Time to go to work

Although teaching has been my main focus the last two years, I try not to lose sight of what got me into this mess in the first place: I love doing mathematics. As such, I will make my annual sojourn to Europe to hobnob with people much smarter than me in the hope that they will drop a crumb off of their plate that I might legitimately turn into research. I’ll spend a week at Oxford working with über-genius David Craven and then make my way up to Lausanne, Switzerland for the conference celebrating the 60th birthday of algebraist Jacques Thévenaz, author of one of the five books I’d take to a deserted island, G-algebras and Modular Representation Theory.

Anyway, this little trip won’t be about teaching or blogging, but actually doing mathematics. Sweeet. So I’ll be back at the beginning of July and ready to tell you all about the class (or possibly classes) I’ll be teaching (one is actually for high school students!).

Rest well blog-o-sphere,

Adam

P.S. In case you’re wondering, the other four books would be Gorenstein’s Finite Groups, Ayn Rand’s Atlas Shrugged, Heller’s Catch-22 and the U.S. Army Survival Manual.

Stupid Factoring Trick

After the up tick in hits from my last post, WordPress asked if I’d post something to decrease the load on their servers and so I bring to you the fourth installment of my widely panned series on mathematical shortcuts. Today’s topic:  Factoring.

Tricks of the Trade

(with Professor Glesser)

Quick: what do the following polynomials have in common?

If you said, in less than 10 seconds, that 1 is a root to all of them, you can probably stop reading now! I did it in less than 3 seconds, so I should stop reading, but given that I’m writing this, that would be problematic.

The trick isn’t really a trick. It is just plain obvious once you see it. If you plug into any of the polynomials, you are left with 0, so it is a root. But, if that is what you do, then you won’t see the trick. When you plug 1 in, the parts simply vanish and you are left adding up the coefficients. For example, if you notice that 5 – 6 + 1 = 0, then you know immediately that 1 is a root. In fact, that is how I came up with all the examples: I made up all but the last coefficient psuedo-randomly and then made the last coefficient the opposite of the sum of the other coefficients!

Is that it?

No, there’s more. Already, we know that has as a root, but we can also immediately find the other root. How? Let’s factor. Since 1 is a root, we know that is a divisor of and so . Equating coefficients, we get and . In other words, the other root is . Hmmm, that wasn’t so immediate; that took effort. Fine, lets start over with an arbitrary quadratic such that (implying that 1 is a root). We can factor and, equating coefficients, we get and . Ah, hah! So under our assumptions, the roots are 1 and .

If we go back to our second example from the beginning, we can now immediately see that the roots are 1 and .

Pretty good. That it?

No way. If you start with a cubic whose coefficients sum to 0, then we can factor it as before:

from which we get and (that last one takes more work than the others). This isn’t quite as nice as before (not even close, actually), but if you’re into obscure formulas, this might be your cup of tea.  Let me give another example of a cubic, though, that is a little more fun.

Consider . At this point, your root detector should be screaming like my two-year old when he spots a caterpillar on the ground. But, if you quickly compute , you also get 0. Meaning that is a double root (i.e., divides ). It also means that the third root is !

I know, I know: there’s more.

That’s right, Diane. Part of the reason this trick will work so often in math class is that textbook writers are lazy. Why make up problems with realistic roots when you can make them all have the same roots? Anecdotally, the three most common roots of polynomials in textbooks are . Checking for 0 as a root is pretty straightforward and I’ve just shown you how to find as a root; how about ? It is only slightly more complicated.

Let’s start with our general quadratic again: . If we substitute in for , every even power of will simply vanish, while the odd powers of will negate the sign of the coefficient. That is, you get . If this equals , then is a root. For example, knowing that implies that is a root of . As before, we can factor: and so and . Thus, the other root is . This tells us that the other root of is .

As before, there is no reason to stop at quadratics. For a general cubic , you simply check whether .

Put together, you now have a really good way to spot check whether the three most common (textbook) roots occur in a given polynomial.

Phew, we’re done.

Not quite. I have one last bit for the interested. My high school algebra teacher taught us the rational root test (do they still teach this in school?) which says that the rational roots of a polynomial of degree (rational here means an integer divided by a non-zero integer and the degree of a polynomial is the highest power of in the polynomial) of the form (where I don’t care at all about the terms in the middle) are all of the form where divides and divides . In the special case where , then there are only two possible rational roots: and . So, if given

we can immediately see by adding the coefficients that is not a root and by negating the odd power coefficients and adding that is not a root and so, by the rational root test, this quintic polynomial is irreducible (i.e., it can’t be factored¹).

