What comes after "four"?

Deep not-specifically-mathematical thought for the day: it's been a long time since I've been at the same school for more than four years without some major milestone (e.g., receiving a degree) marking a significant turn in the road. I was at UIUC for three years only, and at Vandy for four years before that...there were five years in Denver, but I earned the B.S. at the end of the third year and at that point entered the wide, wonderful world of graduate school, so I may as well have been on a different planet.

I'm about to enter my fifth year here (and my fourth year of blogging about it). I wonder how it's going to feel? Steady? Stable? Institutionalizing?

We'll see.

Everything's more or less in place for the REU to start in a little over a week. (A week from right now is when I expect the first of the students to arrive.) Aside from the personnel, little is changing from last year's program: if things go well (and they did), why bother messing around? I've managed to put together a list of 39 at least moderately tractable questions to start with, we'll see where we go from there. Obviously the REU is going to take up much of my summer, and what little work time I've got outside of that will go toward prepping for the fall semester's courses (Calc I, with a new text, and 280 again).

In other news, my colleague Euterpe agreed to help read through last year's students' weekly reports, in order to assess the development of the students' writing skills along various axes. Thanks, Euterpe!

Plans for the weekend? Finish up the revisions on the paper on mathematical poetry that I wrote for Math Horizons. The editors gave me some great suggestions, and I hope that after I've hammered out the last remaining details the article will be more engaging and enlightening still.

Independence

What shape of mountain might you be,
what manner the curve
about your single upward-arched umbilicus,
what steepness the slope of your coefficients' climb,
that after convolving
and contorting
and clever recursion,
crushing you
and crashing you
against your brother's craggy scarp,
a more vertiginous mount remains,
a peak more sharp, more stately still,
looms loftily above your head?

A half-dozen deep(ish) thoughts

I never cease to be amazed by the simultaneous simplicity and utility of guided free-writing.

Here's the skinny:

1. Choose a topic on which to write, or let someone else choose a topic for you.

2. For five minutes (set a timer for yourself), write without stop on the topic you've been given. If you get stuck and can't think of anything more to say on the matter, just write "I'm stuck I'm stuck I'm stuck" or "what in the hell am I thinking right now?" or whatever you'd like to, over and over again until you become unstuck and refocus on the chosen topic. Don't stop writing, and don't correct yourself, grammatically, orthographically, or otherwise. And don't hurry. You don't have to write quickly, but be sure to write continuously.

3. When your time is up, stop writing.

4. Now review what you've written and select a few words or phrases you find startling, surprising, or important in some way or another.

5. Choose one of those words or phrases, copy it at the head of a new piece of paper, and...

6. ...begin anew, writing for five more minutes, without stop, on the key word or phrase you've selected from your first piece of writing.

7. If desired, repeat.

I helped put together another writing workshop for my colleagues today, and my colleague Euterpe, director of our First-Year Writing program, all-around wonderful teacher, and majorly cool individual, led the workshop participants in a guided free-writing exercise centered on the topic of writing assignments. Through the exercise, she hoped, we'd be able to more fully develop our vision of a writing assignment we hope to pitch to students in one of our writing-related courses.

I don't think I was very successful in that effort, but the discoveries I made more than offset my lack of progress towards constructing a meaningful writing assignment. I made no fewer than six major revelations in the course of my writing, four of which I realized right away, even as I was writing, and two of which I realized only later, as I was transcribing my handwritten work onto the digital page below.

Where'd it all come from? With the MATH 280 "equivalence class" exercise about which I blogged the other day fresh in mind, I began thinking about how I could ask students to further interrogate the idea of "equivalence class" through writing, and my output from the free-write is as follows (underlined handwritten text has been replaced by italicized text; everything else is verbatim):

***

Given a set of objects, how is it that we can make some sort of mathematical sense of them? How can we group like objects together, and what does it mean to be "like"? How can we put objects in order, from smallest to largest, and what does it mean to be small or large? This activity will help you to accomplish this task. By presenting you with a seemingly chaotic pile of objects you'll be asked to provide structure where structure is not immediately apparent, and in so doing will learn to recognize what it is that defines structure in the abstract: what properties does structure encompass, and how can you recognize these properties?

You'll be asked to come up with a short list of characteristics that define what it means for the sorting of a set of objects to be an "equivalence relation": that is, the way you sort the objects should be in such a way that two objects are sorted together if and only if they're "equivalent" in some meaningful way. What properties much such a means of sorting have? That is, if I asked you to say, "if x and y are paired together, and y and z are paired together," what can you say about y and x? about x and z? What about x and x?

