Reflections, Part I: Textbooks

I had a really thought-provoking conversation close to a month ago with friend and colleague Justin Lanier. It was meaty and far-ranging. Much of it has evaporated since I wrote down just the barest of notes immediately afterward, but I wanted to consolidate whatever insight remains from it, so here goes. (Justin if you’re reading I invite you to set me straight if I am misremembering anything you said.) (I’m at a conference this week. I have a healing but broken ankle, and everything that goes on after hours seems to be a walk away, so I expect to have time to write at least two installments digesting this conversation for myself. This is the first.)

By the way, if you ever have the opportunity to teach a math class to a small group of serious, reflective educators who are also friends, do it. All spring, this group theory class I’ve been teaching has been the most professionally exciting thing I’ve been up to, just in terms of the pedagogical task. (Challenge: take a subject you know well but learned in definition-lemma-theorem-proof-problems-repeat order, and try to re-envision it completely so that not only the theorems but the definitions are natural answers to natural, interesting questions.) But every so often I realize that in addition, I now have this collection of people, whose experience I have some access to since they’re friends, and who also happen to be deep, thoughtful math teachers, and they’ve been in my class. Have you ever wanted to know exactly what a class you taught felt like to be in?

(I’ve previously experimented with test-driving workshops on friends, and this has been massively illuminating in terms of being able to hear in forthright and intimate detail about what the experience was like. Actually, I learned an incredible amout from that; thank you to all who participated. But this is the first time I’ve done it in an extended way i.e. beyond a one-shot deal, and the first time the participants have some reason for being there other than to help me out, and the first time they’ve been math teachers. I’ve been a little shy to pick their brains about what the experience is like, except forJesse Johnson’s. It doesn’t feel appropriate with the whole group, and in any case is kind of scary for me. But now that I’ve also spoken to Justin, I just have to take a minute and appreciate the gift it is that these serious educators have taken the time and the trust to tell me something of what the experience I’m trying to craft is actually like. So thank you, Justin and Jesse; it’s really awesome.)

Textbooks

The conversation started when Justin asked me about a stack of about 9 notebooks sitting on a shelf. These notebooks are the product of the intense personal project that taught me algebra: working page by page and problem by problem through Michael Artin’s Algebra, over the course of two years. (2007-9. The project is temporarily on hold.) This connected to something Justin had been thinking about:

He and I have both been disinclined to use textbooks in our classroom teaching. First of all, we have both been very concerned with making sure students understand the authority of their own reasoning; therefore, we’ve been inclined to remove from them the opportunity to rely on external authority (in all kinds of ways; avoiding a textbook being just one of them). Secondly, we’ve wanted our content to unfold, like a story, and therefore we’ve wanted to keep tight control over the flow from the information spout. (Forgive the mixed metaphor.) Textbooks are in the habit of giving you everything up front, creating the very real danger that you’ll watch the climactic scene before the rest of the show. (You know I hate that. Thanks, Kate, for the big ups, btw.)

Lest you get the wrong idea, let me hasten to add that I did usually make a textbook available. I always gave out textbooks, and in my Algebra II and Calc classes I assigned problems from them as well. I virtually never used the exposition in the book as a teaching tool in these classes, but in any case it’s really my Algebra I class I had in mind. In that class I spent four years developing the curriculum from scratch with the help of my amazing colleagues Jess Jacob (now Jess Flick, still teaching math in NYC) and Mike Jenkins (now in France – miss you Mike! BMJ 4eva!). I didn’t refer to the book while teaching the class. Finally, in my fourth year, I got it together to create a syllabus that gave book references for each day of class, but since my teaching never referred to the book and the work never referred to the book, the kids – 9th graders – uniformly ignored these references.

