Monday, June 20, 2016

Thinking as pruning, shaking together, weighing, imagining

I am wearing a wonderful t-shirt today, created for me by Gili Rusak, an awesome alumna in our math circle. I am wearing a wonderful t-shirt today, created by a math circle alumna Gili Rusak for me last year.

Descartes famously wrote "I think, therefore I am." Of course, he wrote it in French (his native language) and later rewrote it in Latin (the language of scholarship at the time), rather than in English. And Latin has two distinct verbs, "cogito" and "puto" that both translate into our single English phrase "I think." The two Latin words connote very different ideas of what it means to think.

The etymology of the word cogito literally means "I shake together."  The root word is related to our English word agitate.  So we could illustrate this with a Descartes shaking ideas together with a cocktail shaker.

Although Descartes actually chose cogito, my favorite image of thinking is actually associated with the alternative word puto, because it derives from a root word meaning "I prune."  Thinking involves a lot of pruning, of cutting away clutter.  I love the image of Descartes with pruning shears.

But before he wrote in Latin, what he actually wrote in French was "Je pense, donc je suis."  The etymology of the French verb penser has nothing to do with either shaking or pruning.  It derives from a word that means to weigh or balance.  So we could illustrate  that with Descartes holding a balance scale with competing ideas piled up on either side.

He didn't write in English, of course, but it is also interesting to consider the etymology of the English verb think.  From the Online Etymology dictionary, we have:

think (v.) Old English þencan "imagine, conceive in the mind; consider, meditate, remember; intend, wish, desire" (past tense þohte, past participle geþoht), probably originally "cause to appear to oneself," from Proto-Germanic *thankjan (source also of Old Frisian thinka, Old Saxon thenkian, Old High German denchen, German denken, Old Norse þekkja, Gothic þagkjan).
Our ability to think is an amazing and multifaceted superpower to reflect upon.  We need to shake ideas together, to prune and simplify, to weigh ideas against one another, and to imagine and conceive.






"We are made of dreams and bones"



This song always inspires me--and I am much in need of inspiration at the moment.

Inch by inch, row by row
Gonna make this garden grow
...
Picking weeds, pulling stones
We are made of dreams and bones

Tuesday, July 14, 2015

Discrete Math with Ducks is awesome!

"The guiding pedagogical principle behind the style and tone of this text is pretty silly.  I mean that literally:  I believe that students are more likely to absorb mathematics that is presented in a goofy way.  Bizarre situations help students separate the abstraction of the mathematics from the presentation of a problem and thus give students practice in recognizing the mathematical essence of the problems they find in other contexts.  Students who are enjoying the weirdness of problem presentations are also focusing on the mathematics.  It's easier to remember a zany concept setup than to recall a straightforward statement.  And there's no reason to be serious when there's an opportunity to have fun." --from the preface to Discrete Math with Ducks by saramarie belcastro

This book may win the prize for the highest variance reviews on Amazon, but I love it.

Saturday, May 23, 2015

words from Persi Diaconis

This is a delightful article about Persi Diaconis,who is now working on the mathematics of "smooshing" cards. https://www.quantamagazine.org/20150414-for-persi-diaconis-next-magic-trick/ But the most memorable take-away for me is the following:
When Diaconis returned to college after a decade as a professional magician, his first three grades in advanced calculus were C, C and D. “I didn’t know you were supposed to study,” he said. His teacher told him that he should write down the proofs and practice them as if they were French verbs. “I said, ‘Oh, you’re allowed to do that?’” Diaconis said. “I thought you were just supposed to see it.”
Write down the proofs and practice them as if they were French verbs! I love that analogy. And it makes so much sense. Math is a language and a certain amount of rote practice is helpful for those of us who are learning it in later life. (Persi Diaconis had a somewhat unorthodox and nontraditional educational trajectory. I guess mine may be more unorthodox still.) The following paragraphs from the article were also fascinating:
When it came to smooshing, instead of just trying to “see it,” Diaconis devoured the literature on fluid mixing. “When we started talking about the connections between cards and fluid mixing, he read the whole 200 pages of my Ph.D. thesis,” said Emmanuelle Gouillart, a researcher who studies glass melting at Saint-Gobain, a glass and construction materials company founded in Paris in 1665. “I was really impressed.”

While Diaconis grew more conversant in fluid mechanics, Gouillart benefited from his unique insight into card shuffling. “It turned out that we were studying very similar systems, but with different descriptions and different tools,” Gouillart said. The collaboration led her to develop a better way to measure correlations between neighboring particles in the fluids she studies.

Diaconis, meanwhile, has developed a mathematical model for what he calls “the sound of one hand smooshing.” In his model, the cards are represented by points scattered in a square, and the “hand” is a small disk that moves around the square while rotating the points under it by random angles. (It would be easy, Diaconis noted, to extend this to a two-handed smooshing model, simply by adding a second disk.)

