Casting Out Nines

I was looking at the web sites of a few colleges the other day which use a “Great Books” curriculum. This is an approach to a core curriculum in which students work their way through a listing of the great books from the past, across a variety of disciplines. Here’s an example from Thomas Aquinas College, a highly-regarded Catholic liberal arts college in Santa Paula, California. St. John’s College is probably the best-known example; I remember getting a mailer from them when I was a senior in high school, and I was fascinated by the idea of attending a Great Books university at the time.  There are also a few public universities which offer a great books curriculum as an option within the larger curricular structure of the university, for example as part of an honors program. 

Apparently Mortimer Adler is credited with coining the concept of the Great Books, and he gives three criteria for a book to be a Great Book (taken from the Wikipedia article): 

• the book has contemporary significance; that is, it has relevance to the problems and issues of our times;
• the book is inexhaustible; it can be read again and again with benefit;
• the book is relevant to a large number of the great ideas and great issues that have occupied the minds of thinking individuals for the last 25 centuries.

Via Vlorbik, here’s a letter to the editor (PDF) of the AMS Notices by Seymour Lipschutz extolling the virtues of Schaum’s Outlines as course texts and giving some suggestions for those choosing textbooks.

I agree with Lipschutz’ feelings about Schaum’s Outlines, up to a point. I’m a big fan of Schaum’s Outlines; they cost less than $20 and are loaded with precise, succint summaries of course material and worked-out problems. I
survived college physics and advanced calculus largely because of my now-battered Schaum’s Outlines for those subjects. I ordered the latest edition of the differential equations Outlines as I was considering using it for my DE course next semester, and I liked what I saw very much; and the publisher sent me a gratis copy of the beginning calculus Outlines and it was very good as well. I will be suggesting these outlines strongly to the students in those courses.

But to use them as the textbook for a course? I’m a little skeptical.  They are, after all, outlines. I think that students in the lower-level courses like calculus, and to some extent mid-level courses like DE’s or linear algebra, would benefit from having a more fully-featured textbook.

On the other hand, a carefully-written set of course notes made up by the professor, augmented by Schaum’s Outlines and hand-picked resources from the web, make up a pretty good blueprint for a cheap, portable, and effective package of course materials that I think students would get a lot more out of than a single monolithic textbook that they can’t carry around easily and never read.

I got a nice surprise in the mail this morning — a review copy of the fourth edition of Marvin Greenberg’s classic text Euclidean and Non-Euclidean Geometries. It seems like this book has been in the third edition since time immemorial. I used the third edition in my first year of teaching after graduate school, 10 years ago, and loved the depth and clarity of the writing. That much seems not to have changed. There are some significant rearrangements and updates to the material, and overall the book just looks a lot nicer (And the color scheme matches my blog, to boot!) There don’t seem to be a lot of good intro-level geometry texts out there — and there are a lot of bad ones — so a new Greenberg is a nice early Christmas present. It’s the kind of book that makes you want to sit down and work through it just so you can learn geometry from back to front.

Freeman textbooks are on a roll these days, what with this new edition of Greenberg and with Rogawski’s excellent new calculus text. (Disclosure: I was a reviewer for Rogawski.)  I don’t advocate for textbook use often, but if you have to use one, use a good one!

I’m teaching differential equations next semester, and I’m changing the course in some fundamental ways since the last time I taught it — so much so that I needed a new book for the course. (I’ve ruled out the textbook-free option for this class for reasons I explained here.) After some searching, I ended up going with the Boyce/DiPrima text. But I gained a lot of respect, and found a lot of affection, for Tenenbaum and Pollard’s classic text on the subject from 1963.

First of all, the textbook is a giant brick of a book, loaded with great exposition, clear examples, and challenging problems. And being a Dover paperback, it’s only a measley $16.47 through Amazon. But the thing I love about it, which is something I love about all math and science books from this era, is its tone — clear, precise, tough-minded, and no-nonsense. And yet inviting and enjoyable at the same time. (Which precisely describes what I’d like the students in the course to be.)

A great example is the following quote that appears at the end of a solid review of functions and just before they start looking at differential equations proper. Note that there is no intervening review of calculus between these two sections. That’s because the authors expect students to actually know calculus upon entering the course. They say:

In the calculus course, you learned how to differentiate elementary functions and how to integrate the resulting derivatives. If you have forgotten how, it would be an excellent idea at this point to open your calculus book and review this material. [emph. added]

I actually cheered when I read that. Differential equations is an extension of calculus; calculus is a prerequisite; you had calculus once; so if you forgot how to do it, get off your duff and crack a freaking book. End of story.

It’s the exact opposite of most modern math textbooks that start by assuming that the reader is five years younger and 30 IQ points dumber than s/he really is, and scared witless of math and unable to read past a 5th-grade level on top of that, and which proceeds to hand-hold and touchy-feel its way through whatever subject it is supposedly about. But not so with this book, nor with any other post-Sputnik era math and science books I’ve seen. That softening up seems to have occurred sometime around 1980.

Ironically, this 1963 text is superbly written with great clarity, vivid illustrations to motivate the material, and plenty of useful examples. When books started softening up — supposedly in an attempt to help struggling students — the things that actually help those students such as clarity and completeness actually went away.

