It happened again yesterday. This time I wandered into one of those small conversations in the theater lobby at intermission. “I was fine until trigonometry,” B was saying. “Everything made sense up to that point, and then I just didn’t get it.” M turned to me and said “Isn’t trigonometry basically about relationships in triangles?” “Yes…” I paused, trying to find a succinct way to capture the essence of trigonometry. How might I convey it in a way that would be meaningful to B? But too late – she had disengaged.
“So sorry to bore you with my life’s calling.” I didn’t say that; it’s a bit over the top. Still, there have been moments. For a sixth grade project, my friend Susie and I wrote the numbers from 1 to 100 in careful alignment on a poster board. We circled 2 in blue, and then used vertical blue lines to cross out the other even numbers. Because 3 was the next uncovered number, we circled it in red and used diagonal red lines to cross out multiples of 3, and so on for 5, 7, 11… This was our version of the Sieve of Eratosthenes; the numbers that escape and get circled are the prime numbers. To me, though, In Mr. Taylor’s sunny classroom upstairs at the end of the hall, the beauty was in the patterns of the colored lines. A number at a crossing of two colors has two different prime factors (meaning divisors), so 6 has both blue and red lines through it. You have to go down to 30 to find three colors in one place, and you’d need a bigger board to get four. I was enchanted. The allure of such moments of insight drew me, ultimately, to graduate school. I hoped to replicate the enchantment and have a chance to guide students toward it. But between acts in the lobby with M and B, I could not express my visceral sense that a close friend had been maligned.
Nor did I suggest that B had missed a profoundly enlightening experience, or rather a whole bunch of them. Yes, trigonometry is about relationships in triangles. It connects angle measure with length measure, which is clever as well as convenient. For one thing, you can use it to explore patterns that repeat over time or space. Consider the passage from one solstice to the next. From December 21 to December 22, there’s an increase of only 3 seconds of sunshine in a Vermont day. From March 21 to March 22, we experience an increase of over 3 minutes; by mid-June, the difference is negligible again. Because this pattern results from the earth’s rotation, in combination with its tilt, the triangles here sit inside a circle; see the Circle of the Year diagram. The color-coded vertical triangle legs measure the deviation in day length from that at the equinox (a bit more than 12 hours), as indicated by the scale on the left.
Notice that from March 20 until April 20, we go one-third of the way to June 21 – that is, we’ve gone through an angle of 30 degrees around the circle — but the day length has already gotten halfway to its maximum. (The yellow vertical segment is half as long as the red one.) From March 20 until May 5, we’ve gone halfway in time from the equinox to the summer solstice, but the day length has gotten about 70% of the way there.
If we unroll the circle to view the passage of time along a straight line, keeping track of the vertical leg lengths as we go, we get the same information displayed differently in the Hours of Daylight diagram.
This is a sinusoidal curve, also known as a sine wave; its wavelength is one year. Lengths of sound waves, by contrast, are measured in small fractions of seconds.
The waves on a heart monitor, the vibrations of a violin string, and that unsettling forward motion at the top of a Ferris wheel are best described using ideas from trigonometry. Yet I did not manage to convey any of those ideas to my friend who had fallen off the trig truck.
No, I was silent. Is there a good response to that simultaneous glazing and diversion of the eyes?
How did we get here? How have we math lovers managed to alienate and even terrify so many perfectly intelligent people? I hear you, fellow mathematicians, saying, “it’s not our fault!” — and it isn’t, at least not completely. We members of the American Mathematical Society, who have devoted our professional lives to the subject, are responsible for advancing new knowledge and educating post-secondary students. We can’t also be expected to influence entrenched attitudes developed in childhood and reinforced by peers, parents, and — sadly enough – some teachers.
Then there are journalists. Consider the following openings to recent stories:
“Although many school-going youth might disagree, a new study finds that geometry is an intuitive subject…” (The New York Times)
“As camps go, the Summer Program in Mathematical Problem Solving might sound like a recipe for misery…” (The Times again)
“The drudgery of solving for X flew out the door of a Presidio Middle School classroom Friday as the giddy students traded in their back-breaking algebra textbooks for an iPad…” (San Francisco Chronicle) (just one iPad for all of those kids?)
“Few subjects are as likely to generate a collective groan as math….” (USA Today)
Do they teach this in journalism school? “When writing about mathematics, start with a contemptuous swipe.”
