I've decided to try something new this semester for my Calc I final.

As in semesters past, the final exam will be a cumulative take-home exam, which leaves me having to do something with the class during the scheduled final exam time (the university mandates that the class meet to do something during this time, even if there is no exam).

Instead of having a review session, I'm going to try out some sort of collaborative component to the exam itself, offering the students a low-stakes place in which to try to put their heads together to solve some trickier extra credit problems.

I'm not sure exactly what it'll look like, but I think I'm going to pitch four or five problems to them and let people work together in whatever way they'd like to to try to find a solution.

We'll see how it goes.

Meanwhile, in the 280 we're spending most of our remaining time together polishing the textbook. Today all four revised chapters got another re-reading, and I'll spend much of tomorrow cleaning up their new suggested edits and getting ready for our collective editing sessions during Wednesday and Friday's classes.

The end of the semester brings another topic to the fore: course evaluations. I'll be asking my students to fill those out on various days this week (Wednesday for 280, and either Thursday or Friday for Calc I), and the task is made a bit less onerous this semester by the fact that the university is using a newly revised evaluation form which I feel is vastly superior to the old one. The new form actually asks students to respond to more meaningful, measurable, outcome-based items such as "the instructor encouraged you to develop your critical thinking skills" and the like, as opposed to the thinly-veiled popularity contest items of the old eval forms (some of which were as inane as "This course was a good course: strongly agree/agree/disagree/strongly disagree"). I'm looking forward to seeing how well it handles in practice.

## Monday, November 30, 2009

### A test with a twist

Posted by DocTurtle at 8:06 PM 0 reflections

Labels: assessment, Calculus I, Foundations, MATH 191, MATH 280

## Sunday, November 29, 2009

### Survey says...

I've heard back from about ten students so far concerning the "survey" I sent out asking for feedback on my plans for designing next semester's courses. About half are students in Calc II next term, and about half will be in Topology.

Not too shockingly, the students from whom I've heard are among the respective courses' strongest and most dedicated, so some of their views should be considered accordingly.

Nonetheless, there are patterns emerging:

1. Like it or not, even the best students are highly motivated by getting good grades. While most of the students admitted they knew they shouldn't be compelled by the desire to get high marks, most of them owned that getting high marks eggs them on and gives them what they feel is an accurate measure of their progress. It's hard to say whether these students, among my strongest, are more or less likely to be motivated by grades than their peers who have to work a bit harder to keep up.

2. Nevertheless, I'm heartened that the students who have responded are all up for something new. Though the idea of their work being assessed in some sort of "portfolio" system is an unfamiliar one to them, they seem open to the possibility.

3. Most of them are very happy with the way their current (or past) courses with me have been run, and are up for more of the same. This is also heartening to me, as ultimately I think I do a good job in most of what I do for my classes, and much of what I do now will remain in next semester's courses, largely unchanged.

More to come, as more responses come in. For now, I'm off to bed.

Posted by DocTurtle at 1:29 AM 3 reflections

Labels: Calculus II, MATH 192, MATH 431, portfolios, Topology

## Thursday, November 26, 2009

### Obligatory Thanksgiving post

The fourth Thursday of November has rolled around again, and comes the day on which everyone is busy reflecting on their lives and feeling gratitude for what they there find. Obviously I've got a number of things I'm grateful for (health, relative wealth, a wonderful life-companion [and a couple of delightful furry four-legged hangers-on], a boatload of fantastic friends, and a few close family members), but I thought I'd update the ol' blog with a post concerning those aspects of my academic life for which I'm most thankful.

In no particular order, then, I'm thankful for

1. My strong students. Almost without exception the students with whom I work on a day-to-day basis are smart, hard-working, and tireless in their efforts to strengthen themselves intellectually. Having taught at schools where students' senses of entitlement often dwarf their receptiveness to new ideas, teaching at UNC Asheville is an especial joy: the majority of my students are clearly here to learn, and to learn how to learn, and they're up for being challenged. Though many (if not most) of them are frantically juggling work rosters and preschool pick-ups and and day care schedules and doctor's appointments and god knows what else, they somehow manage to find time for my class. I can't thank them enough.