For another cool factoring trick that I don’t use (because I only just learned it), but think I might try teaching is David Cox’s Bottom’s Up! method.

¹This means “can’t be factored over the rationals” which means that you can’t factor so that the coefficients of the factors are rational numbers. The polynomial can be factored as , but are irrational numbers.

Story Telling

I wrote, in what is now (thanks to Jason Buell) the most famous of my small collection of blog posts, that:

Our students read textbooks, refined over the years to be ruthless, efficient and deadly. The story is missing, the context is missing and the connections are missing. The textbooks are a reference, not a teacher. It is then the teacher’s responsibility to add the missing ingredients, to tell the story, to explain how experts actually think about these things and, most importantly, to teach the students how to read (or understand) a subject non-linearly.

Recently, my attention was brought to three excellent posts on story telling by Grace Chen, Dan Meyer and Dan Meyer (man, what are the odds those last two guys would have the same name!), all of which you should read before this post. In fact, if in the course of reading those three you get so tired or inspired as to not want to read my follow-up, then consider yourself suitably enlightened and come back later; what I’ll write certainly isn’t earth-shaking or particularly useful (nor will I be brief). However, I will tell stories and I will use bold fonted section names that arbitrarily oscillate between my voice and the voice of the reader held hostage in my head.

Part 1

8-Bit Education

At some point, my wife and I decided we would homeschool our children. Thankfully, we’ve had a good deal of time to prepare; deciding between the plethora of homeschooling structures and then choosing the specific curricula is a daunting task. We finally settled on the classical method popularized recently in The Well Trained Mind. Part of this program is to break up the child’s education into three phases: grammar, logic and rhetoric, roughly corresponding to 1st-4th, 5th-8th and 9th-12th grades. Within each phase, you study world history from the ancients to the present. So as to be practical, you probably don’t start at the beginning of history, but at around 5000 B.C.

While my oldest son is only four and so a little young to start the world history curriculum, he is old enough to begin his video game curriculum (and if you doubt the importance of this aspect of learning, please do watch Jane McGonigal’s informative TED talk). Sure, I could start him on modern gaming systems (he does enjoy Wii Sports quite a bit), perhaps spend $50 on Super Mario Galaxy 2, but it really wouldn’t give him a full picture of the gamer universe, now would it?

The answer: start at the beginning. Well, not really. Much like ancient history, we don’t want to start too early. Although I did enjoy playing games on the Commodore 64, the TRS-80 Color Computer and the Atari 2600, to me gaming as a civilized hobby begins with the Nintendo Entertainment System. Naturally, we started with the original Super Mario Bros. The exploits of plumber brothers Mario and Luigi quickly enamored my son, Alex. Soon, we were watching episodes of the Super Mario Bros.  Super Show! and now both kids want a Mario themed birthday party. Good work, Dad. During one episode, there is a trailer for the Legend of Zelda animated series. My kids were hooked and Alex wanted to play the game.

Your 4-year old is playing Legend of Zelda?

No, not really. He sits on my lap while I play. So here we are, a few weeks later, just one level away from entering Death Mountain to fight the evil wizard Ganon for the Triforce of Power and control of Hyrule. I remember playing it as a kid and it took months for me to unlock all of the hidden loot, destroy all of the bosses, get the magic sword, figure out the NWSW maze trick, etc. It is a wonderful game that kept my friends and I occupied for a long time. Ah, but now I am older and the age of the interweb is upon us! I don’t have the time to search through Hyrule for everything again. No, I simply googl’d “Legend of Zelda walk-through” and found this delightful site which not only explains an efficient way to work through the game, but also has videos of the author doing it.

Isn’t that cheating?

Well, of course it is cheating.

So, you’re teaching your son to cheat?

Wha-buh-guh!!! Yes, that is the point. If studying math has taught me anything, it is why struggle to learn things when others have struggled before? Stand on the shoulders of giants and pick the (now) low-hanging fruit. It took people thousands of years to come up with modern mathematics; why reinvent the wheel?

…seriously?

Blow the Whistle

Part 2

Is it better to know a few things really well or a lot of things reasonably well? I suppose it depends on what you want to do with your time. However, either is preferable to not knowing anything at all. Here is an experiment:

1. Repeat the number 8 one hundred times in a row to somebody. At the end (assuming they didn’t leave or punch you in the nose) ask them what number you were saying. Chances of success: extremely high.
2. Now ask (for your sake) a different person after repeating 38502 twenty times in a row (still 100 numbers spoken), what five numbers you were saying. Chances of success: high.
3. Find a third person and tell them any random string of one hundred numbers and ask them to repeat it. Chances of success: essentially zero.
4. Ask them about the first number of the sequence. Chances of success: low.
5. Ask them about the first five numbers of the sequence. Chances of success: essentially zero.

