Students really are responsible for their own learning (WIL #1)

It’s time for me to start justifying the “What I Learned” part of the title of this blog. I’ve written an article with a similar title, namely “Thoughts on Teaching: What I Learned about Why My Students Didn’t Learn More“, centered on a list of 10 specific things I learned. Below is number 1. I’ll post the other nine in the near future.

It’s important to note that in some sense I knew most of these things very early, certainly after my first month of student teaching; but I didn’t know any of them deeply enough to apply them effectively. By the time I started this essay, 18 months later, I could apply all of them (pant, pant). In addition to the education courses I took and my experience in the classroom, I learned a great deal from outside reading, especially the superb books by Polya (1957), Holt (1982), and Lockhart (2009), the latter with its scathing and eloquent critique of math-education practice.

Naturally, these ideas don’t apply equally to students of all ages; in general, the younger the students, the more relevant they are.

And without further ado, here’s the first thing I learned.

1. To a great extent, students really are responsible for their own learning. To be crystal clear, I mean that students are in control more than anyone else is—certainly more than their teacher—of how much and what they learn. This may be the single most important thing I learned. This idea can be paraphrased in many ways, for example, a statement that’s often attributed to Einstein: “I never teach my pupils; I only attempt to provide the conditions in which they can learn” , and Holt’s (1982) statement:

“I doubt very much if it is possible to teach anyone to understand anything, that is, to see how various parts relate to other parts, to have a model of the structure in one’s mind. We can give other people names, and lists, but we cannot give them mental structures; they must build their own.”

But I’ve tended to feel that, if the students aren’t learning, it’s “my fault”, and one day last fall, I said something to a far more experienced instructor that made that clear to her. She told me in no uncertain terms that it was a mistake to have that attitude, but I still didn’t get it. Last spring, I taught high-school math as a maternity-leave replacement. Near the end of my time there, a class (Honors Algebra II) went especially poorly one day, and I thought of five or six reasons why. All or nearly all were things I’d done or failed to do. The next day, before that class, I talked to Mrs. W., the teacher next door, about my frustration that students weren’t “getting” what I was teaching, and that very few were taking advantage of my repeated offers of help outside class. To my amazement, she told me that Mrs. P. (the teacher I was filling in for) typically had 10 or 15 students in her room getting help before school! Hearing this, I suddenly realized that my teaching was by no means the only reason my students were having trouble. I might have one student come in before school and one after school, and often not even that. But clearly they weren’t coming in outside of class because they weren’t motivated; wasn’t this in itself a problem with my teaching? It’s hard to tell, but—being as honest as possible—I’d say it was mostly because of factors that I couldn’t control, for example, the temporary nature of my position. (I told my class what Mrs. W. had said and again urged them to get help from me outside of class; I didn’t see any increase in help seekers, but I felt considerably better about my teaching after that.)
Thus, the question above “Did I teach the average student as much as Mrs. P. would have?” is seriously misleading. A better question is “Did the average student learn as much as they would have with Mrs. P.?” The answer is almost certainly no, but there were mitigating circumstances.


Holt, John (1982). How Children Fail, revised ed.  Addison-Wesley.

Lockhart, Paul (2009). A Mathematician’s Lament. New York: Bellevue Literary Press. Part I of the book (the “Lament” proper) retrieved May 4, 2012, from the World Wide Web:

Polya, George (1957). How to Solve It, 2nd ed.  Princeton University Press.


Antidisestablishmentarianism the Dog

Warning: While I’m putting this in the “Lessons” category, I doubt if you can use it without major surgery!

I taught math at a small-town high school early this year, as a maternity leave replacement. I decided from the beginning to cultivate an eccentric and playful persona in front of my classes; this was easy to do because I really am both somewhat eccentric and decidedly playful! For example, my first day, I told them my family has a dog named “Antidisestablishmentarianism”: with 28 letters, that’s allegedly the longest word in the English language. (See photo. I’m pleased to say he’s registered with the American Kennel Club under that name, though we usually call him just “Tarry”.)

I explained where the name came from; the reason has minor mathematical implications. Specifically, I was inspired to call him that by the children’s song, “Bingo”. The first verse goes like this:
There was a farmer had a dog and Bingo was his name-o.
B, I, N G O;
B, I, N G O;
B, I, N G O,
and Bingo was his name-o.
The second verse is the same except that each “B” in the 2nd through 4th lines is replaced by a hand clap. And in each of the four remaining verses, one more letter in those lines is replaced by a hand clap, so that by the last verse those lines consists entirely of claps.
Thus, “Bingo” has one verse per letter of the dog’s name, and the length of each verse is also a function of the number of letters in the name. But I’ve always been amused by thinking about a version in which the dog’s name was Antidisestablishmentarianism! (And when my family gave me the opportunity to name our new puppy that, some years ago, I didn’t say no ☺ .)

