Preface

In writing previous self-evaluations, I've always organized the material around the Four Cardinal Virtues of Teaching Effectiveness, Quality of Mind, Service the the Community, and Congruence with Earlham's Institutional Identity.

As I started to think about this evaluation, though, I naturally found myself starting with some of the things I had achieved in what seemed to me to have been a productive summer; and I found that I was having difficulty knowing under which category to place some of my activities, even though they seemed to me to be the sort of things Earlham should be happy to encourage.

I knew that the paper on the Kazhdan-Lusztig representations that I and a colleague at U. Mass. had completed fell under Quality of Mind; but I wasn't sure about things like reworking my weaving software. This had started out as problem in algorithm design suggested by Tekla Lewin, which meant that it probably should be discussed under Quality of Mind; but the fact that the solution of the mathematical problem eventually was manifested as a web page that weavers tell me has been a great resource means that maybe it's really Service to the Community; and the fact that Tekla and I used the Mathophiles Seminar as a venue for working through some of the ideas embodied in the code meant that maybe Teaching Effectiveness was in there, too.

What about the hundreds of pages of calculus lectures and labs that I had made available in print and in Braille? These began as a response to a calculus student with special needs, but they proved useful to many of my students; so they clearly belonged under Teaching Effectiveness. But preparing them for use by my colleagues at Earlham and for others needing Braille resources in calculus felt like an act of Service to the Community.

These were the questions that met me concerning the past few months; but as I thought back further, I found myself with more questions of the same sort. A few years back, Mic Jackson and I did some consulting work looking at the mathematical problems involved in using ground-penetrating radar to discriminate between unexploded ordinance and harmless metal junk. The hope was that this might reduce costs and add effectiveness to the hard task of taking former military reservations and turning them into housing developments. (People actually do this.) Was this Quality of Mind, Service, or Congruence? I thought it was all three (possibly weaker on the first, since in the end we were unable to do what we set out to accomplish, though on the bright side, we could prove that it wasn't at all easy).

In view of these questions, I've decided to organize this self-evaluation slightly differently from the others. I'll talk about

• Computer Science
• Mathematics
• Teaching
• Other Features
• Bugs

Computer Science

Since receiving tenure, I've more or less had a full career as a computer scientist. That is, I've gone from teaching the occasional course in CS to teaching half time in the Program. I've then managed to produce consensus for a hire in CS, and have helped us to make that hire, leaving me out of the computer science business. So in the last seven years, I've learned enough CS to teach half a dozen different courses in the business, have helped lead the Program, and have turned myself into an Associate Professor Emeritus in computer science. I'm proud of all those things.

I've been working with computers in one way or another since 1969, but it wasn't until the last few years that I really undertook anything like a systematic study of computer science. Seven years ago, I had taught a single course cross-listed between Math and CS. Now I've taught Advanced Programming, Algorithms and Data Structures, Theory of Computation, and Programming Languages, and I've been involved with Charlie in teaching both Robotics and Operating Systems. With Charlie, I taught a really innovative pair of courses combining Programming Languages and Operating Systems and based around a common language/OS environment, Oberon. All three times I taught Programming Languages, the course was completely different. I've developed and continue to teach Math Toolkit, which is intended as a course on the basic mathematics needed in computer science.

In the course of all this teaching, I've gotten fairly comfortable with a pretty fair variety of languages and programming paradigms. I've written code like a Lego robot simulator and an algorithmically interesting weaving utility. The last of these projects was a particularly interesting stretch, combining the work of interacting with weavers to learn their language and needs, and the design of a rather subtle algorithm, and the new task of building the web front end. It's been pleasing that this weaving tool has gained a small but enthusiastic community of users. One of my more hyperbolic supporters wrote about it, "I don't know if you can really understand the impact your program has on the weaving world.....being a non-weaver, but it is certainly one of the best innovations to come along in the past century and one of the most useful additions to the weavers 'toolbox' I have ever seen.......you are most appreciated!!" The sort of e-mail that would have made even a frustrating project - which this was not - worthwhile.

In all this, I've learned a great deal, and I've had a stimulating and exciting time.

For me, one of the most significant parts of my foray into computer science was the opportunity to work with Charlie Peck, not only in teaching but in dreaming and designing the College's CS Program. Compared to Mathematics, Computer Science is obviously a new discipline; and the right nature and shape of a CS program at a liberal arts college is still very much in flux. On the other hand, as a discipline some of whose roots lie in engineering, CS has a collective interest and focus on standards that mathematics lacks. As a result, in trying to shape Earlham's CS Program, Charlie and I had both a wide open field and also a wealth of sometimes competing standards documents about which we could try to structure a solution.

CS is also an incredibly rapidly evolving discipline, as is shown by my Programming Languages class, an alternate year class which had to be redesigned from the ground up every time I taught it. It therefore seemed as though every time Charlie and I had worked through the direction in which we thought CS would evolve and had agreed on the conceptual organization of our classes, we found ourselves a year later revisiting those decisions, sometimes to reaffirm them and sometimes to decide that we needed to change with the times. For someone used to teaching courses in which little of the content is newer than the oldest work in computer science, this seemed at times invigorating, and at times merely exhausting.