In the face of all the groaning and misery flies Keith Devlin, a mathematician now at Stanford. His All the Math That’s Fit to Print, a collection of columns for The Guardian, has been followed by a steady stream of books for general audiences. Devlin is NPR’s Math Guy on Saturday mornings. He is not alone; plenty of mathematicians reach out with books, articles, and videos. Others give talks at libraries and senior centers. On campus, college students of all academic persuasions have access to courses like “Math for Liberal Arts” and “Math and Society.”
When I was in graduate school, where the mathematics was pure and the faculty focused squarely on research, I had no idea how many mathematicians at other institutions were involved in K-12 education, and those efforts have grown visibly since then. They include education for future teachers, from individual specialized courses to complete programs such as Math for America (not to be confused with Teach for America; MfA offers a full year of graduate-level course work followed by four years of mentored practice and expects a long-term commitment in return). Some professional development programs for current teachers were developed and are staffed by college faculty; the Park City Math Institute and the Vermont Mathematics Initiative (VMI, where I now work in the summer) are just two examples. Mathematicians are contributing in large and small ways to the Common Core State Standards, an effort to bring states together in determining detailed learning expectations for school children. Several members of my own math department regularly give presentations, such as “The Monty Hall Problem” and “The Mathematics of Symmetry,” at schools around the state.
And yet. If I’m honest with myself, I know that I went too long thinking that I was done with elementary mathematics. This was based on two assumptions: that everyone learns arithmetic the same way I did, and that there isn’t much to say about fractions (for instance) once you’ve moved on to other things. Wrong, as was confirmed when my kids brought home a technique I’d never seen for multiplying two-digit numbers. It turns out that there are lots of methods for this task, and it’s interesting and instructive to consider how they are related and why they all work. I am grateful for the eloquence of a few mathematicians (Roger Howe, Ken Gross, Bill McCallum, Hyman Bass, and Sybilla Beckman, for example) and a lot of teachers (I hope you know who you are) on the value of thinking deeply about foundational ideas.
In that earlier age, as I stayed on my side of an illusory wall, I had an incomplete understanding of the problems and challenges of pre-college mathematics education. I now have a less incomplete understanding, but more importantly, perhaps, I’ve gained an appreciation for the complexity of the issues. This is where I think we – the professional mathematicians, loosely defined – have been prone to missteps. More than a few times I’ve listened to talks delivered at mathematics conferences, where speakers present proofs of new theorems to trained skeptics and answer detailed questions about what is known and what is not known, only to hear sweeping statements at coffee breaks about the sad state of math education across the land based on the complainers’ own perceptions of their own children’s experiences. This is understandable and human, and not without some useful information; these complaints often describe advanced students being under-challenged or even shut down, and it’s worth looking closely at whether this is a widespread phenomenon. But too often we abandon our own professional commitment to rigorous analysis in our attempts to understand what is – or should be – happening in schools.
Sloppy reasoning isn’t a big deal at a coffee break, but by not knowing what it is that we don’t know, we may be putting our limited energy for education issues in the wrong places. We can make a big fuss about specific curricular programs, but there is strong evidence that teachers have a much bigger effect on student learning than curriculum does. (A math specialist at an elementary school in a threadbare Vermont town once explained to me exactly how she and the other teachers supplemented their district’s curricular materials to make up for deficiencies. Never again will I assume that teachers slavishly follow whatever texts are on their shelves.) We can envision the perfect classroom for our eight-year-old-math-whiz selves, but there are other kinds of learners out there who might benefit from a different approach than the one that worked for us. We can distance ourselves from schools of education, but there is good work being done toward figuring out what makes for effective teaching. (Read something – anything – by Deborah Ball on mathematics education. Her work is clear, remarkably free of jargon, and compelling. For just one example, I heard her say this at a conference: “Test results are hugely dependent on things that shouldn’t matter.”) Whether or not we holders of advanced degrees bear responsibility for our society’s troubled relationship with mathematics, it is our problem (must that student tour guide outside my office window gleefully proclaim his loathing of the subject?), and we could be more careful in our attempts to address it.
As for the rest of you, oh experts in other wonderful things, it’s your problem too. We live in a culture that accepts math phobia as a given, perhaps even the norm. My mom was a classic case. When she was a girl, her father told her that girls couldn’t do math. To her credit, she didn’t believe that about her own daughters; in fact I was enlisted as her tutor when she needed a basic math course for her bachelor’s degree, which she finished when I was 20. But in her mind, mistakes in balancing her checkbook confirmed her own ineptitude. I wish I’d thought to point out that mathematicians aren’t always good at arithmetic either, and more to the point, that her abilities to adapt knitting patterns and map out perspective in her landscape paintings were on the plus side of the ledger. In any case, Mom’s deep discomfort with numbers meant that she depended on others for help after her marriage ended and her children moved away. She would decide that Elizabeth the financial adviser was a good person, so she’d do what Elizabeth told her to do, because she wasn’t able to evaluate Elizabeth’s advice on its merits. This strategy worked out much of the time, but there were major glitches now and then. More troubling was her inability to assess her overall financial situation, which led to undue anxiety on some days and reckless spending on others. I’m certain that she chose credit cards for reasons unrelated to the interest rate or other charges because she could not begin to guess what each would cost her down the road.
