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.

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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, 2nd 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.

“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.]

Gamow, Bread Rationing, and the Normal Distribution

"angelic" baker & suspicious professorYou 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

GamowStoryDrawing2

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

TableOfSmallPowersOfSmallIntsSo 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.

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”.