Spring 2009 Reflections

I see that I didn't make any comments about Fall 2008. The only thing I would have to say there is that I decided that I didn't really like the Math 093 textbook.

Math 093 - I'm probably about 30-40% done with a textbook/workbook for this class. My primary concern is that the content is not well organized in the current textbook, and that it does not emphasize the points that will help students succeed in Math 097. One of the frustrating things about this textbook is that I feel like I'm giving the students something other than what they really need, and consequently propagating a systemic problem with mathematics education out here. I hope to have this textbook completed in the next four weeks so that I can get it to print with some help from the bookstore.

Math 330 - Linear algebra is a notoriously difficult class because it's very easy to mess up in one of two extremes: Too much computation without enough theory or too much theory without enough computation. I don't know how well I did on this balance. The students came to the class underprepared as a majority of the students are ARL students who don't necessarily have a good mathematics background. In particular, the students really lacked several basic notions of proof and had some problems with basic manipulations at the beginning of class that I wasn't expecting to see.

Math 466 - Numerical analysis is a class that I never took as a student, so I was sort of making it up as I went. The book was pretty good, and I put the emphasis more on the computational methods than the theory. The problem (again) is that the students' background is not as strong as it could be. Their calculus and general proof-writing backgrounds were limited, so the emphasis on methods made sense to me. I found the material to be somewhat interesting, though the derivations were somewhat tedious.

"Opposites" of numbers

I'm taking a little break from writing lecture notes to complain about things that textbooks do that are really stupid. I'm going through the Math 097 textbook, and I have reached the third or fourth time where I just shake my head at the textbook and hope that somehow either the textbook or the author will feel my frustration and it will be changed in the next edition. I suppose I could write a letter to the publisher or the authors, but I'm not going to do that now.

Why does it matter? I have an expectation that the students will read the textbook. As a consequence of this expectation, I am reading the textbook as well to make sure what they read actually makes sense.

As an aside, in what classes do students pay $100+ for texts that they have no expectation of reading? As far as I know, it's only the math classes.

Anyway, something that really bothered me was the discussion of negative numbers. They talk about 1 and -1 being "opposites". This random terminology that math people would never use reminds me of some terminology I ran into in high school with the CPM program. Apparently, somewhere along the line someone felt that a "common denominator" should be called a "fraction buster". If you don't believe me, just Google "fraction buster." Anyway, I suppose the idea is that when solving an equation involving fractions , you can "bust" the fraction by multiplying by the right number... or something like that. Besides the absurdity of nonstandard terminology, it also means that when you add fractions with different denominators that you would have to introduce the "common denominator" as a separate concept, even though it's the exact same thing.

But back to "opposites". The standard terminology for this is "additive inverse" for the simple reason that the relationship between them is additive, namely that if you add them together and you get zero (4 + (-4) = 0), and zero is the special number that doesn't change the result if you add it to something else (10 + 0 = 10). Then there's a link between this idea and the "multiplicative inverse" of a number. The relationship between a number and it's multiplicative inverse is that when you multiply them together you get one (4 * (1/4) = 1), and one is the special number that doesn't change the result if you multiply it by something else (10 * 1 = 10).

It just got worse as I pushed onward into the following section. Here is a quote from the text:

A symbol like -x, which has a variable, should be read "the opposite of x" or "the additive inverse of x" and not negative x," since to do so suggests that -x represents a negative number.

Furthermore,

As you read mathematics, it is important to verbalize correctly the words and symbols to yourself. Consistently reading the expression -x as "the opposite of x" is a good step in this direction.

If you talk to anyone who does math (this includes chemists, engineers, physicists, and many others), and pointed to -x and told them to read it, they would all say "negative x." So while it would make sense to a student brought up in this way, to the rest of the world, it's nonsense.

But it gets worse. Consider the following sentence "Multiply x by the opposite of x." If given that sentence outside of the context of this subject, I would consider "opposite" to be related to "multiplication" and end up with x * (1/x) instead of x * (-x). The book will likely introduce a different word, "reciprocal" for 1/x. Unfortunately, all this does is introduce more words for students to memorize, and those words are not particularly informative to the nature of the relationship. Why is "opposite" additive and "reciprocal" multiplicative? It just is. Memorize it. (At least "reciprocal" is a common terminology to people who use math...)

I think that's enough of a rant. I need to get back to writing up lecture notes. I won't get into the "Rules for Addition of Real Numbers" table right now...