In these discussions and in our courses, whether taught separately or together, I think Charlie and I made a surprisingly effective team. We looked pretty bad on paper - a mathematician with no formal training in CS and a half-time faculty member who was really just an organic farmer with a bachelor's degree. In reality, though, my facility with the mathematical and theoretical end of the discipline neatly complemented Charlie's applied perspective. Together, we were able to construct the core of what I think was a balanced and exciting and effective program - and certainly a growing one. (Of course, the salaries being paid at the time in the computing business may have had something to do with the last of these attributes.)

After all this, it may seem odd to say that one of the most important things I did to help grow Earlham's CS Program was to arrange to get out of it. I had lots of fun in my time teaching computer science, and Charlie and I formed a nicely balanced team. In the long run, though, a CS Program anchored by the two of us was not the right solution for the College. Since Charlie was the only faculty member in the CS Program whose primary focus was on Computer Science, and since he was the only person at Earlham able to teach many of his courses, a Program whose core was the two of us was critically dependent on a single person, a person who, given the economic realities of computer science at that time, might literally have proven impossible to replace. Institutionally, it was essential that the CS Program have deeper roots.

As I looked at my own future, too, it was impossible not to wonder about my own long term ability to maintain my level of activity in CS. I had found the job of learning a whole new discipline immensely stimulating, but as I looked ahead, I could see that staying current in CS was going to require an enormous investment of time and energy. To commit to making this investment for the next twenty years was daunting. I love doing CS, and I told students in my last class in the Program that leaving the discipline left an RJ-45 shaped hole in my heart. Computer Science isn't the center of my intellectual life, though - mathematics is. I'm also not terribly happy about many of the directions in which computing seems to be evolving, industrially and societally. To commit to spending the rest of my Earlham career running like mad to stay up-to-date in a field that was not my first love did not seem wise. That CS was not only not my first love, but (to stretch the metaphor a bit) that it was also starting to spend too much time partying with folks of whom I was suspicious, made me nervous about our long term future together.

I and my Department therefore undertook a course of action that could probably only have happened at Earlham. After obtaining approval to hire a faculty member in the Mathematics Department, we petitioned to be allowed to attempt instead to hire in CS. We weren't at all sure we'd be able to make this hire, but it seemed to us to be the right thing to do institutionally, and the right thing for me personally. We were somewhat startled at how long it took to persuade CPC that this was a reasonable course - perhaps even at Earlham the idea of one Department petitioning to be denied a hire in favor of another Department aroused unaccountable suspicion - but in the end we managed. To my own considerable surprise, we were also able actually to make the hire, and to bring Jim Rogers to Earlham.

This course of action has substantially strengthened Computer Science at Earlham College. It leaves us with a second faculty member whose main focus is computer science. Jim's interests are largely in the formal areas of the discipline in which my own strengths lie; so he continues to complement Charlie as I did. In watching Jim, too, I've come to realize how much deeper than mine is his understanding of many issues in the discipline. There are plenty of areas in which I'm a perfectly competent first year of grad student in CS, and in which Jim actually understands what's going on in depth.

One of the most satisfying things I've been involved with during the past few years has been this effort of learning a new field and helping to grow a program until it needed resources I could no longer offer it, then leading the effort to replace myself. I learned a lot from the students; I learned a lot from the subject; I learned a lot from my colleagues; I left the students and the College in a much stronger place than they started out. That's why I've started out this evaluation by talking about about a field in which I was only marginally active at the time I received tenure, and in which I have no current plans to teach again.

Mathematics

One of the things I learned anew during my last sabbatical was the degree to which mathematics remains what it has always been - the heart of my intellectual life, my first love and the home to which, despite my academic Wanderlust, I continue to return. After thinking I would devote much of my sabbatical year to deepening my understanding of computer science, I instead spent most of it learning more math. I won't spend a lot of time here going into details, but a web page I wrote early in that year and discussing some of the central ideas I've been thinking about lately can be found in this Introduction to Analytic Number Theory.

Another piece of pure mathematics I've worked on has finally come to fruition in an odd way a year or so ago. Let me tell the story briefly here, apologizing in advance that most of the details won't mean a lot to many of you, as they would not mean a lot to many mathematicians. Compared with other natural sciences, mathematics is a very bushy tree, and this fact, combined with the lack of a concrete, physical subject matter underlying many of our ideas, makes it uncommonly hard to discuss mathematics, even with other professionals in slightly different parts of the discipline. The fact that abstraction is one of our most powerful tools doesn't help matters, either. Still, FAC has asked in the past for more information about what I was thinking about; so with advance apologies, let me give you a little detail.

At the time I came to Earlham, I had been thinking about some things called the Kazhdan-Lusztig (KL) representations of the symmetric group S_n. This is a method found in the late 1970's for producing matrices which behave under multiplication the same way as the rearrangements of n objects behave. In fact, Kazhdan and Lusztig had a much more general problem in mind than this, applying not just to the group S_n rearrangements of n objects, but to a large class of related algebraic structures called Coxeter groups. I don't understand this more general setting very well, though; so there's no reason you should either. (Just on the off chance you do happen to have this stuff wired, could you let me know? I'd like to talk about it.)

