Oh blogosphere, I miss you so! There are so many cool science-y items and fascinating debates to tempt me away from the task at hand, and I hate being cut off from it all for so long, but I must stay the course and finish The Damn Book, as it has come to be known. (Yes, I have reached the dark night of the writer's soul, but we shall overcome.) I'm nearly done, the manuscript is due September 18, and after that, I will return to blogging (and life in general) in full force, rather than the intermittent appearances I've been making for the last few months. In the meantime, we've been debating two options for The Damn Book's eventual title, and I figure, why not let my readers weigh in?
1. The Calculus Diaries. This was my original title, way back in the book proposal stage. It's direct, high-concept, and captures the nature of the book pretty well -- i.e., it's about me learning calculus by seeking it out in the real world. The question is, does putting "calculus" right there, front and center, turn off the potential general reader (as opposed to those who are already mathematically inclined)?
2. Dangerous Curves: How I Learned to Stop Worrying and Love the Calculus. This is the current working title. It's clever, playing on the "area under the curve"/"face of the function" aspects of calculus; and it's more subtle in its appeal to the general reader. However, it might be a little TOO clever, in that it's not as immediately apparent what the book is about. You kinda have to know a wee bit about calculus already to get the title.
I could go either way at this point, so feel free to cast your vote in the comments, with a brief explanation as to why you'd make that choice.
I have also reached that point of the writing process where I must sum up my conclusions/lessons learned from the long journey into calculus, and condense it into a compelling, readable epilogue. Much soul-searching has taken place during the last couple of years. I have talked to dozens of people (scientists, educators, my fellow "mathogynists") about their attitudes towards math (calculus in particular), how it is taught, how they learned it (assuming they did), etc., and compared it to my own experience. Here are some of my random musings thus far (much of which probably will not end up in the book, although it all feeds into the final product).
First, where does this knee-jerk dislike of math come from? Two years later, I can only say, who the hell knows? There is no one single factor, as far as I can tell. For many people, their struggles with math set in with high school algebra. Co-blogger Allyson performed so well in her other high school subjects that she was placed in advanced math classes. Alas, she was ill-prepared for that level, and the placement set her up for failure. “I distinctly recall the humiliation of my eyes welling in frustration at algebra,” she said. By the time her friends were taking calculus, Allyson had been demoted to what one might charitably call “math for dummies,” learning how to calculate compound interest and how to do her taxes – useful skills, no doubt, but she remains haunted by the memory of her failure.
Another co-blogger, Lee, had a similar experience, with more dire consequences: her inability to grasp algebra – despite top grades in all her other classes -- kept her from becoming a marine biologist, and she has a visceral hatred of mathematics to this day. “It wrecked my self-confidence in a way nothing else ever did, and still knots my stomach,” she told me. “I’m not totally innumerate, but anything that looks like an equation makes me break out into a cold sweat and run screaming in the other direction.”(My bloggy buddy Brian over at Laelaps has written extensively about his struggles with math, which keep getting in the way of his desire to be a scientist.)
On the surface, at least, I have no good reason for my own negative reaction to mathematical symbols. I did very well in my high school geometry and algebra classes, yet somehow I never self-identified as someone with a penchant for math. The truth is that fear and loathing in math class does not necessarily arise from a lack of aptitude, but from a belief in a lack of aptitude. Where did I acquire that belief?
No doubt part of it stems from gender bias. There is a well-documented prejudice against women in math and science dating back thousands of years, although history gives us the rare exceptions, such as the plucky Sophie Germain. The century before, a French noblewoman named Emilie du Chatelet translated Newton's Principia and became the lover of Voltaire before dying in childbirth in her early 40s. Victorian England had Mary Somerville, another self-taught female mathematician who defied the cultural stereotypes of her age. Her passion for math was so strong, she was revising a paper the day before she died at 92.
Such women often have been dismissed as mere statistical anomalies, but evidence is mounting that there is, in fact, no innate difference in the mathematical ability of girls and boys. Any gap in performance is due primarily to sociological factors. This is a controversial statement, as evidenced by the heated comment threads that ensue whenever someone in the blogosphere dares to bring up the touchy subject of women in math (and science). We would prefer to believe that the overt sexism experienced by Sophie Germain et al are a thing of the past, and simply not an issue in this enlightened age, but the reality is that these attitudes persist. Women have come forth with innumerable horror stories ranging from mere discouragement to overt sexual harassment.
