In college, I was one of those compulsive read-everything kids. I even felt pangs of guilt when I skipped optional reading. But there was one gaping hole in this policy of mine, large enough to squeeze a whole degree through.
I never did the reading for math. You know, my major.
I’m not proud of it, but I know I’m not alone. As students from primary school to PhD have discovered, mathematical writing is a different beast. It’s not just a matter of jargon, equations, or obscure Greek letters. It’s something more basic about the way mathematical texts are structured and paced.
The trick is this: In mathematics, you say things precisely once.
(And no, I’m not going to repeat that.)
A talented colleague of mine once asked in frustration why her students refused to read the textbook. Her background was in biology, where the book—dense and difficult though it may be—is an irreplaceable source of learning. Now she was teaching Algebra II, and was losing patience with her students’ incapacity to glean anything, anything at all, from the text. “Why do they need it all spoon-fed to them?” she asked.
“I see what you’re saying,” I said. “But I’m not very good at learning math from a book myself. It’s a skill for 21-year-olds more than 15-year-olds.”
You see, ordinary writing has a certain redundancy to it. It needs redundancy, because English (lovely language though it is) can never capture a complex idea with perfect precision. In any phrasing, some shade of meaning is lost or obscured. A subtle, complicated thought must be illuminated from many angles before the reader is able to sift reflections from reality, or tell the shadows from the thing casting them. Thus, typical prose is full of pleasing repetition—paraphrase, caveats. You can skim, and even if you miss a few details, you’ll walk away with the gist.
Math is different. Unlike English, mathematical language is built to capture ideas perfectly. Thus, key information will be stated once and only once. Later sentences will presuppose a perfect comprehension of earlier ones, so reading math demands your full attention. If your understanding is holistic, rough, or partial, then it may not feel like any understanding at all.
Single words are saturated with meaning; immense focus is required; the diction is exactingly precise… more than anything, reading mathematics is like reading poetry.
This is why good mathematicians always read with a pencil in hand. Passive perusal of mathematics is pretty much worthless. You need to investigate, question, and probe. You need to fill in missing steps. You need to chew for a long time on every sentence, fully digesting it before you move on to the next course of the meal.
My indifferent, shrugging approach towards reading math in college may explain my struggles with a certain topology class. That class—a good simulation of first-year graduate school—demanded that I learn from the book, rather than from a professor’s lectures. I was unpracticed and unready for such a challenge.
I hope to equip my own students a little better.
Wisdom from the comments:
Phil H. proposes “a math-reading class, where you take a paper and walk through it, elucidating on the steps and answering questions.” He points out that reading slowly takes – among other virtues – humility. Reading fast is the token of a clever mind; but reading slow builds the wealth of a wise mind. (hackneyed aphorism mine)
John Golden points towards a relevant cartoon: http://abstrusegoose.com/353
Ariel finds math harder to read than physics or biology, and makes a wonderful observation: “In biology… you do the science in your lab, and just describe it [in the paper]. In math, you do the science in the paper [itself].”