## 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?

Filed under computing, gender, learning, mathematics

## Update: Inventing Kindergarten will be back in print

In my last post I wrote about the book Inventing Kindergarten. I’m happy to say that, after a successful Kickstarter, author Norman Brosterman is working on putting the book back in print. I hope it will be available for purchase sometime in 2014, although Brosterman has not announced a date. Meanwhile, if you are in the Boston area between now and April 19, 2014, you can see some of the Kindergarten materials from Brosterman’s collection on display at Northeastern University. The show is called Learning By Design. Nice title!

Filed under early education

## Urgent: An Outstanding Book Needs Our Support

In my post “Richard Feynman and Froebel Kindergarten” I praised Norman Brosterman’s book Inventing Kindergarten. I’ve been buying up used copies ever since it went out of print (because everyone who sees my copy wants to own one of their own). Now there is a chance to republish the book and get your own copy by supporting Brosterman’s Kickstarter Campaign. I hope you will make a pledge! But do it now–the campaign ends on July 12.

A beautiful coffee-table style book with plentiful illustrations of early learning materials, Inventing Kindergarten tells the story of the Kindergarten system developed by Friedrich Froebel and its spread throughout Europe and the United States. Later chapters make a dramatic and compelling case for the influence of the Kindergarten materials and Kindergarten system on the art and architecture of the 20th century. Architect Frank Lloyd Wright and Nobel physicist Richard Feynman, among many others, were the sons of teachers trained in the Froebel system. Many, many big names of the past century, including Buckminster Fuller, Josef Albers, Paul Klee, Piet Mondrian, and a host of other 20th century modern painters were educated using the Froebel blocks, paper weaving, and other materials–and their work, beautifully reproduced in this book, shows the influence clearly.

The book first appeared in 1997 and became a New York Times Notable Book, was called “Revelatory,” by The New Yorker, and won an American Institute of Architects award as one of the best books of the year. It has been out of print for several years, and Brosterman now has created the digital files necessary to republish it. However, money is needed to meet the minimum costs of an initial print run.

Brosterman’s Kickstarter campaign can be found here: http://www.kickstarter.com/projects/1335652536/inventing-kindergarten

For a \$50 pledge you can have a copy of the book, plus some nice supplementary note cards. Since the book initially retailed for around \$50 (and I’ve been snapping up used copies on the Internet for about that price) you can do yourself a favor and help the rest of us at the same time. If Brosterman’s goal is met the book should be ready in time for holiday gift giving, too!

I hope you will support the reissue of one of my favorite books. And finally, please share this message with others you think might be interested in this wonderful work.

Filed under arts, creativity, early education, embodied cognition, learning

## Diabolical Classroom Seating

In January, I saw the chairs in my college classroom featured on the front page of The New York Times. This was not your typical story from the Times’ design section, touting the virtues of the new and flashy. Instead, it was a profile of the perennial “super stacker,” a chair whose use is widespread because of its durability and low cost. When I saw the photo, I knew immediately that these are the chairs I see my students fitting themselves into every day. I have dubbed my classroom “the sensory deprivation chamber” for the absence of engaging materials; this article added new fuel to my fire by describing the discomfort of pupils who spend their days in these unbending marvels. Leading environmental educator David Orr hit the nail on the head when he said in the article, “The chair…originated in the industrial ordering of education. It is maintained by profit-seeking school suppliers and unimaginative administrators who see no other possible arrangement of the body, or bodies, or any possible downside to the lower back from six hours of enforced sitting.”

The more I discover about learning and cognition, the clearer it is to me that these and other human activities are enacted. The passive butts-in-seats model of instruction is anathema to the exploring, interactive body optimized for action and inquiry. Of course it’s possible to learn when sitting in this Procrustean chair, but other arrangements would be better for learners, who currently bear the cost of the “cost-saving” super stacker.

I dashed off a letter to the Times in response to the article, and happily they saw fit to print it. Here is the text of my letter as it appeared on January 9, 2013.

