Friday, October 28, 2011

I'm new here

Parker J. Palmer's and Arthur Zajonc's The heart of higher education: A call to renewal (transforming the academy through collegial conversations) (the centerpiece of the most recent faculty Learning Circle in which I took part, and about which I've posted somewhat recently) gave me more than its fair share of things to think about. Many of its insights offered theoretical, even spiritual, enlightenment regarding teaching, but other insights were more practical and practicable.

One of the more down-to-earth suggestions offered up by one of the coauthors' colleagues (Patricia Owen-Smith, Professor of Psychology and Women's Studies in the Oxford College of Emory University) was a means of encouraging contemplative practice in the classroom simply by playing music to being each class period. Owen-Smith (pp. 157-161 in the above work) describes how several years ago she began the practice of playing 7-9 minutes of music at the outset of each class. She encouraged her students to "go within, be still, and listen to the self." While she admitted the difficulty of tying this contemplative practice to improvements in students' achievement of cognitive learning goals (as if this would be the purpose of playing the music in the first place!), she reports that

over time, I learned that the music and meditative moments had an impact on many students. Some students began to ask for guidance with their contemplation and reflection....By midsemester several students per class would mention that they looked forward to this nine-minute period of music. Some students began to bring music from their own collections that they found inspirational and important. As we neared the end of the semester, the structure of the class had changed from a group of individuals reluctantly gathered together for study to a community of friends and partners who were creating a space of introspection, quiet, and respect for the process of study and the development of self.

Once or twice a week for the past few weeks I've begun a similar practice in one of my classes. It began with my reading of an excerpt from Rilke's letters to Franz Kappus (a reading which moved one student so much she had to leave the room), and continued with a passage from Dick Leith's history of the English language. At one student's suggestion one morning we watched a scene from Harold and Maude, and the next day I read one of my own recently-written poems ("Ode to Ned Maddrell," which I penned for the last-living speaker of the Manx language). To open off on Monday (at the suggestion of another student) we'll take in one of Gil Scott-Heron's last videos.

This practice has developed naturally, in an easygoing fashion, and perhaps unsurprisingly it's happened in the most natural and easygoing section of any of my courses this semester. From Day One nearly every person in that section of Precalculus has worked well together, helping each other out and asking for help when help is needed. This natural ease with which we've worked together all term has made it hard for me to tell whether the contemplative practice has had a real effect on the esprit de corps of the class...

...but I'm not going to take any chances. The practice has definitely helped me to make connections between the intellectual and the personal, between the scientific and the humanistic. It's helped me to remain focused on what really matters, and it's helped to remind both me and my students that we do well to look for mathematics' usefulness...and to look for its beauty. Much like well-chosen low-stakes writing activities, this practice is well worth the few minutes of class time that it takes.

Beginning right away I'm going to introduce this practice in all of my classes. It can't hurt.

What effect can it have in a class like my current Abstract Algebra class, a room full of stressed-out work-wearied students representing, honestly, probably the greatest range of mathematical ability I've seen in one section since I began teaching at UNCA over six years ago? I admit here and now that I find myself frustrated at how I've managed this course. For some time now I've felt the need to slow its pace to accommodate the most modestly quick learners, but without sacrificing the true nature of the subject, a nature laden with often-abstract proofs. I don't know how much slowing the pace down has helped: several students are still struggling, and as much as I hate to leave them behind (I hate hate hate people who teach to the top ten percent), I simply must move on. Moreover, I sense that the slowness has led to frustration on the part of some of the class's quicker students, and I can feel cliquishness setting in...not a nice way to end the semester.

Maybe it'd be wise at this point to remind everyone that we all have a right to say "I'm new here": we're all free to make mistakes from time to time. After all, no one's lived this life before, and the future hits us all at the same time. But no matter how far you go, you can always turn around.

What say, folks? What are we going to make of our last five weeks or so together?

Research update

Ned and Ino are chuggin' along. Their research project, the culmination of which will be a pamphlet detailing the means of making a week's worth of healthy, affordable meals is flying forward. We just overcame a major obstacle today: we had to figure out how, once we'd found the ideally affordable combination of foods needed to obtain the right amount of each nutrient in our analysis individually, to combine this information to give us the most affordable combination of foods to give us the right amount of all nutrients at once. We made it over that hump, and I truly think it'll be the last big one before our projects reaches its end.

So proud!

Wednesday, October 26, 2011

Partners in crime

I know a number of people (many former and current students, a good number of colleagues, and assorted folks I've never met) read this blog, but not many often comment publicly. Quite often, though, I get comments about it on Facebook or in my in-box.

This past week I got a note from a fellow who's teaching precalculus at an inner-city high school in Boston, using inquiry-based learning. He wrote me asking about the methods I'm using in my own precalc classes right now and shared some of his own (he's asked me not to post his notes, as they're very much works in progress). I'm very impressed! His notes are clever and engaging, offering students a scaffolding students can use to climb from the barest basics up to properties of advanced functions, logs and exponents, and trigonometry. You can read about his exploits here.

By comparison, my methods that are considerably less purely inquiry-based...he's doing pretty much straight-up Moore method with high-school students! Inventive and impactful.

Monday, October 24, 2011

Dead Zone

Today I felt like every one of my classes was a bit stuck in the mud. (Oddly enough, my morning Precalc class, often my quietest, was the most lively today.) It was all we could do to keep making forward progress in a couple of the classes, and I felt like I was beating a dead horse more than once.

