Fermat’s Last Theorem on the Greenboard

In 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?

John–

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!  🙂

–Don

Mathematically Correct; Pedagogically Questionable

One important aspect of the ongoing struggle over K-12 math education is over how much rigor is desirable. I’m among those who believe the typical U.S. high school goes way overboard in emphasizing rigor for the vast majority of students. The (college) sophomore calculus textbook I used as a student long ago has something to say about this that I think is well worth bearing in mind. This is from the Preface for the Teacher to A. W. Goodman’s 1969 Analytic Geometry and the Calculus, 2^nd ed.  (I corrected a typo in Theorem R: the righthand side was printed as “f ‘(u₀) u’(x₀)”.)

.                                                                                                          .

…The student who is well prepared and who is interested in pure mathematics for its own sake may be able to understand and appreciate a rigorous course in the calculus. But the majority of students are still a little insecure in their algebra and trigonometry, and are far more interested in learning what the calculus can do and where it is going than in following a purely logical argument… I contend that one should not try to state all the hypotheses in a theorem, because the statement can become so long as to be incomprehensible to the average student… As an illustration, consider the following:

Theorem S.  If  y = f(u)  and  u = g(x) , then the derivative of the composite function y = f (g(x))  is given by

dy/dx = dy/du • du/dx

Here is a statement that is brief and simple, and the average student has a reasonable chance of understanding it. Now let is look at the same theorem when stated in a rigorous fashion.

Theorem R. Let f and g be two real-valued functions of a real variable and suppose that the range of g is a subset of the domain of f. Let  h = fg  be the composite function defined over the domain of g by setting  h(x) = f(g(x)) for each x in the domain of  g. If x₀ is an interior point of the domain of g, and g is a differentiable function at x₀, and if f is a differentiable function at u₀ = g(x₀), where u₀ is an interior point of the domain of f, then h is a differentiable function at x₀, and further the derivative is given by the formula

h’(x₀) = f‘(u₀) g’(x₀)

There is no doubt that R is the correct statement and S is full of gaps. However, the average student can learn and use S, but when R is presented he will either fall asleep or totally ignore it. It is just too complicated for him to master at this stage of his mathematics study. The presentation of R rather than S does real harm because it serves to repel many students who are originally attracted to mathematics and who might turn out to be capable technicians or teachers (perhaps even creative mathematicians) if they are given a reasonable chance to develop.

.                                                                                                          .

It’s interesting to compare the above with what appears in the 4th edition of Applied Calculus by Hughes-Hallett et al, a popular textbook these days for “brief surveys” of calculus for non-science students. Applied Calculus simply states the above theorem as the “Chain Rule”: it does not call it a theorem, nor does it provide any more than a vague intuitive argument for it. In fact, I don’t think it includes anything beyond handwaving arguments for anything. In other words, Applied Calculus is even less rigorous than Goodman’s Theorem S approach. Yet, based on my own experience, it’s all most college business majors (for example) can handle—and yes, an intuition for what calculus can do really is useful to them! The other extreme, as represented by Theorem R, is undoubtedly perfect for some students, but I’m sure Goodman is correct: it won’t do anything for “the average student” except convince them calculus isn’t worth the trouble.

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

I 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:

Since 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.

Gamow, Bread Rationing, and the Normal Distribution

You might be familiar with George Gamow, the mid-20th-century physicist who, with his student Ralph Alpher, came up with the Big Bang Theory long before there was any experimental evidence. Gamow is also the best writer for the layperson on science, and one of the best on math, I’ve ever read. In particular, his book One, Two, Three… Infinity is a masterpiece, jammed with fascinating ideas presented with absolute clarity. (Though much of it — mostly the non-math stuff — is kind of out-of-date now; the revised edition came out in 1961.)

A few years back, my friend Doug Hofstadter sent me a short article of Gamow’s about how the normal distribution was once used to expose a dishonest baker — apparently a true story. But true or not, it’s a fascinating story of mathematical probability in real life, one that I think would interest even a lot of apathetic middle-school and high-school students! A PDF of the story is available at

http://www.informatics.indiana.edu/donbyrd/Teach/Math//Gamow_BreadRationingStory.pdf

Terms, Notations, and (Mostly Needless) Confusions

I wanted to call this “Logician General’s Warning: Confusion about Terminology is Hazardous to Your Understanding”, but it takes too much space…

Needless Confusion Over Terminology and Notation

A friend of mine who has a degree in statistics commented a few years ago that he couldn’t understand why people were confused about the terms random variable, probabilistic variable, and stochastic variable; after all, they all mean the same thing. I instantly realized that I myself had been confused because I didn’t know that. Or, quite likely, I once knew but had totally forgotten! I’ve seen confusion — usually needless confusion — over terminology cause serious problems many times, both inside and outside the classroom.