This suggests the question I asked my students: how much longer would it take to sing if the dog’s name was much longer than “Bingo”—say, the 28-letter word “Antidisestablishmentarianism”? Removing the reference to a farmer to make it easier to squeeze all the syllables in, it might start like this:
There was a dog and Antidisestablishmentarianism was his name.
A N T I D I S E S T A B L I S H M E N T A R I A N I S M;
A N T I D I S E S T A B L I S H M E N T A R I A N I S M;
A N T I D I S E S T A B L I S H M E N T A R I A N I S M;
Antidisestablishmentarianism was his name-o.

There was a dog and Antidisestablishmentarianism was his name.
[Clap] N T I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap] N T I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap] N T I D I S E S T A B L I S H M E N T A R I A N I S M;
Antidisestablishmentarianism was his name-o.

There was a dog and Antidisestablishmentarianism was his name.
[Clap clap] T I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap clap] T I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap clap] T I D I S E S T A B L I S H M E N T A R I A N I S M;
Antidisestablishmentarianism was his name-o.

There was a dog and Antidisestablishmentarianism was his name.
[Clap clap clap] I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap clap clap] I D I S E S T A B L I S H M E N T A R I A N I S M;
[Clap clap clap] I D I S E S T A B L I S H M E N T A R I A N I S M;
Antidisestablishmentarianism was his name-o.

Good luck with the dozens of consecutive hand claps near the end… Anyway, most of my students gave the obvious but incorrect answer, 28/5 times as long. The next most obvious answer, (28/5)2 is closer, but also wrong. Of course the problem isn’t really well-defined, and a very interesting discussion of what the best answer is might have ensued, perhaps considering expressions involving the duration of a hand clap vs. saying the name of a letter, etc.; but I didn’t want to spend a lot of time on it. The main thing I wanted to accomplish with this was to connect with my new students, and I did!

Paul Lockhart’s “A Mathematician’s Lament”

“A musician wakes from a terrible nightmare.  In his dream he finds himself in a society where music education has been made mandatory.  ‘We are helping our students become more competitive in an increasingly sound-filled world.’  Educators, school systems, and the state are put in charge of this vital project.  Studies are commissioned, committees are formed, and decisions are made — all without the advice or participation of a single working musician or composer.

“Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the ‘language of music.’  It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory.  Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school…”

So begins Paul Lockhart’s “A Mathematician’s Lament” (Lockhart 2002; expanded version, Lockhart 2009), the most thought-provoking (and, in a way, depressing) article on teaching math I’ve run across in a very long time. The opening hits me especially hard because I’m a musician myself. I’ve spent many thousands of hours working with what’s technically called “Conventional Western Music Notation”: my undergrad degree is in music composition, and — long ago — I spent years developing a music-notation score editor. What Lockhart describes is indeed a nightmare! He goes on about music for several more paragraphs; describes a painter with a similar bad dream; and then says “Sadly, our present system of mathematics education is precisely this kind of nightmare.” I regret to say that I have to agree. He goes on:

“If I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.

“Everyone knows that something is wrong.  The politicians say, ‘we need higher standards.’ The schools say, ‘we need more money and equipment.’ Educators say one thing, and teachers say another.  They are all wrong.  The only people who understand what is going on are the ones most often blamed and least often heard: the students.  They say, ‘math class is stupid and boring,’ and they are right.”

I’m tempted to quote more, but you really should read it yourself. Don’t miss what Lockhart calls “the first ever completely honest course catalog for K-12 mathematics”! Ouch.

Though written years earlier, Lockhart’s Lament became widely known only after Keith Devlin wrote about it in his column on the Mathematical Association of America (MAA) website (Devlin 2008). Devlin commented that the Lament is “one of the best critiques of current K-12 mathematics education I have ever seen”; me too. It’s all pretty sad. Well, Lockhart exaggerates some. While he gives math teachers credit for good intentions, I think he underestimates how well a lot of us understand what mathematics is really about. Much of what he says has been said before, and some of it has been said better — for example, by G. Polya in the very first section, “Purpose”, of his classic How to Solve It (Polya 1957). And at one point Lockhart says that people are “under the gross misconception that mathematics is somehow useful to society!”; that’s pretty silly. Still, by and large, Lockhart’s writing is brilliant, and I think his scathing criticism is justified.

“But so what?”, I can hear you saying. “In the current political climate, making radical changes in the math curriculum in the vast majority of school systems is hopeless.” True. Of course your school might be an exception. Regardless, to paraphrase what Devlin wrote in his Foreword to the book edition, the book is a “landmark in the world of mathematics education that cannot and should not be ignored… You should read what he says and reflect on his words.” I shudder to think of how I taught geometry, especially proofs, not so long ago; if I’d read and absorbed Lockhart, even under my school’s requirements, I could have done much better.

One more thing. Have you heard of this essay? It was written in 2002, and it seems to be very well-known in some circles — but, unfortunately, mostly college math teachers (e.g., MAA members), not secondary-school teachers or math-teacher educators. Well, in-service teachers are generally (and understandably) utterly focused on short-term goals like getting through the week, month, and semester; but math specialists in teacher-training programs can afford to step back quite a bit. I completed an M.S. in math education just a few months ago, and I think the program I went through was a good one. But I don’t recall Lockhart was ever mentioned, nor do two of my fellow graduates I asked! And one member of the secondary-math-education faculty I asked had never heard of it. That’s interesting, too, isn’t it?

