Wednesday, November 11, 2009
On September 10, 2009 British Prime Minister Gordon Brown issued a public apology, on behalf of the British nation, to Alan Turing for that country’s treatment of him. Alan Turing was one of the great mathematicians of the twentieth century. He worked on fundamental problems in mathematical logic in the 1930s in the process inventing some of the seminal concepts of computer science, including the basics of the theory of computation. During the Second World War he served at the British code-breaking school at Bletchley Park and was largely responsible for breaking the German U-boat code. After the war he worked on the design and construction of two first-generation computers and was an early advocate for, and theorist of, the future of artificial intelligence. Turing was also gay and in 1952 he was convicted of “gross indecency” and sentenced to a course of treatment with female hormones designed to eliminate his sex drive. Not surprisingly, this chemical castration had profound physical and psychological consequences. In 1954, at the age of 41, Turing took his own life.
The extraordinary drama and tragedy of Turing’s life—war hero and mathematical genius driven to suicide by persecution for being gay—has provided inspiration for several artistic interpretations of his life. (Turing did his best to increase the drama by choosing a method of suicide inspired by Disney’s Snow White, a favorite of his—he ate a cyanide-laced apple.) There have been at least three plays, one novel, and two movies based on his story, as well as an extraordinary biography by mathematician Andrew Hodges. That biography reveals Turing as a good and decent man more or less bewildered by the barbaric treatment he received from his countrymen. These retellings of the Turing story have contributed to Turing’s status as something of an icon in the gay community. His profoundly original scientific contributions—in computer science he is memorialized in Turing Machines, the Turing Test, and computer science’s Nobel Prize, the Turing Award—have similarly preserved his iconic status among mathematicians and computer scientists. In fact, the Brown apology was provoked by a petition drive organized by a British computer scientist, John Graham-Cumming, inspired by admiration of Turing and dismay at his treatment.
Once Brown issued the apology the news, and reactions to it, zipped around the Internet. Most folks believed that the apology was long overdue and constituted a genuinely positive development, especially as a small contribution that might help chip away at still existing homophobia. And, to Gordon Brown’s credit, he understood that this should not be just about Alan Turing. Turing is just one particularly egregious and notorious example; Brown explicitly expresses regret over the “many thousands of other gay men who were …treated terribly” and even recognizes the “millions more who lived in fear.” Most fans of Turing seemed pleased; one friend told me he felt “elated.”
I didn’t feel elated and I wondered if there was something wrong with me. Oh sure, I recognized that this was a good and necessary step, but I couldn’t help but feel that it was not proportionate. The British government, in the name of the British people, tortured this good and decent man (and thousands of others) because they disapproved of his sexual habits. Now, half a century later, they offer only words of regret. Maybe I’d feel better if Gordon Brown vowed not to rest until gay marriage was legal in Britain. Of course in Britain today the legal status of gays is a thousand times better than it is in the US, so maybe Brown could work on educating America? How about passing a heavy tax—the Turing Tariff—on all computing hardware and software imported into the UK from countries, like the US, that still discriminate against homosexuals by banning gay marriage? The proceeds of the tariff could be donated, in the name of Alan Mathison Turing, to the leading gay rights organizations in the exporting country.
I know that’s not going to happen. Nothing really dramatic is going to happen. Still, I admire John Graham-Cumming and the thousands who signed his petition and am grateful for their efforts. I’m also appreciative of Gordon Brown’s understanding and grace. I do feel badly that I’m unable to celebrate; a terrible maltreatment of one of my intellectual heroes is being publicly recognized as such. But it just seems that this goodness and benevolence—and maybe all goodness and benevolence—are slow and discreet and progress in tiny incremental steps while hatred and injustice bash and roar and wreak enormities. And we don’t seem to ever learn. Elated? No, I’m not elated; I’m just very sad.
Wednesday, September 23, 2009
I am coming clean. I do not have a passion for mathematics. And I don’t think that’s a bad thing. Here’s why.
I was recently reading an article about a friend who solved an interesting open problem and was rewarded with some well deserved publicity. This mathematician is quoted as saying, “I get a problem like this and I don’t sleep at night.” Articles about mathematics and interviews with mathematicians seem to always include comments like this. There is an indication of a passion for problem solving and a related inability to function in the rest of the world while the math question remains open. I do not recall ever reading an article where a mathematician discusses a recently solved problem and simply says, “Yes, that was a nice result. A good day at the office. That’s what I’m supposed to do. It’s my job.”
Does a person have to possess an extreme level of commitment to be a good mathematician? To choose mathematics as a major in college, should one feel that it is a calling? How about for success in graduate school in one of the mathematical sciences? Can a person forge a happy career in our subject without passion, or is something approaching devotion a vital ingredient?
I think in many circumstances we have, sometimes almost unwittingly, made passion a pre-requisite to entering the major, much to the detriment of the field. There are some of us who have had successful and rewarding careers in mathematics and related fields without having been on the high school math team, without taking the Putnam exam as undergraduates, and without losing sleep over problems. I enjoy mathematics, and I think a day at a math meeting with interesting talks is a great way to spend my time. But at the end of that day when I’m at dinner and a colleague grabs a pencil and a napkin and says, “Here’s a cute problem,” count me among those who try to change the subject.
