary at the beginning, and that takes pressure off students.

It's very exhilarating for an inexperienced student to get a lucky guess early on, occasionally. It's also empowering for an inexperienced student to see veterans getting some unlucky guesses wrong initially.

Real mathematical research often involves a fair amount of trial and error experimentation, and Hangmath is a nice way to ease students into that idea. Sometimes you just have to experiment somewhat in the dark in order to get insights
that will lead you where you want to go.

That's a really profound and important idea to transmit to students early on. The whole notion that a certain amount of intelligent trial and error is a legitimate problem-solving strategy is pretty radical to reinforce. When you don't know what
to do, guess and see what you learn from your guess is a very powerful idea. It's one I never learned in school.

When I was in school (a very traditional and rigid and mechanical rote school, with 56 students per classroom!), the idea was that you solved problems by applying rigid and deterministic recipes. If you didn't know the recipe you couldn't solve the problem, and you were supposed to wait until someone taught you the recipe.

When I run this game, I usually start out by asking each of the individual students to take turns with the early guesses, but then later I have them work in teams figuring out what logical inferences they can make about the best numbers to
guess next. It was really good to hear the math teachers working in teams to figure out strategies for what numbers to guess next.

I'd never asked teachers to play this game before, only students. They did a great job of efficiently reasoning through the guesses, and coming up with strategies to make it easier to be systematic with their guesses.

It should be noted that, at one point, I made a mistake in running the game. I failed to notice that there were two "2's" in the hundreds column. I only wrote one of them after a team asked me "Are there any 2's in the hundreds column?" I wrote down one of the two's in the hundreds column and they then began reasoning about the remaining numbers on the assumption that there were no other 2's in the hundreds column. Eventually, they started to realize something was wrong--and I could tell there was a problem, so I rechecked my copy of the problem, and realized that I'd neglected to fill in the second 2. It's good for students to see that even the "authority figure" running the game makes mistakes. (However, it's also helpful to have a second person with access to the answer key checking for such oversights.)

Hangmath also provides a good opportunity to talk about the value of reasoning from negative, as well as positive, information!

Tuesday, July 21, 2009

History and math walks in Schenectady

Albany Area Math Circle's advisors are delighted that the New York City Middle School Math Teacher Circle will be coming to Schenectady to hold their summer immersion problem-solving retreat at Union College next week.

I have volunteered to lead two optional early morning walking tours to help our visiting colleagues from the Big Apple discover some of the beautiful historic areas on and near campus.

Tuesday's walk will take us through the Union College campus and then eastward toward a beautiful historic neighborhood General Electric Realty Plot. The college campus is exceptionally beautiful and provided the setting for the college scenes in the movie The Way We Were with Barbra Streisand and Robert Redford. The GE Realty Plot is filled with an eclectic collection of elegant mansions built by GE executives circa 1900, including revival styles of Tudor, Queen Anne, Georgian, Dutch Colonial, and Spanish Colonial. The neighborhood is on the National Register of Historic Places.

Wednesday's tour will take us westward to the historic Stockade neighborhood, filled with many beautifully restored houses and gardens, some of which go back to the Dutch settlement along the Mohawk River in the 1600s. After its restoration, this neighborhood was recognized as New York State's very first historic district.

We'll be surrounded by Schenectady's mathematical traditions as well as history and beauty on our two early morning walks.

The campus centerpiece is the Nott Memorial, a 16-sided building that has been called a "Pythagorean temple." The overall campus design has very marked geometric and symmetric elements, originally conceived by noted French architect Joseph-Jacques Ramée in 1813. Union College was the first planned college campus in this country and influenced Thomas Jefferson's design for the University of Virginia campus four years later.

Both day's walks will take us through the former stomping grounds of a truly remarkable and legendary Schenectady mathematical wizard, Charles Proteus Steinmetz, "who could generate electricity from the square root of negative one." [UPDATE: The picture below shows NYC Math Teacher Circle early morning walkers at the monument marking the site of Steinmetz' former home in GE Plot.]

My two posts below will provide more details for the curious.

Monday, July 20, 2009

Charles Proteus Steinmetz: mathematical wizard of Schenectady

The figure above illustrates an example of a Steinmetz solid, a shape formed by the symmetric intersection of two or more cylinders. You can think of the solid shown on the right as the amount of metal you remove when you use a drill of a given radius to bore a hole through a solid pipe of equal radius. A good place to learn more about Steinmetz solids is Martin Gardner's wonderful recreational math book The Unexpected Hanging and Other Mathematical Diversions, which features a Steinmetz solid on its cover.

The Steinmetz solid was named after the legendary Charles Proteus Steinmetz, a mathematician, inventor, electric engineer, educator, and political leader who started up the first GE research lab in the back of his home in a historic neighborhood we'll see on Wednesday's walk, and who later moved it to the GE Realty Plot neighborhood we'll walk through on Tuesday.

