Two out of three ain't bad

I felt good about yesterday, for the most part. It's always a bit awkward getting back to work after a long weekend (like the one handed us by the Spring Undergraduate Research Symposium last Friday), but neither of my first two classes missed a beat.

As I said before and I'll say again, Newton v. Leibniz came off without a major hitch, and all participants were lively and engaged. Moreover, the 280 folks bounced back from their Homework Set 5 debacle and made real headway into the jungle of equivalence relations. I had a really good time in that class, and I feel a lot was learned.

But Abstract II felt a bit...forced?...flummoxed?...flat?...some other suitable and non-scatological f-word?

Maybe you're all tired, which I can definitely understand. Maybe you're all a bit overwhelmed by the notation and the terminology associated with quotients of polynomial rings and with finite fields, which is admittedly dense. Or maybe, as I suggested in class, it's a combination of spring fever and senioritis (if I'm counting right, 5 of the 13 are graduating in May, with three or four more to follow in December).

Whatever the reason, y'all looked yesterday as though someone had cranked gravity up by a factor of two.

Is there something I can do to get you guys unstuck from this rut?

I tried to slow things down a bit yesterday to make sure we were all on the same page with the current proof, but maybe that's not what's needed.

Maybe we need rather to gun the engine and red-line it down the highway for a class or two, to burn off some of the junk that's cluttered up the engine valves?

Or maybe we just need to have a stock-taking class where we look back over everything we've done and fit it all together?

Maybe we just need another day off, like the one this coming Friday offers while I'm out of town.

Or maybe I just need to bring in donuts again.

Unlike my "younger" classes, I know most of you MATH 462 people read this blog regularly, so I encourage you to comment: what can I do for you right now? Help me lead us out of this quotient-ring quagmire!

Newton v. Leibniz: sneak preview

Just a quick thought before I get to the next pile of work on my desk: this morning the Calc I students did a fantastic job in their enactment of a civil trial between Isaac Newton and Gottfried Wilhelm von Leibniz. Both legal teams were well-prepared and had solid, cogent arguments; all witnesses were similarly well-prepared and versed on the ideas they were to represent, and the jury was attentive and respectful.

Well done, all! I'll have more to say about the trial once the jury's decision has been rendered on Wednesday. (I don't want to sway the jury one way or the other, and I'll pridefully assume they might be reading this...)

The big stick

Maybe it's the fact that this last homework set was the most "computational" of the semester's assignments (and therefore the most like the math these folks have seen in earlier coursework), or maybe it's that I scared the crap out of them with the low grades and soapbox speeches that accompanied the last homework set...it's likely a combination of both...but for whatever reason, the 280 students did much better on this last go-around than they did on the previous one.

Kudos, kiddoes! Keep it up.

While I wait

Mathematica's taking a particularly long time to complete this calculation I need for the talk I'll be giving in Iowa next weekend, and in the meantime I thought I'd check in (ever so briefly!) here.

This week's offered a perfect storm of academic events: the Parsons Lecture (supplied tonight by Prof. Thomas Banchoff of Brown University) collided with our Spring Undergraduate Research Symposium (tomorrow) and the Big South Undergraduate Research Symposium (tomorrow and Saturday), the penultimate Super Saturday class (Saturday morning), preparations for my journey to Simpson College for the Midwest Undergraduate Research Symposium, and a whole boatload of midterms, one for each class, to say nothing of the Newton v. Leibniz trial about to take place on Monday. (This semester's students have put a lot of work into this project, as far as I can tell, and they've been more diligent than any previous course section about running internet sources by me. I'm anticipating a solid trial on Monday.)

Banchoff's talk was marvelous: it was well-aimed, perfectly timed, inherently interesting, and executed with good humor, grace, and aplomb. He did a superb job at answering a broad array of audience questions, some of which were particularly difficult. I'd have to say that his public lecture was the best given by a Parsons Lecturer since I've been here.

I know a smattering of students presenting talks and posters in the university's symposium tomorrow and the nearly coincident Big South symposium that'll run the next two days (and into which I've put a good deal of, I don't mind saying, rather thankless effort), and in between those I hope to get a chance to get ready for Monday's classes this week's episode Super Saturday. There once again it's time to put together models of Euclidean, spherical, and hyperbolic space as the students experiment with bending space itself. My thanks go to my Calc I and 280 students for cutting out hundreds of posterboard polygons!

I've got a small handful of additional computations to complete before my MUMS 2009 talk is ready. I'm feeling pretty good about it. A week ago I was quietly panicky: this is my first plenary talk at a conference big enough to have plenary talks, and I want it to go well. I've made my talk a mix of "classical" results from the theory of random graphs (mostly theorems due to Erdős, Rényi, and Bollobás) and more modern results due to myself and my colleagues here. I hope that it will give a nice sense of the sorts of things one can look at in random graphs, complete with pretty pictures.

I'm tired.

Thankfully, as my plane touches down in Asheville next Sunday afternoon, the busiest part of the semester will slide behind me.

Helpful hints, volume 2

More tips from the experts on handling homework assignments effectively! I've gotten another response to my call for advice from the students who're doing pretty spiffily on the MATH 280 homework. Please let me know if you find this information helpful by responding in the comments section. (Oh, yeah, and this is post #250. Staying power!)

***

3. Write. Right? Right!

So i get a new homework (yay!). Here's what i do and typically in this order:

1) Understand the class notes! The concepts of the homework usually correspond to the class notes, so understanding them will help in understand the homework.

