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].”
This suggests that maybe there is value in a math-reading class, where you take a paper and walk through it, elucidating on the steps and answering questions, keeping as many members of the class up to speed as possible.
It took me many years to be comfortable with reading science papers really slowly. I’d always been used to the ego-boost of running through prose at speed – maybe I was praised too much for fluent reading as a child? It takes a certain amount of commitment, concentration, respect and humility to read slowly, looking up the unfamiliar and always keeping an eye on that feeling that you’re not totally following any more.
There should definitely be a class to (try to) teach you that.
Nice one! Reminded me of this Abstruse Goose comic of an Erdos quotation: http://abstrusegoose.com/353
Looks like it was a different great Hungarian mathematician, Halmos, not Erdos, who is being quoted in that cartoon. Still, great point.
Precisely what I was thinking about math books, only expressed with much more virtue than I could ever hope to achieve. You enlighten me once again. Thank you for that.
I noticed a while back that physics or biology papers are much easier to read than math papers. And I realized that in biology, what’s going on is that you do the science in your lab and just describe it. In math, you do the science in the paper. I wonder if that’s a factor as well?
not really. in math u do the science all over: reams of paper, maybe computer simulation, draw pictures, go for a walk, in your head, etc… then the proof is an exceedingly distilled poem of that. reading a math or science paper u still have to use the clues in the each paper to try the actual work out on your own to see if it’s real.
Not being an expert in either discipline, perhaps the difference is that in science you CAN’T always run the experiments yourself? Like, I’m not going to find a bacterial population and go introduce the right plasmids; I’m just going to take the authors’ word when it comes to the results. But in math, you CAN run the experiments for yourself. And in fact, you sort of HAVE to.
i guess it depends on whether it is ur field or not!!! if it’s ur field u can check it (if u have time). if it’s not… with most math papers in the worl, i’m not gonna run the experiment myself either!!
Differential topology was always the worst.
Seriously. I love me some basic point-set topology, but man, I was not a patient enough 21-year-old to wade through those mile-long differential topology definitions.
(And at 201 years old, I may STILL not be patient enough…)
Far too many definitions, but that’s the real mathematics.
For commiseration re topology definitions https://m.youtube.com/watch?v=SyD4p8_y8Kw
I got plunged into the whole “reading math” thing in my freshman year of college, in an honors calculus class that used Apostol’s Calculus book. It’s no friendly introduction – it’s maybe sixty years old, dense, structured in terms of theorems and proofs (which was new to me), and has few illustrations. I believe my professor said, “If it takes you less than an hour to read a page, you’re probably going too fast.”
I constantly had to ask myself “confused” questions like, “What is this saying? How is it not redundant with the last thing that was said? Isn’t X a counterexample? Is this condition really necessary for this proof?” I think struggling with that book was the best possible thing for my mathematical ability. It’s STILL not easy to read, though!
That kind of sounds like a perfect freshman year experience. Builds lots of good metacognitive skills, early enough that you’ve got time to practice and benefit from them.
for me, it was plane geometry with proofs–at the age of 13.
After that, all the HS math courses had carefully written proofs.
That’s how it was 50 years ago.
We used Courant and John in math major calculus.
i don’t know how to relate to this. ever since i was a kid as far back as i remember i was always reading science and math books on my own. kids books, science fiction, field guides, then all the books in the library, scientific american, science news, i even interlibrary loaned a buch of science and math papers when i was in high school on subjects i was interested in. dunno how that happened.
since i was always doing math puzzles,(before and after being on math team) it never occured to me when reading a math book or text NOT have pencil and paper in hand to try to puzzle out solutions before they were given or proven.
granted i did not learn to do ANY of this in school. i guess school still sucks.