Now, both Justin and I stand behind the pedagogical values that led us to ignore our textbooks; but a certain irony is at play. Our own mathematical lives are infused with textbooks! I mean, I feel like in a way, I became who I am as a mathematical person substantially as a product of books. These are the books:
What Is the Name of This Book?, by Raymond Smullyan
Calculus, by William K. Morrill
Geometry, Relativity, and the Fourth Dimension, by Rudolf Rucker
Chaos, by James Gleick
The Calculus Gallery, by William Dunham
and, as I mentioned above,
Algebra, by Michael Artin
To be fair, there were definitely also some teachers involved (a shout out to my dad, Judy Lazrus, Mary DiSchino, Dan Klemmer and Toby Kaplan, Steve Barkin, Bob Kelley, Peter Mili, Igor Pak, and Alex Martsinkovsky), along with at least one movie (Stand and Deliver). But books, and these six in particular, are definitely a big piece of the story. Two of them (Morrill and Artin) are actually textbooks, and two more (Smullyan and Rucker) share some features with good textbooks (in particular, rich and deliberately organized problems).

Justin has a similar tale to tell. He’s the one who got me into the excellent books Visual Complex Analysis, by Tristan Needham, and A Radical Approach to Real Analysis, by David Bressoud. He also owns, and seems personally identified with owning, multiple copies of Euclid’s Elements. All three of these are textbooks.

So who are we that we aren’t sharing this part of mathematical experience with students?

Justin’s evocative phrase: “I’m worried that for my students, I’m becoming the only font in which math is written.”

To add a wrinkle: this year the two of us took a topology class together. In that class, the professor didn’t use a textbook, and it kind of drove us (at least me) a little crazy, in at least two ways. One is that it meant that if for any reason you had to miss class, the best you could do was to copy someone else’s hurriedly scribbled notes and try to understand them, or arrange to meet with the professor, which was hard to do. Another, perhaps more fundamental issue is that if I left class unsatisfied in my understanding, and couldn’t figure out how to satisfy myself, there was really nowhere to turn. I could (and did) arrange meetings with the professor, but I couldn’t always get satisfaction that way because the arguments he gave in class satisfied him, after all. I couldn’t always get him to see what was bothering me. In my experience learning math from textbooks with no teacher at all, you might think I would run into the same problem all the time, but oddly enough I haven’t so much. First of all you can get the book to repeat itself as many times as you want it to without requiring multiple email exchanges to arrange meetings, or the feeling that you are imposing. Secondly, my experience with (good) textbooks is that they tend to have an extremely useful ploddingness, deliberateness, slow-and-steady feel. Even the hard ones. They cover all the bases. The proofs are airtight. The details are dealt with. Nothing gets skipped. And, they have lots of problems, usually way more than you need to get a good feel for the content. (Artin’s Algebra, for instance, has just the right number if you want to get an awesome feel for it. I attempted every problem in every chapter I worked, and this made me a beast of elementary abstract algebra, thank you very much.) If a step in a proof is omitted, it’s in the problem set, don’t worry.

How to reconcile all this wonderfulness of textbooks, with the reasons (to which I continue to be sympathetic, to the point of nearly beholden) why both Justin and I abandoned them in the classroom? I have a few thoughts, but I’m interested in yours too.

My thought experiment: after all the above thoughts, suppose I’m about to teach an Algebra I class again, starting tomorrow. How do I think about the book question?

If I use a book, it’s paramount to avoid letting it become a source of authoritative information to be absorbed unquestioningly. The students must understand the methods and results of the class as logically justified. The fundamental dilemma is this: the most powerful and most intellectually satisfying way I know to get students to understand this is to get them, by posing provocative and compelling questions, to generate the results and methods of the class, by actively behaving like mathematical researchers. They have to be cutting a path through the jungle, not walking a pre-cut path. The textbook, precisely because it’s already been written when the class begins, has already cut a path. This is the problem. (You might be thinking that the solution is a “constructivist textbook,” by which you really mean a constructivist curriculum, like IMP or CMP, that just puts up a few signposts but leaves the vines for the kids. This is not the solution because except for well-thought-out sequences of problems, these books lack all the advantages of an actual textbook, such as that deliberateness and thoroughness I was talking about above.)

So, answer #1 to the thought experiment: This thought involves my Algebra I class meeting every day for 5 years but is otherwise excellent. For starters, we use a textbook; a good textbook.

