Tuesday, May 19, 2009

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"equivalence class" through writing, and my output from the free-write is as follows (underlined handwritten text has been replaced by italicized text; everything else is verbatim):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!

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