Also, the techniques used to solve the class examples are typically useful for the homework problems. I acutally try rewriting many of the proof examples from the notes until i understand them because let's face it: i would never have come up with those things on my own. also, rewriting them helps me understand them. it took me two weeks to understand what union and intersection meant. i still don't understand the inductive proof about the size of a power set, |P(S)|=2^n. but once i do it will become useful for proving many other things.

2) understand the homework question. I learned from my first homework that it does no good to attempt an answer when you don't know what is being asked. if you don't understand the question, ask. if you still don't understand, ask again.

3) when actually trying to solve the problems it helps to write, write, write, write, and write some more. literally play around with it. i usually have a couple pages of calculations with any given problem because i need to see the pattern and how to use it (like the 3x+5y problem? i had like 4 pages of lists of sums! but sonuvabitch, i got that problem right!)

have you ever learned to play a musical instrument before? you don't just learn scales, you play around on it. you ever take a drawing class? you don't just draw fruit in a bowl, you doodle. you ever played on a basketball team? you don't just run practices, you play around and goof off. same with math (or at least it should be). just fiddle around with it until you start to see what you want.

it can help to literally write out "What I know:" and then near it "What I need to know:" Sounds trite but it can help.

and sometimes i will literally write at the top of the page "Theorem: blah, blah, blah. Proof: we need to show etc." then i write at the bottom of the page "Thus, conclusion." then i go back and just try filling it in. the point is to write, write, write.

and keep all your notes and attempts! don't throw them away. you may try something at first, realize it's wrong and toss it. only to realize later that you were right and you need that page to refrence back. (don't you hate that? you thought you were wrong but it turned out you were right? happens to me all the time.)

4) make a complete rough draft. remember the 4c's? Correctness, completeness, clarity, composition? forget it. just do correctness and completeness. just make sure you get ALL the information you need on paper and make sure its CORRECT. after you have all that done, run it by someone. Patrick, or maybe a math lab person, or anyone who knows this stuff (just make sure they don't give you the answers). THEN you can go back and make it more clear and pretty. I once had a rough draft of a single problem that was a page and a half. a page and a half! my final revision was less than a paragraph long. And it was nice.

If you can finish all this stuff before it's due you can always ask a teacher to double-check it. Most teachers are willing to read final drafts if you present them well enough before the due date (you do have to ask first, of course).

i only spend about 6 hours total on each homework set. Maybe 2 hours by monday, by wed (and another couple of hours later) i have my drafts written and looked over, and thurs i finish typing them. i don't do it all at once because my brain won't figure it out if i try that. and believe me, i've tried.

and if all else fails, become an english major.

Telling

What might it mean that my students automatically set about returning our Super Saturday classroom's desks to rank and file format after today's class had completed?

What an awesome bunch of helpers: I had almost as many "big kids" today (8) as I had "little kids" (13). It's the place to be, man.

Helpful hints, volume 1

A couple of posts back I asked some of the tip-top students in my 280 class to write some hints on homework completion for their peers: what do they do that's made them successful so far this semester?

I've received a couple of responses so far, and I thought I'd share these two folks' suggestions in this post. I've left the suggestions as is, and I'll keep them anonymous.

Students (in any of my classes), please feel free to reply in the comments section.

***

1. A chronological checklist

A) I read the problems the day I get them or soon thereafter. I usually have to read them a lot of times before I even understand what is going on in them. But half the battle is knowing what's going on, so I try to get that done as soon as I can. I look for words I've heard before and think about whether I've seen anything similar. I usually write a "translation" to the side of the problem, something that helps me remember what the problem is asking when I go back to it later.

B) I scrawl. I write ideas down, even if they're stupid (and they probably are at first) or messy. They are usually very unorganized. I scrawl all over the homework paper and when I run out of room there, I scrawl all over notebook paper. This is just to get my ideas out, and if I don't have any ideas, I write out definitions that might be relevant, or really anything at all that might be relevant. For example, if the problem is to show two sets are equal, I write down what I know has to be shown for that to be true: containment in both directions. Just by writing that down and seeing it, I might think of the next step. For that matter, sometimes I just write down the entire problem in my own words.

C) I organize my scrawlings. Still on notebook paper, I write the problems out in order and compile all the ideas I had for each problem into one place. The answers might not be complete, but the ideas are at least organized.

D) I get frustrated and leave it alone for a while. Depending on how much time is left, I stop thinking about it for a few hours or a day so I don't burn out. These homeworks are really hard and if I think about it too much all at once I start getting mad and thinking "When am I ever gonna use this stuff???" Not productive. I take a break and do something completely different.

E) I come back fresh. I take my organized scrawlings to the math lab and crank it out. If no ideas ever came, I ask whoever is around if any ideas ever came to them, including Patrick, who lives right across the hall from the math lab, conveniently! I write out a dress rehearsal of my homework (the whole thing the way I want it to look Latexed, just on notebook paper).

F) I Latex it. I won't lie, this takes me forever. But I'm getting better at it and it comes much much easier than it did at first. I usually copy and paste an old homework into a new document and fill things in. That way, I don't have to start from scratch. Latexing it makes it SO much clearer and I can find mistakes more easily.

G) I revel in the beauty that Latex spits out. So lovely!

That's it mostly. It does take a lot of work but it feels so good to turn in a complete, correct, Latexed answer. That is, it's worth it.