Let's consider the example of the random objects sorted by color...

Provide structure where structure is not immediately apparent

What does it mean to provide structure? What is structure? Maybe it's a way of organizing things so that they make "objective" sense to someone other than yourself: you of course understand what you mean by an assortment you've made, but how can you help others to see your thought process? "Structure" provides a "user-independent" means of organization: you agree with others to establish a set of rules or properties that define what you'll mean when you declare a certain kind of structure exists. For instance, in the case of an equivalence relation, we speak of the following structural characteristics: reflexivity, transitivity, and symmetry. These are the defining characteristics of this particular structure. Thus if you tell someone, "oh, this relation is reflexive, symmetric, and transitive," they know that whatever structure stemming from that relation will "look like" an equivalence: every object will be equivalent to itself and so on.

How to best get students to recognize these "atomic" properties on their own? They'll be asked to sort, but can they understand their own method?

How to best get students to recognize these "atomic" properties on their own?

In a sense we have to first move students from intuition to mechanics before we can get them to go in the opposite direction! Students inherently recognize that a structure is present when they're faced with it; they just have a hard time articulating what it is that that structure encompasses. That is, how can we bridge the gap between "oh, I see it!" and "Ah! Here's what I see!"? It seems like the same problem we just discussed regarding good writing: students know good writing when they see it, but can they explain why it's good? Brainstorming about what makes good writing might help students with that recognition task, so maybe a similar brainstorm about equivalence relations and other structures is a good starting off point? From the fruits of a brainstorm session, the students can be asked to reflect and decide which are the ripest, the sweetest, the most delicious and worthy of keeping? Can students then make mathematically precise what these fruits are? I use a first day exercise...

***

Ready for my revelations? In order, they were as follows:

1. "...provide structure where structure is not immediately apparent..." Isn't this, at the end of the day, what math is all about? Isn't this all I'm really doing when I'm going about the business I've selected for myself? Is this what I'm asking my students to learn to do, ultimately? If it's really that simple, can I convey the basic notions of mathematics to my students more successfully if I pitch it to them in those terms?

2. "What is structure? Maybe it's a way of organizing things so that they make "objective" sense to someone other than yourself..." Isn't this, at the end of the day, what is meant by "mathematical structure"? In this case, isn't mathematics really little more than an elaborate metaphor, a linguistic convention, a highly human and humanistic mode of communication used to convey often abstruse and technical ideas from one human individual to another or to others? This is hardly the first time these things have been thought (hell, it's not even the first time I've thought these things), but I feel as though the free-writing exercise helped me to think these things more clearly: I was successful at writing to learn, and writing to discover.

3. "In a sense we have to first move students from intuition to mechanics before we can get them to go in the opposite direction!" At the college level (good) math teachers are always trying to get their students to transcend mere mechanical computation and instead develop good mathematical intuition: it's far better to understand precisely where a formula comes from than to merely memorize its concomitant parts. (If nothing else, with true understanding of a formula's provenance you can rederive it from scratch.)

How funny, then, that I realized through this exercise that in order to develop the most basic building blocks of mathematics (relations, functions, sets, orders, et cetera), one really does have to begin with an intuitive concept and work backwards from there, axiomatizing our intuition with rigorously defined concepts like "reflexivity" and "transitivity." Moreover, every time one adds new ideas to the existing mathematical corpus, one must develop new axioms and new definitions: new mathematical discoveries almost always come about through intuition, which is then succeeded by the axiomatization of the newfound ideas.

I find it ironic that I made this revelation today in particular, as just this morning I found myself facing the unpleasant task of writing to a colleague to disrecommend a student who shows profound inability to make the jump from mechanical computation to intuitive understanding.

4. "From the fruits of a brainstorm session, the students can be asked to reflect and decide which are the ripest, the sweetest, the most delicious and worthy of keeping? Can students then make mathematically precise what these fruits are? I use a first day exercise..." As I was about to point out to myself, my current first-day exercise in 280 challenges students to develop a theorem from scratch: beginning with raw "data" concerning the sums of certain pairs of numbers, students first posit a couple of definitions (of "odd" and "even"), then make observations about numbers having the properties of "evenness" and "oddness," then make a claim based on their observations (there's the theorem), and finally prove their claim carefully.