Digression – what is a good textbook?

By “good” I mean a book with (a) motivation for everything, (b) clear explanation of everything, (c) justification for everything, (d) coherence and logical flow, and (e) lots and lots of good exercises (i.e. practice of the key skills) and problems (i.e. questions that make you think hard about the content in order to answer). Do I even know of an Algebra I textbook like that? My Algebra I books are in storage so I can’t check, but my memory is that UCSMP is decent on (a) and (b), middling on (c), and poor on (d) and (e), Dolciani is strong on (e) and maybe (d) but weak on (c) and very weak on (a), and Prentice Hall is weak on all 5 counts. These are the ones I know. Do you know of an Algebra I book that does what I want?

On the topic: why have I found it so much harder to find high school textbooks meeting these criteria than more advanced books? The books I mentioned Justin getting me into – David Bressoud’s and Tristan Needham’s books on real and complex analysis – do a solid job on all 5 counts. Artin’s Algebra makes an inconsistent but respectable effort with (a) and is reasonably strong on everything else, including excellent on (e). I think George McCarty’s Topology also pretty much hits it on all 5 counts. Why can’t secondary-level publishers get it together? Overwhelming and varied state standards are part of the explanation but not the whole explanation.

/End digression

Anyway, my idea is basically, take a good textbook, and then take every idea in an Algebra I class, and
1) Ask provocative questions leading students to recognize the need for the idea.
2) Inflect the questions carefully to create as fertile a ground as possible for students to generate the new idea. Keep at it till they do.
3) Pose problems that can be solved with the new idea, giving students the opportunity to familiarize themselves with it through practice.
4) Once all this is done, and only then, have students read the textbook’s presentation of the idea and the questions it answers. Ask them to discuss how the textbook’s development is related to and different from their own.
5) Have students work copious exercises and problems from the textbook.
6) Repeat.

Like I said. 5 years. And some danger of steps (4) and (5) feeling like beating a dead horse. But otherwise excellent, right?

Answer #2 to the thought experiment: (Still requires a good textbook in the sense above. Where they at? Also, may still require 2 years.) Alternate. Start the year making students do active mathematical research to develop the content. (I.e. something like steps (1)-(3) above.) Once they know what it is, develop the next important topic by having students study the textbook’s exposition of it. Have them read the book in class together, make sense of it with each other; have them do lots of exercises and problems from it. Afterward, have them compare the experiences (on-our-own vs. from the book) out loud. Next unit, back to the original-research approach. Next, back to the book. Maybe for the next unit after that, have the class pick which approach they’ll use. This is an awesome segue into the next topic of my conversation with Justin, but that’ll have to wait…

Late on the reblog, but better late than…

Riley Lark is my homie right now.

I’ve written a couple times about the usefulness of generating some cognitive dissonance, i.e. some sort of crisis of knowledge, to push students to think deeply. In both contexts I was talking about developing their skills with proof in particular, but actually I think shaking kids up / freaking them out / giving them something to look at that provokes them to question their assumptions is pretty much a great thing to do whatever you’re teaching. (It must be boring by now to also hear me say that I think proof should also be part of whatever you’re teaching, but there, I said it anyway.¹)

Anyway, if it wasn’t completely clear what I meant by “cognitive dissonance” or “creating crisis,”² this is what I meant.

Check out how in the last paragraph, the students have started to develop features of the language of mathematical rigor (“assuming that…”) spontaneously. No one told them to do that. But when your understanding is being shaken up, you don’t have a choice.

On the very same day that Riley posted this lesson description, he also started a conversation I’m very happy to see people having. He is leaving the full-time, full-pay job to hew an entrepreneurial path (designing SBG-supportive software). I did something like this as well. Like Riley, I’m not naturally comfortable with self-promotion, though the last three years have given me a lot of practice and really stretched me on this front. Anyway, Riley wants to hear your thoughts on the place of self-promotion in the math edublogosphere. I do too.

