Fermat’s Last Theorem on the Greenboard

FermatsLastThmOnTheHSGreenboardTRIn the first week of my very brief career as a regular full-time high-school math teacher, my students and I somehow started having a conversation in the lower-right corner of the greenboard at the rate of one statement by each party per day. Their first two comments were very complimentary, and I responded as shown above. Of course the math in the first sentence is Fermat’s Last Theorem. Not that I expected them to understand it (despite the fact that I had all honors classes), or even to be interested in understanding it! But they definitely were interested. So I had the delightful experience of talking to my Algebra II students about number theory for a few minutes.

This is one of several experiences that convince me that a significant fraction of high-school math students — at least of honor students, but probably of ordinary students as well — haven’t had all their natural curiosity about things mathematical squeezed out of them, despite the worst efforts of the American math-education system. It’s well worth a few minutes of classtime now and then for a teacher to try to take advantage of it.

Why does 0! = 1 ?

Below is some email I recently exchanged with my nephew John, an  adult whose unusual upbringing left him as a beginner at math. But he’s intelligent, and very enthusiastic; how many students at this level would try to prove anything?

Subject:      Re: Factorials… Why does 0! = 1?


I applaud your curiosity about this! The basic problem here is that — I’m pretty sure,
but I’d be very interested if your instructor or TA disagrees — what 0! is isn’t a
matter for proof; it’s a matter of definition, just like the definition of factorial
for a positive integer. The real question is what is a useful definition of 0! And, in
math, an essential feature of a useful definition is that it will never lead to a
contradiction. If things in math are defined in such a way that you can get contradictory
results, it’s a disaster. That’s just why dividing by zero isn’t allowed. Now, read on.

On Wed, 5 Mar 2014 16:32:42 -0500, John Doe <jdoe@geemail.com> wrote:
> I rechecked my study materials regarding 0! and it was not explained. I
> emailed the head TA and she confirmed 0! = 1. However, I wanted to “prove”
> it so here is what I came up with:
>           *Solve:*            0!
> *Answer*:          1
> 1. Factorials always equal a natural number.
> 2. 0 is not part of the number set of natural numbers.
> 3. Negatives are not part of the set of natural numbers.
> 4. Without a 0 or a negative, 0 can never be found in an
>    equation given a positive value.
> 5. Then by definition 0! = 1.
> I don’t know how to do a proof, but this was how I can logically reason the
> answer.

There’s nothing wrong with steps #1 thru #4, but #5 does not logically follow from
them. They don’t establish that 0! factorial isn’t 953 or any other positive integer.
What forces us to define 0! as equalling 1 is that any other definition will lead to
big problems. For example, given n people, the number of different pairs of them
(ignoring order, i.e., we’re talking about combinations and not permutations, if you know
those terms) is

n! / 2(n-2)!

If n is 3, that’s

3! / 2(1!) = 6 / 2 = 3

And if n is 2, it’s

2! / 2(0!)

Well, that expression had better turn out to equal 1 — and it won’t unless we agree that
0! = 1.

Finally, from the Wikipedia article on natural numbers: “There is no universal agreement
about whether to include zero in the set of natural numbers: some define the natural
numbers to be the positive integers {1, 2, 3, …}, while for others the term designates
the non-negative integers {0, 1, 2, 3, …}. The former definition is the traditional
one, with the latter definition having first appeared in the 19th century.” Wolfram
MathWorld (http://mathworld.wolfram.com/NaturalNumber.html) says basically the same
thing. But this is a very small point; it sounds like your instructor is using the
positive integer definition.

Keep thinking!  🙂


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.

“The Heart of Mathematics” and the Hearts of Students

HeartOfMathV2I recently ran across the book The Heart of Mathematics: An Invitation to Effective Thinking, by Edward Burger and Michael Starbird. Burger and Starbird are both math professors at well-respected colleges, and they’ve both won multiple awards for their teaching. But The Heart of Mathematics is hardly a conventional textbook for college math classes! The publisher’s website (http://www.wiley.com/WileyCDA/WileyTitle/productCd-EHEP000304.html) says: “Infused throughout with the authors’ humor and enthusiasm, The Heart of Mathematics introduces students to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking.” I’d hesitate to say anything contains THE “most important and interesting ideas” in any subject; but that’s a quibble, and otherwise I agree completely.

I’ve always felt the best hope for engaging most students in math at almost any level is to expose them to what I call “wild and crazy ideas” — i.e., to go for the gold medal of intrinsic motivation, not to try for the consolation prize of external motivation by attempting to convince students (without much justification, and and usually without much success) that knowing math will eventually be useful to them. (If you’ve read Nicholson Baker’s piece in a recent Harper’s Magazine, “Wrong Answer: The Case Against Algebra II”, you won’t be surprised to hear that I agree with 90% of what he says.) After hearing me rant for a few months about such things, Frank Lester loaned me his copy of The Heart of Mathematics, saying he thought it was very much my kind of book. He was right. His only real reservation, Frank said, was that it makes things too easy by letting students see the answers to the many challenges they pose — but that’s hard to avoid with paper. I think he’s right about that, too.

I’ve been working for years on a list of wild and crazy ideas for teaching math, and a lot of the topics The Heart of Mathematics covers (different sizes of infinity, the Monty Hall problem, Simpson’s paradox, Möbius bands, etc.) are on my list — and, it’s clear to me, a lot of the others should be! Frank, thanks so much for exposing me to this book.