I will say though, that in college i kept VERY few of my math texts (most of them didn’t seem like works of art worth keeping) but i usually had favorite alternate math books…
i remember reading and having mind blown by simmons’ topology and modern analysis. I read first few chapters in highschool and every know and then i’d peruse it.
http://mathematicslibrary.blogspot.com/2013/05/george-f-simmons-introduction-to.html
there’s another book i had who’s name fleas my mind. i remember it had towers of hanoi and proof that there were only 5 platonic solids. fun stuff
i think in general i was good at hunting out books (advanced or not, science or math) in which the author spoke diirectly to the reader as a peer as opposed to professor (or publishing committee) vs student.
In later years as a math tutor (arithmetic to calculus) i tried finding books of exercises for my students to use. NADA. there’s such crap out there!
I envy/admire people who had that kind of far-reaching curiosity at a young age. I always played it down the middle of the fairway: soaked up what the teachers taught, but never really taught myself anything. I’m only sort-of-learning that as an adult.
Any other book recommendations?
for students? i’ll look around see if i have some shreds of lists somewhere… like i remember there was a book i learned to do induction proofs from, there was a small 100 page paperback with a big pi on it, i was into computers and astronomy too so there was crosstalk…
there was gardner’s mathematical games section of scientific american… dunno if that section is still around, i think sci amer tanked some time ago..
i seem to have read a bunch of biographical material for kids in middle school about early scientists/mathematicians (often same people) and that gave me perspective, i remember galileo measuring the period of a 20 foot censur in church with his pulse1 clever.
i’ll try to wrak my brainz
Martin Gardner wrote several books, so you don’t need to scavenge old Sci Ams.
Books I liked, in random order and from memory, so author’s names might be a bit off:
– Measurement by Paul Lockhart
– Diff Topology by Guillemin and Pollack
– Algebra by Michael Artin
– Financial Calculus by Remmie and Baxter
– Eisenbud and Harris, something about schemes, I saw a preprint, so name maybe changed
– Most anything by Joe Harris
– Spivak Calculus on Manifolds
– Feynman lectures
– Shafarevich Basic Algebraic Geometry (basic, ha!)
– Rudin Real Analysis
– The Art of Mathematics series is still a work in progress, but looks very promising (https://www.artofmathematics.org/)
Haven’t read, but comes highly recommended
– How not to be Wrong by Jordan Ellenberg
– Zometool Geometry
– Davenport Higher Arithmetic
Once you’ve read all those . . .
Stating each key idea only once is just bad writing, especially in a textbook. There is no reason other than custom why math books, especially those that aren’t aimed at math majors, have to be written this way. I’m a biologist helping develop a math course for bio majors (mostly nonlinear dynamics with some calculus and linear algebra), and I’ve had a student say, “I can’t learn from textbooks but I can learn from yours”. Deliberately keeping the pace and density from getting too high may be one of the reasons for this.
That’s interesting – any way I can see experts from your textbook?
Your phrase “deliberately keeping the pace and density from getting too high” resonates with what a teacher needs to do in the classroom, too. Just like they do when they’re reading, students in a lesson need elaboration, repetition, and thinking-time to apprehend an idea.
The book is still a work in progress, but you can see it and other course materials at https://ccle.ucla.edu/course/view/15W-LIFESCI30A-1 . (Of course, a lot happens in lecture that doesn’t necessarily happen in the book.) I’d be interested in your thoughts!
I wonder if AlanGarfinkel is a relative in my mother’s side.
but u know in college i loved some of those dense sequence of proof books ( have to sweat each proof out) they were like poems!
I agree with Jane. I loved my first year calculus book – Taylor/Phinney, IIRC. Each chapter started with a page of real world applications. The one that sticks out in my mind was using multiple integrals to find the force on a sail on a sailboat. That one aspect of the chapter made reading the proofs and rest of the relatively dense text much easier.
There probably should be a stronger effort to move textbooks from being collections of mathematical papers. Introduce the concept of math writing and reading in the third year or so once students really commit to a major?
Agree with your comment on math textbooks. Repetition and reinforcement are key to learning.