[1] Because I’ve been misunderstood before, let me just clarify that by “proof” I do not mean an overzealous exercize in rigor (mortis); I just mean that no matter what you’re teaching, everybody in the room should be talking about (even, accountable for) why they think it’s true. (In my humble opinion, class should never move on without this. Mathematics as a whole never moves on without this; why should math class be any different?)

[2] For the record, “creating crisis” is actually dy/dan’s vastly superior paraphrase of what I said. What I actually said was “giving them a crisis,” which sounds a great deal like “giving them a complex,” which is not what I had in mind.

Five Proofs of the Irrationality of Root 5

Recently I’ve had the pleasure of teaching a series of Math and Dinner workshops for the New York Math Circle. The series was on the unique factorization of the integers, aka “the fundamental theorem of arithmetic.” I opened the first session with a problem set intended to get the participants (mostly math teachers) to see how heavily their knowledge about numbers depends on this theorem. One of the problems was to prove the irrationality of root 5. The problem served its purpose during the workshop; participants came up with one proof dependent on unique factorization, and I showed them another as well. But to my surprise, when we went to dinner after the workshop was over, participants showed me three more proofs. Two of them were totally independent of unique factorization. The third involved it, but not in any way I would have expected.

The experience was just a mathematical delight – and I’ll share the proofs in a moment – but it got me thinking about teaching too (doesn’t everything?). I’ve repeatedly made the case (along, I suppose, with NCTM and everyone else) that proof needs to be a much more central part of math education than it is, at every level. This is just an elaboration on that theme. How powerful and illuminating it is to see, and consider simultaneously, multiple proofs of the same result; how each proof shines light on the result from a different angle; how different proofs of the same result may generalize differently and show that several large principles might be at play in one tiny case. How in school I’ve rarely given students two different arguments for a major result, and never asked them to compare the arguments to each other. What a missed opportunity that is.

The closest I came was that when I used to teach Algebra I, I would make three different cases that $a^0$ should be defined as 1: to fit the pattern of multiplication and division; to fit the equations we derived for exponential growth; and because an empty product ought to be the identity since multiplication by it ought to have no effect. However, I never asked kids to entertain the three arguments simultaneously, or to ask if they shed different light on the conclusion, or if they generalized in different ways.

Proof number one: the Euclid-esque one

This is the proof given by a participant in answer to the problem I posed.

If root 5 were rational it could be written as a fraction $a/b$ in lowest terms, i.e. in which $a$ and $b$ are integers and do not have a common factor other than 1. Then it would be so that

$\left(\frac{a}{b}\right)^2 = 5$
and thus that

$\frac{a^2}{b^2} = 5$

i.e.

$a^2 = 5b^2$

This would imply that $a^2$ is a multiple of 5. Since 5 is prime, this implies $a$ is a multiple of 5. Thus $a = 5c$ for some integer $c$ , and

$5b^2 = a^2 = 25c^2$

Dividing by 5, this means

$b^2 = 5c^2$

So $b^2$ is a multiple of 5, and, just as it did for $a$ , this means $b$ is a multiple of 5. But $a$ and $b$ were presumed to lack a common factor other than 1, so this is a contradiction, and the fraction $a/b$ for $\sqrt{5}$ must fail to exist.

The covert reliance on unique prime factorization came out when I pushed the participants to justify the step “if $a^2$ is a multiple of 5, so is $a$ ” and they told me it was because $a^2$ ‘s prime factors are exactly $a$ ‘s duplicated; thus if 5 is not among $a$ ‘s prime factors it cannot be among $a^2$ ‘s. This reliance can be removed from the proof by finding another way to argue this point, but this is how the participants did it.

