Category Archives: mathematics

Math Anxiety and Me: A Tale of a Lost Opportunity

Numbers have never been my friends. I remember weeping over my “math” (really, arithmetic) homework in fourth grade. The assignment was to fill in a twelve-by-twelve grid with the multiplication table. I was in my room laboriously multiplying each pair of numbers when my father came up and found me. He pointed out that I could easily fill out each column by simply adding the number at the top of the column to the number in the current cell as I worked my way down (5 times 1 is 5, 5 times 2 is 10, 5 times 3 is 15, etc—each time I am simply adding 5). It was astonishing to me that I could do this task with simple addition, and I was actually sure that he was showing me a way to cheat. I took the hint, of course, and was quickly finished, but I never shook the feeling that numbers were opaque, frustrating, and needed to be approached by devious means.

In seventh grade I discovered logic, and it fascinated me. I spent a lot of time thinking about logic, paradoxes, syllogisms, and oxymorons. I didn’t know that this was a field of study, or that I would be able to find books on this. I just played with it myself and enjoyed the sort of thinking it made me do.

Some of my greatest moments of suffering in school were in math class. I was totally unengaged, had no intrinsic interest, and had mostly unpleasant teachers. When I think of high school math the image that comes into my head is of Mr. Groninger, a large, sagging, grey man who droned on and on and never seemed to notice me or my disaffection. My High School grades in math were pretty good, but won at the cost of drudgery.

During my High School years we adopted a pet monkey, and late in High School my career ambition was to be a primatologist. This was the culmination of a series of career ambitions: first, at five, a witch; second, at nine, a vertebrate paleontologist (I loved the bones at the anthropology museum); third, as a teenager, a doctor. I remember when I was five lying in bed designing the piece of hardware that would be needed to fasten my child’s umbrella to the stick of my broom—but why I did not become an engineer is a story for another day.

Between my junior and senior years in High School I got a summer job working for a young primatologist, Tom Struhsaker, at Rockefeller University in New York City. I lived in my grandparents’ guest room and spent all day, every day punching cards, putting my supervisor’s data on the activity cycle of vervet monkeys into machine-readable form. The following summer, the summer of 1969, I was back again, but this time he handed me a book on Fortran, gave me the formulas necessary to perform statistical computations on the data, and sent me over to the computer area. The Control Data computer filled a whole large room, and I remember my astonishment when I learned that when the university obtained the computer, a man came with it, issued by the company. I don’t remember being unwelcome there, though I was probably the youngest person and the only female around. But the space was made for the computer, not for the people, so it wasn’t really a place you could hang out.

I taught myself enough Fortran to perform the calculations my supervisor needed, and I actually enjoyed myself and got a sense of satisfaction. I was doing something useful in a field that interested me, and what I liked about programming was that I never had to do any arithmetic! I got to do all the thinking about how to set up the problem, and then the computer took care of the unpleasantness of calculation. It was an experience I remember with fondness. At the end of the summer, I even duplicated all the data cards on my own time, partly to have a record of all my hard labor, but also so I could continue to work on the data set in college.

My first year at Harvard, I took calculus, chemistry, and a primate behavior course with one of my idols, a primatologist whose books I owned and had read. Wipe out! I did not do well in calculus, despised chemistry (lecture course of 500, no visible relevance to things that interested me), and could not raise a glimmer of interest from the Great Primatologist, despite writing a paper for which I used the data on vervet monkeys. Oh well. I was interested in philosophy and sociology and got a far warmer welcome there, so I crafted a joint major and had a good time with my classes from then on.

Right after college in 1974 I worked at the Boston Children’s Museum as an intern. I was one of a crop of interns, all or almost all female, and I volunteered to work in the computer area because none of the other interns could imagine doing so. The museum had a PDP-11 and one guy, Bill Mayhew, who sat all day in a closet sized office. Bill put some special programs on the computer that only I could access, so that when I was working with kids I could use things other than the idiot-proof software that ran on the public access computers. I had a good time and the kids and adults, most of whom had never seen a computer before, were amazed.

When I decided I wanted to go to graduate school, I had been out of college for three years or so, so I thought I better brush up a bit before taking the Graduate Record Exam. My scores on the math section of the test were actually higher than my verbal scores (a first for me) because that’s where I’d put my studying effort, and because I finally had a reason—an extrinsic reason for sure, but a reason nonetheless—to study math: I really wanted to get into a good graduate school.