An immigrant who had arrived in this country penniless, he brought with him the treasures of a rich mathematical education, including participating in a math circle in his native Germany. (Interestingly, he also added the middle name "Proteus" after coming to this country, choosing a somewhat mathematically-themed name after the Greek god who could change his shape!)

A prolific inventor with 200 patents to his name, Steinmetz once said, "I want to say that absolutely all the success I have had has been due to my thorough study of mathematics." He liked to work on his mathematical calculations on papers spread out on a board across his canoe as he went up and down the Mohawk River.

Steinmetz is best known for a mathematically beautiful formula he invented that uses complex numbers to elegantly simplify the computations needed for alternating current applications. Because of that beautiful formula, he is known as the "the wizard of Schenectady who generated electricity from the square root of negative one."

Despite all his inventions and discoveries, he realized that his greatest power to make a difference in the world lay in his ability to educate others, so he left General Electric to become a professor at Union College and also to lead the Schenectady School Board, providing much needed educational leadership at a time when the city public schools were bursting at the seams. His campaign slogan was "a seat for every child."

Here's what science historian George Wise has to say about Steinmetz.

In the life of the great electrical engineer Charles Proteus Steinmetz (April 9, 1865- October 23, 1923) truth really is stranger than fiction. He's best remembered today for the many myths that grew up around him. Most are pure fiction, or at best gross exaggerations. For example, it's a myth that Steinmetz invented alternating current. Or that when one of his bosses pointed first to Steinmetz's ever-present cigar and then to the "no smoking sign" and told him to put the stogie out, Steinmetz replied: "no smoking, no Steinmetz". Or that Steinmetz was so valuable to his employer, General Electric, that GE gave him not a salary, but a blank checkbook. Or that GE had bought a whole company just to acquire Steinmetz. Or that when Steinmetz, who suffered from scoliosis so severe that he was cruelly called a "hunchback" by the insensitive, was asked why he accepted the post of professor at Union College rather than at a famous Midwest university, he answered" "The choice was simple-- it was either to become the electrical genius of Union College, or the hunchback of Notre Dame."

All that is fiction. The truth is better. The scoliosis, its destructive effects on his stature, and the taunts it inspired were real-- and so was his triumph over disability, a story deservedly celebrated alongside that of the physicist Stephen Hawking. Growing up in Germany, Steinmetz studied mathematics and philosophy-- but also acquired a devotion to socialism that led him to leave the country one step ahead of the police and shortly before completing a Ph.D. thesis in mathematics. Steinmetz really did arrive in America with no money or reputation, but he had letters of introduction and an academic background from the University of Breslau that quickly won him a job in the infant electrical industry. There he made his name by developing his Law of Hysteresis, an important tool for predicting energy losses in equipment using magnetic fields, such as electric motors. GE acquired him by good luck, not design. But his GE bosses were smart enough to recognize his greatness and soon made him their chief consulting engineer. He did not invent alternating current or even invent the mathematics for analyzing alternating current. But he educated the world about that mathematics. His ability to define and solve complicated problems involving the generation, transmission and control of alternating currents in a pre-computer age earned him the nickname "the Supreme Court"-- the final authority on such matters. His own recognition that his greatness was as an educator rather than as a scientist or inventor led him in 1902 to join the Union faculty. He rejuvenated Union's engineering program, and was the first chair of its Electrical Engineering Department. In his spare time, he founded the first research laboratory in the US (now GE Global Research), pioneered environmentalism and electric cars, helped organize a successful Socialist Party that won control of Schenectady city government and installed such improvements as better schools and a park system, and, daring the unpopularity of a German-American appearing to take the German side in World War I, chaired a mass meeting in 1916 calling for neutrality and peace.

His mathematical and educational leadership legacy is a great inspiration to contemplate as we walk through his old stomping grounds on our early morning walking tours of Schenectady.

Our Chapter MATHCOUNTS competition actually takes place on the huge and sprawling GE Global Research Campus in nearby Niskayuna. That 525-acre facility traces its origins to the rather modest quarters in Steinmetz home. I think Steinmetz would be very happy to see all the enthusiastic young problem solvers who come together in those facilities each February.

A Pythagorean temple

The Nott Memorial pictured above is one of the mathematical highlights of the early morning walking tours I'll be leading for members of the New York City Math Teacher Circle who are coming to hold a problem-solving immersion workshop at Union College next week.

Here's a description of some of the mathematical symbolism employed by Edward Tucker Potter, the architect who designed the Nott:

Potter seems to explore symbolism to discover and be fascinated with proportionalities, among these the ad quadratum, the Golden Section (the ratio of 1:1.618), and the significant yet delicate positioning of hexalphas and pentalphas, using Victorian-Gothic as the vehicle of his expression. The nature of his work may be viewed as more symptomatic of an even larger and all-encompassing plan -- that of the universe as an orderly, integrated macrocosm. Potter had a systematic, Pythagorean approach to his architecture...