A geometry teacher tells the entire class that the girls would probably do the worst in his course because they lacked spatial reasoning ability. A guidance counselor shunts female students into “practical math” classes where they learn how many ham slices each guest would need at a wedding. A physics professor insists on checking his female students’ work before they can leave the lab, yet doesn’t feel the need to check the work of his male students. A computer science professor dismisses any questions from female students as “lazy little girl whining.” And a calculus teacher thinks it’s perfectly appropriate to measure his female students’ bodies and use those measurements as part of his volume calculations in class. One woman on a comment thread over at Tiny Cat Pants last year told of her high school math teacher who made the three female students sit in the front row, “because girls have a harder time with math than boys do.” It was really a flimsy excuse to ogle their cleavage and brush his crotch up against them suggestively during exams. Quoth the commenter: “Guess which three people in that class were not about to be stuck in a basement computer lab with that dude?”
While I have no doubt those things happened (and still do), I never experienced anything so horrific; my math teachers were kind and, if not openly encouraging, they certainly were not discouraging or hostile, nor was I ever sexually harassed. My parents were supportive of my intellectual pursuits, if a bit bemused by my headier inclinations. Nobody ever told me explicitly that girls weren’t as good as boys at math, yet somehow I absorbed that message anyway. Carol Tavris, a cognitive psychologist and author of several popular books (The Mismeasure of Woman should be mandatory reading for young women), explained to me that there are subtle, situational social cues that seep into our consciousness, like osmosis, even if we never encounter overt negative messaging about gender.
The phenomenon is known in psychological circles as stereotype threat, and it has been confirmed in more than 100 scientific articles. For example, a 2007 study in Psychological Science found that female math majors who viewed a video of a conference with more men than women reported feeling less desire to participate in the conference, and less of a sense of belonging, than female math majors who viewed a gender-balanced version of the video. The male math majors were immune to those subtle situational cues.
That’s stereotype threat in a nutshell. The 2004 film Mean Girls perfectly captures the subtle influence of cultural factors. Home-schooled in Africa for most of her early life by her anthropologist parents, Cady (played by Lindsay Lohan) didn’t absorb the subliminal message that women can’t do math, or that liking the subject is uncool – in fact, she appreciates the universality of math, because “it’s the same in every country.” Then she begins attending a regular high school and the inevitable peer pressure kicks in. She is urged by her peers not to join the high school math team (“It’s social suicide!”), and pretends to be bad at math to win over the cute boy in her calculus class. This being a movie, she gets over it, and ends up copping to her love of math and snagging the cutest boy in school. Alas, peer pressure doesn’t excuse me, either. I was a painfully shy, socially awkward, brainy sort in high school, and pretending suddenly to be bad at math would not have transformed me magically into the homecoming queen.
Tavris also cited our fascination in the US with the notion of innate ability. We are born with certain built-in talents, this reasoning goes; you either have a gift for math, or you don’t, and no amount of hard work can make up for that lack of innate ability. The reality is much more complicated. My friend Deborah self-identified as being good at math early on in her education. Her fourth-grade teacher held multiplication table competitions in class. Deborah was highly competitive, so she worked very hard on memorizing her multiplication tables and practicing at home. As a result, she excelled in these competitions and became known as being “good at math.” This had a significant impact on her later on: whenever she struggled with an especially tough problem, she pushed through, thinking, “I should be able to do this because I’m good at math.” Yet her belief in her innate ability, and success at math, were actually the product of a lot of hard work.
Another part of the problem has to be the way the subject matter is presented and/or taught. Guest blogger Alex Morgan offered a clue when it wrote about his own daughter's ambivalence towards math, despite having a certain aptitude for the subject. She just doesn't like it! And after perusing her math homework, Alex found he didn't much blame her. The material was dry, uninspiring, and completely divorced from any real-world experience. (Hence my use of real-world environments in which I seek out possible calculus problems in The Damn Book.)
Students need to feel inspired, particularly when it comes to a difficult subject. While I was at the Kavli Institute for Theoretical Physics last year as journalist in residence, I got to know UC-Santa Barbara mathematician Bisi Agboola, who generously shared his own story with me. Bisi was educated in the UK and failed most of his math classes through their equivalent of high school. “I found it dull, confusing and difficult.” As a child, he was determined to find a career where he wouldn’t need any math, finally announcing to his skeptical parents that he would be a woodcutter. He was crushed when they pointed out that he would need to measure the wood.
