Monday, July 31, 2006

Fun 'n' Games

I'm almost ready to start "integrating" the components of the design into a coherent teaching strategy built around a thematic structure for the course and meaningful learning activities.

Like I mentioned in an earlier post, I've had a hard time "categorizing" learning activities, but I have managed to come up with a few general schemata:

  • Dialogues: students pair up and each pair constructs a script for a dialogue by means of which they explain to one another a tricky concept just defined, or work their way through an application.
  • "What If..." exercises: students are asked to consider the effect of hypotheses on various mathematical statements. By adjusting these hypotheses (injecting a "what if" into the problem), students can explore generlizations of the problems already posed.
  • "Calc Revisited" exercises: calculus topics related to linear algebra, such as Jacobians, change-of-coordinate transformations, cross-products, area formulas, and tangential approximations, are revisited in order to highlight the connections between the two subjects.
  • Logic Games: various standard games involving mathematical logic (like the Prisoner's Dilemma) but no explicit mathematical formulas can be used to encourage participation by students who may be a little more unsure of themselves in performing math in front of others. By removing the formulaic, "formal" mathematical aspect of a problem, those who perceive themselves as weaker at math might step up to the plate with more confidence. (It's almost like tricking them into doing mathematics!) These'd be great for team-building exercises, too.
  • Resource Scavenger Hunt: I've not worked up an exact format yet, but I hope this to be an exercise which encourages efficient and effective use of reference tools (on-line and off-line). The way I envisage it right now is something along the following lines: students are given a short list of topics on which they are asked to find information...maybe an article, website, or textbook...and are let loose for 24 hours to find the necessary material. The more relevant the sources they track down, the more appropriate, the better they'll do. Admittedly, there's some tweaking to do here, but I think I can make this a fun exercise.
  • "Write a Bad Paper" exercise: perhaps the best way to learn how to do something right is to try to do it as poorly as possible. As I see it, you've gotta work pretty hard to write a truly awful paper, and my guess is that most people will notice palpable problems in composition, coherence, correctness, and citations as they try to make their papers bad.
  • "Correct a Bad Paper" exercise: this exercise can be paired up with the previous one. Exchanging intentionally bad papers with a partner, each student can try her hand at righting wrongs in peer review.
Ouch! Gotta go teach...that's all for now! More later!

Friday, July 28, 2006


Another day, another "hmmm..."

I've spent the morning (you guessed it) putting together some more material for MATH 365.

The latest two team projects involve analysis of RLC circuits and computer graphics. The RLC project's not fully written yet, but I'm quite happy with the computer graphics project. Yay me.

By the way, I'm trying to decide if there's any purpose served by posting these project descriptions here, for the benefit of my wide readership (which at this point, confirmed...but she's been a supportive one!). I'm leaning towards not posting, since it's entirely conceivable that one of the folks who'll be in 365 could download the project description, read it ahead of time, and (quel horreur!) put more work into the project before class even starts! Okay, so that's not likely, but I'd rather not risk it. If you have any interest whatsoever in obtaining copies of the project descriptions, or any of the other course materials, drop me a line.

Meanwhile, I'm fine-tuning grading schemes. I'm not sure how that's going to work out yet.

I'm off to do some non-Linear-Algebra-related stuff (yes, I do have a life...).

Thursday, July 27, 2006

Teaming up

I've been poking around in the literature on building and managing teamwork in the mathematics classroom, in both cooperative and collaborative settings, and as a result I feel much more comfortable in refining the methodology I'd had in mind for putting together the permanent research teams for MATH 365.

While Day 1 of the course will (through the Markov Dance) introduce the students to elements of the subject of linear algebra itself, Day 2 will be devoted to the foundations of team work. The students' first assignment, from Day 1 to Day 2, will be to read the syllabus (and be prepared to be quizzed on it, both in pairs and individually) and to fill out the introductory questionnaire, which I've just now redesigned to allow me to construct teams more effectively.

Teams will be put together based primarily upon ease of convening, both in time and in space: I will make every attempt to build 3- and 4-person teams by grouping together folks who live close to one another and who are available to meet at similar times.

On the second day, these teams will meet to introduce themselves to one another (should they not know each other already) and to take part in a discussion of various elements of collaborating on long-term research projects.