One drawback of the Kazhdan-Lusztig representations is that computing the entries of the matrices is not completely trivial. There's a formula all right, listed in Theorem 6 of this reference, for instance, but the formula isn't simple. It involves computing a family of polynomials, one for each pair of permutations. Further, each polynomial is determined by a complicated recurrence that involves summing a large number of other polynomials. Combine this with the fact that the number of pairs of permutations in S_n is (n!)², and you have a problem. The size of this problem might be hinted at by the fact that for n = 16, for instance,

Now, in general, it's known that the entries in the KL matrices are non-negative integers; for most Coxeter groups, not much more can be said. For S_n, though, there appeared to be more going on - every entry of every KL matrix that had been computed turned out to be either 0 or 1. Further, Alain Lascoux and Marcel-Paul Schützenberger at the University of Paris had conjectured a simple method for computing KL matrix entries for S_n which guaranteed that all the entries would be 0 or 1. Lots of combinatorialists got excited about this, since Lascoux and Schützenberger's methods seemed to take the KL representations and embed them in a collection of ideas we understood reasonably well. Since Lascoux and Schützenberger are two of the really big names in algebraic combinatorics, and since they initially announced that they had proven their conjectures, their results came to be widely believed.

In 1988/89, I was working on the Lascoux-Schützenberger (LS) conjectures, in part by doing systematic computer searches for counterexamples. At that time, it was known that the conjectures were true through n = 6, and some exploration had been done for the case n = 7. Moving from S_n to S_n+1 increases the number of polynomials by a factor of (n+1)², though, so the work was slowing down. It took several months of work improving the algorithms for computing KL polynomials and finding ways to eliminate polynomials that couldn't be counterexamples to the LS conjectures before I began finding that in fact there were counterexamples. For the 0-1 Conjecture itself, the first of these examples appeared in S₁₆, much farther out than anyone else had looked.

Here, I made a mistake. I didn't know how to be certain that the calculations I had carried out were really correct. The math was really complicated; the code needed to be pretty subtle in order to be efficient enough to run in my lifetime; the counterexamples emerged at the end literally of several months' of computer calculations; there wasn't much I could check by hand. Feeling that my results were shy of proofs, not knowing how to make them into proofs, busy with everything I was doing at Earlham, I therefore dropped the project and forgot about it.

I didn't think more about the problem until a couple of years ago, when Greg Warrington, then at MIT and now at the University of Massachusetts, contacted me to ask what I knew about the KL polynomials. It was rather a surprise to me to learn that the LS conjectures were still an open problem 13 years later; I had just assumed that as computers got more efficient, someone else would have replicated my calculations. Instead, it turned out there was genuine excitement among the community of folks working on the combinatorics of group representations that what was regarded as a difficult problem had been resolved. I told Greg what I knew, and he was able to confirm with independent calculations that my results from 1988-9 had been correct. He then has carried the work somewhat farther, and we can now replicate some of the results by hand. Anyone interested in the serious details can look at the abstract of our paper in the AMS electronic journal Representation Theory or at the full paper. Finally, one can get a sense of how this work was received by looking at an embarrassingly friendly referee report.

I tell this story here for several reasons. It's an illustration of the fact that I'm thinking about mathematics again after a few years focusing on CS. It shows one of my strengths - that I'm both dogged and really good at taking complicated algorithms and making them more efficient. It shows a real weakness - that I'm too much of a recluse. It's also just an interesting oddity. How often does one have the experience of getting documentary proof that one is 13 years ahead of the rest of the world in working on some problem?

A third piece of scholarship I should mention is a small bit of consulting work Mic Jackson and I did several summers ago. Mic was involved with a company in Oak Ridge which was trying to use ground-penetrating radar to discriminate between harmless metal debris and unexploded ordinance (typically mortar rounds, 55mm howitzer shells, and RPGs) in areas that had been used as firing ranges and that were now being released by the Army for civilian use (typically housing developments). There is currently no effective and even vaguely economical way to do this, with the result that in the eventual housing developments, frost heaving sometimes brings live and not necessarily stable high explosive ordinance to the surface, inconveniencing the residents.

By walking over an area with a ground-penetrating radar unit on a self-propelled lawn mower carriage, one can detect essentially all metallic anomalies in the soil down to a depth of a meter or so. At each point in the radar unit's traverse above an anomaly, one records the distance from the radar unit to the surface of the metal object. The problem Mic was asked to consider, and that he asked me for advice on, was how to take these measured distances and infer the shape of the metal object. If we could do this and reliably distinguish things like hubcaps and shrapnel from things like mortar rounds and RPGs, then the former could be ignored, and the latter "excavated archaeologically" as the professionals seem to say; I had never realized how exciting archaeology could be.

The final report I sent Mic, and that became part of his report, is available as a Maple Worksheet and as a PDF file. (The page breaks are not ideal in the PDF file, but it should be perfectly readable to those not having Maple on their computers; to read the Maple worksheet itself, you'll need to save the link as a Maple worksheet, and then to open it with Maple.) Unfortunately, what we were able to show was that a single pass with a radar unit did not provide enough data to obtain useful images of buried objects. It appears as if multiple passes might provide this capability, and I suspect there is more information in a radar trace that could be exploited if we knew more about the hardware. The firm we were working with was not interested in paying more to get us together with people who really understood the hardware, however, which left us in the frustrating position of having to send them what I think is a carefully argued proof that what they wanted us to do is impossible without more data.