So what?

So what? This is what our math curriculum feels like sometimes: a seemingly random ordered sequence of topics with often arbitrary repetitions that leave an unmotivated student with, essentially, zero chance of success. Our smarter students can see through the games and learn to hate math; our weaker students just learn to hate themselves.

Standard Based Grading to the rescue!

No! SBG isn’t a panacea. It is like watching sports in high-def: yes, you get to see Tom Brady’s nose hair, but a 42-10 rout still isn’t much fun to watch. The problem is with the game, itself. At several points, standards have come down from various organizations as if they were commandments from Mount Sinai. At best, I gather that someone took a reasonable modern calculus text and backtracked to decide what and when students should learn. Of course, one can’t simply do that. This wouldn’t take into account current practice, teaching training, etc., so you would have to create standards that compromised a bit to deal with reality. Does this make a good story? Hardly. The student is told: follow these instructions to the promised land. The student hears: Wa wa wa wa. The student learns: Nada Nada Nada.

A Tale of Two Stories

Much like the Star Wars Trilogy (x2) where there is an overarching storyline as well as individual episode with subplots, teachers also need to be aware of the overall story of mathematical education along with the day-to-day practice of teaching concepts, methods and algorithms. While the posts of Grace and Dan are geared at addressing the second part, the first part is equally important.

Currently, we teach from a walk-through. There is no sense of exploration and no sense of importance. Everything is deemed important except for those things that are difficult to test. While I think of mathematics as a subject that should bring order, we teach nothing but chaos. Ask a high-school student what precalculus is about. Hah! You might as well ask them what the dictionary is about. It seems we’re afraid to make the difficult decisions, afraid to cut down the quantity for fear that they’ll need it somehow, somewhere, sometime.

But here is the funny thing…

At some point, a student will ask their teacher: what is this all good for? One variation of the answer is to say: while this specific topic may not ever be relevant to you, the skills you learn from the process are. Okay, so some of us think that mathematics helps develop critical thinking skills and that the material is simply a catalyst. On the one hand, we’re almost willing to concede that some specific things they learn probably aren’t relevant to their lives, but, on the other hand, are terrified that they might worry about how awful it would be if we didn’t teach it. But, if mathematics really does help develop critical thinking skills, then it really shouldn’t matter if we skip some material: they can learn it later if they need it.

Part 3

I play a lot of board games and one of my favorites is Race for the Galaxy. Really, I love hand-management type games. There is nothing more frustrating than having to choose between two awesome cards, knowing you’ll have discard the one you don’t choose. But, the decisions must be made and the game will go on. Educators in charge of curriculum design might first play these games and get used to making these types of choices. Imagine if you could only teach a student three things. What would they be? Obviously, your subject and level dictates this choice a bit. You wouldn’t teach multiplication to students who already know calculus, for instance. As someone who trains mathematics majors (many of whom will become mathematics teachers), I choose the following three facts (along with the appropriate discussion of what those facts mean):

1. 1 is not a prime number
2. 0 is an even number
3. Derivatives describe rates of change

If you let me teach math majors about just the first two, I think I could put together a pretty solid class; the third is gravy.

Where is the story?

The story begins with a choice: what do my students need to learn. Right now, there are too many characters and not enough character development; too many storylines and too few stories; too much mathematics and too little opportunity to appreciate mathematics. Throw out what you would like them to learn or what you think future teachers would want them to know. Focus on the essentials. Now teach it, tell it, break it down, build it up, grade it, shake it, bake it, use it, love it, leave it, come back to it and, finally, they will know it. Let the students explore and let the students struggle. Don’t help them, help them, undermine them, create doubt, create certainty, destroy the certainty, add characters, conflict, irresolution, resolution, chaos, and, then, order. Every class should have an answer to the question: What is this class about? Help them answer this question.

But, I can’t throw out the national standards.

Yeah, I know. This is where Standard Based Grading comes to the rescue. You see, SBG is a panacea…

Hold on a sec…

Wait! I’m on a roll here. Set your own standards, find the most important things you could teach a student. Teach it; teach it well. And then when they come and ask why your students don’t know about topic xyz, tell them, “I forgot.”

???

It’s Steve Martin, damn it.

???