Part of the homework for Day 3 will consist of reading over (as a team!) the research project descriptions. For Day 6, I will ask the each team to have written a proposal indicating that team's top three choices of research projects (ranked from first to third), the reasons for choosing those three projects, and the team's justification of their qualifications for working on those projects. (NSF, can you hear me now?)


Before I forget, much thanks goes to the MAA publication, Cooperative learning in undergraduate mathematics: issues that matter and strategies that work (ed. Elizabeth C. Rogers, et al.), for much of the information that's gone into the team design sketched above.

Quick notes...

We just wrapped up the final installment of the summer Learning Circle. How sad! I look forward to finding out how others' attempts at facilitating significant learning proceed this semester...perhaps as these folks' experiences take shape, they'll feel comfortable sharing those experiences (hint hint)...

Note to self: regarding Gertrude's closing comment on the inaccessibility of student development in certain of the categories of Fink's taxonomy, I feel the need to include a few items on the semester's opening "questionnaire" with the goal of assessing students' confidence in (a) speaking in front of others, (b) doing math in front of others, and (c) engaging in a long-term mathematics research project.

To be continued...and no, Gertrude's not her real name (see my first post)...

Wednesday, July 26, 2006

Random crap

I had perhaps a single contiguous hour to put toward work on Linear this morning, along with a few minutes scattered here and there amidst the hours that went into helping the Calc III folks study for their exam tomorrow I spent much of that time brainstorming in-class activities, and coordinating them with the stated learning goals to which they correspond. It's really starting to come together: on the back of a page from a desk-sized office calendar (October 2005, I think) I made a table linking each activity with its stated learning goal, together with suitable assessment/evaluation methods and resources needed to pull it off (thank you, Fink!). It'll make great office art.

I'm about halfway through my 14 stated learning goals, and I'm finding that many of the goals are to be addressed and assessed in similar fashions. For instance, I find myself writing "hands-on exercises" and "research journals" and "final projects" quite frequently...I'm hoping this is a symptom of a well-planned battery of feedback and assessment methods and not just a lack of creativity on my part. ("Gee, I dunno how I'm s'posed ta tell if they're learnin' that...garsh...")

I'm also finding that it's hard to design in-class activities without linking them to a specific topic. I have dozens of content- and subject-specific activities in mind, but I find it difficult to distill from them a general essence that would allow me to sort them into meaningful larger units: it's those overarching categories (corresponding to given learning goals, for instance) for which I'm having a hard time finding labels: "what activities will help develop develop analytical skills? I don't know what you'd call them, but they look like this in the context of eigenvectors: ..."

There have been a few activities that seem sufficiently flexible to be applicable in a number of different instances. For instance, expository dialogues might come in handy in helping students to ensure deep knowledge of foundational material: if they have to plan how to teach it to each other, chances are they'll understand it better. And "What if...?" exercises will be a great way to get students used to the idea of generalization: "what if we didn't assume the system is first order...?" This is the kind of probing question I'll ask folks to expound upon in journals as well.

I also spent a good amount of time looking for confidence-building activities, in order to address the stated learning goal of "achieving confidence in carrying out a large-scale research project in mathematics." (This goal falls under the Human Dimension component of Fink's taxonomy.) I found some promising exercises on several websites dealing with improvisational acting (not surprisingly), one of which reminded me of the fantastic parlor game, Werewolf. I also found some great large-group math games. I think that by taking part in math games which are not specifically linear algebraic, which do not even necessarily have a "mathy" feel to them, some students who are less sure of their math skills might be convinced to leave their apprehensions behind and feel more comfortable in doing math with and in front of others. (The Math Forum's game Toss and Sort seemed one of the most promising.)

Meanwhile, our Learning Circle on Fink meets for the last time tomorrow...I'm going to miss this group! Of course, I'm sure we'll all stay in touch on the topic (those of you who read this, leave a comment, let me know you're out there!) and will reconvene to share the thoughts and actions which have come of this particular meeting of the minds. The adventure's just beginning!

Tuesday, July 25, 2006

Vectors 'n' such

I've spent my idle time this evening putting on paper some ideas for learning activities...for the last few days I've been stalled at this point in the planning process, trying to come up with ideas for engaging classroom exercises. I've been spinning my wheels, I think because I've been looking at it the wrong way: I've been trying to create, ex nihilo, "brilliant" activities that will enrapture the students, and then trying to jam those squarish pedagogical pegs into round holes by struggling to find the foundational material those activities might embody.