While it's obviously a pity the final result of our work turned out as it did, I think both Mic and I learned a lot from the experience, and that we enjoyed working together. It was a pleasure to support Mic and to support Earlham as a place where science related to the environment and to peace was taking place. I'd be happy to do more consulting like this in the future.

One of the consequences of doing more mathematics on my own has been that I've found myself with lots of ideas to share with students. I've probably been more active than anyone else in presenting material to our weekly Mathophiles gatherings. I've talked about analytic number theory and about weaving and about unexploded ordinance; some of the other fun things we've done in these discussions are suggested by the following ads for sessions on

• Continued fractions
• Catalan numbers
• Computing pi
• Computing pi again
• A bizarre approximation to pi
• Transfinite turtle races

I've also intermittently posted Problems of the Week in Dennis, in hopes of exciting students to work on interesting mathematics outside class. Typical examples include

• A sine identity problem, solution.
• A formula for primes problem, solution.
• A pseudo-Pascal's triangle problem, solution.
• Perfect square patterns in Pascal's triangle problem, solution.
• Stirling's formula problem, solution.
• Mersenne and Fermat primes problem, solution.
• An identity of Eisenstein's problem, solution.

Finally in this section, I should mention one other piece of mathematics I've worked on without in the end really understanding all there is to be said about it - an amazing formula for pi in terms of nested radicals found by Bryan Fay in a Calculus A lab, and mentioned in this ad. There's no reason this formula could not have been found by Archimedes, but I haven't found direct mention of it anywhere in the literature. Slightly modifying the infinite nested radicals gives rise to a family of interesting functions including the arccosine plus infinitely many others whose properties I am still trying to work out.

Teaching Mathematics

It's hard to try to summarize everything in seven years worth of teaching evaluations. It's particularly difficult to know how to assess student comments in a department like mathematics, which is the humorist's perennial paradigm for academic horror. In some sense, a mathematician should perhaps be content not to be universally despised. As I write this, I'm looking at going in for a root canal at 7:30 tomorrow morning; I know that many students facing 8 AM math classes have similar emotions. For myself, one of the principal things I find myself feeling right now is sympathy for the people who'll be working on me tomorrow. A mathematics teacher has a pretty good idea of what it feels like to meet someone at a party and to have to tell them one is an endodontist.

Let me therefore start this section by discussing a couple of classes in particular. I'll pick two elementary classes, since most of the Department's teaching load is obviously in classes of this sort.

Calculus A

In the past few years teaching calculus, I've made a number of changes in the way I teach the subject. Most significantly, encouraged by Tekla Lewin, I've gone from courses that meet 5 days a week for an hour to courses that meet three days a week for hour lectures and one day a week for a 2-hour lab. For me, this has been a fantastic improvement. On a trivial level, it breaks up the week so that students are not ground down by daily 8AM classes. (Having written that sentence, one has to pause for a moment and ask how many jobs they'll have that allow them to sleep past 7:30 each morning; but that's neither here nor there.) More importantly, the lab sessions get students working together to discover many of the central ideas in the class. Students get the experience of looking at numerical and geometrical data and trying to produce for themselves some formal description of what they are observing. This gets them thinking about questions from a variety of points of view before some formula comes along and resolves the whole issue. This way, they are more able to see the meaning of the eventual formulas, and they are able to be involved in the proofs of the formalism as arguments for their own conjectures or as alternatives to or refinements of their own arguments.

One might think that some of the sorts of labs we do - getting students to conjecture the Product Rule or the Chain Rule for derivatives, for instance - would be seen as completely trivial by the large fraction of the audience that has seen calculus in high school. It's interesting to me that this isn't so. I've had students who had studied calculus in high school badger me to tell them what the name of the formula we were investigating was, so that they would know what to do. When I patiently refused, and asked them to think about what they were doing and about what formulas might apply, or to work through the labs de novo without the formulas from school, these students often proved to be at no advantage relative to those who had not previously studied calculus. It seems clear that for a large number of students, having taken a course in calculus in which topics like these were lectured on and in which they presumably did boatloads of problems does nothing to help them rediscover these formulas in practical settings in which the names aren't mentioned.

Now, it's hard to know whether discovering the formulas for yourself by working numerically and geometrically, and then struggling to validate your discovery with an algebraic argument lets you take away from my class a level of facility these students didn't take away from their previous classes. I'd like to think that it does, but then, I only see the best of these students back for more advanced courses. Even if the success of the approach is not universal, though, it seems to me that the very effort results in focusing the class in appropriate directions. We're asking students to discover for themselves - to see that mathematics is a discipline invented or discovered by human beings, and that they, as human beings, can engage in this process. We're asking students to think for themselves about problems, and to relate for themselves the symbolic, geometric, and numerical aspects of the calculus. This is hard, and lots of students don't like it. ("Why can't you just give us the formulas?") It's also what a math course ought to be about if we are to do more than just train human beings to be uncomprehending replacements for computer algebra systems.