Diaconis has been able to show — not just for a 52-card deck but for any number of points — that if you run this smooshing model forever, the arrangement of points will eventually become random. This might seem obvious, but some shuffling methods fail to randomize a deck no matter how long you shuffle, and Diaconis worried that smooshing might be one of them. After all, he reasoned, some cards might get stuck at the edges of the table, in much the same way that, when you mix cake batter, a little flour inevitably gets stranded at the edges of the bowl and never mixes in. But by drawing on 50 years of mathematics on the behavior of random flows, Diaconis proved that if you smoosh long enough, even cards at the edge will get mixed in.

His theoretical result says that the smooshing model will eventually mix the cards, but doesn’t say how long it will take. The model does provide a framework for relating the size of the deck to the amount of mixing time needed, but pinning down this relationship precisely requires ideas from a mathematical field still in its infancy, called the quantitative theory of differential equations. “Most studies of differential equations focus on what happens if you run the equation for a long time,” Diaconis said. “People are just now starting to study how the equation behaves if you run it for, say, a tenth of a second. So I have some careful work to do.”

Diaconis is optimistic that the work will lead him not just to an answer to the smooshing question, but to deeper discoveries. “The other shuffles have led to very rich mathematical consequences, and maybe this one will too,” he said.

Thursday, May 14, 2015

Avoid Einstellung (getting stuck)

MathJax TeX Test Page On the recommendation of another older learner of mathematics, whom I discovered as a guest poster on the blog of Miles Kimball, a former student of mine, I have begun reading a quite wonderful book.


Despite the subtitle, the book is actually designed for "both math experts and mathphobes." I myself am somewhere in the middle of that spectrum. I am not a "math expert," since I have a lot to learn (and, in some cases relearn) in math, and I am not your typical "mathphobe" either. I love math. And I certainly never "flunked algebra". But at the same time, there are times I find learning new things scary--so I do understand what it is like to be a mathphobe.

And, in fact, I found myself feeling pretty phobic in my final exam in my graduate class this past Monday. I got impossibly bogged down on what was probably intended to be a simple problem #1 on the exam.

What happened to me on Monday was a classic example of the phenomenon Barbara Oakley refers to by the wonderful German word "Einstellung", which means getting stuck. As she writes:
[t]he Einstellung (pronounced EYE-nshtellung) effect. In this phenomenon, an idea you already have in mind, or your simple initial thought, prevents a better idea or solution from being found.
I was so stuck on a particular approach to the problem (based on a misconception) that I couldn't get the perspective to see I was just wrong and had to abandon it for a different approach.

Part a of problem 1 was actually kind of fun, calculating the Fisher information for a random sample drawn from a normal distribution with unknown mean $\theta$ and variance 1. It looked kind of ugly at first (since it involved taking the second partial derivative of the log of the joint pdf of the random sample), i.e., it was:

$$I(\theta)=-\frac{\partial^2 \log(\prod_{i=1}^n\frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}(x_i-\theta)^2})}{\partial \theta^2}$$

but it was kind of beautiful that the ugly looking expression above ultimately boiled down to 1, and as soon as I saw that the answer was 1, it made complete sense that 1 would be the answer (for reasons involving the Rao-Cramér lower bound, which I won't go into here.) And I thought,wow, what a great choice for a first problem, because it was one of those "lightbulb going off in my head moments.)

But then came part b of problem 1, where we had to compute the efficiency of an obviously inefficient estimator for the unknown parameter. That meant that I had to compute the variance of the estimator, which should have been an easy piece of cake.

The estimator was $Y=\frac{1}{2}X_1+\frac{1}{3}X_2+\frac{1}{6}X_3$. It was easy for me to prove that $E(X)=\theta$, so I knew the estimator would be unbiased. So then all I needed to do was to compute the variance of the estimator. And I was used to using the following clever trick to find the variance of statistics: Variance(Y)$=E(Y^2)-[E(Y)]^2$. So, I thought to myself, well, I already know $E(Y)=\theta$, so all I need to do is to find $E(Y^2)$. So I wrote:

$$E(Y^2)=\frac{1}{4}E(X^2)+\frac{1}{9}E(X^2) + \frac{1}{36}E(X^2) =\frac{7}{18}(1+\theta^2)$$

But looking at that answer, I realize that could lead to a negative variance if $\theta$ is sufficiently large and I knew THAT couldn't happen, so I must have done something wrong, but my brain was starting to go "deer in the headlights" and freeze up and I was so stuck on my approach to computing the variance as $E(X^2)-[E(X)]^2$ (which can often be a clever approach) that I couldn't see that my equation above was fundamentally flawed.