Then, while I was lying on the couch in between chapters of my leisure reading (ah, summer!), I just thought for a few minutes about vectors: where do they come from? Where do we see them? What's their natural habitat look like?

My first thought was of the RGB vectors which describe colors in computer displays. They're vectors, after all. And it's not difficult to tell Mathematica to paint a unit cube full of colored pixels which vividly demonstrate the meaning of these vectory incarnations. RGB analysis lends colorful (ha ha) insight into linear combinations and orthogonality.

Then I thought of balanced chemical equations...every student's going to have some experience in that, right? And viewed in the right way, we can learn a lot about linear combinations, linear independence, subspaces, and so forth, just by pulling apart molecular formulas.

Once they've done a little work with these applications, I can set them off on an exploratory analysis of the usual representation of vectors in the plane.

We'll see if I'll have time tomorrow to refine these activities, and coordinate them with the aspects of "vector" with which I want to begin the semester.

Time ain't on my side

More than anything else, that's what's impressed me so far about this damned class: it's proving much more time consuming to put it together than I ever would have dreamed it would.

I've only just now started thinking about what I'm going to do with my Calc II sections in the coming fall, a sin mitigated only by the fact that I just got done with two sections of Calc II this past spring, so it feels fresh. Nevertheless, I know that I won't be able to put the same effort into "overhauling" Calc II as I am into giving Linear Algebra a makeover. That's not to say I'll teach it poorly: as many times as I've taught the course before, and as much success as I've had with it in the past, I've no doubt it'll go smoothly...but I can't help but wish that I'd have time to treat it as closely and as carefully as I'm treating Linear.

To get the ball rolling in Calc II I'm going to start off with an exercise much like the spring kick-off, estimating gumballs. Maybe even the same project?...

Friday, July 21, 2006

JPEG files and seismographs...

...One more project down, maybe three to go: this class'll give me the perfect chance to learn more about wavelets, a subject into which I've always wanted to look more deeply. I hadn't even thought about their connections to linear algebra (which are deep!) until I was browsing through some harmonic analysis texts this morning while looking for an interesting project for an independent study student.

This oughta be good...

Thursday, July 20, 2006

The more things change...

...well, the more things change.

I'm spending a good deal of time today scouring the literature for in-class examples and research project ideas for MATH 365. Just a few minutes ago I opened the text I'd used to teach the "Introductory Matrix Theory" course at the University of Illinois a few years back, and out fell a copy of the department's standardized syllabus for the course.

Perhaps the only things sadder on it than the remarks from on-high, "each section will be covered in about one class hour" and "an hour exam should be given at the conclusion of Chapters 2, 4, and 6" are the scrawled notes I'd left myself which indicate tentative dates for quizzes and exams. Nothing wrong with those notes per se, it's only that this is the kind of behavior that I used to associate with solid "course preparation": if I spent a day at the beginning of the semester matching boxes on the calendar with lines on the syllabus, I was ahead of the game!

I'm also a bit amused about how the textbook is called "Linear Algebra and its Applications." To its credit, it includes far more robust coverage of realistic and real-world applications than many math texts...but it's a bit disingenuous in that of the hundreds of applications cross-referenced on the inside cover, most of them get one-line treatment at the point in the text to which the reader is directed.

In search of linear models...

Monday, July 17, 2006

More meta, and then some

Well, I've got the first day's class clearly in mind now:

We begin with a very active "interpretive dance." The students will be divided into three groups of roughly equal size, and each group will be given a corner of the room to occupy. When the "music starts," a set percentage of the folks in group A gets herded off to group B, another set percentage shuffles off to join group C. The rest remain. Simultaneously, set percentages of groups B and C either stay put or head off to each of the other groups.

The percentages will be chosen so that one of the groups (Group A, say) ends up receiving far more than it grants.

Then, in computer science parlance, "wash, rinse, repeat." After a few more time steps have passed, the students should take note of how many people are standing in each corner: where'd everybody go? There's got to be something to the fact that most people wound up in Group A.

Try it again, this time starting out with everyone in Group C. What happens after several steps have been made?