As a first year student in the Department of Urban Studies and Planning at MIT, I got the top grade in the introductory statistics class. But I still remember a discussion with the instructor, as we walked down the hallway after class. He told me he didn’t believe in this “math anxiety” stuff and thought it was all a lot of silliness. But I, his female star student, considered myself someone with math anxiety. To me, though, statistics was about looking beyond number to patterns, and using data to answer questions I was interested in. Finally I understood why I was studying what I was studying in the number-world, and I wanted to learn it because it was a tool I could do something with. The professor was not a good teacher for people without an intrinsic interest in and aptitude for math (in other words, for most of the students who go into a planning program) but he had two really good section teachers who became longtime friends.

I also took a macroeconomics course from Nobelist Robert Solow. Because he was teaching MIT undergraduates, about 80% of the class was equations. I zoned out. But after the math, he would always say in English what he had just said in mathematics. I got an A in the course, without ever writing an equation, because I had understood the material in a different but valid way.

So what’s the moral of the story? I was pretty good at philosophy, programming, and logical thinking. I hated “math.” I think I learned a lot of the  skills relevant to many computing fields by doing things other than math. I think I could have been a good programmer, though perhaps not a good computer scientist. Can we afford to risk losing people like me by insisting that math and computer science are joined at the hip?

 

 

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Feeling Your Way Into STEM

I have the pleasure of being a part of the SEAD (Science, Engineering, Art and Design) Network, a group funded by the National Science Foundation to promote communication across thinkers in the sciences and arts. “Innovation stemming from interdisciplinary creativity is a major contributor to the development of new, sustainable economies and harmonious, cooperating societies,” their statement reads in part. Joining science and engineering with art and design is brilliant, and a trend I hope will continue to grow. These disciplines are clearly related, but our culture separates them at birth. Under the banner of “STEM (Science, Technology, Engineering and Math) to STEAM (adding an ‘A’ for Arts),” many scholars, activists, and policymakers are beginning to recognize and reconstruct the connections we’ve allowed to atrophy. SEAD has solicited, peer reviewed, and posted online a collection of White Papers, which I recommend for browsing.

My own SEAD White Paper is called “Thinking With Things: Feeling Your Way Into STEM,” and is a more extended treatment of some themes that will be familiar to readers of this blog. I advocate for the unity of “STEAM” and give some examples of how we might get to STEM topics through embodied, engaged learning that recognizes and celebrates the emotions, aesthetics, and the whole person. The SEAD Network leaders insisted, rightly, that White Papers include specific recommendations, and my paper has several, including: Select and create things to think with; Create on-campus spaces that are ecosystems for learning; Create and support “maker spaces;” and Create “labs” in art institutions and “studios” in science centers.

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Hyperbolic Crochet Web Site now open!

I promise I’ll write about things other than hyperbolic crochet, although I do find these planes to be great things to think with, and terrific conversation starters. My new web site on Hyperbolic Crochet, designed by Lowell photographer and web designer Daniel Coury, is now up and running. I hope you’ll take a look.

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Feeling Your Way into Computing and Math

I’m still obsessed with the many, many layers of meaning that I see in crocheted hyperbolic planes. Math (and recovery from math anxiety), systems theory, gender, materials, comfort, tangibles, emotion…the list goes on. I gave a “Flash Talk” (20 slides in 5 minutes) entitled “Feeling Your Way into Computing and Math” at the National Center for Women in Information Technology’s (NCWIT) annual Summit in Chicago in May. I had a great time, and got lots of positive feedback afterward. I would really appreciate your comments and suggestions! What do YOU see?

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Richard Feynman and Froebel Kindergarten

In the same essay I discussed in my last post, “What is Science” by Richard Feynman, the great physicist describes his childhood introduction to science.

My father did it to me. When my mother was carrying me, it is reported–I am not directly aware of the conversation–my father said that “if it’s a boy, he’ll be a scientist.” How did he do it? He never told me I should be a scientist. He was not a scientist; he was a businessman, a sales manager of a uniform company, but he read about science and loved it.

Feynman’s father bought “a whole lot of rectangular floor tiles from someplace in Long Island City.” Father and son played with the tiles, and Mel taught his son to make patterns with the different colored tiles. In telling this story Feynman makes his assertion that “mathematics is looking for patterns.”