2. My clever colleagues. I am surrounded by a family of dedicated, deep-thinking, and tremendously supportive scholars. Each of my colleagues has her or his own unique talents, and each plays a different role in my work with them. Some motivate me as a teacher, offering new ideas to me and challenging me to perfect the ideas I've already dreamed up. Others egg me on and support me in my research, catching my mistakes, and helping me to see my ideas from new perspectives. Still others offer moral support, friends with whom I can sit and bitch about this and that whenever inevitable frustrations get me down.

3. The appreciation I get for the work that I do. I am endlessly delighted by the fact that I get to spending my life doing something that I love doing, and something at which I'm quite good...and that I get paid to do it. Moreover, hardly a day goes by without some sort of indication that I'm doing what I ought to be doing, and that I'm doing it well. Whether it's as simple as an e-mail from a student letting me know how much she enjoyed a particular class activity or as grand as an award for excellence in teaching, I'm grateful that my efforts are noticed, as it makes it that much more pleasant to keep doing what I do.

4. The academic freedom granted me. I speak here not just of the ethereal and ill-defined "academic freedom" to which the academy is ultimately dedicated, though I'm thankful for that as well, of course, but also of the particular freedom my institution and my overseers (chairs and deans and the like) grant me that allows me to apply often nontraditional and sometimes downright revolutionary techniques in my classes. I know very well that at some schools poetry in the math classroom would be anathema, and that at just as many schools I couldn't even dream, as I am now, of moving towards a portfolio-based grading system for my courses. (Hell, at many schools my calculus students would be asked to take a standardized final exam written by a committee of the department's faculty!) I am most decidedly grateful for the freedom I'm given to pursue the methods I think are the best ones for helping my students to learn.

Posted by DocTurtle at 11:30 AM 0 reflections

Labels: theory

## Monday, November 23, 2009

### Ennui

I've got a cold.

And it's late in the day on the last day of classes before Thanksgiving Break.

Needless to say no one (me included) is motivated to do much today.

Blech.

Posted by DocTurtle at 1:06 PM 0 reflections

Labels: bitching

## Saturday, November 21, 2009

### Good advice

I've just send e-mails to all of the students currently enrolled in my Topology class next semester, and all of the students in one of my Calc II sections who've met me before, asking them for their input on course design.

We'll see what comes of it. Even if I only get about 25% or 30% response rate, I'll be happy.

Posted by DocTurtle at 11:31 PM 0 reflections

Labels: Calculus II, MATH 192, MATH 431, Topology

## Friday, November 20, 2009

### Desperados

Yesterday was a long day, and after working pretty much nonstop (class prep to research meeting to class to research to grading to another class to more class prep to another meeting to home to grade and grade and grade) from about 6:30 a.m. until 8:30 or so p.m., nothing could have made it seem longer than discovering at that latter hour that a couple of students had cheated on yesterday's Calc I exam.

Ugh.

Given that I've not much time to write right now (class beckons with a gently arching arm), I'll merely reference an older post which, written in a time of much more leisure, says all I feel like saying right now.

On the positive side, I've found from informal conversations that a number of my Calc I students continuing on to Calc II with me in Spring 2010 are open to the idea of portfolios and other outcome-based assessment methods.

Posted by DocTurtle at 1:04 PM 0 reflections

Labels: Calculus I, cheating, MATH 191, portfolios

## Wednesday, November 18, 2009

###
*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 *L*_{8}:

And finally, here they are basking in the glory of the first *L*_{12} 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.

Posted by DocTurtle at 11:06 PM 0 reflections

Labels: Algebra al Fresco, pictures, Super Saturday

## Monday, November 16, 2009

### Help wanted: apply within

So here's the deal:

As regular readers know, I've been kicking around various ways of putting together next semester's courses, with ideas ranging from a simple tweaking of homework assignments all the way through out-and-out adoption of portfolio-based (and therefore virtually gradeless) assessment.

No matter what, there are certain elements of both courses I'll be teaching that will be essential.