What to do at such times?

The classical response, at least from instructors in content-driven disciplines: "well, we've got so much to cover that we've got to keep going..."

...but this is wrongheaded. If you press on without pause, it's not like students are going to get any more engaged, and it's not like they're going to get much out of whatever you do together, anyway. In pressing on you'll be doing so only for the sake of pressing on, and any progress you make will be illusory.

This is exactly how it felt in my second section of Precalc, and in Abstract Algebra, in both of which classes we worked with (what I thought were) some pretty neat mathematical ideas: in Precalc we solved a nontrivial optimization problem involving rational functions, and in Abstract Algebra we looked at the subgroup lattices of a couple of groups and examined the asymptotic behavior of Euler's function φ. I don't feel that either class picked up on the subtle beauty in a way they would have had they been in a more receptive mood.

Don't get me wrong: I'm not blaming my classes. Both of these classes are full of wonderful students who are generally unafraid of working together to make a healthy and supportive learning environment. (I've bragged on my Precalc peeps enough this semester for you not to know how much I care about them.) Rather, I think we've just hit that point in the semester, about 50-60% of the way through, where everybody's just DEAD.

It's the Dead Zone. Wearied by exams and essays and due dates and deadlines, overburdened by homework and quizzes and lit reviews and response papers, we're tired and jaded and not having much fun. (Believe me, kiddoes, this "we" often includes me. I apologize if I'm sometimes snappy around this time of the term; it's hard to be irrepressibly chipped every hour of every day!)

On reflection, I've thought of a two-fold healthier response to these Dead Zone doldrums:

1. Play. Put down the pen or pencil, put the paper away. Let's just think of something fun we can do with whatever it is we're working at this very moment. Optimization problems? Let's come up with some kind of crazy variation on the theme, whether we have any idea how to solve it or not. Let's set it up and see if we can work it out, like Ariadne weaving a web through the labyrinth. Are we sick to death of subgroup theorems? Let's break it all down with an explicit or example, or two, or three...let's dissect the dihedral groups until there's nothing left but individual elements...let's take it apart!

2. Reflect. The next step comes as no surprise to those who know me well: once we're done playing, let's take a minute or two to write to ourselves, if only to reflect on what we've been able to discover. Did we reach an end? How? Did we run aground? Why'd we lose our way? What's our play got to do with other problems we might encounter in mathematics and beyond? Write about it, write about how we feel about it. Hell, write about how we feel in general: why are we so dead today?

Too often affective learning goals get lost when we focus too heavily on cognitive learning goals, and that goes double for content-laden quantitative sciences. Let's try not to lose sight of our humanity, and the fragility that comes with it. Let's take care of each other as we come together to learn.

By and large all of my classes this semester are doing a marvelous job at this, and I admire them for it. I never cease to be amazed at the quality of students with whom I get to interact and learn. You're terrific people, all of you!

Saturday, October 22, 2011

A somewhat schizophrenic conversation

[Note: this post includes a homework assignment for my readers, toward the very end. If you're a teacher, student, or alumna/alumnus, please take a moment to respond when you're done. Thank you!]

A few years back we graduated one of the brightest students I've yet to work with at UNC Asheville. Sedgwick was a soft-spoken and deep-thinking environmental studies major with whom I had only one chance to work, in a Calc II course he enrolled in just before he graduated. He and I shared some pleasant conversations during his studenthood here, but we've shared many more (often from afar) since his moving off to broader pastures.

He wrote a few days back indicating that he'd had some thoughts (which he'd written down) about my CRTF-related posts, and wondered if I had any interest in reading them. Knowing his perspicacity, I knew they'd be well worth the read, so I told him to send them along, by all means, asking if I might repost them here, as I've done in the past. He's granted permission.

Sedgwick's comments concern the ILS Topical Clusters in particular, which are considered by many (myself included) to be the weak point of ILS as a whole. Here's what Sedgwick has to say:

I will preface these comments by saying that I was one of the last students to graduate under the old General Education requirements, so I have no first-hand experience with ILS, despite being a recent alum.

I did take a look, however, at the clusters on offer. Currently, a cluster appears to be nothing more than an arrangement of existing courses that fit some nebulous theme. This situation seems to be the functional equivalent of forcing all students to declare a 'mini-minor,' albeit less useful because the promise of interdisciplinary depth seems hardly fulfilled, given how little time one can commit to a cluster relative to other requirements.

The main part that confuses me is the fact that there is a theme at all; it seems like a needless restriction. The structure of a student's education comes from their major, which offers the technical, career-focused classes they need. Asking the liberal arts portion of the curriculum to follow a cluster's pre-determined path is like asking a journey for directions: clusters rely on the false premise that students can (or should) connect the dots in the present.

To illustrate that last point with an example (that will no doubt become cliche): the reason that the original Macintosh debuted with multiple fonts and typefaces was because Steve Jobs took a calligraphy class at Reed College years earlier, a course that interested him but had no real-world usefulness to him at the time. My concern is that by requiring students to adhere to a theme in the 'liberal arts' part of their studies, they could be missing out on experiences that may be of use one day, in a manner that's impossible to conceive of while still in college. The way Steve put it: "you can't connect the dots looking forward; you can only connect them looking backwards." UNCA's advantage, as our state's liberal arts institution, should be in providing the broadest array of dots for students to connect in the future, as they need them. In this light, restricting the ILS experience to a small subset of available courses does not make much sense.