Proof number two: straight from unique factorization

Once we saw the secret reliance on unique factorization in the above proof, I offered this simpler proof (lifted more or less verbatim from Hardy and Wright’s compendiously awesome An Introduction to the Theory of Numbers):

if $a/b$ is the square root of 5, then $a^2/b^2$ is 5. If $a$ and $b$ are integers, then you can prime factorize the numerator and denominator of this fraction. The whole denominator needs to cancel because the quotient is an integer. Thus all $b^2$ ‘s prime factors are found among $a^2$ ‘s. But prime factorizations for $a^2$ and $b^2$ contain exactly the same primes as the prime factorizations of $a$ and $b$ (only twice as many of each prime). So all $b$ ‘s prime factors are found among $a$ ‘s, and $a/b$ is an integer. This is a contradiction since 5 is not the square of an integer. This proof generalizes in a natural way to show that it is not possible for any integer that is not a square to have a rational square root.

That’s as much as happened during the workshop. Everything worked perfectly in terms of my pedagogical intention to have the participants recognize how much they needed unique factorization to know what they know about numbers. This turns out to have been sheer luck. At dinner afterwards…

Proof number three: less than

This one was shown to me by Japheth Wood, a math professor at Bard College, who helped to organize the series.

Suppose $a/b$ is the positive square root of 5 and as in proof 1 suppose $a$ and $b$ are positive integers and the fraction is in lowest terms. This means $b$ is the smallest possible denominator for a fraction equal to root 5.

Now, 5 is between 4 and 9 which means that $a/b$ is between 2 and 3. Multiplying by $b$ we find that $a$ is between $2b$ and $3b$ , and subtracting $2b$ we find that $a-2b$ is between 0 and $b$ . In other words, $a-2b$ is a positive integer less than b.

You may be able to see where this is headed. The assumption that $a/b$ is a square root of 5 is going to lead to a representation of root 5 as a fraction with $a-2b$ in the denominator, which is impossible because $a/b$ was assumed to be in lowest terms, so that $b$ was the lowest possible denominator. How this happens is just some algebraic tricksiness:

$\sqrt{5}=\frac{a}{b}=\frac{(a/b)(a/b-2)}{a/b-2}$

This last expression was gotten by multiplying $a/b$ by 1 in the form of $(a/b-2)/(a/b-2)$ . But

$\frac{(a/b)(a/b-2)}{a/b-2} = \frac{(a/b)^2 - 2(a/b)}{a/b-2} = \frac{5-2(a/b)}{a/b-2}$

since $(a/b)^2$ is 5, that’s the whole point of a square root. And now we can multiply numerator and denominator by $b$ to find

$\frac{5-2(a/b)}{a/b-2} = \frac{5b-2a}{a-2b}$

and poof! All this was equal to root 5; but now we have represented root 5 as a fraction with a denominator lower than the lowest possible denominator for such a fraction. This is a contradiction so our representation as a fraction in the first place was impossible.

Proof number four: way less than

This one was passed on to me from a participant named Barry. (Don’t know the last name. Barry if you happen to read this, claim your credit!)

$\sqrt{5}-2$ is a positive real number between 0 and 1. Now, consider what happens when you raise it to powers.

On the one hand, since it is positive and less than 1, it will get arbitrarily small. (I.e. given any positive number, if you raise $\sqrt{5}-2$ to a high enough power, it will be less than that number.)

But also, it will have the form (integer) + (integer)* $\sqrt{5}$ .
For example:

$\left(\sqrt{5}-2\right)^2 = 5 - 2\cdot 2\cdot\sqrt{5} + 4 = 9-4\sqrt{5}$

Or,

$\left(\sqrt{5}-2\right)^3 = \left(\sqrt{5}-2\right)\left(9-4\sqrt{5}\right)$

$= 9\sqrt{5} - 18 - 20 + 8\sqrt{5} = -38+17\sqrt{5}$

This is happening because every pair of $\sqrt{5}$ ‘s in the product make an integer. Thus, $\sqrt{5}-2$ to any power must have the form $m + n\sqrt{5}$ , where $m$ and $n$ are integers.

All this is actually true about the square root of 5.

Now suppose root 5 could be written as a fraction $a/b$ with $a$ and $b$ integers. (This time we don’t have to assume lowest terms!) Then any power of $\sqrt{5}-2$ would have the form $m + n\frac{a}{b} = \frac{mb+na}{b}$ .