For instance, Calculus II will involve the same sort of miniprojects it always has (we'll start things off, as usual, with A Confectionary Conundrum, and we'll keep the Funky Function Festival and various other small-scale applied projects and in-class activities involving delicious comestibles), and I'll still require that some sort of computational work be submitted each week...but much of that work will likely be ungraded, in order to engender a low-stakes atmosphere of exploration. But beyond these basics, I'm open to negotiating just about every other aspect of the class.

Topology, too will have its sine qua nons: however it's done, peer review must appear as a regular, fundamental, and well-structured aspect of the course; as in Calc II numerical grades will be de-emphasized in favor of ungraded, proficiency-based projects, and in-class discussion and discovery will form the basis of most of our class meetings. I will also be offering unlimited revision/resubmission of all coursework to be completed during the entire semester. Beyond this, I'm open to negotiations regarding the finer structure of the course.

I'd like to conduct such negotiations, in order to be sure that the courses I put together really do suit the academic needs and the learning styles of the students with whom I'll be working. Therefore what I'd like to do is ask those of my current and former students who will be joining me in one of these two courses next semester to help me put the courses together. (I'm ideally positioned to do this right now, as 27 of my current Calc I students and 4 of my Calc I students from Spring 2009 will be joining me in Calc II, and all but 4 of the 24 students enrolled in Topology have had me for at least one course in the past few years. I'll likely never have a better chance to engage so many students' opinions ahead of time.)

If you are currently enrolled in one of my courses, or if you have been a student in one of my courses in the past couple of years, and if you will find yourself in either Calc II or Topology next semester, you will soon be getting an e-mail from me asking you for your input on what our course should look like next semester.

I'm open to suggestions regarding just about everything:

1. The nature of homework (graded? ungraded? from the book? invented by me? some combination thereof?)

2. The structure of in-class activities (handout-based? note-based? suited to small groups? suited to the individual? some combination thereof?)

3. The nature of written assignments (papers? original research articles? expository articles? reflection papers? textbooks?)

4. Grading schemes (numerical? low-stakes? portfolios? how heavily is each component of the class weighted?)

5. Topics to be addressed (this applies more to the folks in Topology than those in Calc II; sadly, there's a pretty clear-cut list of topics we'll need to get through in the latter course, since it's not an elective) and the order in which we address them

As you respond to me, I'll ask you to think about what works for you: what do you need to get out of our course, and how can we put together a course which best helps you to learn?

I can't promise that I'll incorporate every suggestion, as I'm sure the range of opinions will be incredibly broad. Moreover, whatever structure results will almost certainly be guided in part by my own recent shifts in pedagogical theory, and there are a few things that will simply not fly. (For instance, I'm firmly and fundamentally opposed to the very idea of grading curves, and have been so opposed for quite some time.) Nevertheless, I'll try to take into account every word of input I receive and cobble together course plans for both of my classes next semester.

So, if you've had me for a class before and if you'll be having me again in Spring 2010, expect to get an e-mail from me soon. If you'd like to get a head start and don't want to wait for an e-mail, please feel free to respond to this post in the comments section (anonymously is fine). Let's get a discussion going.

I sincerely hope that you'll help me out here. Your education is more important to you than it is to anyone else, and I hope that you'll help me help you by taking a hand in scripting the acts of your education in which I will play a role.

Posted by DocTurtle at 6:31 PM 0 reflections

Labels: Calculus I, Calculus II, Foundations, MATH 191, MATH 192, MATH 280, MATH 431, Topology

### Rollspotting

Registration for first-year students began today at 7:00 a.m., and I find myself sitting in my office squealing with glee at seeing the names of the students with whom I'll be working in my Calculus II classes...I've got great carry-over from this semester's Calc I sections, and a handful I taught last Spring in Calc I.

I love these people!

For the record, y'all: Calc II is my absolute favoritest class to teach. I love teaching it so much that I make up words like "favoritest" to describe the experience.

Stay tuned!

Posted by DocTurtle at 7:58 AM 0 reflections

Labels: Calculus I, Calculus II, MATH 191, MATH 192

## Sunday, November 15, 2009

### Philosophical frustrations

This semester's been a trying one for me, and it was only this morning that I figured out just why this has been the case: I'm currently in the middle of the most fundamental shift in my teaching philosophy since graduate school.

As regular readers will know, I've been questioning basal aspects of course design, including assessment, grading, and basic course organization.