If all ILS wants to do (or can do at the moment) is force exposure to other departments, then get rid of clusters and just say that students should take X number of courses outside their major. However, I think UNCA's goal is to emphasize the 'integrative' part of ILS. Clusters were an important first step, but I believe the 'integration' was too high-level to have the intended effect. Ultimately, integration needs to permeate the coursework itself, which why I would suggest 'cross-up' courses instead of clusters.

A cross-up course would be a deliberate collaboration of at least two departments. What would these cross-up courses look like? It's hard to say: I trust faculty to have a better eye for how their chosen discipline can interact with another. I know the synthesis of mathematics and writing is an important part of your teaching, so that seems to be a natural fit. My own background and interests can easily see collaborations between Computer Science and Environmental Studies. I think the possibilities are only limited by the interests of faculty and their willingness to work together.

There could be a simple rule that each department must form a cross-up with at least X number of departments. With a pool of cross-up courses available, just have students take X number of them to fulfill the ILS requirement. That's it. With the Intensives requirements still in place, the curriculum would not suffer in rigor. Like custom clusters, cross-ups could also be student-initiated with proper coordination. What better way to give UNCA students an edge in cross-disciplinary work than by taking classes that are actually cross-disciplinary by design? I cannot help but imagine that this type of setup would also confer a degree of market separation from peer institutions.

In short, exposing students to several unique instances of cross-disciplinary work seems to be a more pragmatic use of the limited time students have to devote to ILS electives. Cross-ups can also align faculty more towards collaboration than rivalry, encouraging departments to think about what cross-disciplinary experiences will work for students once they leave the academy and face an unforgiving job market. Of course, given the fiscal situation up there, asking faculty across the campus to design and teach a dozen or so new courses is likely a non-starter. But doesn't it sound exciting, something that really fits with the purpose of UNCA?

My open-letter response (which is almost identical to the response I had to Sedgwick's last letter to me, linked to above): I agree...ideally. I actually think the cross-up courses are a great idea, and if implemented would lead to a much more flexible, manageable, and student-authored learning experience that would replace (and substantially improve upon) the current system of topical clusters. The primary problems I see (as does Sedgwick himself) are logistical.

Namely, cost, in person-power and faculty time, if nothing else, is a prohibitive factor. Given our current budgetary climate (if I had a dollar for every time I've typed that word in the past few months...), we quite literally can't afford to ask all faculty to take time out of their schedules to design new interlinked courses. Moreover, we lack the administrative power to begin giving faculty appropriate credit for leading the many team-taught courses the cross-up system would entail...

...But wait a minute...Even as I was typing those last two sentences I began thinking to myself..."what?!?" As Sedgwick pointed out in the post I linked to above, universities are, though many who staff them would be loathe to admit it, among the most conservative of institutions around today, and change comes very slowly to them...I often think that we often make up excuses (too expensive, too time-consuming, administratively infeasible, etc.) for doing things we, institutionally, simply don't want to do.

To the first point: how does maintenance of the admittedly flawed and unpopular ILS Topical Cluster system demand any less faculty time and resources than would implementation of a new program that would likely require considerably less oversight and administrative overhead? On reflection, the faculty claim "I just don't have time to sit down and design this course" is wholly ridiculous...faculty are designing new courses all the time. Who among us isn't thrilled and filled with pride when first given the chance (in, maybe, our second or third year on the faculty) to teach a special-topics course related to our research? And how many of us, especially those of us in our first, second, and third years of teaching, find ourselves teaching one or two new preps every year? Though these courses are often not new, they're new-to-us, and take a fair amount of time to tweak and tone as we make them our own.

The last paragraph points out an obvious "in": our newest faculty are likely to be the most willing and able to implement a new curricular component like cross-up courses. Not only do they expect to have one or two new preps any year anyway, they're also less entrenched in their disciplinary positions and are more likely to be open to cross-disciplinary fertilization. I may just have to talk to a few of my younger colleagues about these ideas...

To the second point above: the argument is often made that we don't team-teach much here because it's simply too difficult to give faculty the appropriate "credit" for teaching such courses. The system as it exists, supposedly, allows us only to give credit for teaching half of a course for such courses, and in order to meet various benchmarks for faculty activity (the infamous Delaware study among them) faculty teaching such courses would have to teach far more than an acceptable load to appear on paper as though they're being productive. I can't buy this argument; if I did, I'd be as shortsighted as the folks I've been ranting about in my recent CRTF posts.

I can't buy it because I'm just not sure I've shopped around enough yet: might it be that the problem is one of "vision"? Several of my colleagues on the Curricular Sustainability subgroup have remarked that resistance to change may be predicated on a lack of understanding of other ways we could do things than the way we're already doing them. That is, maybe we're encountering so much insistence on doing things the way we've been doing them because folks just don't know how else these things can be done. Our response on CRTF has been to try to come up with models. Just this past week I asked the folks on my subgroup to identify institutions offering "model" majors and degree programs in their respective disciplines, suspecting that these programs will likely prove more sustainable (e.g., more flexible and less prescriptive) than their cognates on our campus.

Maybe what we need is more models. This brings me to the homework I mentioned at the outset of this post:

1. For those faculty reading this post, it would delight me to no end if you could comment on this post with a paragraph or two (or more, if you wish) about the nature of team-taught interdisciplinary courses at your school. How are they organized? How are they overseen and assessed? How does the administration grant faculty credit for teaching in these courses? How are they received by the students? Are they required, recommended, or simply part of the body of electives students might opt to take? All of this information would give me ammunition I could use to make the case for these courses.