In a parenthetic note, Feynman continues:

The fact is that this education had some effect. We had a direct experimental test at the time I got to kindergarten. We had weaving in those days. They’ve taken it out; it’s too difficult for children. We used to weave colored paper through vertical strips to make patterns. The kindergarten teacher was so amazed that she sent a special letter home to report that this child was very unusual, because he seemed to be able to figure out ahead of time what pattern he was going to get, and made amazingly intricate patterns. So the tile game did do something to me.

I read this, but it wasn’t until I was waking up the following morning that I realized that ‘paper weaving’ rang a bell. I sprinted to my bookshelf and pulled down one of my favorite books, Inventing Kindergarten by Norman Brosterman. Brosterman describes the educational thought and innovations of Friedrich Froebel, the visionary German with a background in crystallography, who invented the original Kindergarten system. Active during the first half of the 19th century, at a time when children younger than seven rarely had a formal education, Froebel developed a series of physical materials and activities designed to expose young children to fundamental ideas of form and relationship. Best known today are the beautiful wooden blocks in geometric shapes, but there were many other materials as well, including the “peas work” with it’s small spheres and toothpick-like rods (an inspiration to the young Buckminster Fuller) and paper weaving.

The first half of Brosterman’s book is a fascinating and thoroughly-researched account of the development and spread of Kindergarten, first under the inspired and committed hand of Froebel, then under the leadership of his disciples, who established not only Kindergartens but also teacher training programs. But it’s the second half of Inventing Kindergarten that is truly revelatory: Brosterman makes an extraordinarily compelling case, in words and images, for the impact that the Kindergarten system had on art and design in the 20th century. Many of the top figures of art, architecture, and design attended or were exposed to, as Brosterman documents,  Kindergartens: from the pioneers of the Bauhaus, to architectural titan Frank Lloyd Wright, to the creator of “design science” and the geodesic dome Buckminster Fuller. In text and in remarkable images, which place the work of anonymous Kindergarten students and teachers side by side with pictures of the strikingly similar work of leaders of Modernism, Brosterman creates a tour-de-force argument for the impact of Froebel’s system.

By the time Feynman was born in 1918, Kindergarten was very widely established not only in Europe but also in the United States. His attendance at Kindergarten, and his instruction in paper weaving, are directly attributable to the remarkable innovations of the man who was active a century earlier. Brosterman’s focus is on innovators in the arts; can a similar argument be made about 20th century scientists who are known to have gone to Kindergarten? Suggestive evidence is probably all we will ever have, but I would argue that in Feyman’s case the suggestive evidence is strong. And there is a crucial piece of evidence whose significance is invisible to biographer James Gleick as well as to Feynman himself. Early in his book on Feynman, Genius: The Life and Science of Richard Feynman, Gleick mentions in passing that before her marriage, Feynman’s mother Lucille trained as a Kindergarten teacher at Felix Adler‘s Ethical Culture School in New York. Eureka!

Happily for me, Norman Brosterman is easy to find on the Web. I sent him an email asking him his thoughts about the influence of Kindergarten on scientists. His gracious reply included the following:

I always assumed modern physics was influenced by Froebel but never had proof…If Feynman’s mother was a trained kindergartner you can be 100% certain she used the gifts, the system, and the philosophy with him at home when he was a boy. Remnants of the original system were still widespread in public schools but would not have been as “pure” as what he got from his mother.

Two things strike me immediately: The first is the complete absence of Lucille and her influence from Feynman’s account. Her only appearance in Feynman’s 1966 talk is as the wife who says to her husband, “Mel, please let the poor child put a blue tile if he wants to” (instead of following the rigorous patterns Feynman’s father was determined to teach.) I’m sure Mel and his aspirations had a profound impact on his son, but Feynman’s gift at paper weaving that so amazed his Kindergarten teacher surely come as much from his mother’s influence. Here again, as in the previous post, we witness the invisibility of women’s intelligence and women’s minds to both the young and the mature Richard Feynman.

The second striking thing I have already foreshadowed. Was the remarkable, visual, unorthodox Feynman’s way of seeing the world profoundly influenced by the Kindergarten system as he encountered it in his own home? Feynman was the first-born, and a boy for whom his parents clearly had ambition. It’s hard to imagine that he was not decisively shaped by a way of thinking and doing that had attracted his mother, even before his birth.