The numerator of this fraction is an integer. The denominator is $b$ . This means the smallest positive number it can be is $1/b$ (or, if for some crazy reason you decided to use a negative $b$ , then $-1/b$ ). Thus $\sqrt{5}-2$ to any power would be greater than or equal to $1/b$ . But we already know it is capable of getting arbitrarily small by taking a high enough power, since $\sqrt{5}-2$ is positive and less than 1. This contradiction proves that the square root of 5 can’t be a fraction.

Proof number five: unique factorization bonanza

This proof was relayed to me by Ted Diament. It’s more technical than the others; sorry about that. It’s using equipment much more powerful than is needed for the task. It is extremely cute though.

This one relies extra much on unique factorization. In fact, not only unique factorization of the integers, but unique factorization of the set of integer polynomials! Like integers, integer polynomials have factorizations into prime (irreducible) polynomials that are unique except for sign. (This fact follows from Gauss’ Lemma.) For example,

$6x^2 - 18x+12$

factors into

$(2)(3)(x-1)(x-2)$

with each factor irreducible in the sense that it can’t be factored further (except in pointless ways like $3=-1\cdot-3$ ). This factorization is unique in the sense that if you try to factor the original polynomial a different way, you will end up with the same set of factors, except possibly for some pairs of negative signs.

Anyway. Suppose root 5 were rational and equal to a fraction $a/b$ in lowest terms. Then $a^2$ would equal $5b^2$ , and so the integer polynomial

$b^2x^2 - a^2$

would equal

$b^2x^2 - 5b^2$

Now the first of these factors into

$(bx+a)(bx-a)$

while the second one factors into

$b^2(x^2-5)$

But since they are equal, this violates the unique factorization of integer polynomials. Things would be fine if we could keep factoring both sides till they were identical (up to sign), but this isn’t possible: since $a/b$ was in lowest terms, $a$ and $b$ lack a common factor, so that $ax-b$ and $ax+b$ are irreducible. Also, $x^2 - 5$ is clearly irreducible since 5 is not an integer square. So the two factorizations are irrevocably distinct. Since integer polynomials only factor one way (up to sign), it must be that root 5 wasn’t rational after all.

I hope you enjoyed these. When they were shown to me I was just delighted by how different they all feel. I’d never seen anything except the first two, which feel very number-theory-ish to me. The last two feel much more algebraish. Proof number four even has a whiff of calculus what with all that “arbitrarily small” business. And all that is going on in the single fact of root 5’s irrationality!

Addendum, 3/25/13

Since for some reason this post continues to get a fair amount of traffic, I’ve got one more to add! This one occurred to me the other day as a purely algebraic reformulation of proof number four above (“way less than”). It’s higher-tech than any of the above, so apologies for that. I am adding this note in a hurry so I am not going to try to gloss the advanced concepts, so apologies for that too.

The ring $\mathbb{Z}[\sqrt{5}]$ is a finitely generated module over $\mathbb{Z}$ . In fact, it is generated by $1$ and $\sqrt{5}$ . This is just another way of saying that everything that can be produced from integers and $\sqrt{5}$ out of $+,-,\times$ has the form $a+b\sqrt{5}$ with $a,b$ integers. This is a slight generalization of what was noted above in proof 4.

Meanwhile, if $a$ is any rational number that is not an integer, the ring $\mathbb{Z}[a]$ can never be finitely generated as a $\mathbb{Z}$ -module, because it contains arbitrarily large denominators since $a$ has some nontrivial denominator and $\mathbb{Z}[a]$ contains all $a$ ‘s powers. Again, this is essentially what was noted in proof 4. Putting these two paragraphs together, it follows that $\sqrt{5}$ can’t be rational.

Ultimately the reason $\mathbb{Z}[\sqrt{5}]$ is finitely generated as a $\mathbb{Z}$ -module is that $\sqrt{5}$ is an algebraic integer; thus this argument shows that an algebraic integer can never be rational unless it is an actual ordinary integer.