my unpromising (?) beginnings in mathematics

Unlike a lot of mathy people, I did not discover my love of mathematics or have any desire to explore it deeply until I got to college.

Indeed, my mathematical beginnings were quite unpromising.

I attended a Catholic elementary school with very dedicated and hardworking teachers who labored under conditions that would be considered unthinkable today.   In those baby boom days, the school was bursting at the seams.   Every K-8 classroom in my school had 56 students (7 rows with 8 students in each row).  There were no teacher's aides, just a single nun presiding over each classroom.  A volunteer "lunch mother" relieved her for about 15 minutes in the middle of the day, but otherwise she was on duty during the whole school day, including supervising recess.  The nuns were extremely dedicated and hard working, but the classroom conditions and hierarchical church authority structure meant that there was an understandably heavy emphasis on rote memorization and recitation, practicing penmanship and neatly legible work.   We blindly accepted the hierarchical authority of the teacher, who in turn accepted the hierachical authority of the archbishop, whose end of year exams dictated our curriculum.  I remember laboriously extracting square roots by applying an algorithm analogous to long division without understanding what the heck I was doing or even having any notion that it was possible or desirable to understand what I was doing.  Certainly, it never occurred to me to ask "Why?" questions or to explore alternative ways to solve a problem.  The idea of asking such a question was akin to heresy.  Once we hit fourth or fifth grade, I remember that we were required to do all our schoolwork--including math--using fountain pens!

But--at home--my brothers and sisters and I had parents who loved inquiry and encouraged us to argue with them, to question everything, to use the public library and a variety of freely available resources (parks with nature trails and nature centers, informal community center classes, museums, monuments, etc.) to learn on our own outside of school.  My dad was a librarian, curious about everything and he liked nothing better than helping people find the resources to answer their questions.  I was fortunate to grow up in Washington DC in a polyglot neighborhood with many immigrants who exposed me to a wide varieties of cultures, languages, and religions.

My mom did not have a lot of formal education (just one year at a now-defunct Catholic women's college run by the same order that taught in my school) but she was a brilliant practical problem-solver, an important asset in a household whose income did not exceed the poverty level until I was a junior in high school.  I was in awe of her spatial abilities and resourcefulness, which enabled her--for example--to wrap leftovers with an absolute minimum of waxed paper.  She apparently inherited those spatial abilities from her father.  Despite having only an 8th grade education, he had been superintendent of buildings and grounds at the US Naval Observatory and oversaw the design, repair, and construction of domes and other spaces there.

I was in awe of those spatial abilities, but convinced that it was hopeless for me to aspire to them.  I was clumsy and uncoordinated.  Before moving to the Catholic parish school in first grade, I had attended a public school kindergarten where the teachers had wanted to retain me in kindergarten for another year because I was so uncoordinated, and in particular, because I could not "skip sideways."  Left-handed, with crossed eyes, and a condition called "right-left agnosia" in which I had difficulty telling my right from my left, and born on the very last day of the year to be eligible for school entry in DC back in those days--I can certainly understand why my teachers had wanted to retain me in kindergarten for another year.

But I muddled through, meticulously following directions in a highly regimented Catholic school classroom.  I did well enough on the archbishop's end of year tests (in all subjects) to make everyone satisfied that the decision to advance me had been fine, but I certainly did not take any particular interest in math.  There were many other subjects in which I chose to read voraciously and explore, primarily in the humanities and languages, but the idea of browsing through a recreational math book or investigating a math problem not assigned for homework was not one of them.

I was the oldest of five children.  Moreover, I was one of the older children on my city block, which was filled with tightly packed rowhouses teeming with younger children and parents in need of an occasional babysitter.  So I was quickly pressed into service.  At the now unthinkably tender age of 8, I began babysitting for my own younger siblings.  By the age of 9, I was babysitting for neighbors as well.