***

2. Facebook is your friend

Timeline:

Upon receiving my homework sets, I usually read quickly through the problems. I do not really think about them critically or read very closely, but just to get an idea about what I am going to be asked to do in the upcoming week. I do pay special attention to the committee problems since they are due earlier.

When I find time over the weekend, I try to go ahead and get at least the committee problems completed. Usually these are due on Monday, and it’s not hard to get something down on paper that is constructive in two days, even if you work on weekends (I do!).

Next comes the bulk of the homework, which I try and get started on as quickly as possible. Usually I know that it will take about three or four hours to make my first run through the homework, not including waiting back on email responses from questions I have. I try and set out about that amount of time in split up in a couple of days. I don’t really think anyone could be sane if they tried to do all the homework at once. Four hours of induction is not really something my brain can take. However, I do try and work at least an hour or even two at a time. It keeps my mind from running off track and forgetting what I have already completed. Once I get that “draft” done, I will usually try and wait a little while, and then come back to it. By this time my mind has cleared and I am ready to get back to work. I read over each of my solutions to the problems, and then start doing corrections. I usually try and correct the simplest things first, because the more I have done on a homework set, the more anxious I am to get it finished. After I have got all of my changes done, I read over it again, and see if there is anything I can reword or make sound better, and of course make sure my proofs make sense. After that I am pretty much done! I do all of my work in LaTeX, so I usually don’t have more than the two “drafts,” the original and final, even though I may do numerous changes to the file.

Approach:

When I first start doing my homework; I pick out the problem I think will be easiest to prove. As I said before, the more I have done, the better off I am. I do everything in LaTeX, just because for me, I don’t have to worry about my chicken scratch handwriting, and the entire proof starts out organized, which helps me. Many times, I will have two different ways I think the problem may work out. Therefore just start typing the ideas for one, and if I get stuck start typing the other ideas. Eventually things will start to work out, or one idea will play off the other, or the ideas may end up combining. Who cares if I spend an extra ten minutes writing down an entire problem and getting the wrong answer, if it helps me understand what is going on, isn’t that what counts?

Committees work beautifully in one of two ways. In the first case, suppose I have absolutely no idea what is going on, or am just not really sure how to go about setting up the problem. I usually go as far as possible, even if it’s only two lines of LaTeX, and then I write comments asking how it should be set up, or how to get moving again on the problem. In the second case, suppose I nail it and know I have it right, or at least I am pretty sure I have it right. What is more encouraging that turning in the paper, and two days later seeing comments on your paper from friends saying “Awesome Job!”?

What happens when I really get stuck? Well may I first say that in every homework set, about two of the problems I end up getting stuck. My first method of attack on this is to take a break. Go grab something to eat. Let your mind wonder to the GPA killer, Facebook, for a little while. If you smoke go get a cig and make your way to the closest designated smoking area. Even play a round of Gears of War 2. Whatever you consider as something to just chill you out, do it. But while taking a break, I always try and keep in mind what I am stuck on. How on earth can this possibly work? Usually with this I can get at least one problem figured out. Taking a break is good too, because as stated before, trying to do all of this at once is insanity. If I can’t tackle all of my problems this way, I next EMAIL Patrick. I do stress email, because in fact going and talking face to face I have found doesn’t work for me. The reason is, I get back and start doing my homework, and I will forget exactly what was said. But if I send an email, I have it in writing right in front of me to stare at until it’s no longer needed. Usually after receiving my response (Which kudos to Patrick for the average 15 second wait time it takes to receive a response) I read the email a few times. Sometimes I may look at an email literally 20 to 30 times, constantly switching between my TeXmaker and my email windows. It’s what works for me. The emails are always helpful, usually contain examples, and are usually encouraging. I am pretty sure I have never got an email describing my idiocy, though I don’t know how Patrick resists. This is just what I do. The math lab really doesn’t work for me, but if it works for other people then that’s awesome! Use that! Find whatever works, Find someone that you can understand and get constructive comments out of. I assure you though, at the least shooting an email to Patrick won’t hurt, and usually gets you a fast response that you can then study.

A couple of things I always keep in mind while I am doing the homework:

• The problem will always work out. No matter what. There is a solution. If the problem asks you to prove a theorem, you know that the theorem will hold. It’s just a matter of getting to that solution. You got this! Just work it out!
• It all comes back to, and sometimes I start my proofs by actually listing these: What you know, and what you want. The theorem is obviously what you want, and the definitions of the parts of that theorem you can usually count on being what you know.
• I always go back to the definition if I start going in some horrid direction to nowhere. Usually between the notes from class and what is given in the question, you have exactly what you need. Since we are trying to prove something in general, it almost always goes back to the definition.
  

LaTeX really seems to help me get my homework done neatly and correctly. The PDF is REALLY easy to look over and it’s REALLY easy to see if you have made any mistakes. Also, after you do it enough, it is not hard at all and is faster than writing it down in most cases. The code is extremely easy to figure out, because it’s all logical tags. If you have to take CSCI 201, you have to learn java, and that’s much harder because not all the code makes sense. But in LaTeX, if you want a fraction, its \frac{}{} or a union sign its \cup, which the union sign looks like a cup. LaTeX is very logical and only takes a very small amount of practice to learn and get good at. Also, if you use TeXmaker, it will actually predict the code for you, and it’s also got buttons on the side that will insert the code right into the document! It’s so EASY! Once you do learn it, it makes the homework so much easier to work on, and then you don’t have to worry about keeping up with three or four drafts that is barely readable. You just have one file that you can edit anytime you want with ease.