Most of my upper-level courses are already taught in a fashion that's quite squarely in line with my newly emerging philosophy on pedagogy, but the realization that many aspects of my first-year course organization run contrary to this philosophy has caused a good deal of dissonance.

I'm frustrated by this philosophical shift, as it's caused me to reassess, mid-semester, the way in which I've put things together, and to plan ahead, looking forward to next semester already.

The frustration is not fruitless, and I'm not regretting it: for the most part I think the changes are good ones. However, they've meant some awkward adjustments for my students in Calc I, and I apologize if these adjustments have thrown anyone off. If any of my Calc I students are reading this entry, I'd like to tell you how much I appreciate your patience and understanding, and the enormous amount of work you all put forth to ensure that our classes as strong ones. I've enjoyed working with you all immensely, and I hope that I'll see many of you again next semester in my fully-redesigned (and silky-smooth!) Calc II.

If you are reading this and you have any comments or suggestions regarding what you think might make Calc II a good experience for you, please let me know.

Posted by DocTurtle at 7:29 PM 0 reflections

Labels: Calculus I, MATH 191, theory

## Thursday, November 12, 2009

### Hall conversations

I spent a lot of time in the Math Lab with my students today, and in the hall outside of it.

Most of that time was spent with my Calc I students, helping them out with the volume maximization problem they're working on right now. It's a barely-unprettified problem demanding a good deal of careful computation and innovative use of derivatives for optimization. They've got to find the least costly means of constructing a collection of dumpsters designed to hold 2000 cubic meters of material, knowing that the dumpsters' shape has to fall within certain parameters. It's a tough nut to crack, whose precise solution involves techniques from Calc III. They students are either loving it or hating it, for the most part. One thing's for sure: they're spending more time on it than they'd ever spend on a set of textbook problems, and they're learning a lot. As I was leaving campus just after 5:00 p.m., two or three of the groups banded together to throw an extemporaneous "dumpster party" in the classroom in which we meet. I almost wish I could have stayed.

I also spent an hour or so talking to a couple of my 280 folks about combinatorial and topological graph theory, and just straight-up topology. I taught Uriah and La Donna how to decompose a torus into a disc with identifications, and showed how this could be used to easily find an embedding of a complete graph on 5 vertices in the torus. It was good fun.

At one point soon after that I mentioned to several current and former students who were there assembled that I'd love to put together an informal reading group (much like the RAP [Research Among Peers] groups we ran at UIUC while I was a postdoc there), and they were all game. I mentioned Herb Wilf's generatingfunctionology, a freely-available text of which I've never read more than a few chapters and into which I'd love to get deeper. I think it would be accessible to some of our stronger students, and they could help the not-so-strong ones along. It could be a fantastic learning experience for us all.

Might could be we could swing that in the Spring.

I also talked to a couple students in the hall outside the Math Lab about my plans for Topology next semester (one's registered already, and the other plans to as soon as she can tomorrow morning). They're both regular readers of this blog, so both were familiar with my portfolio plans, and I asked how they think it'd fly.

Sidney (a student in MATH 280 in Spring 2009) is all for it. "It'd definitely motivate me. What motivates me is proficiency, and you'd be measuring proficiency at achieving learning goals for the course. I'm all about that."

La Donna (a current MATH 280 student) thought it might work well, but was a bit more reserved in her acceptance of the idea. "I have to admit that I'm a little motivated by grades," she said. "A good grade is a signal to me that I'm doing well and getting it."

I suggested that perhaps, as I've posited elsewhere recently, she's motivated by grades because she's been systematically trained to be motivated by grades. She admitted this possibility.

"In any case," I told them, "whether I grade by portfolio or not, whether I hand out numerical grades or not, I know for sure I'm going to permit unlimited revisions. I'm going to let people revise and resubmit, revise and resubmit, and so on, until they're one hundred percent satisfied that they've made their work as good as it can get." Both were excited about this idea.

They're both passionate students, and a blast to have in class. En route to lunch with our speaker the other day Sidney admitted that his mind had been blown on the last day of 280 last Spring when we'd talked about the existence of infinitely many different sizes of infinity.

Crazy shit.

