Send me all your money TODAY! There’s never been a better time!

You’re probably thinking, “Is this some kind of scam, or has this guy lost his mind?” I don’t think so, though I must admit the title is kind of a dramatic way to make my point. (I’d originally intended to call this “Kill your self today! There’s never been a better time!”, but I thought that might be a bit too much.)

I saw an ad on the Google homepage recently: “Fast. Smart. Safe. It’s never been easier to put _Firefox on your Android phone._” This strikes me as yet another example of a kind of statement that’s been annoying me for years. The statement, nearly always in an ad of some sort, exhorts you to do such-and-such now because “there’s never been a better time”. But it’s clear that what they want you to think is that you should do it because “this is the best time ever”! A much stronger statement — and one that (in my experience) really is often justified by the facts. But why would advertisers ever use a weaker argument than is justified? The only reason I can think of is carelessness.


It pisses me off (please excuse my language. I didn’t realize I felt that strongly about this until I wrote that phrase!) that people are so careless about the fact that “less than” and “greater than” aren’t the only possible relationships between two ordinal values. I suggest bringing this up to students to raise their awareness of mathematical thinking in the Real World. A possible assignment: find 20 random examples of statements that “there’s never been a better time” for something and report on how many really mean “this is the best time ever”.

If you’d like to see more examples yourself, there’s never been a better time! At the moment, Googling “there’s never been a better time” (with quotation marks) gets “about 3,360,000 results”.


Price’s Math Myths

The late Justin Price was for many years a professor of mathematics at Purdue University. Price was particularly interested in math education, and one of his students passed this list (dated 1991) on to me. It certainly captures the way a lot of the apathetic math students I’ve known seem to think about mathematics—though for #8, five minutes might be more accurate than 30 seconds! I’ve put this list on my class websites. I don’t know if my students paid attention it, but I think it’s worthwhile for math teachers to think about these popular misconceptions. (Thanks to Paul Weedling for sharing this list with me.)

Justin Price at blackboard

1. Math is a bunch of formulas and rules.  Doing math means plugging numbers into formulas and moving symbols around using the rules.

2. Only a genius knows where the formulas and rules come from. So just accept them and use them.

3. Memorize everything and you’ll be OK.

4. Know how mathematical operations work. You never need to know why they work.

5. The most important thing is getting answers. If you get the right answer, that proves that you understand the math.

6. Every problem has an answer.

7. To solve a problem, just follow the steps in the book. Any problem that is not exactly like one worked in the book is unfair.

8. Any problem that takes more than 30 seconds is either impossible or unfair.

9. Story problems are unfair.

10. There is one way to do a problem—the book’s way. No other way is allowed. Guessing the answer is strictly illegal.

11. Common sense is OK in real life, but it has nothing to do with math.

12. Reading and writing are for English classes; they have nothing to do with math.

13. Math comes in separate subjects: algebra, geometry, calculus, etc. There are no connections between the subjects.

14. Math can never be interesting or fun.

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.)

Syntax of Math Notation: the Anatomy of a Term

At every level I’ve taught at, I’ve found a lot of students are confused about the syntax of algebraic notation. Here’s a handout I gave to my middle-school, high-school precalculus, and college calculus students, showing the coefficient, a single variable, and an exponent, and saying what the defaults are if anything is missing.

I think many of my students found this helpful, but it could be used as the basis for something more engaging, maybe even perplexing, to use Dan Meyer’s word (Ten Design Principles For Engaging Math Tasks). Let’s challenge students to think about what the defaults should be — no, what they must be — if parts of the term are missing, as they often are. If there’s no coefficient, you must assume it’s 1. Why? Because multiplying by 1 doesn’t change the value; multiplying by anything else does. And so on.

Feel free to use these however you like (though I’d appreciate it if you’d give me credit). Higher-resolution versions are available of both the “challenge” version and the “reference” version.