Nearly three weeks in, and still going strong!

The past week has been a good one, and not just for the vacationlet in Virginia Beach, where after the half-marathon Maggie and I and friend/ex-student Mariposa (now teaching middle school in Fredericksburg, VA) hit a local pizzeria called Pi-zzeria, whose theme is the letter pi and from whom I bought a wickedly cool shirt with a pi on the front. I've brought a few fun activities into all of my classes, and I'm particularly happy with some new ideas I've incorporated into Calc I.

There, last week, we pieced together a mathematical jigsaw puzzle, an exercise I thought up on my way into campus that morning. Here's the recipe:

- Print out a somewhat familiar picture (I used
*Mona Lisa*for one section, and a detail from the ceiling of the Sistine Chapel for the other). - Subdivide another sheet of paper into a number of rectangles, grouped in fours, equal in total number to the number of students in your class.
- In each rectangle so created, write an unreduced expression involving exponents and logarithms, in such a manner that the group of four rectangles contained in any given "region" of the paper holds equal values.
- Photocopy the picture onto the backside of the grid you've just created.
- Cut the rectangles apart from one another and shuffle 'em up.
- Distribute them to the students, and let 'em assemble the picture by first piecing together the local regions with similar values, taping these together, and then fitting these regions into one another.

The first section took about 9 minutes and change to put their picture together, while the second section came in around 9 minutes.

That was last Thursday. Then they had their first team quiz on Friday, and everyone did very well (between the two sections only a couple of teams missed a perfect score, and even those got 4/5). For the quiz I gave them a problem which likely would have been rather hard for my Calc I students from last semester, and these kids just ate it up. I can tell I'm going to have to challenge these folks with some tougher open-ended problems. From what I could tell, most of the teams collaborated smoothly, too: as I walked around the room, I heard a good deal of explaining, cooperating, clarifying. I don't think there are any truly indomitable personalities in either section. (I do have to say, though, that one student, Tallulah, did mention that she was a bit disgusted with the nattering negativity coming from a pair of her peers in class the other day. I hope this was just a blip on the radar, not to be repeated. I'm doing all I can to create a classroom environment in which people can feel free to pose possible solutions to the problems we discuss, even if they're not entirely sure of their answers; careless critiquing of those brave enough to venture such solutions is hardly appropriate. I don't know of whom Tallulah was speaking, but if you're reading this and you recognize your own behavior, shame, shame!)

What else? Yesterday towards the end of class I asked each student to provide me with a pair of topics discussed so far in class, one of which she or he understands thoroughly and a second on which she or he feels fuzzy. I took some time last night to match each person up with someone else from the class, pairing people off who expressed the same uncertainties in understanding: two folks who felt iffy on inverse functions might have gotten grouped together, or two who reported feeling lost with logarithms. For next Friday I'm asking the pairs of people so matched to work together to construct a dialogue in which they help one another through their mutual difficulties with the topic with which they both expressed confusion. My hope is that in addition to understanding the relevant mathematical concept more clearly, they'll all uncover something about their own learning styles as they examine what it is they're unsure about. Moreover, hey, it's a great way to get them to do a little writing. (Boy, I am the WAC nerd, aren't I? Speaking of which, I've still gotta finish up an abstract for Austin...)

Finally, before and after class yesterday I approached the students who had done particularly well on the most recent homework sets and asked each if he or she wouldn't mind sending me a short e-mail indicating the way he or she completes the homework assignments, in the hopes that I can glean from the ensuing comments some helpful hints I might compile in a handout to give to these students' colleagues who are having more difficulty with the work. I hope they might answer questions like: what do you do when you do your homework? Do you work alone, together with friends, in the Math Lab? Do you make use of a solutions manual? How do you use it, if you do? Is there a way you approach certain problems, a particular way of viewing them? Do you have any specific techniques you recommend, tips for your peers? I didn't ask these questions specifically, hoping to receive unprompted and unfiltered responses. So far I've heard from Magdalena and Xavierina (whose homework, by the way, is some of the most beautiful I've ever seen: it's clearly written, organized, well-documented with an appropriate amount of work shown, and almost entirely correct; Xavierina, if you're reading this, kudos!), and I've had hallway conversations with a few of the others who promise to send me their comments soon.