It is worth quoting at length from Inventing Kindergarten (but you should also read the entire book):

In effect, the early kindergartners created an enormous international program designed specifically to alter the mental habits of the general populace, and in their capable hands nineteenth-century children from Austria to Australia learned a new visual language. While focusing on kindergarten’s many educational and social benefits, these pioneers overlooked a potentially radical outcome of their efforts that is obvious in retrospect: kindergarten taught abstraction. By explicitly equating ideas, symbols, and things, it encouraged abstract thinking, and, in its repetitive use of geometric forms as the building blocks of all design, it taught children a new and highly disciplined way of making art. Like spokes on a wheel–separate at the rim, but connected at the hub–every lesson of the original kindergarten led from diverse vantage points to a central truth. Simple linear thinking was to be superseded by a more sophisticated, genealogical approach to knowledge that valued relationships as much as answers. The grid of the kindergarten table was symbolic of a type of inquiry that drew from multiple sources, cut across and connected seemingly unrelated data, and had the potential to result in more than one ‘correct’ conclusion. By emphasizing abstraction, kindergarten encouraged the value of unconventional reasoning. (p. 106)

Although Brosterman’s emphasis is on the Froebel system’s impact on the arts, it is no reach to think that Froebel, a trained scientist, would have been drawing at least as much on the fundamentals of science and nature as he developed his system. Was Feynman, the unconventional and deeply visual thinker, inventor of the abstractions known as Feynman diagrams (in addition to many other important contributions) influenced in essential ways not only by his father’s doting tutelage but also by the Froebel system in which his mother was steeped? The shoe fits; let’s walk a mile in it.

Feynman diagram (source http://en.wikipedia.org/wiki/File:Feynman-diagram-ee-scattering.png#file; accessed 18 March 2012)

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Richard Feynman and Women’s Invisible Skills

Like so many others, I am a great admirer of Richard Feynman, the great 20th century physicist, Nobel Laureate, and overall “curious character.” Known for his brilliance, he was also known for being an extremely visual thinker. This is probably why I like him: for me, he is an exemplar of a great mind that got to very deep ideas by an unusual and often-overlooked route.

A recent comment by my friend Herb Lin sent me to one of Feynman’s essays, “What is Science,” in the collection The Pleasure of Finding Things Out. The essay is the text of a speech Feynman gave in 1966 to the National Science Teachers’ Association (shout out to NSTA, whose e-newsletters often point me to useful resources). In it, Feynman tells a couple of tales from his early years, as a way of describing how he learned “what science is like.” After describing his very early education (more on that great story in the next post) he continues:

When I was at Cornell, I was rather fascinated by the student body, which seems to me was a dilute mixture of some sensible people in a big mass of dumb people studying home economics, etc., including lots of girls. I used to sit in the cafeteria with the students and eat and try to overhear their conversations and see if there was one intelligent word coming out. You can imagine my surprise when I discovered a tremendous thing, it seemed to me. I listened to a conversation between two girls, and one was explaining that if you want to make a straight line, you see, you go over a certain number to the right for each row you go up, that is, if you go over each time the same amount when you go up a row, you make a straight line. A deep principle of analytic geometry! It went on. I was rather amazed. I didn’t realize the female mind was capable of understanding analytic geometry. 

She went on and said, “Suppose you have another line coming in from the other side and you want to figure out where they are going to intersect.” Suppose on one line you go over two to the right for every one you go up, and the other line goes over three to the right for every one that it goes up, and they start twenty steps apart, etc.–I was flabbergasted. She figured out where the intersection was! It turned out that one girl was explaining to the other how to knit argyle socks. (Pgs. 175-176)

I have quoted at length because this section is such a rich trove of things to think with. The first–and it was Feynman’s reason for telling the story–is as an illustration of Feynman’s repeated assertion that Mathematics is Pattern. I find this a wonderful and generative idea. I love pattern, particularly patterns made by ancient and indigenous groups, but I have always associated this love of mine with the arts, especially fiber arts. It never occurred to me that instead of turning right from pattern and getting to the arts, I could turn left and be in the realm of mathematics. I am slowly learning this, first from the wonderful work of Ron Eglash on ethnomathematics, and now from Feynman! It’s like stepping through the wardrobe into Narnia. Why didn’t anyone tell me! I feel cheated by my long, tedious, and painful mathematics schooling.

