Random Walk, Take One

I've got a few photos to share from yesterday's inaugural random walk, sponsored by Algebra al Fresco. In all we had 16 different people take part (never more than 13 at a time), including three members of the general community who found out about the event through local advertisements. Good fun!

First, a demonstration of some of the equipment we used to help us plot our course:


Four-sided dice led the way at each four-way intersection; traditional six-sided dice directed us (with two values for each route) at three-ways. Only once or twice did we have to flip a coin for "either/or" choices.

It being a rather chilly morning (about 30 degrees Fahrenheit at the outset of the walk at 10:15), an early consensus decision was made to stop at Izzy's Coffee Den on Lexington Avenue, where walkers warmed up and partook of toasty caffeinated beverages:


Out on the road again, we stopped at every corner to conference on our next move. One person would roll, and another one or two, before the roll, would call the directions, pointing: "1, 2: that way...3, 6: that way...4, 5: that way..." It was a genuine group effort.

I don't know why La Donna's laughing in the picture below, but she's always laughing at me about something. ("You look goofier than me," she tells me. She's probably right.)

Pauses, though by necessity frequent, were never long, and we were soon on our way again. Below, we stride confidently down College Street, Ino (one of three of my Calc I students to show up...way to represent, y'all!) leading the way.

By my reckoning we took 38 steps, visiting 22 distinct intersections. Of these intersections, 11 were visited once, 6 twice, and 5 three times (none more frequently than this). As anticipated we never strayed off of the map of downtown Asheville I'd printed out for folks to follow as we walked. The furthest-flung intersection we ever reached from our starting point at the southeast corner of Pritchard Park was eight blocks to the east, the roundabout at the corner of College and Oak.

I'd be happy to provide readers with further statistics upon request.

All in all, it was a great social event, a good way to get some exercise on a brisk and beautiful late autumn morning, and a fair bit of nerdy fun. For sure there'll be another in the spring!

The calm at the center of the storm

It's early, early, early on Sunday morning on the weekend between the last week of class and finals week. I've had a good stack of grading to get through, but I'm about two thirds of the way through that, and tomorrow morning brings the most recent Algebra al Fresco event, a random walk through downtown Asheville.

The semester's ending well. I've been worried about how my philosophical frustrations from a few weeks back might have been adversely affecting the students in my classes (especially Calc I), as I've feared they may have been buffeted by ever-shifting winds. But signs point to students' weather the storm rather well. One Calc I student spoke of being "inspired," and one 280 student said that, despite the difficulty and relative disorderliness of the textbook project, he learned "50% of his understanding of the concepts from the project." That comment, coupled with the envy my Spring 2009 students have shown for my current ones' getting a crack at this assignment, has convinced me that the same assignment, modified to fix the weaknesses it's shown this semester, should be a part of the curriculum for next term's Topology course.

I've still not put much thought into the precise structural details of that course, but it's starting to come together. It's not going to be as tightly structured, and it's going to be highly collaborative, and all assignments will allow unlimited revision and resubmission. Beyond that, who knows?

For now, it's late, and I'm off to bed. I hope to post again tomorrow with pictures of the random walk!

Ma vie en film

I've finally had a chance to upload a few pictures from somewhat recent math-themed events, and I thought that by sharing them here I might offer a nice break from the heavy philosophy-laden posts I've been cranking out lately.

First, a couple of shots from the Goombay Festival held near the summer's end, at which I helped several of our students staff the Asheville Initiative for Math table. This street festival helped us to bring our Menger-making to a younger audience than we'd attracted in Pritchard Park during FractalFest '09:


The kid above wasn't nearly so gung-ho as the young (I seem to recall she was 10) woman in the following picture, who stuck around long enough to build her own level-1 sponge in its entirety:



Next, a few shots from the L-tile episode ("Build your own fractal") of Super Saturday this past semester. In the first, the kids are just beginning to get the hang of the iterative construction:



In the next shot, they're a bit more ambitious, busily working away at an L₈:



And finally, here they are basking in the glory of the first L₁₂ I've ever seen Super Saturday students (or MATH 280 students, for that matter) build:



Finally, here are a few shots of the most recent Menger-making, which took place a mere couple of weeks ago on the steps of the campus library, on a beautiful mid-autumn afternoon:



Above, Nighthawk (our school's most talented Menger-maker) sits proudly before a couple of the day's first creations. Below, Ino gets into the action:


Things picked up later in the day, with a number of non-math majors joining us:


By the time the sun was preparing to set, we'd nearly finished Algebra al Fresco's second-ever Level-2 sponge, which now, completed, rests in my office:


More photos as events warrant. Now, I'm for bed.