Anyone wishing to see copies of the labs used in a recent installment of Calculus A can find a complete set in several formats in the labs section of my Inclusive Calculus Resources. There are fewer labs there than there are weeks in the semester because we used some of the lab sessions as opportunities to review or to work on homework together. The lab times were a natural occasion for this sort of thing, and the flexibility to use them in this way is an important tool in responding to student overload when it arises.

As can be seen by looking at my Inclusive Calculus Resources, one of the most interesting challenges I've faced recently in Calculus A was figuring out how to make the course work when one of the students, Bobbie Hughes, was blind. My initial reaction to this project was, rather naturally, panic. How could I make accessible a subject which is to me very heavily visual, both because of its geometric component and because much of it involves rather complicated equations, which I found it hard to believe anyone could follow by ear.

After some very useful conversations with Bobbie, with Donna Keesling, and with a number of experts I met over the Internet, I was able to find hardware and software to let me communicate with Bobbie using Braille and tactile graphics, and using a student assistant as an interface between Bobbie and computer algebra systems, which are still not very accessible to blind users. The resulting course was enormously time-intensive for me. We didn't have the Braille embosser and the software I needed to turn mathematics into Braille until almost the start of the semester; so I started out behind and was never able to get even slightly ahead in preparing course material. The materials on the web page represent about half the mathematics I typeset and Brailled during the semester, and even they feel like a lot. Instead of going into class knowing what topic I was going to discuss and roughly how I intended to present it, I needed each day to have my lectures written up in advance essentially verbatim, so as to have all the equations available to Bobbie in Braille as we were discussing them. I also needed to have all the figures I was planning on drawing prepared in advance as tactile graphics so she could read them at the same time as the rest of the class. More information about how we conducted the class is in this document on course design.

(Parenthetically, I should perhaps add that the phenomenal time commitment required by this calculus class is the reason this review is happening so unconscionably late. It should have happened the semester I was teaching this class, but I had to ask FAC to allow me to postpone it until I could take it seriously, which I couldn't that term. I deeply appreciate FAC's willingness to be flexible at the time! Unfortunately, after the end of the semester, a number of miscommunications and misunderstandings with FAC produced some further delay, for which I am abjectly apologetic.)

The resulting course was deeply rewarding and enormously instructive to me; it was also frustrating. For Bobbie, who had taken calculus in high school without having it really gel, the course seems to have been a big success. She ended the class as one of the top students, with a very solid grasp of the material. As a Spanish major, Bobbie probably will never take another math class, but to succeed so well at Calculus has to have been a nice experience.

For my sighted students, the course was more of a mixed experience. The extremely complete notes I was preparing for Bobbie were made available to them as well, and were seen as a useful resource. They continue to be a useful resource in other calculus classes, and I hope that putting them on the web may provide a helpful tool to other blind students learning calculus, and to other faculty scrambling, as I was, to support them. On the other hand, the high level of advance preparation needed to support the blind students meant both that the class was much more formal and scripted than I would have liked, and also that I was often more tired and cross and unavailable than I would have liked. None of this was a disaster, and I'd be happy to try the experience again; but although parts of the course design worked well for all the students, my inexperience with the situation meant that there were plenty of things that could have been done better.

Discrete

The other course I'd like to talk about in detail here is Discrete Math, which is more or less my favorite course in all the world. It is without any prerequisites as far as content, and it lets a student very quickly discover deep and beautiful results. It is the perfect course for anyone who wants to know what mathematics is really about. One thing that makes Discrete both exciting and challenging for an instructor is that the students are enormously heterogeneous. One gets both music students who are very nervous about math, and math and CS majors who have somehow managed to avoid taking the course until their senior years. I've tried to deal with this heterogeneity in several ways. I give very open-ended homework, and I expect performance commensurate with experience. Some terms, I've divided homework into basic problems I expect everyone to do, and harder problems for the cognoscenti, and I require students to do at least the basic problems and to convince me they are working. I warn advanced students that my primary audience is beginners at math. I work a lot outside class with the less experienced students, sometimes forming daily discussion groups for them, and reminding them that I was in such a group at their stage in my career. This has helped some very phobic students to get a lot out of the class. As a department, we have positioned Discrete to encourage our majors to take it Term I of their first year, trying to cut down the upper class majors who intimidate others. For beginning students and non-majors who are not too phobic and who are willing to be calm and think, Discrete Math can be wonderful; for others, all my efforts at reassurance are not always enough. The course is not too difficult and does not have prerequisites, but the problem of convincing people of this fact is not yet solved.

Despite the fact that not everybody who takes Discrete finds it the transformative experience one hopes all our classes will be for all our students, some of the most rewarding experiences I've had in teaching have happened in Discrete. For many of our majors, Discrete has been the course in which they really saw what mathematics can be, and in which they caught fire. Their thanks for this course have been pleasing. Even more exciting, though, have been the students who didn't come into the course thinking they were good at math, and who came out wanting to see more, and convinced that they could do this stuff. Does anything beat getting homework in which someone has written, "Wow! My conjecture is working! I can do math!"? Of course, I'd like to say I get such homework from all my students, but at least I'm happy to get reactions like this from somebody every time I teach the class.