The homework just takes a little bit of time and thought. It really is a very do-able piece of work. All that needs to be done is to find a system that works, stick to it, and ride that through the rest of the semester.

***

That's for starters. My thanks to the thorough job these two folks have done! (I promise, I've not paid them for their remarks. It's all pro bono.) I've got three or more people on the hook, and I hope they'll chime in with their own advice soon, too.

Anyone else?

Ouch burn!

Jeez Louise.

In rereading my post from last night, I realize how embittered and cold I must have sounded to my students who've happened to peruse the post.

The note I'd hoped to strike was more akin to the one I sang in class today (well, yesterday, technically, but who's counting?)...more of a "please, please, on bended knees" sort of deal: please help me to help you, Jerry Maguire style.

I want you all to know that I'm frustrated, but it's not with you, it's with the inherent difficulty of the problems with which we're all dealing together, and with the frustration you're all feeling in meeting that difficulty head-on.

I guess what I'm trying to say is this: y'all are awesome people who are a genuine joy to work with, and I want to make the most of our time together. Whatever you need from me to make that "most" happen, let me know; I hope I've just done a good deal to let you know what I need from you.

Meanwhile, I have to say that I had a wonderful day this past day: all four classes offered up cool mathematical ideas, from optimization problems in Calc I to cool combinatorial trickery in 280, and from 480's highly successful peer review (folks, chime in in the comments section if you'd like to say something about how things went for you today) to philosophical meanderings and ethnomathematical thought experiments in Abstract II. For homework, try to think about what sort of intelligent alien creature it might be who would have no understanding of discrete quantities.

It was a magnificent day, and I thoroughly enjoyed spending it with all of you.

I hope tomorrow brings as much fun and fulfillment as the past 24 hours have.

With that note of confidence (and with the bulk of this damned NSF report finally behind me), I'm off to a belated beddie-bye. Avanti!

Hendrix ain't got nothin' on these guys

I'm sitting in my MATH 480 class right now, listening to the happy hubbub of 16 senior math majors, paired up and giving each other feedback on the rough drafts of their papers for the course. Eavesdropping has been enormously enjoyable: from what I've heard their conversations are lively, relevant, and to a one respectful.

This is an almost uniformly strong group of students, and I'm looking forward to the next seven weeks of talks. They're gonna be good ones!

Error

I'm frustrated.

I'm smack dab in the middle of grading the latest homework set in 280, and so far it ain't pretty.

They're having profound difficulty in proving a few propositions with which students have had little trouble in the past, things like "the operations of union and intersection are commutative" and "the operation of set difference is not associative." The second it trickier than the first, no doubt: it requires not only the recognition that a counterexample will suffice and is in fact called for, but moreover the construction of said counterexample.

The former proposition is a fairly straightforward one, or so it's seemed before. For example, all one need do to prove that A ∪ B = B U A is use the fact that x ∈ A ∪ B if and only if x ∈ A or x ∈ B, and then use the "commutativity" of the word "or." Generally students overlook this method proof because it seems to easy; this time around I get the feeling that some students are overlooking this method of proof because they've simply not put any thought into their work.

Indeed, several of the students' papers bear evidence of quick completion, as though they've put maybe a half hour's, or at most an hour's, worth of work into the homework set: there are obvious omissions that seem to spring from haste, errors in transcription that belie similar speed, and signs that very few students are drafting and redrafting their work before it's stapled and slid under my door.

What's most frustrating is the fact that the students have had two weeks' time to meet with me and ask about any uncertainties they might have had about the homework. Judging from the quality of the submitted work, uncertainties they must have had, indeed uncertainties galore.

Why not ask?

I don't bite.

Not hard, anyway.

And not at first.

And several students ought to know this. Several of the people in this class (seven, if my mental tally's right) I've had in some other class before. Of those roughly half of those are doing very well (three, with a fourth coming round quite well in the past couple of weeks), while the other half are struggling mightily.

My long-time friends, you know how you are: why've you not yet felt the need to darken my office door? I know you're having a rough time of it, and there's no shame in that. It's a brutal class, hard as nails and heavy as a hammer. MATH 280 is the stone that stands at the halfway point from the major's entry to its exit, with jagged letters chiseled into its rock-hard face, something about abandonment of faith.

Yet like Dante on his trip through Hell, you'll not go unaccompanied. I'll be your Virgil, if you'll let me, and your classmates, for a while at least, will join you on your way.

If you'd asked, what might I have said?

1. Write, rewrite, and rewrite. Chances are your first attempt will be an ugly one: it'll be clumsy, oafish, and bear little resemblance to English, math, or anything in between. Your first draft of a solution (if it's anything like one of mine) will be barely-readable scrawl; it'll be something like pure thought, a bubbling spring of formulas and figures that needs to be bucketed and brought to boil before it looks anything like a coherent paragraph or proof. Only in the drafting and redrafting of your solution will you work away your initial errors and misstatements. If you're only writing one draft, you're almost certainly writing it wrong.

2. Picture this. If you can't get your mind around the definitions and the quantifiers, draw a picture. Draw a graph. Draw a Venn diagram. Draw an arrow diagram. Draw a concept map. Create some sort of meaningful visual representation of the problem with which you're faced, even if the picture you draw means nothing to anyone but you.