At this point it'll be time to get the students to suggest means of making the exercise mathematically precise: what formulas accurately describe the game just played? What information can we extract from these formulas? What techniques might make manipulating these formulas easier? What realistic applications might involve computations like these?

This is a simple hands-on, and very concrete, introduction to Markov processes, a concept involving several of the key aspects of the foundational knowledge the students will be asked to store in their long-term memories: vectors and matrices (and their concomitant operations), linear transformations,, determinants, and eigenstructures. Of course, we won't be getting to all of these ideas immediately, but even on the first day we should be able to talk about the relationship between vectors, matrices, and the Markov game.

I've got some other learning activities in mind. This morning while walking in to campus I thought of an integrative exercise whose successful accomplishment demands that the students understand the interrelationships between the various key concepts of linear algebra: a knock-down, drag-out, no-holds-barred battle between vectors and matrices for the title of Most Important Linear Algebraic Concept. Somewhere near two-thirds of the way into the semester, one small group will take the side of vectors, a second that of matrices, and drawing upon all that they've learned about these concepts and how they relate to linear transformations, vector spaces, bases, and so forth, representatives of each side will do their best to persuade the remainder of the class that their respective point of view is superior. (The undecided hoi polloi will be free to ask probing questions of either party.) A solid defense of either position will require well-rounded knowledge of all related concepts.

I spent the last couple of hours typing up tentative versions of three of the semester-long projects. As planned, one deals with analysis of traffic flow, while a second, slightly modified version asks after applications of linear algebra to games such as Monopoly and backgammon. A third investigates the relevance of linear algebra to the structure of crystal lattices. While the first two draw heavily upon the theory of Markov processes as illustrated above, the linear transformations that arise most naturally in the third have to do with the symmetry of the crystal lattices considered.

Hot dog!

I've got to find four or five more suitable project topics...I'm thinking along the lines of some application to differential equations for one, and perhaps economics for another. Given the number of atmospheric science majors floating around, it would be nice if I can give some of them an introduction to linear models in atmospheric sciences...I'll see what kind of storm data I can find.

For now, it's time to head home.

To be continued...

Friday, July 14, 2006

Meta, meta, meta...

I've spent much of the last several days putting together a lot of the "metamaterial" for MATH 365: assessing what L. Dee Fink refers to as the "situational factors" affecting the course, laying out a comprehensive set of learning goals, brainstorming methods of feedback and assessment. That was today's chore, actually: I managed to describe a pretty robust set of feedback and assessment measures to be used on both individual and team activities during the coming semester. The students'll have to contend with

In-class quizzes
In-class exams
Course journals
Course blog postings
Research notes
Research meetings
Preliminary project reports
Final articles
"Symposium" presentations

The first four of these measures will assess individual learning, and the final five pertain to each of the teams of four or five students into which the class will be divided. Each measure, including the more "traditional" categories of quizzes and written exams, has been tailored to the needs of the course and the very non-traditional way in which it will be taught. I'm happy to provide further details concerning feedback and assessment methods, as I conceive them for this course, if you care enough to drop me a line!

My next goal is to come up with some good teaching and learning activities to address the stated learning goals. At this time I'm searching for the perfect "first day" activity, one that will capture the interest of the greatest possible number of students, and that will incorporate as many of the seven major aspects of the course's foundational knowledge as possible. I'm now envisioning students running around in really nasty traffic patterns...

Thursday, July 13, 2006


One the finest of Monty Python's skits is the "Albatross sketch." John Cleese, clad in a cute little lacey pink number, stands in front of a listless theatre-going audience, attempting to entice their appetites with a giant dead albatross piled uncomfortably in the box hanging around his neck. "Albatross! Albatross!" Michael Palin enters, stage-right, and politely asks for two choc-ices. "I 'aven't got any choc-ices, all I've got is this bleedin' 'uge sea bird!" responds Cleese. Palin pleads futilely for some other sort of treat, but cannot get around the fact that if all Cleese has got to sell is that damned bird, then that's all Palin's going to get.

Teaching college mathematics (or any other subject, I'm sure) can feel like this sometimes: constrained by some generally agreed-upon list of "must-know" facts, I've got an inventory of information I've got to unload on my students, and whether they like it or not, they're not going to get any substitutions. But not everyone cares about real vector spaces and complex eigenvalues for their own sakes. No one gets hot under the collar about mind-numbingly boring change-of-basis transformations and rank equations. And why should they? For crying out loud, I don't get excited about these sorts of things...that's why I hated linear algebra when I first had it! I didn't give a rat's rear end about it until I had to use it.