Why on Earth have I not thought to pattern further 280 exercises on this model? Why can't this model serve not only to develop the ideas of "equivalence" and "order," but also "function" and "set" and "combination" and "universal" and "existential" and...

...now for the two revelations that struck me later:

5. "How to best get students to recognize these 'atomic' properties on their own?" I tell my students in 280 over and over and over again: "whenever you get stuck, whenever you don't know what else to do, go back to the definition."

Why? First of all, often the definition is all you've got: if you're asked to prove something about continuous functions, then you'd by god better know what a continuous function is.

Second, definitions are generally atomic, or at least molecular. A definition concerns first principles, and is free from unnecessary clutter. At the 280 level, at least, if the definition doesn't offer an entirely self-contained description of the object or idea being defined, then unraveling the definition's meaning generally involves no more than tracking back to one or two slightly more basic definitions, the atoms in the molecule.

In a similar fashion, theorems are broken into propositions, and propositions into lemmas. You can't possibly come to a proof of a complicated statement like "the expected diameter of a use-it-or-lose-it tree grows linearly as a function of time" without breaking it down further, into simpler statements about the center of the tree, about its diameter and vertex eccentricities, about the way in which those quantities are likely to change as the tree grows according to the defining process, and so forth.

The upshot of all of this is that I realized why it was I'd been hung up on my own research (into "use-it-or-lose-it trees," in fact) for the past few days: I'd forgotten my own mantra and had been attempting to prove too much at once. I needed to step back and break things down into simple lemmas, the mathematical equivalent of Bob Wiley's baby steps.

I've done that now, and I've made more progress in a couple of hours than I'd made in a week or two before I realized my misstep. (280 students, take note! It works. It really works!)

6. I use a hell of a lot of colons when I write.

Seriously. Go back and count 'em. Nearly every other sentence I write has a colon in it.

I wonder why this is? Is this trademark quirk a function of the way my mind processes what I'm writing about? A colon generally precedes elaboration or clarification: what comes after it is meant to provide an illustration of what's come before it. (See?!)

Maybe teachers, prone to using examples to illustrate their ideas to their pupils, are more apt to use colons than people in other lines of work.

Ya think?

I don't know. I just find it fascinating that I so often use that particular piece of punctuation.

I'm going to end this post in just a moment, as it's been a long one, full of fun things to think about. But I'd like to leave you with an exercise, those of you who actually read this thing (I know you're out there!). Given the great deal I learned about myself today through free-writing, I thought I'd assign you, the reader, a brief free-writing task.

For those who'd like to try it out, please respond in the comments section to this post with the fruits of your labor on the following activity. I really do think it will prove a meaningful and enlightening activity, and I hope that you'll consider trying it out. (Former students: how 'bout it, huh? I know you miss my classes! It'll take you a half hour, tops, and I promise it'll be harmless and fun.)

1. We begin with the following question: "What is mathematics?" Now we follow each of the steps below.

2. Give yourself five minutes (set an alarm on your watch or cell phone), and write, continuously, on the topic above. Don't correct yourself, don't change anything you've written, just keep it intact, word-for-word. If you get stuck, write some sort of nonsense until you get unstuck and refocused on the topic above. You can type or write, whichever you prefer.

3. When five minutes are up, take a few minutes to look over what you've written, and select a word or phrase that strikes you in some way. Copy it to a clean sheet of paper (or a clean file in MS Word) and begin anew, writing continuously for another five minutes, starting from the word or phrase you've selected.

4. At the end of these five minutes, once more select a word or phrase that stands out, and copy it to a clean sheet of paper. Write continuously for another five minutes, starting from the new word or phrase you've selected.

5. What results? I'd be delighted if you could share your personal revelations (even anonymously) in the comments section to this post. You could even share your entire free-write, if you'd like to, but you certainly don't have to. Think of this as a semi-public performance of mathematics, a project undertaken in the spirit of Algebra al Fresco. I wouldn't ask you to do this exercise if I didn't think that in doing it you'd make some meaningful observations about yourself.

Please do give it a shot. (It's also a great activity for overcoming writer's block.)

In closing, let me say to my colleague Euterpe: many, many, many thanks for once again proving yourself an exceptional teacher!