I ended up going with the more modern book because I needed more in the way of computer applications (not many of those were around in 1963). But I will be using this Tenenbaum and Pollard text quite a bit, for my own enjoyment if nothing else, and perhaps as a lesson in how to write mathematics clearly.

We’ve got just 4-5 weeks left in the semester and until the textbook-free Modern Algebra course will draw to a close. It’s been a very interesting semester doing the course this way, with no textbook and a primarily student-driven class structure. In many ways it’s been your basic Moore Method math course, but with some minor alterations and usage of technology that Prof. Moore probably never envisioned.

As I mentioned in this lengthy post on the design of the course, students are doing a lot of the work in our class meetings. We have course notes, and students work to complete “course note tasks” outside of class and then present them in class for dissection and discussion. The tasks are either answering questions posed in the notes (2 points), working out exercises which can be either short proofs or illustrative computations (4 points), or proving theorems (8 points). We have a system for choosing who presents what at the board — I won’t get into the details here, but I can do so if somebody asks for it in the comments.

So the class meetings consist almost entirely of students presenting work at the board, where their responsibility is to make their work clear, correct, complete, and coherent — and ruggedized against the questions that I inevitably throw at them.

I was thinking yesterday that this method of doing class has really done a lot of good for the students in the class, in several key ways.

• Students ultimately rely upon the soundness of their own work. The students can work with others or with print or electronic resources — although with no textbook, they have to learn how to find those resources and tell the good ones from the bad ones, which is a great skill by itself. But it boils down to presenting that work, on your own and with nobody there to bail you out, in front of your professor and peers. I think this is a good antidote to the occasional over-reliance on cooperative learning that we (in education as a whole, and in my department) have. Group work is all well and good, but to be a complete learner you have to be able to rely on your wits and your skills and not just prop yourself up on the strength of peers.
• Students prepare for class in advance, several days in advance, every night. To do reasonably well on course note tasks, students need to plan on successfully completing 15-20 course note tasks throughout the semester, which comes out to about 1-2 per week. Combine that with the fact there are 8 students in the class all trying to do this, and it’s easy to see that working ahead is really essential. You want to get so far out in front of the class that you have no competition for a particular range of problems. Very often in college, there is no sense that you have to get ready for class the next day — unless there’s an assignment due — and we profs reinforce this by running classes that do not penalize the lack of preparation. (It’s not enough to reward the presence of preparation.) The course design here, though, rewards the students who have read and practiced ahead and learned on their own.
• Students become skeptical and tough-minded about their own work. It’s quite common in traditional math courses for students completing an assignment to simply barf up something on a piece of paper, hand it in, and see how many points it gets. When you are presenting work before a class, that route leads only to embarrassment. When most of the class time is spent doing these presentations, students learn something I didn’t learn until graduate school — that if you are going to hand something in or present something with your name attached to it, you had better make very sure that it works. I’ve noticed the students anticipating not only the fact that I will be asking them penetrating questions about what they are presenting, but also what those questions are. At that point they are learning to think like mathematicians.
• Students pay (more) attention to detail, especially terminology and the sensibility of a proof. It’s easy to write a proof or a solution to a problem that has no coherence or sense to it at all — but that incoherence and senselessness vanishes the moment you do something as simple as reading the solution aloud. Which is what these folks are doing every day. Example: A colleague told me a story of a student who was asked whether or not two groups G and G’ were isomorphic. The student answered, “G is isomorphic, but G’ isn’t.”
• Students base their confidence on the math itself, not on an external authority. Students aren’t allowed to ask me “Is this right?” or “Am I on the right track?” To clarify, they can ask me those questions, but I will only greet them with more questions — mainly, “What justifies this step?” or “How do you know this?” It’s not about me or what I like or what makes me happy with regards to their work — it’s about whether each step of the proof follows logically from the one before it, and whether that logical connection is clearly validated. Students know pretty well now when they have got something right and when they don’t, and if they don’t have it right they have a better sense of what’s missing or incorrect and what they need to do to fix it.

A lot of these effects I’m describing are just embodiments of what it takes to be successful in math after calculus in the first place.

Editorial: This is article #8 in this weeklong series of reposts of “classic” articles here at CO9s. The article I’m posting below probably has the most references to it of any article I’ve written. It’s the culmination of a bunch of prior posts about the nature of college textbooks, and it kicked off a pretty major experiment of my own that is currently underway — the design and execution of an abstract algebra course that does not use a textbook. The story of the textbook-free algebra course is still unfolding, and there’s a lot of good coming out of my little experiment.

We hear a lot about “innovation” in education, almost as if it were an end in itself. But I like to think about and write about ways of doing college differently that actually make students’ college education better.

Escaping textbooks

Originally posted: March 28, 2007

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I’ve blogged before about my ambivalence towards textbooks, at least in mathematics (here, here, here, here, and here). But a couple of recent events have really motivated me to think seriously about not using textbooks at all in my courses. And this fall I will be taking the plunge, requiring no textbooks except for my precalculus class (which has to have a book because the course has to be somewhat standard across five different sections). (more…)

It’s been a while since I last said anything about the textbook-free Modern Algebra class experiment. This is mainly because the class itself is now underway, five weeks into the semester, and it’s only now that I’ve got enough perspective to give a reasonable first look at how it’s going. So, let me give an update. (Click to get the whole, somewhat lengthy article.) (more…)