All I’m saying is that once we teach our students the core ideas about how to think about the world in a mathematical way (read “logically creative way” not “creatively logical way”), then and only then does it make sense to start adding things back in. The standards are the problem. But you have to teach them the standards. But the standards are making it harder to teach and harder to learn. But you have to teach them the standards. But teaching them the standards is, in most cases, tantamount to making sure they will not learn any real mathematics. But you have to teach them the standards. I give up. Teach the standards. The story sucks, but so do most movies.

Dénouement

I apologize for wasting your time. I don’t have any practical ideas. I’m lucky: I can get away, in many of my classes, with doing my own thing. If you’re teaching middle or high school, you are accountable to so many people that trying something like what I’m suggesting is probably career suicide. Don’t do it. If you’re reading this, you’re already too good a teacher to waste, especially on this rubbish. Maybe I’ll get back to reading Dan Meyer instead of trying to write like Dan Meyer.

Man, I started this post feeling so enlightened. Now, I feel…defeated.

A symmetry trick for integration

Tricks of the Trade

(with Professor Glesser)

Parentheses in mathematics never fails to impress me. Take, for instance, the freshman dream:

FAIL!

Not only do the parentheses matter, but in nontrivial way. Another of my favorite examples is the difference between and . Just a teeny little difference that makes all the difference in the world. You see, the first function is always greater than or equal to 0. Here are their graphs:

Outside of , they aren’t even close. Of course, this suggests that if you integrate them, you expect to get wildly different answers (there are some exceptions to this: try integrating both from 0 to ). Ah, but there is a little problem when you try to integrate, isn’t there? You can probably handle (possibly with a great deal of effort) finding an antiderivative for , but more about that in a bit. Rather oddly, there is no elementary antiderivative for  . Integrating it from 0 to gives an example of a Fresnel integral, but already this is beyond what most of my students want to hear in calculus. So, let’s talk about what we can actually do.

C’mon, Get to the Trick

I’ve seen two reasonable ways to find the antiderivative of : integration by parts and trig identities. The former method is actually used twice along with a little trick (I’ll get back to this later in the summer when I have a four-part series on integration by parts), while the latter requires you to remember how to convert products of sines into the cosine of a sum. I tend to use the former since I don’t have to remember anything, but the latter is probably a bit easier.

Using either method, we get . Now, say that we want to compute (this is a rather common integral that seems to show up quite a bit in integral calculus and, especially, in multivariable calculus when you start doing coordinate changes). Using the fundamental theorem of calculus, we have:

and after realizing all the terms but one are 0, we see that the integral evaluates as .

Some trick. I already knew how to do that.

Here is the trick. Notice that and have nearly identical graphs on , the only difference is a shift. This implies that if you integrate them on , you should get the same answer, i.e., . If we square both functions, the same result holds: ( you better convince yourself of this before moving on).

From here, we get .

Why did we clutter up our integrand? Because, of course, we didn’t. The integrand is 1 and hence the integral evaluates to the length of the interval. In particular, .

Cool, no?

But that is just one integral

True, but it is an important one. False, because obviously we can use it to compute . Okay, that is cheating a bit. But we can actually go a little further. First, we really didn’t think hard enough about the last example. Consider the graph of on :

Notice that we get a full period of . Therefore, the reason what we did above worked on is that it works on and we just repeated it.  Instead of the observation about , we could also draw a rectangle with height 1 and width and remark that the area under the graph makes up precisely half the area of the rectangle, i.e., . The graph makes it obvious that we could also look at only .

Before you get too excited, though, this will not work on any interval. We don’t get . Generally speaking, you want the interval to consist of integer multiples of .

Is that it?

Not quite. Roger Nelsen, in a paper entitled Symmetry and Integration described the following Putnam problem:

This problem is absolutely ridiculous. Wait—did I say ridiculous? I meant ridiculously easy!!! Consider the graph of :

It appears to have the same sort of symmetry as before. In fact, if we draw in few lines and do some shading, we get:

With just a modicum of thought, we see that .

Whoa! What is going on here?

Actually, a lot. But let me keep it simple (there are nice generalizations of what I’ll write here); I’ll give a heuristic argument for why these things work. The key is that the functions we’ve been dealing with are symmetric about a point. Without going into too much detail, let’s just say that a function is symmetric about a point , which is the midpoint of an interval , if for any such that is still in the interval , the average of and is , i.e., . Truly, then, the average value of the function on is . However, we also know that the average value of any continuous function on an interval is given by . Therefore, or, equivalently, .

In our case, , , and we get . Well, except for one thing. We still need to show the averaging property. This is just a little bit of algebra, thankfully. First, I leave it as an exercise to show that (try converting things into sines and cosines and using the angle addition formulas). Now, for simplicity, we write . The average is now given by

, where the negative exponent comes from .