But this book seems to be almost unknown to secondary-school math people. If it’s so good, why is it that? Probably because it’s explicitly intended for college-level courses for non-science majors, and for that audience, it’s been a huge success: according to the publisher, it’s “the most widely-adopted textbook in liberal arts and liberal studies mathematics and teacher preparation in over ten years”.  But there’s plenty of material here for a book — or, perhaps better, an online course — for high-school students. More important, by the time students reach college hating math and having a hard time learning any, it’s too late!

I’d love to see Burger and Starbird come out with a high-school level version of the book, and Starbird tells me they’re interested. The problem, of course, is that this isn’t a textbook for any of the standard high-school courses, so it would be hard to be confident of its adoption by many school districts. Still, it’s worth a try; we as a nation need desperately to do something about our lame-brained approach to teaching mathematics — something other than pushing the same bad ideas even harder.

[revised January 2014: improved the illustration; updated the last paragraph.]

Baby’s First Infinite Series


Okay, okay, I admit this isn’t suitable material for babies, nor for any normal kid whose age can be expressed with one digit! The “baby” phrase just popped into my head and I found it too cute to resist. A more accurate title might be “Joanie & Johnny’s First Infinite Series”, or “Kids’ First Infinite Series”.

I’m always thinking about ways to get those apathetic middle- and high-school students interested in math again by showing them something wild and crazy, yet simple and concrete enough (for them, of course) that it shouldn’t be too difficult to understand. My latest idea is an infinite series with a surprising and very simple visual “proof”.  Can you guess what it is? Hint:  the “proof” is in each of the figures above (surprise, surprise). What it proves is…


I’m sure each of these informal proofs — really variations on a single proof  — has appeared in many places, but I ran across the one on the left first, in Roger Nelsen’s wonderful Proofs Without Words. (The one on the right is from the Wikipedia article “Geometric series”.) The two volumes of Proofs Without Words, published by MAA, contain dozens of marvelous and miraculous visual “proofs” on a wide variety of mathematical topics; I highly recommend browsing through them. Full disclosure, though: in my opinion, not many are as elegant or as easy to see as this one. Still, I believe that several of the proofs Nelsen has collected, certainly including this one, could be presented successfully on the middle-school level — though probably not with the above notation. Regardless of notation, of course, appreciating this proof requires some ability to do arithmetic with fractions.

I think most people with experience teaching high-school math would agree that (1) not many “apathetic” secondary-school students have much of an idea what a proof is (notwithstanding the emphasis on them in a typical geometry course), and (2) hardly any of them know why they or anyone should care! That may be a problem with my idea, but it strikes me more as an opportunity. I suspect that going through something like this proof would help with both problems; it would also expose students to the important idea of adding up infinitely many numbers and getting a finite sum.

Do you agree? Do you this could work (with a reasonable amount of scaffolding, naturally) as a discovery lesson for, say, 10th graders? If you think it’s worth trying, here’s an idea for “reasonable scaffolding”: Discuss the fact that 1/3 = 0.33333333…, with infinitely many 3’s, and ask the class what 0.33333333… with infinitely many 3’s actually means. Of course it’s a very simple infinite series, one with the same sum as the series I’ve been discussing:

InfiniteSumFormula2Since both series have the same sum, an obvious question might be whether it’s possible to draw a picture to illustrate the above equation, and if so, what it would look like. I leave those questions as an exercise for the reader.

Zeno’s “Achilles and the Tortoise” Paradox and Geometric Series

When I decided to retrain as a secondary-school math teacher and applied for a Woodrow Wilson Indiana Teaching Fellowship a few years ago, I had to teach a 5-minute lesson. It could be on any subject I wanted, and I chose Zeno’s first paradox of motion, the “Achilles and the Tortoise” Paradox. As you may know, the Greek philosopher Zeno (5th century BCE) devised several “paradoxes of motion” that baffled all of his contemporaries. His first paradox involves a race between Achilles (a very fast runner, said to be “the fleetest of foot of all mortals”) and a turtle (a rather slow crawler), with the latter having a head start. Zeno gives a simple but surprisingly convincing argument that Achilles would never catch up: “In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.”

In my brief lesson, I explained that this argument assumes that the sum of an infinite geometric series cannot be finite, and showed why that’s not true, i.e., why such a series can converge to a finite value.

Why, in my interview lesson, did I teach something as advanced as convergence of infinite geometric series? In two words, intrinsic motivation. High-school Algebra II (at least in Indiana) covers geometric series and their sums, and infinite geometric series are very likely to be touched on. Still, students won’t be expected to understand why the sum of infinitely many numbers can be finite until they take calculus. But, while the details require methods of calculus, the concepts are simple; there’s no reason they can’t understand immediately. I’m fond of the infinite, that fruitful source of crazy ideas, and I’ve wanted from the beginning to expose students to the wildest and craziest ideas I could possibly get them to understand. And why did I want to do that? The main reason is that, based on my own experience, I believe that wild and crazy ideas can be a great source of intrinsic motivation for a lot of kids—certainly not the majority, but a lot. And intrinsic motivation is generally agreed to be far more reliable than external motivation. Along the same lines, Sawyer’s fine book Prelude to Mathematics comments “A non-mathematician learning mathematics…often has to plough through routine procedures, which can be extremely dull… An education should also contain elements that perform the functions of a cold bath—to provide a shock and keep one awake.”

I’ve given versions of this lesson three times now, including once to a high-school “math exploration” class, and I think it went over well each time. The latest version — available in narrative form — has two parts: “Resolving the Paradox” (roughly the original 5-minute form) and “Boring Details, A Related Example, and the General Situation” (about infinite geometric series in general and when they converge). The only technology needed is a yardstick and a greenboard drawing.

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!