I hope you are still with me, and have not thrown your iPad across the room in disgust. The second point, of course, is the extraordinary sexism of the passage.  Although I find it reprehensible (and he digs himself in even deeper in the paragraph that follows these; I will leave it to you to read the original essay) I will say in his defense that he was a man of his time. I was a freshman in high school when Feynman gave this talk, and although I know I have repressed a great deal of what I heard, this was a very common attitude toward women. Not everyone was as outspoken as Feynman, but the fact that he so clearly articulated his position, and that he was open to revising his opinion of women based on this experience, puts him ahead of many men in mid-20th century America. A plea to young people reading this: do not forget how far we have come! Do not take these gains for granted–anyone reading the news these days must know that women’s rights are under attack even in 2012, and that it would not be difficult to lose hard won gains. When I hear, “I am not a feminist…” I want to ask “What aspects of patriarchy are you especially fond of, then?”

Finally, for me, this story from Feynman reveals, in the words of a great thinker, the deep connection between mathematics and the fiber arts–knitting, weaving, and so forth. Most people, I think, believe that “women’s hobbies” and mathematics are opposite poles: concrete vs. abstract, female vs. male, informal vs. formal, casual vs. professional. Why do I love crocheted hyperbolic planes and, now, argyle socks? They are emblematic of the fact that it just ain’t so!

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Crocheting Hyperbolic Planes

A hyperbolic plane is a mathematical object. I first made its acquaintance through the work of Daina Taimina, author of Crocheting Adventures with Hyperbolic Planes. The photograph in the previous post is of a crocheted hyperbolic plane I made this past summer.

Taimina’s book won the Diagram Prize for Oddest Book Title of the Year for 2009 (the prize was originally conceived as a way to avoid boredom at the annual Frankfurt Book Fair, and has been awarded annually for more than 30 years.) In her blog, Taimina writes, “Best of all, in The Booksellers announcement I liked what Mr. Bent said about my book:

I think what won it for Crocheting Adventures with Hyperbolic Planes is that, very simply, the title is completely bonkers. On the one hand you have the typically feminine, gentle and wooly world of needlework and on the other, the exciting but incredibly un-wooly world of hyperbolic geometry and negative curvature. In Crocheting Adventures with Hyperbolic Planes the two worlds collide—in a captivating and quite breathtaking way.”

Mr. Bent’s reaction to the book was mine as well. I was drawn to the paradox, the incongruity. I had no interest in math, and really no interest in crochet either, but the idea of the book stayed with me. I don’t recall how I first heard of the book, but I think it was before the award was announced. Perhaps my brother, Nat Kuhn, who has a Ph.D. in Mathematics (his dissertation adviser William Thurston–winner of the Fields medal–wrote the introduction) told me about it. I know I bought a copy in the Winter of 2010 for my students in the Olin College of Engineering extracurricular “Fiber Arts for Engineers.” Then in September 2010, Taimina spoke at the Common Cod Fiber Guild at MIT (fiber guild? a bunch of knitters at MIT? Absolutely!) Of course I had to go.

At her talk, Taimina answered the question you are probably asking too: what IS a hyperbolic plane? Her simple and accessible answer is more or less as follows: an orange is a surface of constant positive curvature; a banana has both positive and negative curvature; and a hyperbolic plane has constant negative curvature. Imagine the ruffled edge on a leaf of kale, my personal favorite hyperbolic vegetable. Or the ruffles found on other natural things, such as some seaweed or the inhabitants of the coral reef (the inspiration for the Crocheted Coral Reef, an effort sparked by Taimina’s work.) These real life objects, of course, are not perfect models of the mathematical ideal–but they bring us close, and are lovely inspirations to mathematical inquiry.

In spite of all this inspiration, it was not until this past February that I finally decided to try my hand at hyperbolic crochet. I bought some cheap fuzzy synthetic yarn at a local big box store and set to work. The directions in the book were fairly minimal, and minimal is all you need–if you already know something about crochet. I struggled with my yarn, and soldiered on, but as I did my misgivings grew. What was I doing wrong? What was I doing right? I had no idea. I didn’t even have much idea about how to get help. At last I came up with the idea to go to drop-in night at my local independent yarn store, and I did get a little lesson on how to crochet. Finally, the penny dropped when I went to an hour-long workshop on “crocheting a hyperbolic dishcloth” (yes, you read that right) at Common Cod’s spring Fiber Camp. Thanks to Rachel, my teacher, who helped me see the light. My skills are still extremely rudimentary, but they are good enough to allow me to churn out the hyperbolic planes at the rate of one every week or two since March! Thank you, Daina!

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