Using our fraction addition trick, this becomes  which is the required value.

Annotated Papers

As I mentioned in my last post, I recently had a research paper accepted (yay!). The process of writing a paper is both satisfying (I’m accomplishing something) and frustrating (won’t this something ever end?). Perhaps, though, the most annoying thing is that the structure of papers that describe mathematics is so far removed from the production of the same mathematics. For instance, the reader will only see my most polished proofs, not the three less elegant proofs that actually contained the motivation. I like to keep my proofs short, so I usually pull out steps and turn them into lemmas. Part of the justification is that I (or a reader) can then easily cite just that lemma if that is all one needs. This is vastly superior to making people cite the proof of a theorem. However, this is what you end up with:

1. Preliminaries

Lemma 1 Something completely random at this point holds.

5 sections laters:

6. Main Theorem

Theorem 2 Wow! What an amazing theorem.

Proof: The only thing cooler than the theorem is its proof. And here is where we use Lemma 1.

There is nothing logically wrong with the order. But how can someone read the lemma and have any reaction other that WTF? Perhaps I add a little remark that the proof of Theorem 2 uses the lemma. This is helpful, but it still feels random.

Now, part of the problem is that I am expecting people to read a paper linearly. The better mathematicians don’t do this; they jump to the good part and only if the idea of the proof doesn’t immediately become transparent do they hop around the paper looking for answers. This, incidentally, is why sticking an important lemma in the preliminary section of a paper can be dangerous: the experts are likely to skip it.

Back on Track

I was reading my favorite group theory text this last weekend and noticed how uneven the exposition is. Early on in the book, the sections jump around from one (seemingly) random topic to another, frequently omitting any discussion of why it is being treated this way or where these topics will show up again. Although the terms below won’t mean much to my readers, I think the point will be obvious. At the end of section 2.7, Gorenstein writes:

The class of [Frobenius groups]…is of fundamental importance in the theory of finite groups, and several basic problems that we shall later investigate stem from this class of groups.

Contrast this with section 2.8 where he writes

The class of [Zassenhaus groups]…is a very important and interesting one, which we shall study in detail in Chapter 13…in a Zassenhaus group, the subgroup fixing a letter is always a Frobenius group in its action on the remaining letters.

Notice that in both remarks he talks about their importance, but that only in the second one does he give an explicit connection to anything. At least with Zassenhaus groups, I know that I can look for more info in Chapter 13 and that they always contain a Frobenius group (which is of fundamental importance in the theory of finite groups, although I don’t have a clue why). I have a bit more to say on this subject, but it probably would behoove me not to get into doubly transitive permutation groups.

No, really. Back on Track.

Historically, we have to write mathematics in such a terse unhelpful way because of two considerations: space and time. Journals don’t want to publish twenty pages of my thoughts just to get 3 pages of mathematical progress. Also, readers (i.e., experts) don’t want to spend an hour wading through explanations of my missteps, wrong turns and false proofs just to get one idea they might already know. On the other hand, people with less experience in the field are likely to wonder at just about every step: why? How in the world did you come up with that unnatural monstrosity? This reader would gladly read some exposition that brings a few things together. They’d also love the reassurance that even experts make mistakes; even better, they would love to see how the experts move from false belief, back to ignorance and finally to enlightenment.

Didn’t I say I had an idea?

The system isn’t going to change any time soon and even if it could, I’m not sure it should. However, perhaps a bit of (would Dan Meyer call it) scaffolding would be useful. Imagine if, in addition to downloading my newest paper off of the arXiv, you could also download a screencast of me going through the paper, telling its story, elaborating on difficult sections and giving hints about how I really understand the subject. I’d be willing to try it myself if I thought anyone would watch, but my research is a bit technical and only a handful of people would follow and even less would care.

So, if nobody cares, why are we still reading?

The thing is, this translates to the relationship between teacher and student. Our students read textbooks, refined over the years to be ruthless, efficient and deadly. The story is missing, the context is missing and the connections are missing. The textbooks are a reference, not a teacher. It is then the teacher’s responsibility to add the missing ingredients, to tell the story, to explain how experts actually think about these things and, most importantly, to teach the students how to read (or understand) a subject non-linearly. Mathematics is structured so poorly K-12 partly because we keep treating the learning of mathematics as a well-ordered system and it isn’t (the axiom of choice, notwithstanding).

Okay, I have so much more to say, but I’ve begun to bore myself. I’ll save the rest for my post on story telling.

Next time: Tricks of the Trade (pt. 3): A Symmetry Trick for Integration