To get a sense of what Discrete looks like on paper, you could look at this bare-bones home page. Really to understand the course, though, you have to see us struggling together to agree on sensible definitions (a matter in which we sometimes get into rather lengthy and impassioned debates), to discern and to articulate the patterns in the calculations we've done, and to craft convincing arguments that those patterns continue. In doing this, we end up dealing with deep mathematical ideas - deeper than many of the students realize - yet there is very little machinery between the student and the ideas. That is, anybody can start messing around with clock arithmetics as we do in Homework 1, yet the ideas we see in doing this messing about end up leading to a whole raft of important ideas in abstract algebra as well as to practical applications like some of the most secure codes known. I love the idea of a course like this, in which those who have seen a lot of calculus and those who have seen none are on absolutely even footing, in which all of us can contribute to the community's knowledge, and in which so much of the course is a conversation in which we all work together to discern and to articulate what the efforts of all have let us discover. It's the most fun you can have with a piece of chalk.

Other Teaching

As I look at my teaching, there are other things I'm proud of. In general, I think I do upper level classes pretty well. For many students, these are hard stretches, but I've watched a lot of students grow and stretch to meet some of the goals of proof and precision and abstraction that mark modern mathematics. The longer I work in the business, the more I think I can identify for the students useful connections among areas of mathematics, central issues in mathematics, and essential tools and habits of mind for a productive mathematical investigator. All these things make it seem to me that the insights I'm able to convey right now in my advanced classes cut a lot deeper than was true a few years ago. I think I continue to grow as a scholar and as a teacher.

Working hard with difficult students is something that has also been awfully rewarding to me. As I start thinking, I come up with too many people to name, but here are some examples, referred to briefly in order to try to avoid violating their privacy.

• C, who was a very talented student of mathematics, whose emotional disorders made him frightening and disruptive to the community and to himself, and who eventually committed suicide after leaving Earlham for graduate school. I suppose starting my list of successes with someone who died in such a way and who frightened so many of us is perhaps odd, but I really think that in this case Earlham managed to treat a very gifted but very troubled individual in a way that was remarkably communally caring and remarkably communally courageous. In the end, C lost a battle with the demons inside him, but we treated him with love, and for a time, we gave him life.
• A, another student with great insight, and another student struggling mightily with very serious emotional issues. Again, I'm proud of how we managed to support him and to offer him the chance to develop his talents. I'm also delighted that now, having left Earlham, he has begun to live again, freed at least for the moment from the intellectual and emotional paralysis that afflicted him during his last months here. More recently, I've had a fairly large number of advisees and other students with substantial mathematical talent whose expression is being partially or completely blocked by their struggles with psychological issues. I wish I were better able to help them work through those issues, but it has at least been a privilege and an education to work with them; and I think in general I have done well at working with them in a way that is supportive and nonjudgmental, but that still challenges them to reach for deeper and deeper mathematical ideas.
• W, an African-American football player with a tremendous work ethic and with a surprising ability to see to the heart of mathematical questions. W is a student I feel both proud of and bad about. I personally had him as a student only at the beginning and the end of his time at Earlham, when, like many of us, he was struggling with proof and with abstraction. At the beginning of his senior year, he really wasn't internally convinced of his mathematical skills; and the lack of confidence was hurting him both mathematically and personally. By the end, he saw just how much he was capable of; and while he never found himself loving proof, he wrote some very touching things about the satisfaction of overcoming that obstacle. There's the satisfaction. The sorrow comes from the middle part of W's time here, when I think we failed properly to nurture someone who arrived with a lot of talent, but who found Earlham Mathematics a scary place. W always had good insight, and he was recovering his confidence and abilities when he left us. I wish very much, though, that we had managed to help him grow in such a way that he grew smoothly in confidence and ability throughout his time here. He's a great kid now, but he could have been mathematically and personally much stronger had we done our job right.
• On the other hand, I think of J, another athlete, a very quiet person, and one I first got to know under unpleasant circumstances involving plagiarism. I don't think J had W's raw talent, but as we worked together, in every class he could do a little more, cut a little deeper, express himself a little more clearly. Watching him grow was a great experience.
• Finally, let me broaden these summaries slightly to include groups rather than individuals. I've worked, for instance, with several students in Biology on independent study projects in dynamical systems, chaos, and mathematical ecology. I think particularly of one year-spanning independent study in which we not only did mathematics together, but also raised Drosophila and modeled their population growth, studying the transition to chaos as the growth rate increased. It was a great project; it has been wonderful to be able to work with students as good as these were at bringing exciting new mathematics into their disciplines.
• I also think with great respect of several students I've worked with who suffered from learning disabilities that made it very difficult for them to do formal algebraic manipulations. For a number of these students, modern computer algebra systems turned out to offer a critical enabling technology by taking over the algebraic tasks of which the students were not capable. This, combined with course designs that focus on creativity and understanding rather than formal manipulation, and in which exams are all take-homes, really let a number of students suddenly open up and succeed in a math class for the first time. A number of these students had truly exceptional geometrical abilities and were remarkably mathematically creative and far-seeing; on one level, they were among the very best students I have ever taught. Perhaps some of their insightfulness had arisen in compensation for their formal deficiencies? In any case, one of the most thrilling educational experiences of my life has been sharing in the excitement of these students who have suddenly been freed to realize that they can think about a subject that they thought was closed to them, and that they can do it well.
• Last of all, let me mention a group that continues to be a challenge to me in math classes - Earlham's CS students. Earlham is unique among schools I've been associated with in the degree of separation between Math and CS at the student level. Since for me, Math and CS seem very closely connected, it has taken me a while to begin to understand and address this mood among students. Many of our Math students are rather pointedly not interested in and not good at computers. I think I understand some of the motivation of this, though I don't really share it. The austere simplicity of pure math, a Zen garden independent of the world, is an image of power among mathematicians. To be free of the need to use tools, free to study the Real in Pure Thought, is an alluring attraction. If it speaks to some among us, I understand.