For a time in college my roommates and I were members of a food co-op. Once, as I scooped clumps of raisins from large tubs into small plastic bags, I overheard a conversation through the shelves of peanut butter and sacks of brown rice. The manager was directing a volunteer in a bookkeeping task. “These prices were marked up 20%, but we need to report the original price, so just take 80% of the new price to get the old one.” Oops. If the original price was $1.00, then the marked-up price was $1.20, and 80% of $1.20 is $0.96, not a dollar. I didn’t say anything (is there a pattern here?), but I suspect the books didn’t balance at the end of that month.
I don’t mean to imply that discomfort with trigonometry is equivalent to a poor understanding of ratios and percentages. It’s also important to distinguish between mathematics, which my dictionary defines as “the abstract science of number, quantity and space,” and quantitative literacy, which refers (roughly) to the ability to interpret and apply numerical information in context. I do believe, however, that our collective surrender to the notion that either you get math or you don’t – a view not shared in many other countries – leads to the glazed-eyes response we American mathematicians know so well.
This can’t be good on an individual or a societal level. On NPR (yes, NPR) recently, I heard “…he scored higher than 73% of the people who were tested. But that’s getting technical.” If it’s a stretch to quote a percentage, then can I trust this journalist’s investigative skills? As soon as high school students stop taking math, they close off whole categories of career options. If they also flinch at quantitative information, they will be hampered in a wide range of roles, from consumer of credit to elected official to parent of math student to consumer – or reporter – of news.
We need to address this in schools, of course, but I refuse to give up on adults. What I’d like to suggest to the most math-averse of my friends is that the practice of mathematics isn’t as alien as you think. I experience mathematics in a variety of ways; perhaps one or more of them will be familiar to you.
The quintessential mathematical moment is when I’m so focused on an idea that I’ve lost track of time and place. This is mathematics as contemplative practice. My mind is relieved of mundane concerns, and though I’m not paying attention to my breath, there can be a rhythm, an ebb and flow, of mental energy as I move outside the realm of words and look for connections I hadn’t seen before. Meditation is new to me, but the state of being simultaneously alert and detached from the world is not.
It dawned on me last fall, while going over a student’s work, that I was reading poetry. Not only did the author adhere to the formalities specific to the genre of mathematical proof; he also used language so carefully and efficiently that a convincing argument took up just a few lines. In elegant mathematical discourse, one word or phrase stands in for many instances. To figure out whether this metaphor makes any sense, I went to the Poetry Foundation website, where I found “Independent Study,” an article by Emily Gould. As she describes her experience with a self-designed poetry course, she writes “A poem requires full attention in a way that prose does not, and worse, a poem is much harder to like because every word matters.” This is exactly what I tell students about reading mathematics. Well, not the hard-to-like part; they’ve already been bombarded with that message. I prefer to say that a particular challenge of mathematics is the importance of every word, which also makes it particularly rewarding.
During yoga class today, our instructor told us to position the back foot at a 45 degree angle. At another point, he used the word “rectilinear.” Later, he told us to try to visualize the whole body in one plane. This, I thought, is a fellow mathematical thinker. It’s not just the vocabulary, it’s the habits of mind that go along with it and affect the way I understand my world. Once a can of Pringles chips slipped past the food police and into my house. You may see a processed food product of little nutritional value; I see a hyperbolic paraboloid. I took a felt-tipped pen to one of the chips and traced a parabola opening up, another one opening down and crossing the first at the chip’s center, and then a couple of horizontal hyperbolas. Naturally, my children (in middle school at the time) were mortified in front of their friends, but I hope they also considered the idea that we all have lenses through which we view our surroundings, and mine have been shaped by the hours I’ve spent thinking and speaking mathematics.