I'm delighted that that delighted him. It's nice to have students like him, and like La Donna. And like Cornelius and Uri and Uriah and Tedd, all of whom were there to voice strong support for the idea of an informal research reading group.

They've got my back as much as I've got theirs.

That's a comforting feeling.

Posted by DocTurtle at 10:36 PM 0 reflections

## Tuesday, November 10, 2009

### 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.

Posted by DocTurtle at 10:01 PM 0 reflections

Labels: Algebra al Fresco, Calculus I, Foundations, MATH 191, MATH 280, portfolios

## Sunday, November 08, 2009

### Night and day

After all of the smack talk about grading I've laid down in my past few posts, I have to admit that I really enjoyed responding to the students' work on which I was working today.

Today's task was to give feedback to the students on the "application miniprojects" on which we'd worked in class for two days late in the last week of October. I'd made up three handouts, each of which led the students through an application of derivatives involving differential equations. One concerned terminal velocity, another population dynamics, and a third capacitance, current, and charge in a simple circuit. I made only the slightest effort to clean up the computational details, making sure to leave some messiness for the students to deal with as they solved the problems placed before them. (I wanted them to see some at least marginally unprettified problems stemming from realistic applications.)

Working in groups in class, the students were asked to complete one handout apiece and then put together a fairly extemporaneous informal presentation on their solution. Those presentations were solid, especially considering the students hadn't prepared much at all. They were then asked, for this past Friday, to complete two of the three handouts neatly as part of their homework for the week.

With no solutions manual to fall back on (they'd only whatever notes they'd scrawled during their peers' presentations to help them out), the students' completed handouts offered authentic examples of their work. They made mistakes, obviously, but the mistakes were real and understandable ones, not like the odd transcription errors that show up when a student is sloppily copying straight from a manual or from a friend's superior solution. (A tip to those of you who rely too heavily on the manual: when you begin a problem on your own and get stuck, ending your work in a messy pile of erroneous figures...yet somehow in the next line the correct answer magically appears after a logical lacuna the size of Texas, I'm liable to suspect that you didn't do the whole problem yourself.)

When the students failed in these handouts, it was because they honestly miscomputed a derivative, and didn't simply miscopy it. Or it was because the wording of their interpretations were clumsy, and not because the interpretations were offered in the stilted technical language peculiar to textbook authors.

In short, without the solutions manual, they really honestly had to do this homework. It was a refreshing experience to respond to them.

I'm going to ask them how they felt about it. Similar assignments could serve as a stepping stone towards a more outcome-based course, something I could reasonably put together for next Spring's Calc II courses. I envision suggested (but optional) textbook problems for computational practice, coupled with weekly handouts challenging the students to apply the principles discussed in class. These handouts could be the basis for in-class presentations, just as were the handouts from two weeks ago.

We'll see. I'm going to get the students' take on these handouts soon.

Further bulletins as events warrant.

Posted by DocTurtle at 11:08 PM 0 reflections

Labels: Calculus I, MATH 191

### Self-questioning

"Why do homework?" I ask myself, after a long and frustrating day (yesterday) spent plowing through somewhat lackluster and clearly lackadaisically-done homework sets completed by my Calc I students.

Why, indeed?

For the opportunity for practice it offers in applying important concepts.

For the chance to experiment with relatively unfamiliar computations.

For the offer of exploration it gives.

Not for a grade.

So why grade it?

Because, like it or not, students are motivated extrinsically by receiving highly idiosyncratic, often arbitrary, and sometimes meaningless numerical scores on their papers...the bigger the numbers, the closer to the onset of the alphabet the letter they can receive for those numbers at the semester's end.

About those letters, at the risk of sounding crude, who really gives a flying fuck?

I don't.

Nor should the students.

I wrote "like it or not" above almost cavalierly, as though I myself am a victim of circumstance, that I play no role in establishing the primacy of those numbers, the hegemony of grades.

Of course, that's nonsense: it's clear from the comments I receive on student evaluations and the feedback I get from them after class that I play a major role in their academic developments. I'm proud of that.

But I can't be proud of building up and bolstering the hegemony of grades.

This shit has got to change.

Those grades have got to go.