If my head had been clearer and I could have gotten unstuck, I could have made the problem trivial by writing:

$$Var(Y)=\frac{1}{4}Var(X)+\frac{1}{9}Var(X)+\frac{1}{36}Var(X)=\frac{7}{18}$$

which is the simple, clear, and straightforward approach to solving what was intended to be an easy softball problem. Somehow I had gotten my brain stuck on an unworkable and incorrect approach. Having done that on the first problem, I was then sufficiently rattled that I strongly suspect I messed up some later problems on the exam as well.

Oh well, live and learn. At my age and stage of life, I don't really care about grades, but I was disappointed because I thought the professor had been terrific and that I had learned a lot in the class and I would have loved to have done a better job of showing him what I had learned. Fortunately, there was also a takehome part of the final, which I felt much better about, plus I felt pretty good about all the homeworks I had turned in as well as the midterm.

But the whole thing was really a remarkable learning experience to me. As an economics professor, I often see students do incredibly silly things on exams and I beat myself up, thinking "Haven't I taught these folks *anything*? How could they be so confused." But having seen the incredibly silly thing that *I* did on the exam and how it didn't really reflect my understanding (but more my nervousness under time pressure), it reminded me again that tests are necessarily flawed measures of learning.

But I also realize that maybe if I knew how to study more efficiently and master a subject more deeply than I had, I might have done significantly better on that exam. And I like challenges, and I feel that following some of the ideas in this book may be helpful to me.

So feel free to stick around if you want and follow the future challenges of a sixty-something "Velveteen Mathematician" who wants to become a "real mathematician."

Struggling to learn math again in my 60s

My last post was written a little over two years ago, shortly before my world turned totally upside down when my husband died unexpectedly.

Ross loved many many things and math was a particular love we both shared. Both of us had undergraduate degrees in math, but we met at the beginning of graduate school in a PhD program in economics, not math. Why? Because in 1975, economics seemed like a vastly more practical choice. There were far more jobs for economists than mathematicians.

But both of us loved math anyway and did it just for fun in various ways. In fact, the year before he died we enjoyed a nightly ritual before bedtime where I would read aloud from a mathematically fun Japanese novel called Math Girls, which I learned about from Math Mama's blog.

And Ross was very supportive of my amateur mathematical adventures like founding and advising the Albany Area Math Circle, doing Guerilla Math Circle outreach activities and other such mathy stuff. In fact, just two days before he died, he was on one of his solo walks around the neighborhood on a fine spring day and a guy came up to him and said, "Excuse me? Are you Mary's husband?" and Ross said "Yes???" and then the guy hugged Ross and shook his hand and thanked him profusely, saying that his daughter was headed to MIT next year and that it never would have happened without our math circle. Anyway, Ross came home and hugged me and told me how good that made him feel.

And then, the next day (which turned out to be the day before he died, though of course we had no way of knowing that was coming), our math circle had its end of year picnic and the students wound up deciding to built a really large tetrahedron and they were so excited about it that they didn't want to take it apart at the end of the picnic, so one of the moms with a large minivan brought it back to our house and a couple of students managed to get it up the winding stairs from the garage (with a bit of minor disassembly and reassembly) and into our family room. And Ross loved seeing the students' excitement about the tetrahedron as they brought it into the house and then every time he walked by the tetrahedron on what turned out to be the last 24 hours or so of his life, the tetrahedron made him smile. He liked living in a house with things like a giant tetrahedron built by math-loving students in it.


Anyway, the next day he left this earth and since then I have been faced with the question of finding a reason to keep on living in as joyful a manner as possible.

I did not want to continue running the ocnsulting firm we had run together. Economics is known as the "dismal science" for a reason. I do feel our work did some good, but I couldn't continue it without him.

So...instead I decided to *seriously* undertake graduate studies in math. It hasn't been easy. I started out with a graduate class in real analysis in fall 2013, studying a subject for which I'd taken the prereq (undergrad real analysis) forty years earlier! There were more than a few cobwebs in my brain, but I loved the challenge. Then in spring 2014, I took graduate abstract algebra--more cobwebs. What they say about abstract algebra is that it is like doing a jigsaw puzzle in a completely dark room. But for me (given the number of years since I'd taken the undergraduate course prereq to the graduate class in abstract algebra), a better analogy is that I felt like I was doing a jigsaw puzzle in a completely dark room WHILE WEARING THICK MITTENS!

But I persevered and learned a ton--not only about math and the process of learning. This past year, I have been taking a two semester sequence in the theory of statistics. It's been challenging but illuminating. One difficult issue for this most recent class is that I really never had the undergrad prereq class, so unlike the previous year (when I was just dusting off cobwebs in my brain) I was largely working with a lot of completely alien theoretical concepts.

I have decided to use this blog to share more of my journey and my insights as I go.

Wednesday, March 27, 2013

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.