Of course passion for mathematics can be a wonderful thing and it may be a necessity for the giant steps that move the field forward. The truly great mathematicians have that trait and I recognize its value. I enjoy reading mathematical history and I delight in accounts of the single minded tenacity shown in solving hard problems, and the euphoric feeling of triumph when successful. Anyone who has watched the wonderful Nova episode about Andrew Wiles and his solution of Fermat’s last theorem can see the great interaction between a scientific breakthrough and a personal victory. But tremendous devotion is demonstrably neither a necessary nor sufficient condition for building a successful career in the mathematical sciences. To use a baseball analogy, it might be necessary for a “Hall of Fame” career, but those of us making contributions as utility infielders have a place too.
Making passion a requisite part of our culture has costs. Here are a few of them:
Requiring passion discourages talented students from studying mathematics.
Recruiting students to mathematics is like a political party trying to decide whether to appeal to a committed base or expand a point of view to broaden participation. A common theme among colleges that have large and successful math majors is that they have put up the ‘big tent’ and allowed lots of students in. Does this hurt the quality of their top majors? It doesn’t appear to. For one thing it gives them enough students to offer required and elective courses more than once every other year!
Passion makes for poor advising.
Advisors who want passion as a pre-requisite often see graduate school as the only winning outcome for students with talent in mathematics. This runs counter to one basic goal of a liberal arts education, namely to encourage people to be thoughtful, adaptable and open-minded in career choices. Mathematics majors can go on to be wonderful accountants, machinists or landscapers and perhaps make unexpected contributions in those areas thanks to their point of view.
Faculty may quit research sooner than they should.
When faculty members insist that their research must be elegant, ground breaking and the result of a deep commitment, a common mid-career result is to give up completely. But there are myriad open problems in other disciplines where we can be of great help. There are plenty of opportunities for pedagogical research. There are consulting opportunities in business and industry. My PhD is in statistics, so finding these opportunities might be easier for me than for people in more abstract fields, but there are dozens of underpublicized success stories of people who make contributions in this more mundane way.
I feel fortunate to have an enjoyable, rewarding and I hope useful career in mathematics and statistics. My nonmathematical friends see me as a math nerd who sees the world differently than they do, and some of my mathematical colleagues probably see me as a slacker who doesn’t really ‘do’ math. That’s a balance I like.
Wednesday, April 1, 2009
Frank Swenton - Middlebury College
The function is well known as one of the most fundamental concepts in all of mathematics, and it is equally well known as one of the most misunderstood by students in mathematics courses. Some would say that this is simply the nature of the student, and others would blame the advent of the graphing calculator. But the root cause of this problem lies much deeper, its seeds planted more than 300 years ago: it is the graph itself, which through the primacy it has attained as a means of dealing with functions, has obscured—and even supplanted—in the minds of students the very concept it was meant to illuminate.
In the Beginning, there were the Domain and the Range, each a distinct set in its own right, and there was the Function sending each element of the domain to one element of the range. Domain was domain, range was range, and never the twain did meet. Fast-forward to the 17th century: René Descartes rotates the range by 90 degrees and clanks it down right on top of the domain; upon these axes he draws a function’s graph: a construction both revolutionary and visually seductive—in fact, enough so to eclipse the function concept itself.
Considering functions via their graphs is a practice so deeply ingrained in the mathematics classroom that it escapes the critical eye given to all other aspects of mathematical pedagogy. Graphs readily provide easy answers (in some cases), but the answers are all too ready, too easy, and—most importantly—quite often incomplete or deceptively facile. X has become synonymous with “input,” and y with “output”; the derivative means “slope”; the integral means “area”; continuity means that a graph can be drawn “without lifting the pencil.” Even the very heart of being a function has sadly devolved into the “vertical line test.”
The fundamental problem is that these rough and ready geometric answers often serve to distract the learner from well-warranted further consideration of the true nature of each concept; they provide just enough of a simple, soothing “answer” with which to be content and to quiet any call for further thought. Moreover, when a learner is presented time and time again with graphs, it becomes far too easy to see only the graphs. When applying a function f to a value, the eyes quickly move directly to the point of the graph at the correct horizontal location, and there they stop. Seldom does the eye ever stray to the domain, at which the action starts, or the range, at which the action ends—the function’s meaning as the link from one to the other is lost, all attention drawn by the immense visual pull of its graph.
The effects of this graph-centric perspective on the function are felt far beyond the topic of the function concept itself. Inverse functions have students immediately flipping the plane or wrenching their necks; that inverses are simply rules sending elements of the range back whence they came becomes secondary. Differential approximation becomes the circuitous unraveling of a triangle tangent to a graph, finally arriving at a formula that is a direct consequence of the definition of derivative. The epsilon-delta definitions of limits and continuity become a crisscrossed clutter of horizontal and vertical lines on a graph, yielding a similar tangle in the learner’s mind, when the domain and range have no business intersecting in the first place. Careful inspection across the gamut of function-related concepts shows that graphs very often complicate or confound in their effort to simplify and explain.