2. For those students reading this post, it would give me similar delight if you could comment on this post with a paragraph or two about how you would receive such courses. Would you be interested in taking them? Among what disciplines would you like to see more collaboration? Would you find it helpful if such courses were required...and would you take them even if they were not? For UNC Asheville students in particular: would you prefer this kind of system to the current system of ILS Topical Clusters? Why?

Obviously I can't require you to respond, but even just a few words would be of such tremendous help to me that I really hope you'll consider writing back.

I realize now that this conversation, once between Sedgwick and me and then just between two mes, has turned out slightly schizophrenic. It's really helped me to get these thoughts out of my skull, though: I'd not before now seen the untenability of the "we don't have time to..." argument. I needed to write it to see it. (It's writing-to-learn, y'all!)

My thanks for those who've read this far! I'll soon be posting on a conversation with another reader, a high school teacher in Boston who's making use of IBL methods in his precalculus course. Stay tuned!

Friday, October 21, 2011


After having started off two different meetings of my second section of Precalc with readings from reflective writing of some kind (see this post and this one), I invited the students to join me in bringing contemplative readings into class. Several have indicated that they enjoy this start to the class, and I look forward to seeing what the others will bring in.

Thursday, October 20, 2011


I noticed the other night, just before leaving campus, that in Abstract Algebra I'm about two weeks "behind" (about three handouts, each roughly equivalent to two days of class) the place where I was at this time in Fall 2008, the last time I taught the course.

My immediate reaction, which, fortunately, dissipated almost immediately: "Holy can I catch up?!?"

I looked over the handouts separating now from then. They were filled, for the most part, with technical lemmas and other minutiae about groups, facts like "the condition that g-1 be a two-sided inverse is redundant" and "to check that H is a subgroup it need only be shown that gh-1 for all g and h in H." I thought about my students, and their aims and ambitions, and I realized almost immediately that they could do without "covering" these lemmas. At best, they'd memorize them for several days, work a contrived homework problem or two meant to test them on the results, and then forget them.

Meh. We'll skip those handouts. Instead, we'll move on to the good stuff: subgroups, homomorphisms, more meaningful structural results that are powerful, intriguing, and beautiful.

Meanwhile, the homework is clearly kicking my students' butts. Even the more experienced students are struggling with it. It was only after thinking about it for a bit that I realized why this is: instead of asking them to pantomime the proof of some result that differs only slightly from something we've talked about in class or put the polish on a theorem I read out loud to them, I'm having them build from scratch most of the canonical examples; I'm having them introduce and analyze the most important definitions.

It would take me fifteen minutes to "teach" them all there is to know about the subgroups of the integers, but in doing so I'd guarantee that nine out of ten of them would smile (or scowl) and nod their heads, take careful notes, commit what I'd said to memory, and not understand a lick of it. I'd rather they take a few hours outside of class to do it for themselves, struggling with every step, pounding their heads on the Math Lab's countertops in frustration, cursing me under their breath, gaining intuition all the way. The best human computers of all time, from Napier through Gauss to Ramanujan, built their skills by living with numbers, loving them, spending their days with them cheek to jowl. It's hard work, but you'll gain so much for it, mathematically speaking. You'll gain intuition, and, ultimately, understanding.

I spent an hour or two with several of these students in the Math Lab this afternoon, and the progress they made was incredible. I'm so impressed by their intelligence and determination.

Keep it up, folks! I'm proud of you all.

Wednesday, October 19, 2011

Grades, schmades

Tonight I talked for several hours with one of my best friends on Earth, and one of my most reflective fellow teachers of mathematics, Griselda. (Tired of this and that, and longing to move closer to friends and family in the Northeast, Griselda recently left her job teaching at a public liberal arts college in a nearby state to teach at a boarding school in Pennsylvania.) As usual, our conversation was broad and far-reaching, and dealt with issues in every corner of education.

My most meaningful self-realization of the evening: the only reason I still give grades in my courses is because I have to; the school requires them from me at the end of the term. And the only reason I give grades throughout the semester is to provide substantive justification for those end-of-term grades.

Don't get me wrong: I love responding to my students' work. My responses make up my half of a conversation with the students about the discoveries they're making. It's an ongoing dialogue, and often an exciting one. They say something to me, I say something back, and after one or two iterations we might make some sense of what it is the other is trying to convey. Eventually we'll come to a consensus regarding the meaning of whatever matter we're faced with. I love these conversations. They're where learning takes place.

But grading? Uh uh. I'm over it. I'm tired of this competitive academic economy based on artificial and extrinsic rewards.

How can I break the cycle? Maybe if I rebel and refuse to assign grades...?

Something to think about...

...Griselda always leaves me something to think about.

Thank you, my friend.

Running in reverse

Recently I've had a chance to feed my undying love of linguistics. I've been reading up on the history of English and its antecedents (like Angl0-Saxon) and victims in the clash of tongues that's taken place on the British Isles since the early common era (like Cornish and Manx). The text I'm currently reading is Dick Leith's A social history of English (London: Routledge & Kegan Paul, 1983), an interesting book offering a glimpse of English's development as a social, as well as a purely linguistic, phenomenon.

