JMM 2008

The Joint Math Meetings for 2008 (San Diego) just finished, and so it's time to start sorting through my notes to figure out what they mean.

Class related:

• Minute Paper - I've seen this used and discussed many times, but maybe I'll actually give it a try. The basic idea is to give the student one minute at the end of class to write down what he thinks were the main points for the day. This forces the student to reflect on the day's work before it gets lost.
• Algebra for Dummies - This book exists (as well as others that present themselves in the same way), but the question is what they try to do to make the math more understandable (and whether it works). Part of me thinks that anyone motivated enough to buy such a book will be the type of person willing to put in the work to learn, whereas not all of the students in my classes will be like that. I tend to believe that the personal motivation plays a huge role in education. However, it might be worth my time to look at that book to see what it says.
• Math Labs - It would be nice if I could get students to do self-directed labs (like science labs) by giving them a handout with some instructions to follow and some mathematical things to compute. The problem for pulling this off right now is that I have no idea what topics would be good to pursue in this way. One example for a higher level class (like number theory) is the "McNugget Problem" (boxes of 6, 9, and 20 -- for what n can you get exactly n nuggets?)
• Technology - I've always kept my distance from using technology in the classroom because the students won't be able to use them on tests or anything. However, I can see some time-related shortcuts with graphing where it would be nice to be able to generate several graphs quickly and have students make observations to get the ideas behind the graphs before we actually go through the details. I need to find an internet resource that will allow this (Sage?)
• Pretests - I need to give my Algebra students a pretest so that they can get a better measure of "progress" throughout the quarter.
• Handouts - I haven't made much use of handouts, but maybe I should go back to that. I made these my first year or two as a TA and the response was strongly positive.
• Hiding grades - I went to talk where someone did a study on student improvement when you didn't tell them their grades, but only made comments. I don't know if this works at the level of developmental algebra because the students may not be mature enough (mathematically speaking) to make sense of it.
• Grading - I had an interesting thought about how I can grade my students. I still don't really like the idea of percentage grading. So perhaps I can make competence grading by passing a series of "Levels." For example, a level 1 arithmetic computation would be something like 45 + 24. Then a level 2 arithmetic computation would be something like 6 - 4 * 5^2 (introducing PEMDAS). Then a level 3 arithmetic computation would be something like 5 * 2^2 / 4 + 3 * 8 (a complicated string of PEMDAS where the only way they would get it right is if they knew how to completely break it down). There would be similar levels for other ideas, such as solving linear equations, graphing, and so forth. Then their final grade would be a measure of how many topics for which they were able to show a high enough level of competence. This still needs to be worked out in greater detail.
• Colors - Maybe I can use black/red for positive negative numbers at the beginning of class to highlight the difference between the minus sign as a binary operation (5 - 2) and as a unitary(?) operation (-2)
• Spoken/written mathematics - This wasn't from the conference, but it was something I've been thinking about. I think I need to make my students write "five minus two" and "negative two" because their words and their written math often say different things (and sometimes the ideas in their heads are different from both of those!).
  

Other ideas:

• Placement Exams - I want to look over the placement exams to see what they are testing and how they are graded.
• NSHS Math Students - I don't know who the bright high school math students are, but we should probably be actively looking for them and encouraging them in some way. We could try to get students to take the AHSME or something like that.
• Other math students in the area - Can we make a presentation to math clubs and that sort of thing at other schools? Will this be a helpful endeavor for advertising? (It would help if we had a math major to offer them!)
• Putnam - There was an interesting-looking book titled "Putnam and Beyond" that I might want to buy
• Minicourse - I went to a minicourse on Departmental Self Reviews. At this point, I'm going to treat it as background information for me to have as I start looking forward into where the department is going. But there is one thing that I thought would be helpful, which is to get a list of who taught which classes for the past few years to see what the teaching distribution has been and learn more about our part time instructors.
• Articles - "The Way We Think" (Fanconnier? and ??), "Where Mathematics Comes From" (Lakoff and Nunez)
  

Miscellaneous:

• Webpage - I need to update my CV and webpage
• LaTex - "More Math to LaTeX" looked like a good reference. Also, I want to see if I can learn how to make hyperlinks in LaTeX.
• Mathematics of Poker Class - Given the current budget situation, this probably won't happen. But it would be fun if it did.