Farewell

As I've said many times to many people in the last few weeks, the class of students who yesterday were graduated from UNC Asheville is in many ways "my" class. Having arrived at this school in Fall 2005, were I a student with a traditional sense of timing, I would have joined them in their sunlit march across the stage in front of the Ramsey library yesterday morning.

My last teaching gig, my first one post-graduate school, was a three-year research postdoc at the University of Illinois's main campus in Urbana-Champaign. I only taught one course per semester at that school with roughly 25,000 undergraduates, and, being given "juicier" teaching assignments like Accelerated Honors Calc III for Engineers, Abstract Algebra II, and a special topics graduate seminar in Coxeter groups, I never once taught one of the "core" courses like Calculus I or II, and thereby my chances of having the same students for more than one semester were further diminished. All told I taught 150 or so students during my stay at UIUC, and not a one of them more than once. The same holds true for my teaching career as a graduate student at Vanderbilt: I worked with maybe 200 students over three and a half years there, with no repeaters (well, one exception...but that's a [not altogether pleasant] story for another post).

So it's been radically and refreshingly different seeing the same students in class after class, some taking as many as 19 credit hours with me, watching them grow from timid (or not-so-timid) first-year students, many still clinging to their high school memories like a tattered spit-soaked security blanket, to mature, clear-thinking, sophisticated scholars capable of creating their own original mathematics.

It's been wonderful.

My class has come of age, and it's time now for them to leave. They leave to undertake new adventures, to seek out new experiences, to learn more, to do more, to contribute more to this world.

I say farewell now, fondly, to Bertrand, Cassio, and Davina; to Deidre, DeWayne, and Farina. I say my goodbyes to Farrah, Katya, Kaytlynne, Leonardo, Nicolette, Nidra, Oswald and Stanley; to Ulrich, and, last and certainly not least to Sylvester, Tatiana, and Nadia, the last three of whom shared 16, 17, and 19 credit hours of classes with me over the course of the last four years.

Though I know that I now have and will continue to have in the coming years many more wonderful students, those to whom I now bid adieu will always be dear to me, they'll always be my class.

I love you guys, and I'm proud of you all.

Good luck in all that you do. Keep learning, keep loving, keep living. Keep doing marvelous and incredible things with your time on this Earth, and with the talents you've gained in the years in which I've known you.

And for Pete's sake, keep in touch!

What's in a word?

Would one be more apt to call the academic community afforded by our department "healthy" or "vibrant"...or both?

A different class

I've always found it compelling to think that our ancestors from thousands of years ago were no less clever, no less smart, than we are today, and that they merely had a bit less experience, had had a few fewer millennia in which to sort things out by trial and error and intentional experimentation, than have we. Given several dozen more centuries in which to try their hands at various critical and computational maneuvers, certainly they'd have come to many of the same conclusions as we have by now. (You must admit that we've been given a distinct advantage by the astute application of printing technology and modern methods of data storage, data recovery, and data transmission.)

One day at some point during my third year of undergraduate study at the University of Denver I was idly toying with some polygons that I'd circumscribed with a unit circle and I noticed it wasn't hard to recover an inductive formula for the lengths of the polygonal segments that made of a circumscribed 2ⁿ-gon a circumscribed 2ⁿ⁺¹-gon instead. With a little basic trigonometry (it turns out that the Law of Cosines works best) you can arrive at an iterated radical formula for the number π.

I was flabbergasted, thrilled by my discovery, and the next day I told my adviser, excitedly, about what I'd found.

His response was something along the lines of "oh, Euler's formula!" I'd recovered a formula first noticed by the great Swiss mathematician Leonhard Euler (the 300th anniversary of whose birth was recently celebrated in the math community), akin to an even earlier formula, the first successful arbitrary approximation of π, due to the French mathematician and astronomer François Viète.

If you're going to get scooped by someone, Euler, one of the most prolific mathematicians in history, is not a bad one by whom to be scooped. Still, that discovery that your discovery is not a discovery at all, or at least not a new one, can be unsettling. Certainly it's happened to us all, and it happens more frequently when you make it your business to ask tough questions. How often do even the biggest names in math research get one-upped by slightly cleverer colleagues?

Asking tough questions is the job of the mathematician, so it's imperative that young math-minded minds get used to tackling tough questions in a controlled environment, one in which the answers are already known to be known, and in which tough but tractable questions can be set up for what they are: challenges and tests of skill, yes, but not traps meant to lure the student into a sense of hubristic invention.