More interesting than Leith's treatment of English per se are some of the observations he makes about the codification of language, and the role of "authority" in the preservation and propagation of language across time and space. A central thesis of his book is that all too often we forget that language is very much dynamic: it is ever in flux, constantly changing...and that in the end that change is not driven by grammarians or the intellectual or economic elite so much as it is by the ways in which every member of society chooses to use the language.

These are points that even the most perspicacious language-lovers among us tend to overlook. The reminders Leith offers have made me think of new (to me, at least) and "subversive" paradigms for poetry (a post on that soon, perhaps)...but they've also recalled for me the central role every member of a learning community plays in that community's advancement of knowledge, while issuing a reminder as to just how dangerous it can be to trust blindly in the authority of a textbook.

The following passage from Leith (p. 68) struck me (cf. the comments some of my precalculus students made on their last exam):

Unfortunately, many people tend to treat dictionaries with reverence: rather than being seen as a record of usage, the are often regarded as the arbiter of it, a source of enlightenment for the ignorant non-specialist. In fact, the traditional arrangement of words in dictionaries gives people a strange idea about language. The alphabetic arrangement disassociates a word from the company it keeps, presenting it as a unit isolated from context and words of similar meaning. More important, many dictionaries give the impression that words have only one meaning, to be found on the right-hand side of the page. Even the fullest dictionary, the Oxford English Dictionary (OED), which shows the whole range of meanings by citing examples of a word in use at different periods in its history, puts the meanings first, then lists the examples, thereby obscuring the process involved in deriving the meanings; for we learn the meanings of new words most efficiently by hearing them in a wide range of contexts....It is not surprising, therefore, that people often misunderstand them.

How often too we ask our math students to use their textbooks in the same way they'd use a dictionary, placing theorems and proofs before (or, more often than not, simply in lieu of) the intuition and arguments that led to those theorems and proofs in the first place? How do our textbooks obscure the many long hours of exploration and discovery that went into the derivation of the theorems that pepper the textbooks' pages? Without access to the discoverer's process of discovery, the reader is apt to feel as though a given fact or formula arises ex nihilo, and that they, the uninitiated, are not privy to its inner workings.

Food for thought. By me, it's better to let the students stumble around a bit, piecing things together for themselves as they author their own textbooks. That's just what I'll be doing when we talk about general rational functions in Precalc tomorrow...strap yourselves in!

More words of wisdom from my precalc students

Yesterday afternoon I worked for an hour with the student consultants at our university's Writing Center, helping them to understand what mathematical writing might look like, preparing them to work with students in the quantitative sciences who might come in with writing assignments from their quant courses. I was happy with the conversations we had together, and delighted by the students' energy and enthusiasm.

I shared with the consultants one precalc student's response to the midterm question in which I asked the students to describe the most meaningful learning outcome they've achieved so far this semester (see here for one response, not the one I shared at the writing center), and promised that I'd share a few more.

Let me throw a few more out there, all with a common theme. Several students, including the three quoted below, indicated experiencing the realization that mastering math is more than just memorizing formulas, and that if you stop trying to memorize every last formula but rather try to break each down and understand its inner workings, you gain immeasurably through your effort. That understanding is strengthened if math is put in a contextual matrix, placed alongside other disciplines so that its relevance becomes more apparent.

Katarina had this to say about her personal revelations:

This course is unlike any other math class I've taken. We actually discuss math in English at a level that I believe everyone in the class can understand....Instead of memorizing formulas, we play them out on the board, multiple ways, so that it is almost illogical not to understand their function and there is no need to actually memorize them....We write our answers in paragraph form, truly explaining the reason for doing them and our end result....It is so unusual to me to have math and other subjects (such as writing) overlap. My initial response was to avoid it but now I'm beginning to embrace the idea. After all, isn't UNCA's liberal arts program all about integrating many subjects in order to have a broader education?

Thomasina (who took a year off from high school before coming back to college) had this to say, upping the ante by acknowledging not just understanding, but enjoyment, and even aesthetic appreciation:
When I graduated high school, I decided that school was pointless....Read, memorize, regurgitate. That is all I was ever taught. But to understand? To break something down to its very core and build it back up, seeing every piece as they’re placed together to form a whole again. It's actually beautiful. I never got what you meant when you’d say math is beautiful, but now I get it. To have the ability to look at something complex and make it simple and tangible - it's art. And it's not just with math. It applies to everything: decisions, work, other people. Everything is a complex formula waiting to be taken apart and understood, and then put back together in a way that makes sense and feels right.

Kurt reports an experience similar to that of Katarina and Thomasina:

I've found more interest and enjoyment in something that I previously found tedious and boring and have found that I actually can relate math (including calculus) to the real world and my everyday life in ways I'd not before considered or imagined....I see this insight as far more valuable to me as a person than any one mathematical concept on its own. This is the kind of insight that changes people's lives, gives the potentially brilliant scientist a view into the potential locked up inside, or even just changes a fundamental attitude or a paradigmatic shift in thinking altogether....It's a great feeling to realize suddenly 'hey, I GET this!' and even better to find 'hey, I actually LIKE this!' I truly wish there were more classes like this one which, if not persuading one to major and work in the field, to at least open one’s eyes to the possibilities.

If a precalc course can move students to wax this rhapsodically about their learning, how much can students get from still deeper and more meaningful courses? It's up to those of us who teach to make our courses as relevant as we can.

Saturday, October 15, 2011


As I mentioned in a post not long ago, I recently asked my Precalc students to draw comics in which the characters explain how to multiply two complex numbers. I'll showcase a few here and there in the next few days. Here's an almost sickeningly cute one by Thomasina and Urban:

Also soon to come: more excerpts from the students' midterm essays on their most meaningful learning experiences so far this course.