3. For example... If a proposition makes no sense to you, consider what it might look like for small cases or small sets. If a claim's made about all natural numbers n, what happens when n is 1? When n is 2? 3? If a claim is made about all sets, what does the claim say when the set is empty? When it's got one, two, or three elements? If a claim asks about a function, might you not try to see what it says when your function is constant, linear, bounded, or continuous? While examples alone don't often serve as proofs (the exception being when a counterexample establishes the falsity of a universal claim), examples go a long way to helping gain intuition and insight into a problem, and once intuition's gained, a proof is often close behind.

4. True or false? Right or wrong? If you've been asked to decide whether a given proposition is true or not, you can do far worse with your time than play around for a little while, considering both sides and trying to come to a conclusion as to which side makes more sense. This is where (1) and (2) can serve you well: sometimes the right picture can suggest a statement's truth, or an easy example demonstrates that the statement's not true at all because a counterexample's lying close at hand.

5. Wash, rinse, repeat. If the proposition you've been asked to prove is very much like one you've seen before, you might as well try to prove the new claim using a proof very much like the proof you've seen before. Why not? Similar problems call for similar solutions, and often a slight reworking of an old proof will yield the verification you're looking for.

And if you ask, what might your colleagues say?

I've asked a few of them just that. A couple of hours ago I e-mailed the six students who I feel are at the top of the class right now, students who've submitted consistently strong homework sets and exams during the first half of the semester, students who've demonstrated some sort of je ne sais quoi so far this term, and I asked them, if they would kindly oblige, to provide us with a few tips for their classmates on meeting the challenges posed by our course's concepts. I hope that they'll indeed oblige.

While I wait for them to write me back, I hope that those of you who are struggling will take a moment to read and reread the advice I've given above. If you're floundering and feel like you're about to go down for the first, second, third, or fourth time, you're not alone: we're here to help, and the only real mistake you can make is ignoring the hand we're holding out to you.

In other (though, as it turns out, not wholly unrelated) news, I've just tonight had a chance to read an article I've been meaning to get to for quite a while now, David Bartholomae's "The study of error," which appeared nearly 30 years ago in the journal College Composition and Communication (October 1980, 253-269). Like most good papers it's made me want to read several others to which it refers, but its own take-home points are many; as it's nearly midnight and I've got another long day tomorrow (peer review in MATH 480!), I'll limit myself to one, but I'll make it a big one.

Bartholomae focuses on "intermediate systems" of meaning evident in so-called beginning writers of college-level prose, witnessed by "intentional structures" that arise in these writers' compositions: certain sorts of errors crop up when students have successfully constructed idiosyncratic but in some way internally consistent grammars that help them govern their writing. As I understand it, while the gap between the writer's text and a conventional text may be great, an understanding of the writer's intentions (gained through interviewing the writer about her writing, for instance) can lead to an understanding of the writer's personal grammar and therewith and understanding of her understanding of the world.

What does this mean for math?

Consider the following passage from Bartholomae: "They [beginning writers] are not, that is, 13th graders writing 7th grade sentences. In fact, they often attempt syntax whose surface is more complex than that of more successful freshman writers. They get into trouble by getting in over their heads, not only attempting to do more than they can, but imagining as their target a syntax that is more complex than convention requires."

Now consider a line from a paper ("Math and metaphor: using poetry to teach college mathematics") I just submitted to the WAC Journal: "Even when asked to use 'their own words,' [beginning math] students' papers are overburdened with jargon, passive phrasing, and misused terminology that has a 'mathy' ring to the students' ear. The writing is stilted and unconfident."

I can't help but notice how well "syntactically complex" matches up with "mathy": just as beginning writers in the traditional sense often sink because they dive into the deep end, beginning writers of mathematics might sink because they think they've got to sound more like mathematicians than mathematicians really sound. (Two examples spring to mind, both common with 280 students: first, students will often say "n number of elements" in place of the more correct but less technical sounding "n elements"; cf. "7 number of elements" versus "7 elements." Second, consider this one, which I saw quite frequently today: students will often say "there exists an x in the set X" instead of simply "x is in X," or better still "x ∈ X.")

So what's to be done? Bartholomae prescribes close reading with the students, in order to encourage them to catch their errors and learn to perform more effective self-editing.

And so close reading I shall ask the students to do. Those of you who've had a struggle with this most recent homework assignment, please expect me to be calling you onto my carpet in the next few days: I'm going to ask that you stop by so that we can go over one or two of your problems together. It's high time we did so, for both of our sakes.

Stop

Nashville trip outstanding, stop.

Homecoming at Vanderbilt pleasant, stop.

Grading at Fido nice change of pace, stop.

Writing short course went very well, stop.

Student talks solid, stop.

Won some long-named award, stop.

Back in town, overly tired, sleep now, stop.

Growth industry

It took Maggie and me about eight hours of opening, printing, sorting, shuffling, filing, searching, and piling, but we've finally got this year's REU applications compiled.

We've ended up with 112 complete applications (a 14-to-1 applicant-to-position ratio), with another 20 or so incompletes. That's a 50% increase over last year's roughly 75 (that figure was little changed from the first year's number). I'll likely start the vetting tomorrow or Monday, but there were three or four who've caught my eye already.

For now, it's bedtime. Giassou!

72

The next seventy-two hours are going to suck.

That's all I've got to say.

If they can put a person on the moon...

Embarrassing.