  What I find more difficult to understand is the attitude of the substantial majority of our CS students who want as little to do with mathematics as possible. For many of them, it isn't a matter of interest - that CS with its practical and engineering focus seems more congenial to them than the abstraction and aloofness of mathematics. Rather, they fear mathematics and feel they cannot do math. Although I like these students and I consider us to be friends, it has taken me a long time to try to begin to understand how one can be adept at analysing a problem and implementing a solution in the abstract, formal language of C++, but find it impossible to understand and to solve a problem in the abstract, formal language of algebra. Yet this is the reality of many of my CS students.

  I continue to work to find ways to equip my CS students with the formal and mathematical tools they need in order to succeed in their business, while at the same time respecting their expressed fear and lack of interest in these tools. It's a hard task for me to understand what it feels like to be one of these students, and so to devise courses that work for them; and I don't always succeed. The new course, Math Toolkit, that is now required as part of the CS major, was an unqualified disaster the first time it was taught; though the students cheerily or sheepishly admit their own significant contributions to that disaster. (The class should have been designed to inspire and motivate them to work; but then, they should have worked in any case, and passed the course.) After a first year in which most students failed the course came a second year in which nearly everyone passed; so somehow both I and the students are managing to do better. They still are learning less than I would like, though; so the process of finding ways to understand and to inspire them obviously needs to continue.

Other Features

For better or worse, at bottom, my service to the College and the community consists largely in teaching and thinking and doing mathematics. It's always a stretch for me to find other places in which my service has been crucial, with the result that this is probably a short and weak section. Here are a few things I can mention, though.

• As mentioned above, I've thought a fair amount about how Math and CS fit together with one another, with other disciplines, and with Earlham's mission as a Liberal Arts College. This has resulted in new courses, in redesign of old courses, and in new degree requirements.
• I've been involved in 3 hires of new faculty - Joy Williams Lind, Jennifer Ziebarth, and Jim Rogers, all of whom have brought a lot to the College. I was also part of the committee that recommended hiring Tom Steffes in his current position, a hire that was not what I expected us to do at the start of the search, but that I think has certainly been a good thing for the efficiency and morale of ECS.
• I've served on committees when asked, though for the most part, this service has been fairly routine.
• I continue to be active in my church, and to maintain an occasional but friendly relationship with ESR and Bethany, talking to classes a few times on Orthodoxy. I've also attended local Yearly Meetings a few times as part of a continuing effort to understand Friends and to do what I can to participate in friendly and Friendlike relations between the College and our associated Yearly Meetings. It's probably of rather peripheral connection to my work at Earlham, but an example of this sort of thing is a recent talk on Orthodoxy I gave to a study group at West Richmond Friends.
• My program to serve the community by raising quality citizens of the next generation continues, enhanced by my successful effort to attract to the community a quality citizen of the previous generation in the form of their grandmother.

Bugs

This document so far has outlined some of what I've done over the past few years; but obviously one of the most useful parts of a self-evaluation is the opportunity it affords to reflect on what one has left undone, or left undone to one's satisfaction. In no particular order, here are some of the areas in which I still need work.