Not the homework: the homework should stay. As should the feedback provided on it. But the homework itself should be the end, and not the number scrawled at its top.

The same goes for quizzes, exams, team projects: they all should stay, sans numerical rankings.

That much is clear.

But it's just as clear that making the transition from a graded to a gradeless introductory mathematics course is going to be a tough task, and I'm not sure it's one I'll be able to tackle between now and January's start of a new semester (and a new Calc II course).

I am, however, willing to try. I've just got to wrap my head around this portfolio idea.

Anyone else up for it?

Posted by DocTurtle at 7:53 AM 3 reflections

Labels: Calculus I, MATH 191, portfolios

## Saturday, November 07, 2009

### A simple observation

Boy, it's been a hell of a week, hasn't it?

Posted by DocTurtle at 11:25 PM 0 reflections

Labels: anxiety

### Subterranean Homework Blues: an open letter to Calc I students

Dear Calc I folks,

Well, it's another exciting homework-filled Saturday night, and I'm about 4/5 of the way through my Calc I students' textbook problems. (I've yet to get into the similarly-sized pile of differential equations applications.)

More than half of my sixty-odd students have clearly spent a fair amount of time during this past busy busy busy week putting together honest and authentic solutions to the assigned problems. They've shown their work, and though they've made occasional mistakes, they've carried those mistakes through to a "wrong, but consistent" end. They may not earn perfect marks, but their work is, as I said above, honest and authentic: they've gotten out of the homework what I'd hoped they would.

The other dozen or so folks who submitted solutions...not so much.

Believe it or not, my young friends, I don't assign homework in order to give myself something to do on the weekend. I'm only human: there are definitely other things I'd rather be doing at 9:28 on Saturday night than working my way through a four-inch-thick stack of calculus papers, especially when a dozen or so of those papers are little more than sloppily copied versions of the solutions manual so readily available in the Math Lab.

Believe this, too: I assign the homework for you. Not for me. Presumably, if you're in my class, you're in it because you want to get something out of it. Maybe it's been your passion to be a physicist, or an engineer. Maybe (I hope, I hope!) you've always wanted to take up serious study of mathematics. Maybe you're not sure what you want to do, but you thought you'd give math a chance and try Calc I on for size.

Whatever your reasons for being with me for 200 or so minutes out of every week, the homework I assign is meant to help you out. It's meant to give you a forum in which you can apply the ideas we discuss in class in order to refine them, explore them, and take them out for a test drive. It's meant as a place in which you can practice. It's meant as a place in which you can learn.

It's not meant to be an eight-hour time-sink for either of us.

Yeah, I'd estimate that I spend something on the order of four to eight hours per weekend grading homework, and I suspect that the most diligent of you spend roughly that same amount of time per week on the homework and on going over class notes, putting together projects, and preparing yourself for the time we share together in the classroom. For these people, the homework serves a real purpose (see above), and it's to these people my grading is dedicated.

To the rest of you, I have the following thoughts.

First, to those of you who take your answers straight from the solutions manual: please give these exercises a shot. The homework is worthless, both for you and for me, if you aren't really doing it yourself. If you've fallen behind in your work a little, now's a good time to catch up again: the sections we're working through right now are pretty straightforward, interesting, and useful ones, and students generally find that they're quite fun. Give them a shot, huh? I promise you you'll get something out of it.

Second, to those of you who've clearly (as evidenced, for instance, by your exam scores) got a grip on the course material but who for some reason just can't seem to find the time to do the homework: wake up. Classes come a lot harder than ours, and you're not going to be able to coast through them not doing the work. You might be able to get by on minimal effort now, but minimal effort will only take you so far.

Finally, to those of you who feel as though you're putting your head into a wall every time you open your textbook, please, please, please come and see me. A little struggle is good: without at least a little bit of struggle, you're not making progress, and you're not learning. But a lot of struggle is bad news: it's distressing and debilitating, and it can sap your confidence like nothing else. (A propos of very little, I hope soon to post on my thoughts on one of Alfie Kohn's essays I just finished reading, on self-esteem.)