It's embarrassing when you can't assume that the computer technology in your classroom will be error-free enough to allow you to ask your students to (1) download a file from your website, (2) open it using software that's already installed on the computer and easily accessible from the desktop, and (3) use the software to solve a few simple problems...in under half an hour.

My students took Team Quiz 2 in Calc I just now, and there were glitches galore, none foreseeable, all technology related. Of the 18 five-plus-year-old Macs in the room intended for student use, 3 were powered down, 3 were frozen, and 1 was (and has been for as long as I can remember) rendered useless by having no mouse.

That left 11, to be used by 8 groups of students. So far, so good.

Another froze soon after the students managed to download the file they needed from the course website.

Three more performed so...damned...slowly as to render them unusable for the Mathematica computations I was asking the students to perform. One poor team had to hop from one computer to a second, to a third in order to find a machine that would actually carry out the incredibly simple command they'd given it without stalling for several minutes. Since completing the quiz required at least six such computations, there's no way they could have finished on the first two machines.

Keeping score? With the aforementioned 4 of the 11 remaining machines by now out of commission, there were 7 student computers left for 8 teams, and fortunately the instructor's terminal at the front of the room proved functional, so the eighth team could hop over to it.

To my Calc I students: I humbly (and I do mean humbly) apologize for today's technology woes. I'm not sure if it's even fair to grade the quizzes, given the way at least two teams were stalled by recalcitrant computer behavior.

Argh. That really was pathetic.

Overkill

What can intermediate-level math students be expected to get out of a highly technical research seminar?

I'd talked up my out-of-town colleague Seymour's visit quite a bit over the past few days, to both students and colleagues, partly because I knew that if I didn't get the word out, Seymour might end up speaking to an embarrassingly small handful of our strongest students and most devoted faculty members. The last thing I wanted was for a friendly and devoted colleague to drive four and a half hours (both ways, in one day!) to give a talk to a pitiful few. I've given talks to such small groups, and I know how disappointing a small turnout can feel.

As it was eleven of our ever-stalwart students (including five from my Foundations course) came to Seymour's talk this afternoon, and several faculty members rounded the audience out.

As it was I ended up feeling bad not for Seymour, but rather for a few of my students whom I'd strongly encouraged to attend the talk, expecting that though they'd struggle to keep up they'd likely be able to grasp a good bit of the presentation. It turned out that I had a hard time keeping up with the bedazzling details, so I'm certain a number of the students were having a bear of a time.

Nevertheless, by the talk's close I remained convinced of the truth of what I'd told my 280 students in class yesterday: "even if you don't understand every twist and turn of every argument, going to seminar talks early and often prepares you well, as it exposes you to the language, the notation, the terminology, and the conventions of mathematical communication. The earlier you start going to math talks, the quicker they'll start to make sense to you, and the stronger a student you'll be."

To help students get the most out of the talk, I typed up and sent out a "companion piece" indicating to the students who attended what I thought were the important points to keep in mind about the presentation. Leaving aside the technical details of Seymour's highly convoluted argument, I encouraged students to stay focused on the talk's big picture and to try to understand

1. the ways in which Seymour's use of notation for operations in unfamiliar settings suggested analogous, more familiar, mathematical operations;

2. the culture of mathematics evident in Seymour's "humanizing" commentary during his talk, highlighting the importance of collaborative research and the ability to ask good questions as well as answer them; and

3. the mathematician's penchant for ranking, sorting, and ordering that formed the basis for much of Seymour's work.

Intermediate-level students hoping to hone their seminar-watching skills will do well to look for these general "trends" and meanwhile gloss over inconsequential details. Remember that no one should go into a talk expecting to understand every last jot and tittle of the presenter's argument; just because a theorem's proof involves computations you can't possibly understand, not being an expert in a particular field, doesn't mean you're barred from getting a lot out of the talk, provided you know how to take the talk in.

By the way, my thanks go, again, to Seymour, for his unselfish trek up to UNC Asheville this afternoon. I'd also like to give a shout-out to Tomassino, who complained that he hadn't been able to figure out his pseudonym on this here blog. (I've mentioned you before, man, although you'd have to search the blog to find out the context in which your name appeared.)

To be continued, I'm sure!

The seduction of induction

Here's a question I hope some of my 280 folks will be willing to answer in the comments section (other onlookers are encouraged to reply as well): what's so hard about inductive proofs?

I've noticed that every time I've taught a proofs class, three times here now, and once at the University of Illinois, the homework set the students found most challenging was the one dealing with induction. Fellow teachers, have you seen this phenomenon at work as well?

Students simply seem to find induction tricky, intellectually irksome.

Here's one reason this proof method might prove (no pun intended...okay, kinda intended) more difficult for most than others: the method relies on the proof of a conditional and not an absolute statement, and aside from the base case, therefore, one is never proving a statement directly. The very reason the proof method succeeds is the "if it's true for n then it's true for n+1" clause. Understanding the sufficiency of this clause (together with the base case) is itself a major leap, and therefore most beginning provers are uncomfortable with the very validity of an inductive proof, and therefore are less motivated to complete all of its components.

Maybe that's it.

Ideas, anyone? Throw me a frickin' bone here.

As it is, though, this semester's crop did pretty solidly on their induction homework. Of the twenty-five or so homework sets I graded over five hours this morning, although there were only two As, everyone who managed to turn in a completed homework set did halfway decently, and the class average couldn't have been much below 80%. I'm pleased. Though this term's students are a bit shyer in class than their Fall 2007 counterparts, their dogged pursuit of homework perfection is easily comparable to that strong semester's work ethic. Kudos!