But one summer he encountered a Time-Life book on mathematics –- Mathematics by David Bergamini -– that offered “an account of the history of some of the main ideas of mathematics, from the Babylonians up until the 1960s, and it captured my imagination and made the subject come alive to me for the very first time.” It changed his mind about this seemingly dry subject. He realized there was beauty in it. He wound up teaching himself calculus, and told me he is convinced most physicists also do this. Today he is a PhD mathematician specializing in number theory, and exotic multidimensional topologies. Ironically, he still doesn’t much like basic arithmetic: “I find it boring.”
Some students respond well to how calculus (and physics) is traditionally taught, others don't. The Spousal Unit sent me a link to a fairly new blog called Gravity and Levity, written by a physicist with a way with words:
For those of us who immediately liked physics class in high school, physics was a game. It was like a little logic puzzle where the rules of the game were given to you (usually on a formula sheet) and you were asked to use them cleverly to come up with a solution. A friend of mine once put it succinctly: “Physics is all about finding out which variables you know and which variable you want, and then searching through your formula sheet for an equation that has all of those letters in it.” That, more or less, was the physics game. You rearrange some symbols on a paper and you come up with an answer. Instant gratification.
Those of us who went on to study physics in college almost invariably did so because we liked the game. I personally loved it, and I was good at it. But as I went further into physics, it began to be more than a game. Little by little, all the equations and “rules of the game” started coming together into a coherent perspective on the universe and how it works. ... Over years of study my interest in physics gradually but completely shifted from “the game is fun” to “I want to know how to think about what the universe is made of and how it works.”
I hated the game; I couldn't really see the point -- at least until I got interested in physics and realized that calculus was relevant to my world. I'm one of those people who really needs to understand the context, and the "why" of things. (Although I must admit to a fondness for word games. I killed at Boggle and Scrabble in college.)
The point is, different people learn in different ways, and that makes coming up with a standardized educational system particularly challenging. The best model I heard about was from a woman scientist I met at a party, whose daughter went to an elite private school on the East Coast. There were several teachers for each subject, and students could transfer out of a class if the teaching style didn't agree with them, and try another instructor. The daughter's attitude towards her history class did an about-face once she found an instructor who inspired her. Just like science communication, learning (and teaching) science, or any subject, is about making that critical connection. I just don't see how it would be possible to scale up that approach to a nation-wide level; there's a reason this system was implemented at a pricey private school.
So: stereotype threat, gender bias, peer pressure, bad teaching, poor subject presentation -- all of these play a role in discouraging people (especially those of the female persuasion) from taking/liking math. There are countless efforts worldwide to combat these sweeping socio-cultural factors, and we should continue to fight the good fight in that regard. That said, Sophie Germain and Mary Somerville didn't let socio-cultural factors keep them from pursuing their love of math. What made the difference? Personal mentorship helped at some point, but it started with inspiration and falling in love with the subject in the first place.
Ultimately, when I closely examine my dubious history with math, I can't really cast too much blame on those Big Picture factors for my prolonged ignorance/avoidance of math. They discouraged me because I let them discourage me -- because I wanted an excuse to avoid calculus. I can't say I was intellectually lazy, because I avidly pursued knowledge in any subject that caught my interest, and worked very hard in those areas. Like Alex's daughter, I just wasn't all that interested, and hence was more than happy to be pushed away from math and science. What is the cure for willful ignorance and very deliberate avoidance?
I'm not sure there was a single game-changing moment, since the process of thawing towards math was gradual. It started with going to work for The American Physical Society, then becoming a science writer specializing in physics. And one day, out of curiosity, I asked a physicist named Alan Chodos (associate executive officer of the APS) about why objects fall at the same rate regardless of mass -- it seemed really counter-intuitive to me, although I had no doubt it was true. He insisted that I didn't have to take the matter on faith, and walked me (kicking and screaming) through the basic algebraic equation. Suddenly the numbers had relevance to something I'd actually experienced. And my kneejerk defenses started lowering bit by bit. I can't say I love math and calculus, but I understand the basics now, and I no longer break into a cold sweat at the sight of an equation. Believe me, that's tremendous progress.
It just goes to show that real learning is personal and individual. A good teacher can change someone's life and undo years of willful ignorance. I owe Alan a great debt, I realized. The least I can do is say thank you in public.