Off to a better start

I've given tests in three classes, and in all of them the grades were significantly higher than the first test was last year. Why, I wonder?

Perhaps it's because I already prepared the notes for these three classes last semester. This semester I spend more time reviewing and thinking about the topic, rather than learning it. (I spent a lot of time last semeter relearning statistics.) I probably spend more time thinking about the presentation this semester than I did last semester, even though I'm spending less time in actual preparation.

I've also begun "filtering" the topics. Lower-level college mathematics textbooks have a bizarre obsession with throwing together all sorts of unrelated material into a chapter, presenting it out of logical order, then repeating it later. This is so stupid that it stinks of editing by non-mathematicians. I can't believe that actual professors of mathematics could be responsible for such mishmashes of illogical, non-sequential textbooks. Quite honestly, most of them would have more value as toilet paper than as resources for a student's learning, and guides for a professor's curriculum. If publishers would have some mathematicians sit down and think about the best way to arrange the material, they would do students and professors a huge amount of good.

Then again, maybe that's what they do. "Death by committee." Hmm.

In any case, I decided this semester to throw out about half the material in our textbook's first chapter. I also abstained from most homework problems in the textbook, and told my students to do certain drills that I wrote in the Java programming language. I required them to do a certain number of problems in each drill; they could do as many as they liked, or restart the drill any time, and submit their highest score to me. I didn't care, as long as they gave me a score. The one major exception was in setting up word problems. I can't do that very well with a computer program, so I focused instead on a very small group of simple word problems that students can visualize easily.

Last week, I gave them essentially the same test that I gave last semester — only a little harder. Yet the mean and the median were both higher than last semester.

This is progress! :-) What is the cause? I don't know.

I would like to think that the drills helped, since they provide students with instant feedback and encourage them to correct their mistakes to earn a higher score. One simply cannot accomplish this with problems from the textbook: if students do the problem on paper and submit it to me, it will take at least two days before they receive feedback, and there will be no opportunity to earn a higher score by submitting corrections. I have a schedule to keep, after all.

These students really are the weakest of the weak in college mathematics; they need instant feedback, and appear to benefit from it. The online drills also gives students some breathing room to learn "at their own pace"; this is a big buzzword in education, and I've heard that it works very well on students who don't do well in the traditional, structured curriculum.

Indeed, the tests had many, many fewer instances of the technical mistakes that students typically exhibit, and more students reflected an understanding of why things work. For example, most of last semester's students were at a total loss to explain why adding two negative numbers gives a negative number. Most of this semester's students were able to explain it in terms of the number line, which I had explained, and which one of the online drills reinforces.

This suggests (to me) that their minds were freer to remember why a "rule" is true, rather than bog down in their confusion over what the "rule" is. (I don't like the word, "rule". I'll discuss that later.)

It is almost certainly better to focus on a few things deeply, spending sufficient time on the logic and reasoning, than to focus on many things shallowly, presenting them as facts without explanation. However, the filtering out of useless, or at least irrelevant, information may also have contributed to the focus on a few things. The Java drills can't take all the credit.

I didn't filter out as much as I could have. Why exactly do we teach division of monomials, and simplification of negative exponents in fractions? I can't at the moment think of any place where this will be useful to them in Intermediate Algebra, College Algebra, Trigonometry, or Statistics. The only people I can think of who need the simplification of negative exponents in fractions are Calculus students, and I can guarantee you that not one of the students in an Intermediate Algebra course will ever dream of taking Calculus. Remember that the man writing this is unapologetically fanatical about teaching of painful but necessary mechanics such as factoring, and I am openly disdainful of so-called reformers who use the word "reform" as a cover for watering down the curriculum to remove essentials that they don't like (often because they don't understand it themselves).

Another factor could simply be that this is the spring semester. I've noticed over the years that students in the spring semester are more serious about their studies than students in the fall semester. They have survived at least one semester of college; they have heard rumors of, or even known firsthand, students who leave campus because of a poor GPA. They are catching on to the fact that learning is hard work, and failure has serious consequences.

That doesn't mean learning has to be painful. I believe that learning is fun. But that doesn't mean it's a game, or a TV show, or a rock album. Learning requires hard work — especially learning mathematics.