This and that

This week's gotten off to a good start, though Tuesday already feels like Thursday, and Friday will feel long overdue once it's come.

Today I played host to one of my colleagues from Samford University. Having driven seven hours from Birmingham, Alabama, Colin spent last night and today with me and my colleagues here, giving a great talk, chatting with me about REUs and the Sectional MAA, and meeting with various faculty and students from the department.

His talk was fantastic, offering the audience a unique blend of real analysis, linear algebra, and introductory proof techniques. There were about a dozen students present, and many of them are currently enrolled in...well...Real Analysis, Linear Algebra, and Foundations. For the analysts there were metrics, and orthogonal families of functions, and convergence; for the linear algebraists there were opportunities to apply eigenvalues to compute the closed forms for the terms of the Fibonacci sequence. For my MATH 280 students there were both implicit and explicit references to a number of the core concepts from the course: bijections, the pigeonhole principle, induction, proofs by contradiction, and equivalence classes and partitions. The talk was challenging but, I hope, accessible, and there were knowing smiles on a number of the students' faces as Colin reached his deftly delivered denouement.

In the afternoon, after his talk, Colin spent a few hours with me in my office talking about the design and execution of REUs, as he's hoping to submit a proposal to start one up at his own institution. I think I was able to give him some pointers and step through the process I followed as I put my own program together, but I couldn't answer every question. I honestly don't know what in particular about our program, aside from hard work and dedication on the part of the participating faculty and students, has made it so successful.

Colin will be heading home tomorrow; I've already been invited to join him at Samford in April, where he'll return the favor of hospitality he granted him during his stay here.

What else is new?

I realized yesterday that I was so busy bitching about grading over the weekend that I neglected to mention even once that on this past Thursday Algebra al Fresco sponsored the building of our second full Level-2 Menger sponge. (Pictures soon, I promise!) This one came together on the quad, on the steps leading up to the library. Working from 10:45 in the morning until nearly 7:00 that night, last Thursday several different students joined me in making the monster which now rests on a card table in my office, right where this past summer's sponge sat for a few weeks before moving on to the Engineering Department to get shellacked for display (so I'm told...it's yet to reappear).

A single student, Nighthawk, was singlehandedly responsible for about half of the cube's construction. The guy's a born folder. By 5:00, when I had to head home, Nighthawk and my current Calc I student Lambert, having overseen the splicing of 16 of the 20 Level-1s needed to complete the Level-2, decided they'd not rest that night unless they'd finished the sponge, and so they worked away in the Math Lab for a few more hours, wrapping up over eight hours after construction had begun.

Nighthawk swears that he'll be able to set the unofficial world record for solo construction of a Level-2 sponge (current record: 15 hours). I believe he'll be able to do so, maybe after a few practice runs. Speedy construction poses an interesting operations research problem, actually: imagine a team of four builders working together to complete a Level-2 sponge. How best to use their time? All four should start out building Level-0s, and at a certain point one or two should switch to sewing together the Level-1s, and at a later point still one of these should switch over to the making of the Level-2, all while their two friends keep plugging away at the basic building blocks.

But when should the switches occur in order to minimize construction time?

And is there a more efficient means of splicing the lower-level cubes to form the higher levels? (There surely is...the question is more "what is the most efficient method?")

As I said above, I'll soon post some pictures of the construction. Most of it took place on an unseasonably warm and sunny day on the library steps. It was a pleasant Thursday.

What else is new?

Perhaps an update on the Fall 2009 Calc I Homework Debacle is in order.

After a good deal of thought, I decided to make all homework for my Calc I students optional for the remainder of the semester. It's simply not worth my time to grade half-hearted attempts at homework completed (or, more to the point, incompleted) by undermotivated students who are more often than not cribbing their answers from the solutions manual. To those (who I suspect will make up the majority of the class) who still wish to complete the homework, I promised to continue providing the same robust feedback and the same careful attention I've always given. (Not once have I begrudged granting such feedback and attention to deserving students; I'm frustrated only when a dozen hours of my time spent grading sloppy work remains unreciprocated and undervalued.) To these students I also promised to "lock in" their current homework grades, ensuring them that their grades will not fall but can only see improvement between now and the semester's end.