In future posts, I’ll talk about other slants on what’s wrong with American math education and what can/should be done — for example, Vi Hart’s slant. If you haven’t seen her stuff, don’t wait for me; spend a few minutes today watching one of her wonderfully engaging “Doodling in Math Class” videos (Hart, 2012)!


Devlin, Keith (2008, March). Devlin’s Angle: Lockhart’s Lament. Available at

Hart, Vi (2012). Doodling in Math Class. Available at

Lockhart, Paul (2002). A Mathematician’s Lament. Available at

Lockhart, Paul (2009). A Mathematician’s Lament.  New York: Bellevue Literary Press. (Consists of the “Lament” proper (Lockhart 2002) plus a second
part, an “Exultation”, with a Forward by Keith Devlin.)

Polya, George (1957). How to Solve It, 2nd ed.  Princeton University Press.

Meaningful Numbers and “Significant Figures”

. Isn’t the idea of owning a hammer whose weight you know to five significant figures thrilling? Too bad only Spanish and French speakers get that much information, while we English speakers have to get by with only one figure! Ahem. Of course, the additional “information” in the Spanish and French versions is almost certainly meaningless.

Not long ago, I taught Brief Survey of Calculus, a one-semester 100-level course, as a part-time instructor at a university. The only major requiring the course was business, and naturally, I was supposed to teach a very “applied” slant on calculus, heavily emphasizing problems with only approximate solutions. Okay, but I soon discovered that most of my students had no idea of what a reasonable approximation was! I finally wrote something up on significant figures—see the attached document—and I added the following text to the Course Policy. (I realize that changing the Course Policy during the course isn’t a great policy, but I felt this was too important to let it go.)

In this course, please use at least 4 significant digits in all calculations, and give answers to at least 3 but no more than 5 figures. On one problem in Exam #1, looking for a function to fit the data given, one student wrote “2 / 1.5 = 1.3”, which is lower than the correct value of 4/3 by about .0333. That may not sound like much, but it’s a large relative error: .0333 is over 2% of 1.3. That’s enough to lead to the wrong answer in many situations, including that problem!

But why not give answers to more than 5 significant figures? Because in situations involving mathematical modeling, it can be very misleading. A question on Exam #1 said a company’s sales were “$257 million” in a certain year, stated they’d gone up by “at least 5%” every year since, and asked for the minimum sales in a later year to the nearest million dollars. Several people answered “$361,624,809”, or even “$361,624,808.6”. These look like very accurate figures, but how could the last five or six digits mean anything? “$362 million” is a much better answer.

Naturally, when a problem specifically calls for more accuracy or for an exact answer, these rules don’t apply.

I’ll bet I’m not the only teacher who’s run into this problem! Feel free to use some or all of my verbiage, and by all means the picture of the hammer label. (If you’d like a 226.72 gram hammer of your own, I’ve seen this model for sale in a number of places, including CVS drugstores. You can write yourself a reminder to get one on a 5.08 cm. square PostIt note.)

Hello (Math Educators of the) World!

Welcome to “What I Learned About Why My Students Didn’t Learn More”! This blog is addressed mostly to secondary-school math teachers, though I hope and expect some of it will be of interest to other STEM teachers and to parents. Who am I and why should you read what I write?

First, my background.

I had a long career going back and forth between industry and academia, mostly applying computers to solve problems in music, Geographic Information Systems (GIS), and information retrieval, and concentrating on visualization and what is now called user experience design. Well, I’d always loved mathematics, but had never really concentrated on it. In addition, by 2009, I’d come to believe the average American’s general lack of understanding of math was a very serious problem and I wanted to help improve matters. So I decided to try a new career. As a student teacher, I taught at a magnet school, and I had a precalculus classroom in which many students took courses at a college less than a mile away; their schedules went in and out of sync with ours because of our block schedule, so they often missed entire weeks! (What administrators would let this ridiculous situation arise? I can only say that this was one of the Indianapolis Public Schools, schools that don’t have a reputation for good management.) In response to the situation, I created supporting materials, including 2-column notes for each lesson, and put them on a webpage. Then — while looking for a “real” job as a secondary-school teacher — I taught freshman calculus at the same college, and created a webpage for that, with improved versions of some of the same material as well as new material. Then I taught high school as a maternity-leave replacement for a short time, and when that gig was over, I had even more time to put things together.

Why should you read what I write?
1. As the above description of my recent career suggests, I’ve been lucky enough to be able to spend a long time thinking about and — much less common, I believe — writing the stuff I’m posting here.

2. I do okay as a teacher, but I’m not particularly talented at it. But I think I am a very good writer, and I love to think about, and write about, how things can be done better.

3. I’ve seen math education from the parent’s perspective: I have two children, both in college now.

And without further ado, voila: “What I Learned About Why My Students Didn’t Learn More.”