Friday, October 14, 2011

Author! Author!

Every person is the author of her own adventures.

This is a point I try to make to all of the students in all of my courses, in which I downplay my own authority and up-play the students'. "I've got no more claim to the truth than you do. The only difference between you and me is that I've been doing it for a few more years."

It's a point I've tried to make to my colleagues, most recently this afternoon at yet another CRTF meeting. This one was a meeting of the "Big Picture" Subgroup, at which the leaders of the other subgroups (including yours truly) were asked to make presentations on our ongoing work. I had a bit to say about our review of department responses to our "information request," and about our intended review of various ILS components.

I hope that our review will be guided by a handful of basic principles:

1. Our curriculum will function most efficiently and effectively when ILS learning outcomes and departmental learning outcomes (and the means of achieving those outcomes) are brought into fullest alignment.

2. Our curriculum will be most sustainable when the resource demands it places on faculty, staff, and students are minimized.

3. Our curriculum will offer the most rich and most meaningful learning opportunities to our students when they are allowed to plan and pursue their own courses of study, navigating course requirements that are rigorous but flexible.

This last principle places a high value on non-prescriptive curricula, featuring both general education programs and degree programs with relatively few specific requirements...programs that ask the students to play an active role in putting their own academic houses in order. I don't feel that our current curriculum features such programs.

The other day, in a hall conversation with a colleague, I referred to our role in the current system as "helicopter professors": our requirements are structured in such a way that our students' academic careers are micromanaged stringently. Students are tended to carefully, led from year to year in flocks, protected and prepared (for graduate study or real-world employment), but rarely challenged to set out on their own. Based on analysis of student behavior over the past several years, the Research and Evaluation Subgroup of CRTF discovered that only 18.5% of the courses our students take count as "free electives," taken for no purpose beyond academic exploration (such courses satisfy neither major nor ILS requirements). All other courses, all but little more than a semester, go toward putting a check in some bureaucrat's box.

"We need to make sure that our students who want to do graduate work are at least well enough prepared to get into a decent masters program," one of my department colleagues insisted at tonight's meeting. I agree, wholeheartedly. But I disagree with the means he suggests we must use to get them there. Many of our peer institutions offer much more flexible programs, with far fewer explicit course requirements, and still manage to send higher percentages of their graduates into prestigious programs. (The fact that this friend of mine is shortsightedly using graduate school enrollment as the be-all-end-all measure of an academic program's success is a topic for another post...)

More important, students completing more self-directed courses of study gain authority over their own actions. They grow in competence and confidence as they're asked to take on more responsibility for their own lives. They mature more quickly. They learn how to ask and answer important questions concerning their coursework and their careers. Forced to connect the dots for themselves, they become more authentic experts in their own disciplines.

This isn't to say we shouldn't offer our students some kind of guidance: nothing can supplant informed academic advising. Good advising can take the place of stringent requirements. If a student should wish to pursue graduate study, she should be encouraged to take courses that will most well prepare her for that study. If she fails to follow up on the advice her professors give her, she might be sunk...but she might not. She may succeed in her ambitions, but even if she doesn' what? Even if she doesn't end up where she'd originally set out to be, she's had a chance to plot her own path in the meantime, learning from whatever mistakes she's made on the way. Life is what it is, and each of us is who each of us is.

In my second section of Precalc (and again in my Abstract Algebra class), I read an excerpt from Rainer Maria Rilke's 6th letter to the poet Franz Kappus (Letters To A Young Poet, translated by Joan M. Burnham, Novato, CA: New World Library, 1992, pp. 53-55):

You should not be without a greeting from me at Christmastime, when in the midst of festivities your feeling of aloneness is apt to weigh more heavily upon you. Whenever you notice that it looms large, then be glad about it. For what would aloneness be, you ask yourself, if it did not possess greatness? There exists only one aloneness, and it is great, and it is not easy to bear. To nearly everyone come those hours that we would gladly exchange for any cheap or even the most banal camaraderie, for even the slightest inclination to choose the second-best or the most unworthy thing. But perhaps it is exactly in those hours when aloneness can flourish. Its growth is painful as the growing up of a young boy and sad as the emergence of springtime....Think, dear friend, reflect on the world that you carry within yourself. And name this thinking what you wish. It might be recollections of your childhood or yearning for your own future. Just be sure that you observe carefully what wells up within you and place that above everything that you notice around you. Your innermost happening is worth all your love. You must somehow work on that.

Let us reflect, my friends. What is it you find within yourself? How can you make your life your own?

Thursday, October 13, 2011

Seriously, she was not paid to say this

In my last post I mentioned I'd be posting excerpts from my Precalc students' midterm exams (and a few assorted comic strips offering explanations of complex multiplication). Here's the first installment.

Tonya's always challenging me with what I believe is the best question a student can ask in a math class, a question which can be paraphrased succinctly by the words "who cares?" She's always on the lookout for relevance and applicability. "That's cool," she'll say, slightly sardonically, and then add, "but how can that be used?" I love it. Every math class should have three or four Tonyas.

Tonya's response to my midterm question asking students to indicate the most meaningful thing they've learned so far this semester was a near-perfect defense of writing-to-learn. It was a delight to read! I asked her for permission to quote her response in full, and she gave me the go-ahead.