Is the graph logically inconsistent? No. But we must reply equally in the negative to the question of whether a graph properly expresses the function concept in its entirety. One might rebut that the modern definition of the function from X to Y is as a particular subset of the Cartesian product X x Y. However, consider the logical definition of the implication “if P, then Q”: do we properly conceptualize it as “Q or not P,” or is this merely the logical statement that defines its formal meaning? A function, while encoded as a subset of X x Y, is conceptually no more a subset of X x Y than a poem is ink on a page; the difference is that no one forgets that a poem must be read aloud (at least in the mind) to be fully appreciated or understood.
A function acts: it sends each element of its domain to some element of its range; it maps subsets of the domain to their images in the range; it even pulls back sets in the range to their preimages in the domain. A function acts—and a graph simply sits, dead. We must view that static graph as a machine ready to act, or else the graph only serves to gloss over the concept; we must make the crucial effort not to be content with the limited perspective on functions afforded us by graphs alone. The graph is only a valid tool for studying a function when viewed not as the function itself, but as a representation of the function—only in vigilance of this crucial distinction will the concept retain its integrity; and with a healthy independence of graphs, the function concept can properly grow and flourish within the minds of students.
Tuesday, February 10, 2009
Tom Tucker - Colgate University
What is the population of the United States? How far is it from my house to my office? How much is the national debt? All of these questions have numerical answers, sort of, but we all know it would be absurd to answer 307,285,671 people or 1135 feet, 11 inches or $11,245,734,298,635. First, none of these questions is well-defined. For the population of the US, at what instant are we talking about? Are we including US citizens who are living abroad? For the distance from my house to my office, where in my house? Where in my office? As the crow flies (an expression that should tell you we are in trouble here)? For the national debt, again at what instant? Second, even if the quantities were well-defined, they are all measurements subject to error. For most measurements, you’re lucky if you can get an error as small as 1 part in 100.
A few years ago, I served on a committee to revise the K-12 Mathematical Standards for New York State. I had followed the NCTM Standards since the late 1980s and there was one “strand” that always bothered me: Measurement. In the early grades, there were recommended activities with rulers, such as measuring the height of a desk. By grades 6-8, measurement consisted mostly of formulas for areas and volumes. By grades 9-12, the strand petered out. It was always the thinnest of the five strands. As I sat through the committee meetings, it occurred to me that it should be the thickest. My rule is this: If it has units and an error, then it’s a measurement. By that rule, every number of importance in our lives is a measurement.
Mathematics education in the US has done pretty badly on the matter of units, which is mostly ceded to science education. What a shame, since units are enormously helpful in understanding equations, notation, and terminology. Although it now seems generally accepted that mathematics should be taught from the algebraic, graphical, and numerical viewpoint, there is a fourth medium for presenting mathematics: words. And units are words, which reach students in ways that algebra, graphs, and numbers cannot.
The real failure of mathematics education, however, has been its treatment of error, especially relative error. Is an error of a foot a big error? Yes, if you are measuring my height, no if you are measuring the elevation of Mount Everest. Mathematicians know that only relative error makes sense. But relative error is almost nowhere to be found in the mathematics curriculum.
I was on the Mathematics AP Committee for the College Board when calculators were first allowed on the exam. We had to decide how much accuracy we required. Of course, any scientist would give an answer in terms of significant digits. Because we knew, however, that the concept of significant digits was not a standard part of the K-12 mathematics curriculum, we said instead “three digits to the right of the decimal point” (no mention of floating point). A year later, we wrote an exam problem on US soda consumption that entailed numbers in the billions of gallons. If a student chose gallons as their units, rather than billions of gallons, the required answer had 14 digits, more than a calculator could handle at the time. I always thought a clever student should have chosen quadrillions of gallons as her unit and given the correct answer of 0.000. The AP Committee had been painted into a ridiculous corner by the failure of the US mathematics curriculum to deal with relative error.
Estimation skills may sometimes be viewed as just another form of “fuzzy math,” even though when we grow up, we find in our everyday lives that there are two kinds of arithmetic: what we do in our head and what we do with a calculator or computer (everybody who does their tax forms with pencil and paper only, please raise your hands). But I am talking about the role of estimation in measurement, not arithmetic. A proper approach to measurement would also involve estimation. Physicists have long played the Fermi game of trying to estimate some strange quantity—the real estate value of Hamilton NY, the number of people who have ever played Major League baseball, the number of piano tuners in Chicago. All involve seat-of-the-pants estimates based on a few facts, like the population of Chicago. But why aren’t math students playing this game? Why aren’t 6th graders being asked about how many ice cream cones their school eats in a year? Why aren’t 9th graders being asked by their math teacher to go home and come back tomorrow with their estimate of the number of gallons of gas consumed annually by cars waiting at red lights, with a full explanation of their answer?
I understand that the other strands—number and operations, algebra, geometry, data analysis and probability—are important, especially for the scientific infrastructure of the US—but I would feel a lot more comfortable about the public understanding of the costs and benefits of our country’s policies if I knew we could get the measurement strand right.