Put another way, if you know from the get-go that the discovery you're about to make is not a new one you can take your attention from the statement of the theorem on the page in front of you and place it where it really belongs, on the path you're about to trace out that will lead you to the theorem at its end. That same path, you'll know as you walk along it, is the same as or similar to the one taken by hundreds of highly intelligent human beings who came before you...but like they did before, you'll make your way along the path yourself, and the fact that the land at which you'll find yourself at the end has already been mapped out and explored doesn't make that land any less beautiful or wondrous.

Discovery is like that.

While running this morning I thought of a discovery activity I can use in MATH 280 this coming fall when it comes time to rap about equivalence relations, a topic that proofs dauntingly difficult to a large number of students.

I'll gather several dozen small objects of various kinds and bring them to class in a big ol' bag and empty the bag onto the classroom floor.

"Sort 'em out," will be the order of the day.

"How?" I can imagine students asking.

"You tell me." They'll pick through the pile of stuff scattered before them, and after a bit of trial and error patterns will emerge: the Tonka truck matches up with the lemon-shaped lemon juice bottle (for obvious reasons), and by the same logic the wingnut and the nickel get tossed in the same subpile, and the magnolia leaf meets up with the mango. Without realizing it, the students have constructed an equivalence relation, creating classes whose elements exhibit demonstrably reflexive, symmetric, and transitive properties.

"Can you do it another way?" The next iteration takes a bit more thought, and perhaps now inorganic objects are grouped together while once-living things share a different class. Or perhaps size proves to be the most distinguishing characteristic. Somehow a new partition emerges, and another equivalence relation is born.

A similar exercise may well work to demonstrate order relations. Confronted with a disorderly mess of objects, can the students impose some kind of order on them? What properties does this "order" satisfy? What properties does it not satisfy? Does the order need to be a total one?

Surely the students, without formal knowledge of the definition of the phrase equivalence relation will be able to build several such relations of their own, and having done so will be far likelier to recognize such relations when they encounter them in more mathematical contexts. Moreover, they'll have a greater appreciation for the technical definition of equivalence relations when it's given to them.

That's the power of discovery: you're much likelier to remember and understand something you discovered yourself than something someone else discovered for you and merely told you about.

Why in the hell don't we teach like this more often?

I know an answer to that question already (and my colleagues and students should feel free to supply many more in the comments section): because it's difficult to do so. Setting the stage for incipient discovery is far more difficult than describing what discovery looks like.

I admit that, though I hope that my classes set students up for discovery more often than those of less ambitious instructors, I make use of discovery-based pedagogical methods more rarely than I should. I'm trying, my friends, I'm trying to address that. I hope to devote a good deal of time this summer both to my own discovery (during the hours I spend with my REU students and the other students with whom I'll be doing original research) and to developing means by which I can facilitate others' discoveries on their own.

What discoveries, new and old, await us? I'm tremendously excited to set out on this summer's journey.

I am not Euler, and you are not me. Yet we're all human, we're all clever and intelligent, we're all naturally inquisitive, and we're all equally capable of discovery should we put ourselves in positions from which discovery is easily possible. In this regard no one of us is in a different class.

Making sense of making sense

As you've surely surmised, given my incessant ballyhooing about my Newton v. Leibniz analysis coupled with the continued absence of said analysis from this blog, it's proving to be a helluva chore to extract the essence of my students' reflections on the project and distill it into anything simultaneously meaningful and manageable.

I really do hope to have a nice post on the project up before the REU starts (June 8th!) and my life becomes decidedly busier once again.

For now, I've started to catalogue my notes on the students' reflections according to several rough headings, as follows:

The nature of discovery. Reflections on this topic center on the ideas of discovery and invention: is there a difference between the two, and how can one tell one from the other? Who is entitled to discover or invent, and how does one go about doing these things?

The nature of mathematics as a discipline. Reflections in this category include comments on the way in which math is made and the way in which it's communicated. The deepest such comments were purely epistemological, striking at the very heart of knowledge itself.

Personal reflections. What did the students gain from the project personally? What did they learn about themselves as people, and as learners?

Suggestions. How might the project be modified in the future in order to make it more effective? The suggestions I received ranged from minor comments on the process to major structural overhauls.

Given the depth of these ideas (and the prolixity with which my students produced them), I hope you'll agree it's no surprise I've been prevented as yet from digesting them and writing on them.

This week, an abstract; next week, a post. I promise!