Meanwhile, there's 280. I'm still having a bit of ball in that class. As well as last Spring's 280 was received, I feel better still about this most recent installment. I can't put my finger on it, but I feel there's a healthier dynamic in this group of students than there was last semester. It almost feels as though the class is significantly smaller, even though there are only two fewer students now than a few months back. It's cozier, comfier, somehow.

The first committee report was made last week (Monday, I believe?), and the three members of that first deliberative body seemed to work well together. At least, I heard no complaints. Their report was a brief one, doing little more than illustrate a couple of the superior responses the committee received (by the way, participation was salutarily high, with about 2/3 of the class submitting solutions). I think the students might have felt a little uncomfortable about indicating others' errors in front of the class (even anonymously), so they avoided outright criticism, but I hope future committees (two more reports tomorrow!) will feel it's okay to indicate common pitfalls, especially if many people fell into them.

Voluntary committee involvement has been strong, I've had no trouble getting people to offer themselves up, and both committees received submissions from over half of the class yesterday.

I have 280 components like the homework committees at the front of my mind as I make my way through the latest in a long line of teaching-related reads, Peggy S. Meszaros's (ed.)

*Self-authorship: advancing students' intellectual growth*, Jossey-Bass, San Francisco, 2007, the focal text of yet another university learning circle I'm taking part in this semester. So far though I've not found the book thoroughly engaging, it's served to reconfirm much of what in the past few years I've come to know and believe about progressive pedagogy at the university level.

In the opening essay, "The journey of self-authorship: why is it necessary?," Meszaros takes the definition of self-authorship offered up by the now-canonized Marcia Baxter Magolda: "the capacity to internally define [one's] own beliefs, identity, and relationships" (p. 10, from Baxter Magolda,

*Making their own way: narratives for transforming higher education to promote self-development*, Stylus, Sterling, VA, 2001). (Justifiably this concern takes center stage in many of today's progessive college classrooms: time after time we hear that what students in today's universities most need to learn is indeed simply

*how to learn*.)

On the facing page in Meszaros's essay, we find the following snippet: "Becoming the authors of their own lives involved reshaping what they believed (epistemology), their sense of self (intrapersonal), and their relationships with others (interpersonal)" (p. 11). As I read this, I jotted some notes in the margin regarding the role played by a discovery-centered approach to proofs and proof-writing in helping to affect changes of all of these sorts:

- By being encouraged both to construct their own proofs and to thoughtfully critique others', the students gain a deeper understanding of the nature of mathematical knowledge, particularly of the fact that it doesn't inhere in any one person, no matter how intelligent that person is. Knowledge ceases to be "out there, somewhere," but rather "in here."
- By allowing students to take command of both the proof-
*writing**and*the proof-*reading*(in a literal sense) processes, as I'm attempting to do in our class by establishing the homework committees, I challenge the students to take on the role of the mathematical authority: mathematically speaking, anyone who can grab hold of the governing rules of math logic can stand in judgment of the correctness of a given proof. No longer are the students simply vessels for knowledge not yet bestowed; they are the bestowers themselves, they are the experts. They are participants in the mathematical process, not merely spectators. - By cooperating and collaborating in the proof-writing and proof-reading processes, the students come to appreciate that mathematics is a social enterprise, that it is conveyed in a transmittable medium, that it is a part of our shared heritage, ultimately constructed by human beings working in concert with one another.

How successful will this class prove (no pun intended) in easing my students down the road to self-authorship?

I don't know.

Do my colleagues think as deeply about these issues as I do?

I don't know.

I hope so.

I'd really like to see my department develop a more coherent pedagogical philosophy.

But that's another story.

And it's late.

I'm going for now. I'll let you know how tomorrow's committee reports go.

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