Plus many math textbooks are poorly organized and written. And yes, I’ve written a textbook myself.
Math can be surprisingly hard to organize!
What topic is your book on?
Great post–so many topics I’m interested in!
1) I am Queen Book Learner. I started in the “hard” series of calculus in college (bio major), then switched to the “easy” sequence. So, maybe harder math books are different, but the first day of “easy” calculus, the teacher said he taught right out of the book, followed it exactly, so I never returned to class and did fine.
I need to STARE to learn. Incidentally, I’ve never been a fan of reading quickly.
2) I DID run an experiment re: learning, once. Attended 1 chemistry lecture (hm, ochem??) and took NO notes to see if I could actually learn more from the teacher WHILE the class happened. Yes, I did. Totally got it right then, which was new and exciting.
The next day, I had no notes, and NONE of the previous day’s lecture was in my head. Lesson learned (so to speak).
3) Also re: the comment of doing bio labs then describing them v. reading the books. I had VERY few labs in bio, and LOVED learning out of the book. Once reread a section of bio text years after graduation, and in one section I struggled w/understanding what they meant. I was really stuck, then saw my writing in the margins where I’d translated it YEARS before, and PING! Light bulb in brain is lit, and I’m off to the races. =)
4) There are a few popular science writers who treat topics of evolution etc. and I CANNOT READ THEM because they repeat SO MUCH I want to stab myself. Or them. Giving a real world e.g. of a concept is one thing, give FOUR examples makes me throw your book across the room.
Certainly an interesting perspective on a very pervasive problem. I saw the provocative question “Why Aren’t Math Textbooks More Straightforward?” on Quora just the other day. I take much the same view as you (though I religiously read not only my assigned math texts for classes, but also read their competitors in search of increased clarity and beauty.) I spent a few minutes writing up a few thoughts on what is going on in math textbooks taking a broader view of the forest instead of complaining about the trees.
Perhaps some of the tentative will gain a better appreciation for mathematics: http://boffosocko.com/2015/03/16/why-arent-math-textbooks-more-straightforward/
The repetition that a good mathematician (or really any good thinker) reads books with a pencil in hand needs to be drilled down to younger learners.
A parting math textbook joke:
Q. Why did Nicolas Bourbaki quite writing mathematics textbooks?
A. “He” realized that Serge Lang was just one person.
Grant Wiggins had a similar discussion about reading generally: Maybe We Don’t Understand …
Read the introduction quickly, focus on the part with the Kant quote, then go back and read from the beginning again.
The key point: difficult texts in any discipline require active engagement and study to understand. Also, while the definition of “difficult” varies by reader, there are ways to confound even experts (see the questions over whether the ABC conjecture has been proven).
As writers, though, do we plan to write difficult texts? No, if our main purpose is helping the reader understand the subject. In that case, we want to make use of multiple representations, heuristics (possibly with warnings), analogies, examples, and suitable repetition. No, if we are writing a book of puzzles where the purpose is to challenge the reader. No, if we are writing to merely communicate that we (the author) understand the material.
Good post! I find myself hoarding tons of math books (i’m still in college) and my friends chastise me for never reading them; they don’t understand that it’s difficult! Because i’m a pretty skilled math book-finder, i’ve become very picky with the particular texts I purchase; God knows how many times i’ve had to use a text that is just awful (Boyce and Diprima’s book on ODEs come to mind, or worse…Vector Calculus by Marsden and Tromba…*shudders*). The important things when keeping an eye open for good math books are:
1) is it accessible? This could mean cost wise, or the language used. If you find a text that presupposes a knowledge of homology, but you’ve never taken an intro topology course, then it’s probably not the best text!
2) do the theorems contain complete proofs?
3) do other physics/math peers like this book? I find that running an author/book by my other math/physics friends works pretty well, and you get a lot of good hints and tips for reading as well as possible recs for other books.
This is a lot of info no one asked for lol but here it is!