The indestructible classroom chair (“Ergonomic Seats? Most Pupils Squirm in a Classroom Classic,” front page, Jan. 5) is a great example of our “penny wise, pound foolish” approach to learning environments. The student who becomes fidgety and disruptive spending six hours a day sitting in a hard chair may end up in special education or on the streets, simply because we ignore the essential role of the body, not just the brain, in learning.

We know from research that learning and cognition involve the entire body, the senses and the emotions, not just the contents of the cranium. We could make great strides in learning if we were more attuned to how a classroom and its contents can support active, engaged, embodied learning.

SARAH KUHN
Lowell, Mass., Jan. 5, 2013

The writer is a professor of psychology, the University of Massachusetts Lowell.

1 Comment

Filed under embodied cognition, learning

## 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.

Filed under arts, computing, creativity, embodied cognition, learning, mathematics, science

## Classroom Improvisation: a favorite moment from my teaching

I used to teach a research methods class that appealed particularly to students writing Masters and Doctoral theses. My students came from several different disciplines, and from all over the world. What they had in common is that they were all facing the terrifying prospect of having to do an extended piece of original research, and then to write what would be by far the longest research paper—a book, really—that they had ever produced. They were wonderful students to teach, because they were dedicated to making the world a better place, and because fear concentrates the mind wonderfully and made them very motivated to learn.

The class was a three hour seminar, and because three hours of talking can be deadly, I always had the students do something that took them out of their seats and got them using a different part of their brains. On this day, I had hung large pieces of paper around the wall. Each page had a different question or caption, and I asked the students to move around the room and write something on each sheet, in response to the question posed. One page said “The thing that scares me most about doing my research is…,” another said “What I most want to accomplish with my thesis is…” One said “A picture of me as a researcher.” This was the one I most enjoy, because the social sciences, which I teach, are so often entirely verbal and text driven. One of my favorite things to do with students in my classes is to use pictures, Lego, hands on exercises, and other things to get them out of a purely text-and-talking mode.

After the students were finished, I walked around the room, commenting on what they had written on the flipcharts. My last stop was the chart of self-portraits. In the bottom left hand corner was a little cartoon of a pig with wings, surrounded by the legend “when pigs fly” and “a snowball’s chance.” I knew who had drawn this one. I’ll call him Joseph. He was an engineer from a developing nation in Africa who was here in Massachusetts to get a doctorate, and his passion was the recycling and redesign of electronic products to prevent environmental harm. He was new to the US and homesick for his family, but determined to do well. He was struggling more than his fellow students, who had all been in their respective degree programs longer than he had. I had already spent considerable time with Joseph outside of class, at his request, because he was so concerned about whether he could do well in the course and had so little confidence in his ability to understand.

All the drawings showed students who were frazzled, at sea, overwhelmed, but Joseph’s was ostensibly the most hopeless. Although I didn’t look at Joseph when I pointed to his drawing, I could feel him tense as I began to speak. He had not signed his drawing, and I didn’t say whose drawing it was, but watched him out of the corner of my eye. I said “This student is afraid of not being able to do what he has set out to do. He thinks he will succeed only when pigs fly. Well, I have news for him. Pigs don’t fly, and they will never fly.” I could feel, rather than see, Joseph sagging into his chair as I spoke. “But the good news is, that although pigs cannot fly, there are other things that they can do very well. Pigs are great at hunting for truffles, which they detect through careful searching and dig out of the ground, like precious gems. Research is a lot like hunting for truffles—pursuing and finding gems of insight in a vast field of dirt. So for all of you, if you are a pig, don’t try to fly. It won’t work. But do what you do best—do what you are good at, and what you are passionate about, and you should be a good researcher.”

Filed under creativity, learning

## 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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## Thinking With Your Hands

In January of 2011, I gave a short presentation at “Ignite Craft,” an evening of 5 minute talks organized by the Common Cod Fiber Guild, which meets at MIT. The talk is a whirlwind tour of themes I am continuing to think about; it was my first “public airing” of these ideas, and certainly my first appearance on YouTube!

Filed under arts, embodied cognition, learning

## 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)