The same offer I make to you all: come and see me. Talk to me. Ask me questions. I'm open, I'm approachable, and I'm friendly. As one of my 280 students said to me by e-mail today, I'm human. I want to see you succeed. Hell, I want to see you soar. I know that not every one of you is going to be a math major (though I hope a good number of you will!), but whatever your goals, I want to help you achieve them.

With that, my friends, I'm going to get back to grading. Wish me luck.

Posted by DocTurtle at 9:21 PM 0 reflections

Labels: Calculus I, MATH 191

### At the corner of stress and frustration

There was something in the air today.

Everyone (and I mean everyone) I dealt with today seemed beaten, defeated, on the verge of tears.

Including me.

What have the students got to be down about?

There's a bad case of Multiple Exam Syndrome going around campus.

Tuition money's scarce.

Time is scarcer.

And relations are a bitch, aren't they?

They're tricky, they're terrifying...but they're downright beautiful once you start to get the hang of them.

My 280 students did really well on the first go-through of Exam 2, getting caught up on 2 of the 5 questions, both of the bugbears having to do with equivalence relations. Those that took them nice 'n' slow and wrote out everything precisely and explicitly had no difficulties; those who just kind of threw some stuff down on the page fared more poorly.

I get the feeling that several of them were about ready to kill me by about 2:30 this afternoon.

One student caught me in transit as we headed to and from our respective classes, claiming dibs on my time once she got out of her Calc III class. "I've got three advising appointments between now and 5:00," I told her.

"Oh."

"Come on by, and I'll see what we can do."

"I don't know why you keep saying this exam is easy," she told me. "I'm finding it really hard, and you calling it easy makes me that much more frustrated."

"Maybe it's not so much easy as it is basic...or elementary. Meaning that you don't have to use complicated concepts to finish it...everything goes back to the definitions."

She stared at me somewhat icily.

Minutes later, another student cornered me in my office and confessed she'd not started the exam until that morning. She's a senior with a full course load, and had three papers due that week. I thought she was going to cry, and I knew for damn sure that if she started to cry, I'd start crying, too. We shared a candid conversation about how much was expected of us, and I'd like to think that we both left the office feeling a little better about where we are right now. (I did.)

Another student still admitted that she's simply no longer motivated about the class. She's a non-major who's recently come to the conclusion that, once she drops her math minor, she has no reason whatever for taking the class, except for the Writing-Intensive credit she'd be able to get from a major course anyway. She's enjoyed taking math classes, but when faced with the likelihood of being here for more than four years just to finish up her major (minors notwithstanding), she's finding it hard to get into the mathematical swing of things. "I hate to drop all of this on you," she told me.

"I really appreciate your honesty," I said. "I've always liked the fact that you're not going to bullshit me or hand me a line."

Things started to pick up a bit once Calc III got out (several of my 280 students are in that class) and the students drifted on over, one at a time, almost continually until just shy of 5:30. I noticed that the students weren't doing nearly as poorly as they thought they were doing, and I began handing out a few hints here and there to encourage them to keep moving in promising directions. One by one, the exams came in, most of them complete.

"I really, really understand relations now," one student confessed to me. He was fairly glowing with the excitement of understanding. "And that's really cool, because I didn't understand them at all before this exam."

Sweet. What I hope most for my exams is that they'll prove to be effective means of strengthening student understanding, offering yet one more chance for students to explore, to analyze, and to synthesize. Exams in upper-division classes are more for me than merely assessment tools; they're means for making better thinkers of my students.

In that regard, I think this exam succeeded.

As of 12:45 a.m., seven hours later and 40 minutes ago, I finished grading the exams.

They did all right.

Actually, they did quite a bit better than all right. Granted, I gave them a couple of different extra credit opportunities, but even so the class average was higher than I'd expected, roughly 79.1%, before the revisions I'll allow until next Friday. Aside from the two tricky problems dealing with equivalence relations, the exam proved to be a walk in the park.

They'll get the hang of it. It's all good.

And now I must away to bed; I've got to be up in about 5 hours in order to get to Super Saturday tomorrow (on the syllabus: Euclid versus Lobachevsky!) and to get a head start on grading Calc homework.

Posted by DocTurtle at 1:07 AM 0 reflections

Labels: anxiety, bitching, Foundations, MATH 280