Moreover, their LaTeX is coming along beautifully. A handful have really gotten the hang of it (Trixie, Tish, and Siegfried, y'all rock!) already and are executing nearly flawless TeX masterpieces, and several others (Omar's comin' atcha!) are using it regularly and their proficiency isn't far behind. More importantly, three or four people have already commented on how having to type the math up makes them think much more carefully about what they're writing, which is ultimately the primary goal of the class.

Keep it up!

Right wrong turns

Sometimes it just all comes together, y'know? It helps having a class full of kids (and a few older folks) who are willing and able to put the responsibility for their learning on their own shoulders and run off with it, leaving me in the dust.

Several times during the past week I've found myself extemporizing in Abstract II as the students have asked questions the answers to which I had no idea. That's the satisfying/scary thing about teaching these high-end courses: on the one hand, the students are smart and savvy enough enough to ask really deep and interesting questions on the fly; on the other hand, the students are smart and savvy enough to ask really deep and interesting questions on the fly. It ain't Calc I in there: although I will never tire of the thrill of teaching calculus (Calc I folks: I'm not kidding when I say that I never get over the beauty of the definition of the derivative. That's not feigned excitement: I really am in awe of the limit definition, after all of these years, and I'll never get tired of seeing the hs magically disappear!), and though those first-year courses are often jam-packed with smart students, it's no more than once every few semesters that I find myself perplexed by a student's question in Calc I: there's just not enough that's unfamiliar, and the concepts are second nature to me by now.

But Abstract II? Wow. That class is a kaleidoscope, and whatever view we find on a given day seems to depend as much on the mood of the class as it does on the phase of the moon.

A couple of the questions that have come up lately in that course are pretty routine and standard ones I was able to respond to without skipping a beat: "If f(x) is irreducible in k₁[x] and k₂ is a subfield of k₁, then is f(x) irreducible in k₂[x]?" "What would an irreducible cubic polynomial in ℚ[x] look like?" (This last was in response to the proposition showing that the irreducible quadratic and cubic polynomials over any field k are precisely those with no roots in k.)

The really interesting stuff happened when I was momentarily fertumult.

"Why would you want to represent f(x) in 'base' b(x)?" asked Bertrand in class on Wednesday. He's always good for a truly probing question or two.

"Um..." For a moment I rethought my policy of encouraging students to ask good questions like why?

"Um...well...if we know that k is finite..." I hemmed and hawed for a few minutes and waved my arms around in an unconvincing fashion. A few muttered words seemed to make some sense, and reason returned, and I felt better as my mental wheels gained a bit of purchase. "...then if we've got a homomorphism φ:k[x] → k[x], φ is completely determined by its action on the finitely many polynomials of degree less than deg(b) and its action on b itself."

Whew!

I've not done so much tiptoeing as I've done with these kids since my first semester at the University of Illinois when they gave me a section of Accelerated Honors Calc III for engineers to teach...those kids were smart. It hadn't helped me that at the time I hadn't done vector integration since my second year of undergrad.

This past Monday's class was the funnest so far this week, and it came about because I'd been sloppy in setting up an example in my worksheet for that day. The exercise asked the students to come up with two quadratic polynomials in ℤ₈[x] which each exhibit more than one nontrivial factorization, thus demonstrating twice over that ℤ₈[x] is not a unique factorization domain.

One of my preplanned examples worked, the other was verkochte (I'd forgotten that in ℤ₈, -3 and 5 are identical, so the "distinct" factorizations I'd expected were one and the same). Did the students give up? No, no, no! Instead, after a few minutes of floundering about trying to fix the failed example I'd started with, we just set up the equation that would have to hold in order to yield two nontrivially distinct factorizations, and solved away. We soon had a whole passel (not a half of a passel, or even two-thirds of one) of examples.

That's a great class.

While I'm bragging on them for their self-directed successes, I should give props to Uri for coming up with the basis for a fantastic question for the first exam. Though several of the students, when asked last week as part of their homework to posit potential exam questions (with accompanying solution sketches), came up with appropriately difficult and interesting posers, it was Uri's question involving the advanced ring theoretic properties of the powerset ring that proved the most amenable to inclusion on the exam. It was just right.

Okay, off to read a few Project NExT-Southeast Fellow applications before I hit the hay.

Au revoir!

Jacta alea est, and all that jazz

[This post was written at the Reagan Airport in D.C. on Tuesday night...but there was no way I was gonna shell out $7.95 to use the airport's wifi for the sake of posting it...]

Well, it’s done.

We did two takes, as that’s standard practice, but we were all pretty happy with the first one. In fact, after all is said and done, I’m happier with the first: I feel I was clearer, and that I did a better job of explaining myself than in the second. On the other hand, I believe the producers thought I was more animated and personable in the second one, and they thought it was a toss-up between the two.

We’ll see. I’m confident. I should hear from them within the next four months or so regarding where we go from here, one way or another. I let them know of a couple of courses I’d be willing to put together for them, a couple of which (including graph theory) they were quite excited about.

Me? I’m tired. Fortunately, aside from the inevitable backlog of e-mails to slog through, I’ve got nothing to worry about work-wise once I get back, at least until tomorrow. It thus promises to be a relatively relaxing red-eye run.

29 minutes, give or take

Ugh...I slept like crap last night.

Just a few hours before my date with the Teaching Company. I'll be happy to have this damned thing behind me.