A central element of the Pythagorean philosophy is that there is a profound numerical order, unity, and harmony in the Universe (the macrocosmos) as symbolized by the icosahedron and the hexalpha, and in man (the microcosmos) as a refinement, a distillation, an analog of this grand plan.

The hexalpha probably emerged most strongly as a symbol of harmonious duality and in particular the ten primary contrasting qualities of Pythagoras -- the limited and unlimited, odd and even, male and female, one and the many, right and left, rest and motion, straight and curved, light and darkness, good and bad, and the square and the oblong. In essence, the hexalpha and icosahedron represent the union of complementary forces.

In this light, it is highly appropriate for the dome of the Nott Memorial at a college called Union to bear its array of hexalphas and pentalphas. The Nott Memorial may be viewed as a Pythagorean Temple of the Muses and a beacon leading us toward the Truth and the Good.

More information about the mathematical elements in the Nott is available in Decoding the Nott Memorial.

Thursday, July 16, 2009

Richard Feynman books and videos

If you haven't already encountered the legendary Richard Feynman through his wonderful books, they are very much worth checking out. He's not your usual Nobel laureate in physics (even assuming such a concept exists.)

He has a couple of very funny autobiographical memoirs, Surely You're Joking, Mr. Feynmann: Adventures of a curious character and What Do You Care What Other People Think? Further adventures of a curious character. Both books would make great beach reading, as they are full of fascinating and often funny anecdotes that strike the interest of math circle members, including his experiences on his high school math team in New York.

Actually, in all his books, even his classic and famous (but far more technical) Lectures on Physics, Feynman displays a wonderful conversational style. Because of his delightful and engaging conversational style, his books are great fun to read aloud to family members, friends, your dog, basically anyone who'll listen! One of my favorite Feynman books is The Character of Physical Law, which contains the text of seven fascinating public lectures which he delivered at Cornell.

A hat tip to Professor Moorthy for sending us links to newly available videos of the inimitable Richard Feynmann delivering those lectures. The videos are available at the following links:
Youtube version and Microsoft site.

Here's what mathematician Terry Tao has to say about the videos:

Seven videotaped lectures from 1964 by Richard Feynman given in Cornell, on “The Character of Physical Law“, have recently been put online (by Microsoft Research, through the purchase of these lectures from the Feynman estate by Bill Gates, as described in this interview with Gates), with a number of multimedia enhancements (external links, subtitles, etc.). These lectures, intended for a general audience, broadly cover the same type of material that is in his famous lectures on physics.

I have just finished the first lecture, describing the history and impact of the law of gravitation as a model example of a physical law; I had of course known of Feynman’s reputation as an outstandingly clear, passionate, and entertaining lecturer, but it is quite something else to see that lecturing style directly. The lectures are each about an hour long, but I recommend setting aside the time to view at least one of them, both for the substance of the lecture and for the presentation. His introduction to the first lecture is surprisingly poetic....

[Update, July 15: Of particular interest to mathematicians is his second lecture "The relation of mathematics and physics". He draws several important contrasts between the reasoning of physics and the axiomatic reasoning of formal, settled mathematics, of the type found in textbooks; but it is quite striking to me that the reasoning of unsettled mathematics - recent fields in which the precise axioms and theoretical framework has not yet been fully formalised and standardised - matches Feynman's description of physical reasoning in many ways. I suspect that Feynman's impressions of mathematics as performed by mathematicians in 1964 may differ a little from the way mathematics is performed today.]

Wednesday, July 1, 2009

Leonard Mlodinow on probability blindspots

In 2002, psychologist Daniel Kahneman won a Nobel Prize in Economics for work that pointed out that human beings often have problems reasoning through real world problems involving probabilities.

Caltech physicist Leonard Mlodinow has written a fascinating and clearly written new book, The Drunkard's Walk: How Randomness Rules Our Lives, with many beautiful examples illustrating common logical fallacies. You can learn a good deal in a very enjoyable way by reading his book. You might want to start out by watching a talk he gave about his book to Google employees. (See the video above.)

As Mlodinow points out, probability blind spots can cause seriously bad decisions in many domains. Many professionals, including physicians, judges, and investors, make errors in reasoning through situations involving probabilities. (Sadly, it appears that medical schools and law schools don't teach much about probability. Business schools DO teach about probabilities, but it's not clear how much actually sinks in.) I personally think the answer is that students need to grow up thinking hard and deeply about probability. The habits of thinking correctly about probabilistic calculations need to be ingrained deeply in all of us long before we become jurors or adult patients, let alone judges or physicians.

There is a growing consensus about the need for a really sound mathematical education in probability and statistics. Harvey Mudd math professor Art Benjamin makes a good case for it in his TED Talk video below:

Hat tip: Richard Rusczyk