Can one hear mathematics? According to the 17th century mathematician Gottfried Leibniz, “music is the pleasure the human soul experiences from counting without being aware that it is counting.” Indeed, there are entire books devoted to connecting math and music, and they go well beyond the analysis of sound waves. I’ve known plenty of math people who are also gifted musicians, starting with my older brother. But pleasure of the human soul? Well, yes, and it’s more than just counting. Listening to polyphonic music, I hear patterns of different voices and the braiding of those patterns as the piece progresses. There are variations and discontinuities, and often the pleasant wonderment at human ingenuity. Reading through any of the various proofs of the Fundamental Theorem of Algebra, I see an analogous braiding of ideas from different branches of mathematics as the argument progresses. Re-reading, subtleties that had slipped by before now seem louder. To read through additional proofs is to take a tour through some of the most powerful concepts in mathematics.
When people tell me their math stories, they often say, “I liked the puzzle-solving aspect.” (This is usually right after the “I was fine until…” statement.) In my own high-school trigonometry class, we spent weeks proving identities; that is, demonstrating that given equations are true by strategically using previously established ones. I loved this activity. I gather that it has dwindled or disappeared, and I can see why; it can quickly become strictly procedural and detached from the meanings of the puzzle pieces one is moving around. Still, there’s a distinct satisfaction that comes from reaching a clearly defined goal, especially if there’s a level of challenge along the way. The moment when I figure out how to construct a proof can feel similar (in quality if not degree) to that of finishing a New York Times crossword, another thought I wish I’d shared with my mom, Queen of the Sunday Puzzle.
A drawback of the math-as-puzzle frame is that it may reinforce the myth that there’s only one path to the answer of each mathematics question. Implicit in that myth, I believe, is the assumption that one must carry in one’s head at all times entire solutions manuals from earlier math courses. This explains, I now see, why my approach to graduate school was all wrong. It made me reluctant to ask questions (not that they would have been welcome, but that’s another story) or work with my classmates, for fear of exposing the shortcomings of my internal answer books.
Besides, the satisfaction of closure is fleeting; who hasn’t felt a sense of loss at the completion of an engaging task or book or movie? My friend Claudia, who has a literary outlook, introduced me to the term “negative capability” (introduced by John Keats), or as she put it, the ability to “be content floundering in the muck.” This may be where math education, as traditionally practiced in this country, has failed so many. An emphasis on quick recall of facts as an end rather than a means could make floundering feel like failure, rather than a necessary – indeed, productive – stage of learning. It also removes the most interesting parts. On a suburban junior high school playing field, we girls spent gym class after gym class on field hockey skills, but I don’t remember ever playing a game. That was for the real athletes who played on the team after school. No wonder I hated field hockey, and no wonder so many of my peers hate math. (But really, you don’t need to tell me so the first time I meet you at a party.)
Come to think of it, my favorite fourth-grade homework assignment brought home by my oldest child was “play a math game, and then write about your strategy,” which happened weekly. Options included Othello, Battleship (with pencil and graph paper, so I could suggest a smaller board when I was tired), Mancala, and Connect Four. The kids were invited to see math as the engaging and creative social activity that it is, and to reflect on their own learning as well as the value of persistence in the face of frustration (well, not necessarily in those words).
It’s March in Vermont, and the snow is melting on the University Green beyond the marble pillars of the red brick Waterman building. In a seminar room at the back of the third floor, before the formalities start, a third-grade teacher describes her family’s shift schedule that keeps the maple syrup boiler going. For another teacher, the family business is a ferry service, which has seen the strain of long lines and extended hours since a bridge across the lake was condemned. The conversation quiets down, and the first presenter begins her VMI story. She shows us her work from two years ago on the “Potato Race” problem, and describes how her approach to mathematics instruction changed as a result. We see photos of her students working on a lesson she recently revamped, and then the data she gathered to evaluate her results. She tells us that she’s not sure what she’ll do without VMI now that she’s finishing the program.
As the afternoon progresses and the weak sunlight strains through the dusty windows, I sit back in my chair to hear one transformation story after another. “Now I understand the building whose foundation my students are working on.” “I feel more ready for students’ different approaches.” Evidence that adults can grow as mathematical thinkers – there’s hope! Then again, this is a self-selected group, these educators who chose a three-year challenge. Still, some of them were quite apprehensive, and they all know some calculus now. But that woman at the summer party next to the lake was so hostile, years after her last math class in high school… In the midst of my internal debate, J says, “I always liked math, but now numbers are neon.” I imagine her making her way through books and papers, all of the numerals glowing for her with their own fluorescent wavelengths.
Evidently I need to develop my own negative capability, to be content floundering in the muck of our society’s problematic attitudes toward mathematics. This is not a theorem ready to be proved or an equation to be solved, and it’s not even clear what a solution would look like. I’m pretty sure, though, that more neon numbers would help.