There Are No Lazy Students (WIL #8)

How many lazy students have you known? How many stupid students? Maybe none. I’ve never known anyone, student or not, that I was convinced was lazy, and very few I was confident were stupid. Laziness sounds like a fundamental aspect of a person’s character. But how can you know that, especially about someone you’ve known only in the context of school? A much better way to think about someone that doesn’t want to work on whatever they’re supposed to be doing is just that they’re unmotivated. One very experienced teacher I made this argument to commented that “even the laziest people I have known were willing to work hard on things that interested them, and I suspect more than a few were depressed or discouraged.” It may not be possible to motivate students to do what you want them to do, but once you decide they’re lazy or dumb, you’re already most of the way to giving up on them. That’s an easy out for the teacher—too easy.

I think this is a really important point. As just one example, how about an exceptionally talented student who doesn’t work in class simply because the material is too easy and they’re bored stiff? This is important both because it may well be more common than educators realize (how can anyone know?) and because it’s particularly unfortunate to lose talented students for no good reason. Yes, little Ina Albertstein might truly be lazy or stupid, but probably not. As her teacher, it’s not likely you’ll ever know for sure, and assuming she is has far more potential to hurt than to help.

Be realistic about your strengths and weaknesses and adapt! (WIL #7)

It’s important to be realistic about your strengths and weaknesses and adapt accordingly. An important special case: Grading homework and exams can easily overwhelm the teacher. My brief tenure as a regular, full-time classroom teacher was as a maternity-leave replacement, and by the time I started, I knew I was a slow grader. I had four (count ’em) preps, and getting four lessons ready every day took so much time that I quickly realized that I’d have very little time to grade homework. From the beginning I didn’t try to grade as much homework as the very experienced teacher I was filling in for did. But I couldn’t even do what I thought (and had told my students) I could. It wasn’t until I graded the final exam that I realized how slow I was! I can see several reasons for this: wanting to give students really useful feedback, wanting to grade as consistently as possible, my inexperience, etc. If I teach much more, I’m sure I’ll get more efficient, but not by so much that it won’t always be an issue. But — as Ofer Levy pointed out to me later — the teacher usually has quite a bit of flexibility in how much grading they have students generate. One reason is that there are often ways to substantially reduce the load of grading without harming learning, so the teacher can avoid being a victim of a system they created themselves. Some ways that seem appropriate for teaching math, along with many other subjects: assign group instead of individual work; let students correct each other’s work; go over homework and quizzes in class but don’t collect or grade them. I did some of this, but could have done much more… and if I had, not only would I have been better off, my students would have been too because I would have had more time for their more serious problems.

Some students’ problems are beyond what a classroom teacher can solve (WIL #6)

Some students’ problems are beyond anything a classroom teacher can solve. Here are two examples from “Brief Survey of Calculus”, a 100-level college course I once taught—but the same problems could occur at the secondary-school level.

(a) Problem with course content. “Colton” told me he always struggled with math, and he certainly did in my class. He spent as much time in my office as all the other students combined; that seemed to help some, but not that much. And he wasn’t at all stupid. Like so much mathematics, calculus requires a fair amount of algebra, and I already knew many of my students’ problems were more with algebra than with calculus. So I finally asked Colton a very basic question: How much is 3 × (5 × 4) ? I wasn’t too surprised that he tried to use the Distributive Law, as if the question was 3 × (5 + 4) ! Colton was a senior, and older than most—probably 23 or 24 years old, and the Distributive Law is generally taught by 6th or 7th grades (under the current Indiana standards, it’s taught in both). So he’d undoubtedly been confused about this for well over 10 years, and it takes a long time to unlearn something that deep-seated.

(b) Problem unrelated to course content. At the beginning of the semester, “Isolde” really seemed to want to do well, but she did several odd things that hurt her grades enormously. For example, she had a graphing calculator but not a non-graphing one. I didn’t allow graphing calculators on quizzes/exams, so she started the first quiz/exam without any calculator until I noticed and loaned her one. Exactly the same thing happened on every other quiz/exam! She kept missing classes and even exams, and she never came to my office hours; she blamed all of that on transportation problems. And she repeatedly forgot important things. Perhaps she had tremendous family responsibilities, or she might have had ADD, or both. But she never told me what was going on, and, sadly, I couldn’t find a way to help her.

Example (b) is clearly related to my What I Learned item #1, “students really are responsible for their own learning”; but example (a) is something else.