I can't stay mad at these students: for the most part they're hard-working, well-intentioned, bright, and fun to work with. As I said to them in class, I'm not frustrated with them so much as I am frustrated with the process. And as I said to one or two of them in the cozy confines of my office, I'm not disappointed that they come to me seeking ways to maximize their grades, I'm just disappointed that they and I have been caged in a system in which they feel it's necessary that they maximize their grades in the first place.

The students' relatively strong performance on the applications handouts from two weeks back has convinced me that such assignments may be able to form the backbone of a yet more student-centered Calc II course. Next semester's homework schedule might look something like this (assuming a four-day class meeting on MTWF):

Week 1, Tuesday: suggested textbook problems from Section x

Week 1, Wednesday: suggested textbook problems from Section x+1

Week 1, Friday: suggested textbook problems from Section x+2; due for feedback only: textbook problems from previous week; due for a grade, or for inclusion in a student's portfolio: applications handout regarding Sections x-3 through x-1

Week 2, Monday: applications handout regarding Sections x through x+2

And so on.

There's that "p" word again: "portfolio." I've thought a bit more about portfolios, and about what might go in them. Whereas, as I've said before recently, students might be able to demonstrate their achievement of very skills-oriented learning goals (like mastery of derivatives or integrals, for example) through including in their portfolios more traditional exams or quizzes, suitably suggestive applications handouts could provide students with relatively uncomplicated low-stakes writing assignments through which they might demonstrate achievement of some of the harder-to-get-at goals, such as maintenance of skepticism and application of problem-solving methodologies.

Speaking of skepticism, it delighted me to no end to hear Uriah, one of my Foundations students, talk about the ways in which our class has begun to change his perspective on mathematics. "You just can't take anything for granted," he said as we sat at the dinner table with our guest speaker. "I want to question everything, and prove everything to make sure it's true."

His comments reminded me of the Calc I learning goal I recently discussed on this blog: "Demonstrate (through informed question-asking) a healthy skepticism regarding mathematical and scientific arguments." His comments assured me that he, like a number of his peers, is getting a lot from our class.

And speaking of getting a lot from our class, I'm getting more and more excited about the textbook as it begins to come together, and as several of the students are expressing increasing interest in ensuring that it's executed as cleanly, completely, and correctly as possible. "I intend to share it with future 'generations' of students who come through this course, so please keep in mind as you write it that you ought to be writing to help them." It's got tremendous potential, and I hope to share it was as wide an audience as I can. You can bet I'll bragging on it at the Southeast Sectional Meeting of the MAA in March.

Okay, I'm clocking out for the night. I'll leave with a notice of publication: I found out a week or two ago that my article on using poetry in the mathematics classroom, complete with poems by several wonderful students whose work first appeared here and here, has now appeared in The WAC Journal. Let the celebration commence.

Fractal Fest '09

I know it's been a while since I checked in...suffice it to say that the REU's been keeping me busy (no surprise there). That, combined with trying to wrap up the proposal to get continued funding for the program (this year's the last supported year) and trying to think of something meaningful to say about writing programs in times of economic hardship, has kept me away.

I simply had to say a little something about this afternoon's first "math in public places" event, the Fractal Making Festival sponsored by Algebra al Fresco. With roughly 25 students and passersby lending their help, we were able to build a Level-2 Menger sponge out of business cards in about four hours. The actual construction of the cubes and the Level-1 sponges took about three hours, and the assembly of the Level-2 from the Level-1s took another hour of careful surgery.

The finished product is pretty sweet, I have to say; it's sitting in the back seat of our car right now, awaiting transport to the Math Department sometime early next week.

Check it out, pictures below!

This first was from early on, when things were just getting underway:


Not more than an hour or so in, we really started to build up a backlog of little Level-0s, as you can see here:


Pretty soon two of us dedicated ourselves to the sole task of assembling the Level-1s, and a third joined us an hour later, just to keep up with the Level-0s coming our way.

We had help from folks of all ages:


Progress was made. Soon we turned to putting together the Level-1s, and the Level-2 was born, a little bit...

...at a time...


In the end:


We had a bit of fun transporting it down Haywood Street to the parking garage where I'd left the car. The table on which we'd set it while it was under construction made a handy palanquin:


Oohs and aahs accompanied this procession (at least, I like to think they did). In any case we got a good number of stares of curiosity.

Next on the Algebra al Fresco agenda? Pi recitation? Math poetry slam? Some sort of graph theoretical six degrees of separation sort of game? I'm not sure. Stay tuned: further bulletins as events warrant.

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!

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!