I've always found that the most difficult point to get across to college math students is that there really is meaning behind the mad methods of mathematics. Mathematics, from calculus on up, and from calculus on down, plays a pivotal role in thousands of different applications, many of which involve the students in their everyday lives, and which thus can be made instantly tangible to them. There's really no need to shout in their faces, "you oughta buy this, this means something!" (an only slightly more articulate rendering of "albatross") when there are more civilized means of impressing them with the relevance of the mathematical tools they pick up in any given course. It's not always easy to keep this in mind, the way many math courses have traditionally been taught: definition, theorem, proof, definition, theorem, proof, definition, theorem, proof...with an occasional example thrown in to break the monotony.

For the past few years I've toyed with the idea of teaching a course from a purely problem-based point of view: I'd hand the students some real-world posers when they first walk in the door, let 'em tinker, let 'em play, let 'em experiment. I'd let 'em suffer a bit, if the suffering is a useful one, but gradually direct them towards more sophisticated experimentation and understanding by slowly supplementing their bag o' mathematical tricks. Ideally, all questions of relevance will have been answered by the course's end: every student will have played a part in solving at least one substantial realistic problem by applying the techniques learned in the course.


But I've been afraid to take this pedagogical plunge, and why shouldn't I be? I've always gotten good reviews from my students when I teach the way I do, so I'm obviously not doing anything truly heinous. And gosh darn it, I'm comfortable in the way I teach: it's a method that's worked for me for years. I've simply never had the courage to try something radically new until now.

For the past few weeks, I've joined several of my colleagues at the University of North Carolina, Asheville in a Learning Circle focussed upon the book Creating Significant Learning Experiences, by L. Dee Fink. This man challenges the university instructor to take that last step off the high-dive by designing and implementing a college course which will provide students with a meaningful learning experience, one whose effects they will feel years after the course is completed. It's not sink or swim, though: besides laying out a coherent structure for successful course design, Fink gives a lot of hints as to how one might create such a meaningful course. The book is replete with helpful suggestions and illuminating case studies, and lively discussions with my colleagues have supplemented the reading with oodles of ideas, particularly when it comes to "UNCA-specific" issues. The past month that I've spent with my colleagues has given me the little bit of courage (and a good deal of the practical know-how) I was wanting in order to make my "dream course" a reality.

Beginning in mid-August, I'll be leading thirty-odd students (or, just as likely, "thirty odd students") through a problem-based study of introductory linear algebra. From the outset they will approach the topic from a practical point of view, learning linear algebra by working through realistic applications to chemistry, network design, crystallography, cryptography, thermodynamics, economics, aeronautical engineering, traffic flow...whew! Working in small teams, they'll wrestle with unprettified problems and learn advanced mathematical techniques by playing with them and applying them to the questions that arise in their research. They'll take notes on their research activities and keep track of the progress both of their teams and of themselves as individual thinkers. They'll write preliminary reports and academic articles. They'll present their findings to each other in class, and to the department as a whole at an end-of-semester "symposium."

And, I hope, they'll have fun. Why shouldn't math be fun?

From now until the end of the course, I'll be posting all sorts of muck on this site: thoughts, meditations, ideas, course material...I'll likely often use this blog as a laundry list for myself, so I warn the reader in advance that some of what I write will probably either be boring as hell or intelligible only to me. I'll do my best to keep it clean and coherent, though, and in the spirit of fair play, unless otherwise instructed to do so I will betray no one's secret identity save my own: the names have been changed to protect the innocent. (Oh, except Fink's...that one's real...)

I invite all readers to respond to my ramblings, and to offer me their own thoughts, comments, and suggestions. The course (as yet in the planning stages) is now and, until it's over, will be, a work in progress, and I will be delighted to hear from my students, colleagues, and assorted other interested parties, as that progress is made. I'm excited to find out what we can make of the course!

I'll conclude this first post by giving hearty thanks to my colleagues in the UNCA Learning Circle, and in particular to its esteemed facilitator: you know who you are! Without you all, I'm certain I would not have found the derring-do to carry this out.