24 April, 2008

Livening things up until they die

To enliven a topic, I sometimes throw in a colorful comment. "The book lies!" is an example from today's lecture. (It does, but never mind the details. Should you insist, I went on to explain that, "Not all rings have a multiplicative identity." See? I told you not to mind the details.) I also like to throw in historical stories that should be true, even if they aren't. To the best of my knowledge, they are; they come from books on the history of mathematics. (Consider the unfortunate story of what befell the fellow who first discovered that the square root of 2 was irrational: "They sent him on a one-way cruise to the bottom of the Mediterranean Sea." That was from Tuesday's lecture. Amazingly, students who had taken History of Math didn't know this story, or claimed not to. Scandalous.)

This never bothered me much in the past, but lately I've been making comments that, upon later reflection, I regret. Today, for example, I happened to mention a student from my days teaching high school. He had a habit of waiting until my back was turned, then throwing spitballs at posters I had hung in the room, thereby ruining them. The story goes further than that, but you can guess that it had no substantive connection with the ideas that we were discussing.

When does the attempt to liven things up a little become more of a hindrance than a help? It probably occurs more quickly than we realize. Trying to make a topic lively can have the effect of killing it instead. It can sap precious time from the class, and if one is careless or conceited it can lead to controversial or even indefensible statements. (I have been called both careless and conceited at various times in my life, so this is not a trivial concern.)

On the other hand, a dead class can be worse, especially if it results in students who are not motivated to learn the material.

I've been teaching for 15 years now. I should know where this line lies.


Brandon said...

Are you really using a book that says that all rings are commutative? Even I knew that was false, and my mathematical education is patchwork and piecemeal.

I have problems with finding the proper line as well; I don't think I've ever managed to find it.

jack perry said...

I'm glad I delayed replying until I had the chance to look at the book again. Apparently my memory sabotaged me. The book does not claim that rings are always commutative. On the other hand, I did in fact say that there was a lie in the book, and I was telling the truth when I said it.

The actual "lie" is that a ring always has unity; that is, "there exists an element 1 ∈ R such that 1⋅x⋅1 = x". (Concrete Abstract Algebra, Niels Lauritzen, Cambridge University Press, 2006 reprinting, pg. 112, part (ii) of the definition)

I was writing the definition of a ring on the board, and a student who was looking in the book at the time said, "The book also says there is an element 1..." at which point I said the book lies. Later on the page it says "R is commutative if xy = yx for every x, y ∈ R." I don't know how I confused the two, except that it was late in the evening when I wrote this entry!

I was being facetious when I called it a lie. It's acceptable to define a ring this way if one wants, de gustibus non disputandum.

Rotman, for example, defines a commutative ring as having 1 (A First Course in Abstract Algebra with Applications, Pearson Prentice Hall, Third Edition 2006, pg. 219); Anderson and Feil do not (A First Course in Abstract Algebra: Rings, Groups, and Fields, Chapman and Hall/CRC, pgs. 74—5). Rotman acknowledges in a footnote that some people define rings without unity, but Lauritzen does not. Jacobson also defines a ring with 1 and without any comment on other definitions (Basic Algebra I, W. H. Freemand and Co. 1985, pg. 86) while Hungerford does not (Algebra, Springer-Verlag 1974, pg. 115). Artin includes 1, but also acknowledges the possibility of another definition (Algebra, Prentice Hall 1991, pg. 346).

There are also notational differences between books; some use e for the identity of a group and some use 1, which makes things a little confusing if you decide to talk about the integers as a group (since then the identity is 0, not 1).

Do things get like this in philosophy?

I'll change the post to reflect the actual "lie".

jack perry said...

By the way—if anyone reads this, I don't want them to come away with the wrong impression. Concrete Abstract Algebra is my favorite of all the texts listed here, although Rotman's also looks nice.

Brandon said...

Interesting; looking at Neal McCoy's Rings and Ideals (1948) on my bookshelf, I find that he doesn't define rings with unity and uses, if my quick look isn't mistaken, the e notation.

Believe me, it is much, much worse in philosophy.