And while I’m talking about probabilistic things, how about Bernoulli “processes”, Markov “chains”, and Hidden Markov “models”? In my experience, those are the usual terms for the three phenomena; but they’re all “processes”!

The same thing happens with notation. I was guest-teaching a lesson on Zeno’s paradox of Achilles and the Tortoise to a high-school math “exploration” class (see my post about it, https://whymystudentsdidnt.wordpress.com/2012/11/30/zenos-achilles-and-the-tortoise-paradox-and-geometric-series/). As an example of a convergent infinite series, I wrote on the board

A lot of students had trouble with the 1/(2^n) part until their regular teacher pointed out that it means the same thing as (1/2)^n — a more familiar notation to them. And I probably would have used the latter form, if it had even occurred to me it might make a difference 😦 .

Hard-to-Avoid Confusion Over Terminology and Notation

How many students confuse quadratic expressions, quadratic equations, and quadratic functions? Many of my own students certainly did, but at least the terms are as consistent as possible. I’d say the situation with the two common notations for derivatives — dy/dx and y’ — is somewhere between “Needless” and “Hard-to-Avoid”. There’s some justification for both notations, but I wonder if it’s worth it.

It’s vitally important that students understand and remember the terms and notation we throw at them. If they’re mechanically following rules but they confuse widgets and wodgets, they’re dead; even if they’re really going for understanding, confusion about terms and notation can waste a lot of their time, and ours.

Early Number Sense and Success in Math

For the last few months, I’ve been teaching Algebra II to a class of one, a high-school senior who’s struggled with math for years. After spending two or three hours a week together for 11 weeks, I’m happy to report that Ellie (not her real name) has unlearned a lot of wrong ideas and learned enough right ones to get an A- on her first test (factoring quadratics, solving quadratic equations by factoring, etc.). Ellie told me the last time she felt she understood math in school was 6th grade and she hasn’t gotten better than a C or D (she’s not sure which) on a test since then. So she was happy with the A-, too!

I’ve always wondered how strong the connection is between number sense and learning math. The other day I mentioned to Ellie the recent news item (widely reprinted on the Web, e.g., http://www.8newsnow.com/story/21786760/early-number-sense-plays-role-in-later-math-skills) about University of Missouri research showing kids with poor number sense in primary grades have trouble with math years later. She responded that she remembers struggling with concepts like adding 4 and 4 in 1st grade — unless she used her fingers so she could see what the numbers meant. Yep.

Why I’m Better Than High-School Students at High-School Math

So why is the person whose mathematical ability I’m most familiar with (me) better at high-school-level math than nearly all high-school students I’ve encountered? Knowing the answer might help me help the other students! I think the rather short list below pretty much explains things. (Of course I’m better in a lot of cases because I already know exactly how to solve the problem, but of course there’s more to it.)

I’m certain I’m better at:

1. Understanding “word problems”, i.e., translating into mathematical form the kind of verbal descriptions of situations that commonly appear in textbooks and on standardized exams. However, it’s not clear how much this has to do with math skill as such, as opposed to reading skill; Jo Boaler has some interesting comments on this point in What’s Math Got to Do With It?.

Outside of translating words to math, I’m better at:

2. Factoring small integers, especially recognizing perfect squares.

3. Seeing simple algebraic relationships (e.g., in equations), simplifying expressions, etc. To a considerable extent, this is just a matter of being really comfortable with mathematical notation.

4. Intuition for what’s reasonable — intermediate values and results: among other things, estimating numerical values.

5. Avoiding specific common mistakes, e.g., “everything is commutative”, confusing what works for addition/subtraction vs. multiplication/division.

Is this a good list? How would you describe why you’re better than students?

Now, accepting my list as a starting point for discussion, what can be done to help students with these skills? Or are all of these skills truly important, and if not, what can be done to reduce their effect on students’ grades and mathematical confidence?

To keep this post short, I’ll just mention one item in my list that strikes me as both not very important and easily mitigated: #2. I discovered last year that many of the students in both of my Algebra II classes — both honors classes, in fact — were struggling on tests with what I thought were easy problems simply because they didn’t recognize small perfect squares as such. When I realized that, I started putting the above table of Powers of Small Integers on the whiteboard, and that seemed to do the trick.