In classroom management, the more concrete, the better (WIL #5)

For my first semester as a student teacher, I taught 6th grade math. Many of my students had a hard time controlling themselves, and one of my standard tactics was to make a list of offenders I would make stay after class. A very common problem, and a common way to handle it. But it was obvious that the clearer the connection between the behavior and the consequences, the more effective a deterrent it was. So my partner and I started putting a timer on the smartboard to keep track of how much class time the offending students had wasted, and announcing that they’d be staying after class for the same length of time.


To make the connection even more concrete, I looked for a simple, easy-to-use program that would count down whatever length of time it had gotten up to, and preferably one that could run in a Web browser with no installation. But I couldn‘t find one: every program I ran across treated going up and down as unrelated functions, typically calling them “stopwatch” and “timer”.

Well, I know how to program a computer: I spent years working as a software engineer. So I finally (long after my semester in middle school) wrote it myself. Following a suggestion by my friend and colleague Jeremy Sebens, you can even set a time of day — the time the class ends is the obvious candidate — at which it’ll start counting down automatically!


“Don’s Up/Down Timer” is available at

There are also instructions for using (and, if you want, customizing) it there, but it requires no installation and it’s very easy to use.

Another classroom application might be the opposite of the above scenario: you want to reward students for doing something they don’t like — and the longer they do it, the longer they get to do something they do want to do.

I think Don’s Up/Down Timer would be most helpful in middle school, but it could be useful in any situation where you want to measure an unknown interval of time, then set a timer to go off when the same interval elapses again. Prof. Kathy Marrs of IUPUI tells me she’s used it with her college-freshman biology class — I’m not sure for what. Certainly there’s nothing specific to math about it.

(4 Feb. 2013: revised to add the “rewarding” application and update the URL.)

When presenting content, the more concrete, the better (WIL #4)

I’ve learned that, when presenting content, the more concrete, the better. This is something I thought I knew quite well by the time I started my first (and only, so far) regular job as a secondary math teacher, but I found otherwise! For example, many of my Algebra II Honors students were having trouble understanding negative exponents. To make the relationships involved more concrete, I drew the following table on the greenboard, showing how positive integer exponents relate to repeated multiplication:

  x1 x2 x3 x4
  x x· x x· x· x x· x· x· x

Then I observed that each time you add one to the exponent, you multiply by x one more time. I asked the class how this could be extended to non-positive exponents. Not getting an answer from them, I answered myself by pointing out that that rule is equivalent to the rule that each time you subtract one from the exponent, you divide by x one more time. I added a few columns to the left:

x–2 x–1 x0 x1 x2 x3 x4
1/( x· x) 1/x 1 x x· x x· x· x x· x· x· x

I commented further that defining powers of x below the 1st that way is the only way to be consistent with positive powers.

That helped some, but not as much as I expected. It was only much later that I realized I could easily have made things really concrete simply by substituting, say, 3 for x ! (It might also have helped if I’d used a variable other than x, since x looks a lot like a multiplication sign.) I’ve made this mistake of not being as concrete as possible many times. I think the main reason is that I tend to assume that expressions like the ones above are already so concrete, there’s no need to get even more so. But that’s not a safe assumption, to put it mildly—not even with an honors class like mine.

One Tenth of A Picture Is Worth A Hundred Words (WIL #3)


This gallery contains 3 photos.

The old saw, “a picture is worth a thousand words”, applies to teaching mathematics and the other STEM disciplines as much as anything, and I think most STEM teachers are reasonably aware of that. But there’s a discipline called graphic … Continue reading

Appropriate Preparation for Teaching as a Function of Students’ Ages, etc. (WIL #2)

To continue justifying the “What I Learned” part of the title of this blog, here’s item #2 from my “Thoughts on Teaching: What I Learned about Why My Students Didn’t Learn More“.

The appropriate preparation for secondary-school teaching varies greatly with students’ age, maturity, etc. As a student teacher in middle school, I discovered over and over that I hadn’t prepared thoroughly enough. In particular, transitions are an opportunity for 6th-graders to get completely distracted; a transition I didn’t manage well could easily waste five minutes of class time. I found my best bet was to have a written plan for every lesson describing what I’d do in some detail, including transitions. But then I taught high school, and I quickly discovered that high-school students are far less distractable, and generally have no problem with transitions. With them, I found I was better off spending less time on detailed plans and more time on other things, e.g., having written solutions to the more difficult homework problems. That really made a difference. (The experienced teacher I was filling in for could answer almost any question a student might have immediately; I could answer many questions immediately, and most questions almost immediately.)