Before letting Tonya take it home, I should note that several other students indicated the same realization (of the power of writing as a tool for discovery and for gaining understanding) as the most meaningful outcome of the course so far.

Saith Tonya:

As much as I hate to admit it I think the most beneficial thing that I have learned or rather have incorporated into my learning process during this class has been providing sort of narrative explanations for the mathematical concepts in our homework assignments. This practice really brings light to the idea that the best way to learn something is to teach it. Although laborious, time consuming, and even a bit tedious it has proven beneficial to my comprehension.

I think it may be in some way related to uniting the two sides of the brain or the two main avenues in which human beings tend to process information. It seems that people so often separate quantitative reasoning and verbal reasoning as almost dichotomic and even hierarchical in nature. The fact of the matter is however that both approaches to logic are inherent to one another. Both numbers and words are at their most fundamental level simply expressions humans use to describe the world. Thus I have found great significance in the practice of incorporating those two expressions.

In my mind (as is obvious from my questions in class) mathematics bares very little significance independent of some broader meaning or application. Being forced to go through problems step by step and constantly attend to the looming “why?” in explaining the process that leads to the solution has been instrumental in illuminating that broader meaning. Being able to explain why something was done at a certain step in the problem forces you to draw on the most rudimentary understanding of the process and ultimately universalizes the relevance of that action. I believe that this is the underlying principle behind all creative thought. It is the ability to rearrange, expound, and theorize about the world with our little tool kit of axioms if you will.

As I move forward in my professional/academic life I think it will serve me to have been denied the temptation to skip steps or overlook details in order to more readily achieve whatever end it is that I am vying for whether it be the solution to a hw problem or a policy report. It has been an exercise in demonstrating that anything whole is made up of nothing less than the sum of its parts (maybe more but definitely not less).

I love my job.

Saturday, October 08, 2011

But wait, there's more...

I spent most of the day responding to my Precalc students' first midterm exams and their most recent homework. It was time-consuming, but fulfilling: the trick is asking meaningful questions. The exam included a question asking students to reflect on what it is they've learned so far this semester that will most help them meet their own personal and professional goals. The homework asked them to draw a comic strip in which the characters explain how to multiply two complex numbers.

Both of these exercises were answered with truly creative responses. I'll be sharing several of each here, once I get permission from the students to do so. For now, let it suffice to say that in their reflections many of the students report new-found appreciation for mathematics, new or renewed excitement about it, and greater confidence in working with it. As many or more gave great tips on solving problems or approaching weighty matters more critically and with greater skepticism.

I expect great things from these students. I really do have the best job on Earth.

Friday, October 07, 2011

lovin' it

I am so enamored of my second section of Precalc right now. They're so much fun, and so smart, and so engaged! Gush gush's a link to the first edition of their class newsletter (see this post from this past May, in which I talked about this project): To Infinity and Beyond (Issue #1). It's marvelous! I've got a full-color hardcopy on my desk right now, my own smiling face beaming up at me.

In all seriousness: I find myself very calm in that section. They're inquisitive and skeptical; they're not satisfied with pat answers or particular formulas; they think in big pictures and big ideas; they're reflective and resourceful; they're fun.

It's a good learning community, the healthiest class I've had in a long time in that regard. I like to think I've played some part in making it so.

Tuesday, October 04, 2011

NIMBY (academic edition)

[WARNING: what follows is an administration-heavy post that will likely bore most readers to tears. Faculty, please read on only if you're masochistic. Students, please read on only if you want to learn faaaaaar more about the inner workings of your university than you ever dreamed you'd know.]

I've been serving on the Curriculum Review Task Force since its inception back in March. In the half-year since this project got off the ground, I've probably been involved in three dozen meetings (some two or more hours long), written fifty pages of position papers, policy statements, memos, information requests, findings summaries, and spreadsheets. I've read two or three hundred pages of curricular data and carefully scanned the websites of two dozen peer institutions whose programs serve in some way as a model for our own. After all of this work, I feel like we might finally be near to having something to show for our effort.

To be sure, there's more work to be done, but for the first time I feel like we might have reached the point where we'll start rolling downhill again. We've got the information on department curricula we asked for from chairs and program directors, and as we begin sifting through that info we'll be writing our responses to the departments, even while we collect a little bit more data on ILS components before writing similar responses there.

We were intentional in our move to address departments and degree-granting programs first: since several powerful people first got involved in the task force specifically because they perceived its function to be to mount an assault on ILS, we wanted to make clear early and often that we would consider just as carefully the efficiency and the sustainability of departments and programs as well. It would be unacceptable, we wanted known, simply to chuck aside ILS and retreat to the cozy confines of our disciplinary silos.

From the get-go, aside from the usual (and expected) assortment of trolls who got on board the ILS-bashing bandwagon when it first rolled into town, my impression was that folks were behind us. With times as hard as they are (and faculty as overworked), we'd look for efficiency in whatever place we could. ILS, though certainly not a sacred cow, would be considered no more carefully or critically than would the departments.

But when it comes to paying the bills, everyone seems to be busted.

Evidence showed that, expressed in terms of required courses, our degree programs are larger across the board, on average, than their corresponding programs at peer institutions. Our students, not surprisingly, take longer than their peers elsewhere take to complete these programs. Therefore we asked departments to identify ways they might move to make their programs more sustainable: might they remove unpopular concentrations? cut back on course requirements? improve opportunities for "double-dipping" courses?

What we're discovering is shocking: while the curriculum in general is in disarray, every department, on its own, is doing a fantastic job.