1. I'm more committed to mathematics than I am to students. By this I don't mean that I'm ignoring my students in order to do research; I plainly am not. But I mean that when I think about a subject or a course, what excites me is the content of the material and its internal logic. What I want to do by nature is to work out ways to make that logic as lucid as possible, to find as many connections as possible with other fields of mathematics, to develop ways to make the directions taken in the development of the theory seem obvious and natural, so that we can discover it ourselves. I also want to get excited about the discipline, and to convey that excitement. None of this is bad, but all of it is focused on the mathematics, not on the student. As I talk to my colleagues, I hear many people who seem to spend much more time than I analysing what each student is thinking and experiencing, developing courses around the students as much as around the subject matter. I meet colleagues who seem to have deep information about their students' lives that it would never occur to me to explore. I like my students and I care about them a lot, but I'm not good at that level of human-centered analysis of a course. I'd like to be more creative in course design, led by an understanding of my students, rather than an understanding of my subject. None of this is easy and natural for me to think about.
2. In the same vein, I'm a less sensitive advisor than I'd like to be, and than many of my colleagues. Where I'm effective is at the fairly mundane level - what course should I take if I'm interested in thus and such a subject. As I talk with colleagues, though, I often seem to find people who have a deep sense of their advisees' lives and dreams and struggles and who work with their advisees in ways much closer than anything I normally manage. That said, as I look at my advisees and other students in recent years, there have been what I hope is an abnormal number of them experiencing surprisingly serious psychological and emotional struggles. For these most demanding students, I have probably managed to give as much time and energy and to offer as much love and support as anyone could ask of someone who is, in the end, not their psychologist or their priest. With less obviously needy students, though, I'd like to manage to be less distant.
3. One of the most important parts of my job is to get people excited about and committed to mathematics. I spend a lot of time developing and conveying my own enthusiasm. Despite this, I'm not the effective leader and motivator I wish I were. Our informal Mathophiles seminars have never really taken off. The student willing to do more than is absolutely required in a class - or even to do that - is far too rare. Beautiful and tractable problems proposed with enthusiasm too often get polite looks. I need new ways to convince students - at least the best ones - that there are exquisite ideas within mathematics, and that the students are capable of working hard and finding them. What I perceive as my failure to inspire and draw people in is a serious concern.
4. A related concern, and one that affects both me as an individual and the Math Department as a program is our relative lack of success in getting majors to take seriously the senior comps as an opportunity to pull together material from their various classes. There are exceptions, but far too many of our majors end up submitting comprehensive exams that are passing but disgraceful. I'm sure I contribute to that behavior by organizing my classes too heavily around creativity in open-book, untimed settings, with the result that too few people ever actually memorize and deeply internalize anything. I also contribute by failing to inspire people to regard the subject as so exciting that they desire to dig deeply into it, and actually to learn the material in a deep and permanent way. I and we need to work very hard on this in coming years.
5. In recent years, I and we have also failed a number of students because we did not recognize to how great a degree they were failing to rise to their potential. I mentioned W above, but the people I am really thinking of here were a couple of very organized young women who arrived at Earlham with a fair amount of mathematical experience, and who set out to double major in math and another discipline. In both cases, they were the sort of dream advisees who appear at every meeting with a thought-out plan for the semester, and having considered how that semester fits into the rest of their time at Earlham. They were in class, turning in carefully written work on time, and seeming to have it all together. What I didn't observe, and what we as a Department didn't observe, was the degree to which they were also failing to grow during their time at Earlham. Partly this may have been because their energies were going for the most part to their other majors. Partly it may have been because they were good at the algorithmic side of mathematics but were not developmentally ready to function at the level of abstraction and creativity. Whatever the reason, though, we allowed their surface polish and early promise to blind us to how little was happening for them during their time here. We should have helped them either to surmount the barriers that stood between them and real mathematics, or to find another major. We did neither, with the result that students I was delighted with when the arrived here left as extraordinarily weak students of mathematics. I don't want to do this again.
6. Another clear area of weakness, one I'll struggle against but never really overcome, is my tendency to introversion. I'm not good at networking, both because I'm busy and because I'm shy, and because I'm really not at all interested in almost anything social. This has made me a less effective mathematician and a less effective spokesman for mathematics than I should have been.
7. Perhaps related to my social isolation is a general lack of interest and ability to think about the larger political structures of the College. I'm great thinking about just what elements need to go into a proof. I'm not good at thinking about issues of College governance and structure, even in areas like computer technology where I have some level of interest and expertise. At least, I'm not good at shaping consensus on such issues. Too often, I find myself in meeting in which our administrators and my more politically adept colleagues seem enthusiastic and focused, and in which they seem to see important work being done, and I find myself not really understanding what we are doing. Len once made reference to me obliquely as someone with administration in his blood (my father was provost at Macalester), a misperception that truly alarmed me. Perhaps it skips a generation.
8. I'm also pretty poor at paperwork, unit planning, and the like.
9. As the example of the work on the Kazhdan-Lusztig polynomials shows, I'm not getting enough research done, either for my own good, or for that of the students.
10. Like many of us at Earlham, I'm not living a balanced life. As I write this, it's a lovely fall day, a Sunday afternoon on a holiday weekend. My kids are out playing soccer. I'm writing a self-evaluation preparatory to writing up homework solutions. What more can I say?
11. Finally, I don't think I'm being very effective at Earlham as an outlier. As discussed above, I'm hardly a political person, but insofar as I have political and philosophical convictions, they are in many ways far from the Earlham norm. This doesn't bother me terribly much. In past evaluations FAC has been supportive of my notion that although in geometry, figures can be congruent only if they are similar, at Earlham, it is possible to be congruent without being similar. That's me, and I am content. I think, though, that I have not always managed to find useful ways to offer divergent perspectives as a resource to the community. Instead, it is often easier to shake my head and to allow myself to be silenced. I'm hardly proposing that I think either I or Earlham would benefit by having me become an argumentative political and philosophic zealot, a role for which I have no stomach and for which the College has no need. The polite and visible presence of advocates of eccentric positions is, however, useful for students and useful for the community. If not me, then who? If not now, then when? Yet I still choose mostly to take the easy course and just hope to stay under the radar.

Tim McLarnan,
Assoc. Prof. of Mathematics
Earlham College,
Richmond, IN 47374 USA