Good advice

I've just finished typing up my 280 students' responses to the final question on their exam: "Please compose a brief statement detailing any advice you would like to give to someone taking 280, to be given to the student on the first day of class. You can be as specific or as general as you like, but please be sure that your response offers honest and thoughtful advice to a new 280 student." The advice will indeed by posted on the website for the Fall 2009 MATH 280 class, and it will be the first reading required of the new 280 students.

The advice ranges from comical ("I recommend keeping a running total of how many times Professor Bahls yells, throws a chair, drops a table, ...or does anything else that startles or amuses you") to metaphysical ("I also found that meditation helps out a lot"), hitting nearly every point in between. Some of the advice is, shall we say, idiosyncratic enough that it might not prove so useful to anyone but its purveyor; other advice is solid.

One thing I noticed about the advice this group has offered to future generations of 280 students is its general practicality: it's very focused on the basic "mechanics" of homework completion and class survival. For instance, the several students emphasize things like starting the homework early, unfailingly attending class, and asking questions of the professor as the keys to academic success; surely this is good advice, but it's hardly different from that one would offer to a student about to begin any other class.

What is it about 280 in particular that makes it such a difficult class? And what specific advice might one offer to a new 280 student, as opposed to one ready to begin Language 120 (our first-year composition course) or Humanities 124?

The students had a few words to say about this. I'll be posting the full text of the students' advice on the website for the Fall 2009 280 course in due time.

For now, below, I've compiled a laundry list of things I want to be sure to tell next semester's students, early and often; admittedly the list betrays my own personal bias, but I think every item is a worthwhile bit of wisdom:

1. Write. Write to communicate, write to learn. Write knowing that every jot and tittle of every bit of notation means something precise, something definite. Write clearly, and keep an eye on your composition. A proof, even a correct proof, is meaningless if it's not also clear and well-composed. But don't forget that writing is an iterative process, and that rough drafts are meant to be rough. In all likelihood, your first draft will be shit, but it's important to get it out in front of you so that you can work on cleaning it up.

2. When faced with a new problem, write down what you know and what you need. Often 50% or more of the solution of the problem consists of formulating a clear statement of the problem's hypothesis and a clear statement of its conclusion. Once those statements are on paper (literally) in front of you, often the path you must take to connect the two becomes evident.

3. Trace out that path in baby steps, applying a single definition at a time, a single logical inference at a time. Don't combine steps or take more than one at a time until you are fully confident that you've not misstepped. Most of the mistakes you make will be made when you attempt to take big steps or to skip them altogether; the smaller the steps you take, the likelier you'll be to stay on course. If you're not sure about something you've written, read it out loud.

4. When in doubt return to the definition. As noted above, every symbol, every term, every penstroke means something, and something definite: if you're not sure what that something is, look it up.

5. Work together. Very few are the problems in mathematics for which a single solution is known. Moreover, there are manifold viewpoints on any single solution to a given problem, and it's likely your friends' viewpoints will differ dramatically from your own. In working together you can more easily combine your viewpoints to create a richer picture of the problem with which you're all faced.

I'll leave it at that, as the best lists of advice are short lists of advice. For now, it's bedtime.

Coming soon: the follow-up to Newton v. Leibniz, further reflections on the current graduating class. and more!

Thank yous all around

It's 9:00 on the "morning after," and I've just finished grading the Calc I exams. Only one person failed! I don't think anyone will be failing the class, a success by any measure.

I wanted to check in briefly and send a quick thank you to the hundreds of wonderful students I've had at UNC Asheville, and to dozens of my wonderful colleagues, without whose hard work and support I would not have been able to have earned the honor bestowed on me yesterday. Any award for teaching excellence rightly belongs as much to the students whose dedication drives them to academic success day after day with intervening sleepless work-filled nights, and as much to the faculty members who reach out to their fellow teachers with new ideas for class activities, assessment techniques, and innovative learning experiences, as it does to the award's recipient himself, a single person who is the product of the environment in which he does his job.

Once again, thank you, all of you, my fellow-travelers on this neverending intellectual journey.

More soon, on what it's like saying farewell to "my class" of students as they prep themselves for graduation, and on this summer's coming wave of researchers from near and far, and on the dawning of Algebra al Fresco...all in due time, once I've got my head above water.

For now, it's on to 280!

Reading day

One course down, two to go.