The story goes something like this: "our university's curriculum is clearly unsustainable. Students aren't graduating on time, and that's scaring away students, so our retention suffers. My students have a hard time figuring out how to put their schedules in order so that they can graduate in four years, and I have a hard time advising them. But...

" department is functioning fine. We offer a premier program, the likes of which you'll find nowhere else in this state, no matter the price." At this point the story follows one of two paths. Either (a) "our program is efficient and sustainable: the number of hours we require is well below the university average, and our students graduate on time far more often than their peers in other programs" (admittedly, this is sometimes true: there are a small number of very sustainable program on campus) or (b) "we recognize that our program makes excessive demands on our students' time, but these demands are in line with the rigors of our discipline and with the accreditation standards laid out by our discipline's professional governing body."

So where's the breakdown, folks? Who's to blame? Are we going to pick up the switch and start thrashing away at ILS, the same old whipping boy we've been beating since I got here over six years ago? (This approach is puerile, people: though ILS could be better, it's not entirely broken...and as a liberal arts institution that purports to take that mission seriously, we need some sort of ILS-like program in order to maintain our street cred.) Or are we going to waste more time pointing every finger we've got at our colleagues in other corners?

Or maybe...just maybe...we'll get the balls to ask ourselves tough questions: do we really need that extra concentration or those curiously complicated course requirements? Can we serve our students as easily through advising as we can through administrative fiat, thus making their schedules more flexible? Just what do our students gain by being forced to take these specific six courses, and not simply six from among ten or twelve, chosen according to their individual interests and aptitudes?

I can't end this post without a note of thanks to the folks in my own department. I've been at odds with many (most, perhaps?) of them on the matter of reducing our concentrations (we could easily, I feel, stand to combine our pure and applied concentrations into a single one), but I like the moves that were made today in our department meeting to make more flexible the applied concentration. I appreciate their willingness to move in that direction.

Compromise, folks, compromise. It's how shit gets done.

I apologize for the pottymouthery. I've been hard at work since 7:00 this morning (it's now nearly 11:00), and my internal filter fell off a long time ago. I'm off to bed.

(Oh, from the "it gets much more fun from here on in" department: I'm tremendously excited about the next few days in both of my classes. Applications of quadratics in Precalc...and symmetric group silliness in Abstract!)

Good can come of department meetings

Propositions cut
like glass, bright obsidian
in soft poetic flesh

Saturday, October 01, 2011


Yesterday I ran into one of my Precalc students outside of class. I was on my way back to Robinson Hall after teaching my Abstract class over in Karpen; Becky was en route to the library with the rest of her LANG 120 class. There they'd be discussing the use of library resources in conducting research.

We chatted briefly, and I asked her if she was still considering a math major. (She was one of two in that section who, without prompting by me, indicated interest in the major at the outset of the semester.) "I definitely am," she said excitedly, and then a look of worry spread over her face, "do you think I still should?" I was a bit thrown off and only after a few seconds managed to reply with something like "of course!" She went her way and I went mine, but our encounter stuck with me. Why had she asked what she had?

I suspect it may be because she may not feel as confident in her math ability as she did at the start of the semester. Though she's done well on every homework set and on every quiz, like everyone else in the class she's made her share of mistakes and hasn't presented complete understanding of everything we've talked about. Might she believe that only those who can complete Precalculus with flawlessness and perfection are worthy of pursuing a degree in mathematics? (Only later did I think of an apt analogy: as I'm highly unlikely to ever run a four-minute mile, might I just as well give up on one of my favorite hobbies?)

After thinking it over, I found that I could understand Becky's belief, given the traditional structure of mathematics education, home to bell-curve-based grades, punctilious point-based assessment, and lecture-based teaching. There's an air of elitism to the way students are often ranked and ordered, made to fight with one another for a scant few As. The unsaid assumption in classrooms where deep and steep curves guarantee a normal distribution of grades is that only the best need move on, and that the others' services will not be needed. Detailed rubrics with single-percentage-point resolution signal to the student that mastery of fine detail takes priority over authentic understanding. (No wonder students clamber after every point, wondering what it is that separates a score of 8/10 from a score of 9/10!) Fast-paced lectures make sure sure the students who start off slowly get little chance to get ahead; the quickest students (who are often, but not always, the brightest) control the pace in these classes.

All of these factors discourage students who are excited or intrigued about math, but who are put off by the way in which it's often taught. We can't afford to turn these students away. The fact of the matter is we, as a society, need more mathematicians than we can possibly prepare, and we do no good in discouraging anyone who's passionate about the field from pursuing it further. We do well to let as many students through the gate as we can, and to give them all of the support and encouragement they need to develop their skills fully. We do well to eliminate curves and to downplay in-class competition between students. We do well to "coarsen" our grading scales to accommodate "big-picture" thinkers who might miss a detail or two but who grasp complex systems in their entirety. We do well to step away from our classroom's center stage and let students take our place, so that it's not to the top ten percent that we teach, but to the class as a whole.

The bad news is that these practices are not universal; the good news is that they are more popular than ever. They're in use throughout my department and many like it. The youngest math teachers (at every level) are more adept at applying them than their older peers. These teachers are daily developing new tricks and techniques to make these practices more effective, and they're not shy about sharing these tricks and techniques with their colleagues and with their students. The future is bright for math education.

Stick with it, Becky! Welcome to the team. You're in good company. You'll do wonderfully.