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… wait, those strands of words they put in-between the lines of equations are supposed to be read? When did they start doing that in mathematics books?
I often find an alternative textbook to the one suggested in class, reading different versions of a proof adds to my understanding of all versions. Also, like you say, rewriting a proof with your own words helps.
Interesting you say this. I always read the book, because I was too lazy to go to class LOL, so I got used to parsing dense writing. I think in a way, math is easier to read than other things because it’s precise; you can usually back-step to the definitions if you forgot something. It puts you in the habit of “loading” definitions into your memory and organizing your thoughts to handle the chains of definitions, and this process itself is not too terrible (but learning really complicated stuff can be haha). Contrast this with say physics, where they say things over and over but it’s not really clear exactly what they’re talking about…
Nice Post …compulsive reading sounds so familiar 🙂
TeachMathfree
When I took my first course in Analysis, it took me a long time to realize that reading the book before and after lecture was very important (as well as re-reading sections a few weeks later). It took me longer to realize that only reading 5 to 7 pages in an hour and a half wasn’t an abnormal way of doing things and was to actually be expected. The information density is much higher than I was used to. Now I actually enjoy reading a math text with pencil and paper in hand.
I really like how you broke down the proof of irrationality of root two by using different colors. Your board work must be beautiful.
Wouldn’t it be great if all math classes focused more on reading math and developing the necessary skills? I’m currently in a teacher education program where we are learning how to incorporate these literacy skills in our classes. A strategy we a learning to use is Talking to the Text, developed by Reading Apprenticeship, where basically we teach students how to do exactly what you did with your proof and what you talk about in your post. Students learn how to annotate a text based on measuring their understanding of what the text is saying and connecting it with other things they already know. So there is hope for our future math students to be prepared to engage difficult texts, and in particular math texts.
I know that every fraction is rational. But I do not know how is it possible?
Actually, not every fraction is rational. Pi/2 is a fraction but it is not the ratio of two integers, nor can it be simplified into such a ratio. The precise definition of rational numbers is the set of all numbers WHICH CAN BE EXPRESSED AS the ratio of two integers, p and q, such that q is not zero. Thus, 1 1/2 is a rational number, as it can be expressed as 3/2. 0.333. . . is a rational number because it can be expressed as 1/3. But in general, the ratio of two irrational numbers is not rational, and the ratio of a rational number and an irrational number is NEVER rational. Those can, however, be considered to be fractions, which is a less restrictive concept than rational number.
I don’t remember how I stumbled in here but I’m glad I did. Very informative and a unique way to approach understanding math. I’ll take these steps with me next semester in my probability class where the professor isn’t very helpful and students must teach themselves with the book!
I figured this out myself, self-studying AP Calc and AP Chem. I don’t think it is such a rocket idea that you have to work the example problems as the explication goes along, or fill in the missing steps in a derivation (usually even the ones that aren’t crazy opaque, still do have at least some basic algebra movements to verify). That’s actually good, because working it helps you understand and remember the point.
Is this such s novel concept? Note, I am not saying I am in favor of crazy hard books or the like. But many students don’t even follow a pretty accessible text.
Just buckle down and do it.
As someone who has been scared of and struggled with maths most of her adult life (I am talking about real nightmares here), I think teaching maths should start with this blog. For both teachers and students.
Hello! I’m an undergrad at the University of Illinois at Urbana-Champaign studying math with a minor in secondary education. I find your ideas to be intriguing, and I think it would be valuable to include reading in my future general education math classroom. Is there any suggestion you have on ways to do this effectively? How can I encourage student comprehension? If anyone has any ideas, please let me know!
You have the “to be” and “not to be” mappings exactly backwards.
Huh – can you explain your gloss of the sentences?
My understanding is that “to suffer the slings and arrows of outrageous fortune” means to stay alive and just deal with the fact that life is unfair, whereas “to take arms… and by opposing, end them” means to kill oneself, thus bringing an end to the unfairness.