After a few practice runs, I've got it down to about 29 minutes, give or take a minute or two, which is smack dab in the right range, so I'm pleased. I'm going to do one more run-through of the last half (modular arithmetic and the RSA cryptosystem) in a few minutes, and then call it good.

A word about classwork: I'm in the middle of my reading of the Calc I students' final drafts of their derivative "paperlets," and as one might expect there's a broad range of quality. Some are adequate, and others are positively fantastic. The best are distinguished not only by the completeness of their treatment and the correctness of their computations and statements, but also by the composition of their exposition: the ideal narrative offers smooth transitions from topic to topic that highlight rather than downplay the interrelation of those topics.

I hope to finish grading those over breakfast this morning.

Then it's off to the races.

Nearly there, my friends, nearly there!

Breakthrough

Today's the day I'm going to want to remember in the waning weeks of the semester: in the closing classes of the term I nearly always find myself thinking "what could I have done better? Did I really reach them? Did they ever find that spark?"

This morning my Calc I class lit up like a powder keg. Though it's taken a little longer than I would have liked it to, I genuinely feel that at last I've got a sort of simpatico with these students: I get them, they get me, and we're both willing to work our asses off for each other.

(Incidentally, I apologize for the uncharacteristic potty-mouthery in the past couple of posts. It's just been that kind of week.)

What happened this morning? Things just clicked.

I think the restructuring of the homework, basing it on problems of my own devising instead of on the arcane and ethereal exercises offered by the textbook, has helped a lot. It's a no-brainer, really: only I can create the sort of exercises that challenge the students to grapple with the topics as we see them come up in class. Moreover, while the book's exercises are interesting and thoughtful ones, the lessons the exercises purport to teach are wasted if the density of the problem precludes the student's understanding of the lesson's import. For example, the problem that purports to show that the derivative of a quotient is not the quotient of the derivatives (the now-infamous #36 from Section 2.3) would do well to simply say that that's bloody well what it's trying to show, rather than coyly trying to trick students into that understanding. "That's what it's trying to show us?" said several students, one after another, after I'd shown them the point behind the problem. "Why didn't they just say so?" Understatement's all well and good for French cinema, but with first-year mathematics exercises you're often better off being as subtle as an atomic bomb.

Yes, the few extra minutes it takes me in preparing for each day's class are a small price to pay for the benefits that accrue. Several students have indicated that though the exercises I've made up are still challenging and enlightening, they make a hell of a lot more sense than the ones the book had dealt them.

Today's in-class activities also proved exciting. We went on a limit hunt along the lines of an exciting game of Battleships: Having shown that the ratio (a^h - 1)/h tends to roughly 0.693 when a = 2 as h tends to 0, and to something slightly more than 1.098 when a = 3, the students set about trying to find the value that would make the limiting value 1 on the dot, asking Mathematica to graph the ratio for increasingly precise values of a. Several times Hera literally jumped from her seat in excitement as the race to estimate e tightened (granted, she's an exciteable soul in the first place, but still...). The excitement was hardly hers alone: several others were visibly intrigued, and by the time it was revealed that the number we were seeking could never be numerically known, there were warm smiles of understanding on faces scattered throughout the room.

To the Calc I folks who may be reading this: thank you. Thank you for your feedback in leading me to the change in course on the homework, and thank you for the warm and supportive reception of the change once we made it. Thank you, too, for the willingness to work with me in class and to take an active role on in-class exercises. I got more out of class today than I have from any class in a long time, and that success is based largely on the cooperation you've shown in creating a healthy learning environment.

Meanwhile the students were a bit more subdued in Foundations, a class that mercilessly meets just after the lunch hour, in the valley of biorhythmic cycles that tend to pull people bedward. "I'm sorry I was so quiet in class today," Trixie told me later. "I'm quiet in all my classes, but I feel bad that you were getting mad at us for being so quiet."

"I wasn't mad," I said, "I was just trying to wake people up!"

I will credit 280's Nighthawk with the line that brought me the greatest joy today, though: "it's [this course's emphasis on clarity in writing is] bleeding over into my other courses." It would shock me not if Nighthawk's colleagues found him guilty of felonious brownnosery and bullshitting in the first degree, but the fact remains I was tickled by the comment.

Meanwhile, my twelve-member Abstract II class was cozy and familiar. Indeed, I remarked at one point, as I paused to find my place on our worksheet, on how they chatted with one another warmly like old friends. "You are old friends," I said. "You've been in many classes together by this point, and most of you know each other really well." Then I waxed a bit pre-nostalgic. "I've had many of you in several classes, too, and for some of you this will be the last class I'll ever see you in! Many of you will be graduating in a few months, and it's going to be a very different school without you."

"Oh no, I'm going to cry!" Nadia said, hiding her face behind her notebook.

Fortunately Euclid's Algorithm intervened and saved her from her tears.

All told, it was a good day. A mite busy (two hours of unceasing student deluge after Abstract II...can y'all make an effort to finish the homework with a liiiiiittle bit more of a margin than a few minutes before five?) as the day neared dusk, but a good day in the end.

And now it's time to end this day (14 minutes to go!). Tomorrow brings a batch of grading and a little bit more class prep for which I won't have time as I'm winging it to Washington next week...and I hope too tomorrow brings a happy Valentine's Day to one and all.

From a hopeless romantic to his faithful readers, my thanks for your words of wisdom, your help, and your support!