Abstract Algebra II is now wrapped up, all grades successfully submitted by noon today. Overall the students did very well, which is hardly a surprise since the class had a number of our best and brightest and most senior students, five of whom are leaving us after commencement in a few weeks.

So far the questions from students in the other two courses (both sets of students are dealing with take-home exams) have come in a steady trickle.

Yup, two take-home exams.

I was a bit reluctant to make the Calc I exam a take-home exam, after the debacle from Fall 2007, but I decided that what I perceive to be the benefits of granting the students a take-home far outweigh the negative aspects, including the risk that one or more students might again decide to cheat.

1. By affording the students time for meaningful reflection on the concepts learned in class, take-home exams offer a further formative learning opportunity rather than simply a summative assessment of a student's ultimate performance.

2. By eliminating the largely artificial "high-pressure, high-stakes" environment of an in-class test, take-home exams are more apt to measure more clearly students' understanding of (or at least ability to recover and synthesize deep ideas involving) the course's subject matter than simply aptitude in test-taking.

3. As hinted in the previous point, take-home exams more closely approximate "real-world" settings in which students will eventually find their skills tested, in which they will generally have access to resources (books, notes, mentors, and, yes, the outlawed-even-on-take-home-exams colleagues!) in order to better to meet the challenges with which they're faced.

We'll see how it turns out. I did give a little bit of a lecture when I distributed the exam sheets a few days ago, an uncharacteristically stern admonition at the outset, hoping to instill in the students the gravity of the trust I'm placing in them.

I've received one student's exam so far, but I've not had a chance to look over it. We'll see.

For now, I'm working away, one day at a time.

Tomorrow: donuts and derivatives, and a meeting about writing assessment. Oh boy!

One down...

...two to go. Somehow this end-of-semester has lacked the spark with which most terms terminate. Today's last session of Calc I seemed anticlimactic.

I dunno. Maybe it's just me.

Not ones to disappoint

The 462 presentations were solid.

Though they too have yet to get so far in their careers as to overcome the butterflies that bubble up when one's at the front of a darkened room, all six students whose task it was to speak today did very well. One of the presentations was the tiniest bit rigid, another could have stood slightly stronger preparation, and another seemed a little hurried as the speakers tried to fit in all that they had to say, but the slips were only of the slightest kind, and overall the talks were of the quality you'd expect from this set of mostly seniors. It'll probably be a bunch of As, all around.

For my part, I feel I did better in providing the students (both those in 280 and those in Abstract) with the scaffolding they needed to construct their talks this time around. Next semester I hope to be more intentional still in preparing my students for sharing their ideas with one another orally, both in the formal structured setting of end-of-semester presentations and in relatively informal day-to-day chit-chat sessions (both fora have their merits).

As another day draws to a close, I'll say farewell for now.

Jolly good show...jolly good!

I'm halfway into a Friday filled with student presentations: the 280 folks have offered up three inventive exhibitions, and all did pretty well (especially when one norms out for the nervousness and jitters that accompany what's likely the first semiformal mathematics presentation most of these people have ever given).

The first course on the menu was a deep dish of induction (one within another, like two thirds of a turducken): three of the students worked up a careful verification of the fact that the gamma function generalizes the factorial function, complete with gorgeous LaTeXed slides!

The next offering was a quick course on multinomial coefficients and their usefulness in solving a number of enumeration problems. A few typos aside, their presentation was clear and correct, and interactive! Worksheets are never a bad idea.

Finally, we finished things off with a multimedia derivation of the closed form for the sum of the first N cubes, assuming the formula for the corresponding sums of squares and linear terms. Though there was insufficient time to develop the full proof in class, the method these three folks used was identical to the inductive proof one would use to derive the closed form for the sum of the first N nth powers.

All of today's talks (aside from minor bubbles and burps) were clear, correct, and mostly well-composed. I'm happy. Overall I'm mightily impressed by the maturity of the problems selected by the students in this class, especially since several of the choices made were motivated by inherent interest in one problem or another. (I.e., I didn't have to twist too many arms: a lot of people naturally gravitated towards problems they'd thought up themselves.) The level of preparation has also been exceptionally high, as has the mastery of the subject matter. Though this semester's definitely had its ups and downs, I've got high hopes for a number of the students in this class as they move onward in their math careers. (At least three of them will be doing work with me this summer.)

For now, I must say adieu...I'm off to Abstract II, where I'll be hearing all about lattices, fuzzy groups, and the Sylow Theorems.