My siblings and I had free-range urban childhoods and spent many hours each day roaming around the neighborhood or even the city at large on foot starting at an early age.  Our parents were both native Washingtonians who had themselves grown up with the freedom to roam the city during their free time.  My mother told us that a favorite pastime of her childhood was to go with friends and hang out in hotel lobbies, trying to snag autographs of movie stars passing through while simultaneously evading the attention of house detectives.  My dad had frequently hitchhiked to school in order to save on busfare.  Although they didn't encourage us to hitchhike or hang out in hotel lobbies, they largely trusted our judgment (or "our guardian angels") and gave us a lot of freedom.

I remember being sent unaccompanied on errands to the drugstore half a mile away at the age of 5.   By the time I was in third grade, I was considered sufficiently responsible to be the one in charge of escorting my younger siblings to school almost a mile away.  We had the freedom to go parks, playgrounds, the library, even the National Zoo a couple miles away.  The only explicit rule I recall was that by the end of the day, as dinner time drew near, we had to be within earshot of the handbell my mother rang to call us in to eat.  When our country cousins came to visit, they were always astonished by the freedom we had.  I recall one brother treating a cousin to a guided tour of the city's storm sewers, much to the consternation of his mother (and the embarrassment of my mother, when she heard about it afterwards.)   I wasn't aware of it at the time, but we probably developed a considerable sense of spatial problem solving and geometry from all this autonomous ambulation, since we regularly created mental maps in our heads as we figured out how to navigate Washington's famously geometric layout of streets arrayed in a four-quadrant rectangular coordinate grid indexed numerically in one direction and indexed alphabetically and by number of syllables in the other direction and intermittently intersected by avenues radiating like spokes from traffic circles.  Growing up in Washington DC meant growing up in a geometric wonderland, though I was not fully conscious  of it at the time.  I fondly recall my brother's Cub Scout den on our porch carving Ivory soap into models of the monuments and museums and public buildings downtown, and struck by seeing them all laid out on a green felt-covered board.  I also remember their den creating a 3-d topographic map of DC using chicken-wire and paper mache to represent the information encoded in 2-d paper US Geological Survey maps.  Although I myself was all-thumbs and too uncoordinated to help, just contemplating those models as the scouts worked on them gave me more immersion into geometry.

On evenings and weekends, my dad liked to take us on what he called "expos" (short for "expotitions," Winnie the Pooh's terminology for expeditions.)    These made us very aware of the third dimension, as we lived within easy walking distance of DC's "highest hill," and I was fascinated by the views and hearing the law which prohibited any building in the city to reach a higher altitude than the Washington Monument, which was on low-lying ground near the river.  Of course, we had fun walking up that monument too.  Before we were born, he and his library-school buddies used to enjoy hiking up the nearby Blue Ridge Mountains together.  After we came along, he would frequently take us up those mountains, partly to share the joys, vistas, and challenges with us, and partly to provide my mother with some weekend respite from the burden of caring for so many young children.

My dad was a chess player--actually not a very good chess player, but a very enthusiastic and evangelistic one.  He would happily set up a chessboard on our front steps and take on all comers, and he taught anyone in the neighborhood who wanted to learn.  Later, after he joined George Mason University as the assistant director of their library, he became the founding faculty advisor to the first George Mason chess club, and he accompanied students to play inmates in the DC prisons.  He also made a successful bid for George Mason to host the US Chess Open in 1976, and he served as tournament director that year.  This was a very big deal and quite a coup for George Mason as the nation was still at the height of the Bobby Fischer-induced chess mania and George Mason was a very small, new, and unknown small commuter college at the time.    (Twenty five years later, in 2001, George Mason was far better known, thanks to acquiring several Nobel Laureates, and it hosted the 2001 International Math Olympiad, though GMU still probably did not reach most people's radar screens until it made the basketball "final four" in 2006.)

I learned the rules of chess from my dad, but had no particular enthusiasm for playing it myself.  But what strikes me now is how my role as a founding advisor to Albany Area Math Circle is somewhat parallel to his as founding advisor to the GMU Chess Club.    I am--in my own way--a community builder and visionary, just as he was.

My parents were very politically active--and strongly opinionated.  As native DC residents, they could not vote in Presidential elections until after the 23rd Amendment passed in 1961, but my parents loved to invite their incredibly diverse group of friends over for coffee and conversations, which often ran late into the night.  My dad loved to argue (in good natured and respectful ways) with friends from all over the political spectrum.   He was at the right end of the political spectrum (once characterizing himself as "slightly to the right of Louis XIV") but guests ranging all the way to Marxists and beyond were welcome and apparently greatly enjoyed themselves.  My siblings and I sat on the steps near the living room and listened in awe to the heated and lively discussions going on in our living room.   Every now and then we would be unable to restrain ourselves from jumping into the discussion to contribute a point.  As I recall, we always wound up arguing against my dad's side--yet he was fondly indulgent of our occasional interruptions, which greatly amused our adult company.

All this might seem to have nothing to do with math--but I now realize that growing up in a home full of friendly good-natured arguments was an important formative part of my education.  I am struck by the parallels to an anecdote from Sarah Flanery's wonderful book, In Code, where she describes growing up in a home with a mathematician father who loved to argue with his colleagues at the blackboard in their kitchen.  As they pointed out the flaws in each other's reasoning, it was eye-opening for her to realize that grownup professionals she respected and admired were not infallible beings, incapable of making mistakes.

Okay, I am rambling on way too long here.  I was definitely not a "math person" in high school.  I was generally a good student, but math was the school subject that probably interested me least.  I found it pretty tedious.  Other subjects intrigued me and drew me into outside independent explorations and reading, but not math.  It never occurred to me that I would enjoy doing math in a recreational way.

Midway through my high school career (which involved a fair amount of teenage rebellion and turmoil I won't go into here), I convinced my parents to allow me to transfer from the small Catholic girls high school I had been attending to the large public high school in my neighborhood.  My boyfriend from down the block attended that school and his glowing description of the array of advanced classes offered there convinced me I was missing something.  They reluctantly agreed.

The differences between the two schools were eye-opening.  The Catholic school had been extremely disciplined as we had worried about things like demerits for not having our saddle shoes polished properly or skirts hiked up in a way that might reveal we had kneecaps or whether subtle amounts of makeup might be noticed.  In public school, students wore jeans and skirts of all different lengths, and there was a chapter of the SDS and a feminist consciousness-raising group and the entire school regularly walked out and sat in the stadium to protest the war.  There were weapons, drugs, and other contraband confiscated from  lockers, and a guidance counselor was stabbed at a dance he was chaperoning.

My Catholic school had been in dire straits.  The very modest tuition ($200 per year, which is equivalent to about $1,200 today) had been sufficient for times when the school had been staffed by nuns living under a vow of poverty, but women were leaving the convents in large numbers in the late 1960s.   Attracting qualified and experienced lay teachers as replacements was a huge challenge on that budget.  The quality of the lay teachers who taught us left a great deal to be desired since the school was unable to pay competitive salaries.  Many of our lay teachers were likely teaching motivated more by a desire to be exempt from the draft and Viet Nam rather than out of a sense that being a teacher was their true calling in life. Most notably, our French teacher spoke the language with an egregiously awful West Virginia accent. ("Ray-gayr-day lay gayr-sown" for "Regardez le garçon" still rings in my ears.)   A senior who had spent time in France took pity on us hapless freshmen stuck in her class and she organized underground afterschool classes to remedy the awful French pronunciations we were being mistaught.  

My teachers at the new public school, however, were dedicated professionals, all of them women with a decade or more of experience.   They had high academic standards for themselves--and for us.  I was intimidated--and definitely behind.  Although I was officially a junior, I was surrounded by sophomores in most of my classes, including math.

A year later, my family moved to the suburbs and my brother and sister knew they would be joining me in the public schools.  They did not want to be a year behind their classmates.  I was resigned to being a year behind the seniors, but decided to help my brother and sister--because I had always really enjoyed teaching anyone who wanted to be taught.

In retrospect, my decision to spend the summer before my senior year helping my younger sister learn algebra I was possibly the best thing I could have ever done, one of the most transformative learning experiences of my life, far better than trying to somehow catch up with my own cohort by teaching myself trigonometry.  Because she felt totally free to question everything I tried to teach her, I was forced to think deeply about the rationale for all the manipulations and algorithms I had been mindlessly applying by rote.