I don't understand this

I just finished grading my Math 097 exams this afternoon. There were two parts to the exam. The first part was a 125 minute in-class exam with one page of notes. This is the standard sort of in class final that you would expect from a math class. The second part was a two day take-home final, where students could work together in groups, talk to tutors, and do whatever it takes for them to figure out what's going on (except asking me for help). It's also exactly the same test as the in-class exam.

The average for the in-class: 42.5
The average for the take-home: 55.5

13 point improvement... That's not too bad. However, the exam was out of 94 points. That means that even if the students were allowed almost unlimited resources, they were still only able to get about 60% credit. I find this very disappointing. I also don't understand why the students couldn't pull off at least a solid finish. This is even more confusing to me because the take-home part was worth a full 13% of their final grade!

Sadly, this result on the final exam means that over half the class failed. There were a couple of bright spots. I had a couple students who worked very hard and came away with a well-deserved B.

Coming to the end of the semester

Even though there's another week left in the semester, I've reached the point where I have no more teaching to do. At least, no more planned teaching of new material. I have one more class this evening to teach, but we're just doing review. I've got a couple hours right now with no specific plans, so I'm going to spend the time rambling away with my thoughts on how this semester has gone in order to prepare myself for next semester. I'm pretty sure some of this stuff I have mentioned before, but it's worth reconsidering again.

General things

I'm not going to be using WebCampus for my homework postings next year. Doing it this year was too much of a hassle without enough benefits to make it worth my time and energy. I think by hosting it on a normal webpage will make my life much easier. (Hopefully, I can figure out how to get proper FTP access from home to make this part even more flexible and accessible.) Along the same lines, I need to get better posting quiz solutions and that sort of thing.

I let myself get lazy with my bookkeeping and grading, which was a little bit frustrating just because I don't like to have stuff like that hanging over my head. I also found myself about halfway through the semester not having my quizzes ready until just before class. I think if I get myself a little more organized (the point above with the homework posting frustrations comes into play here), I should be able to make that happen a little more regularly (and efficiently).

I made a homework cover sheet that I think I will have my students use. The point of it is to increase the impetus on my students to be organized. It has questions on the front that I want students to answer, which will force them to think a little bit about what they did with their homework instead of just turning it in. I also gave them a space to ask questions. I didn't leave myself any room to make comments in return, so maybe I'll change that.

I want my students to check their answers in the back of the book. I don't feel that grading their homework is actually something that is particularly benficial for them. I think they need to learn how to use the tools they have to assess themselves. Of couse, I need to explain this process to them on the first day of class.

Math 097

I feel that I need to change my approach to this class completely. I almost want to break it into two different sections: One hour lecture and one hour lab. That might even need to be broken down into two half hour lectures and two half hour labs. I want to keep the daily quizzes, though I may change them into weekly quizzes. There's something about giving them a problem to do and forcing them to show what they know that has seemed to help things stick in the minds of some of my students.

I think the first 4 weeks of the class are the most critical as the foundational algebra must be set up by then for any of the rest of the stuff to have a real chance of making sense. But this doesn't just mean the mechanics of algebra, but a sense that the algebra is actually connected to real life (as a way of representing real values). I've got some ideas that I'm working out right now, but nothing solid yet. It has to do with starting the class using just numbers and then introducing variables *after* they have a sense of seeing patterns and *after* they have a notion of what it means to generalize a pattern.

I am also going to be less ambitious about the number of topics I cover in that class. I need to think a little more carefully about it, but I've got a preliminary schedule that feels like it's only about 2/3 the speed as I had this semester. I would like to follow the "teach less, teach better" mentality this time around and see if I can cut back to the core essentials.

There are things I want to do, but I don't know of any textbooks out there that do it. Maybe I'll write my own someday...

Math 124

There is a move right now for us to get rid of this class. The entire math department wants to do it. We've got a proposal in the works to make it happen.

The problem is that the course has turned into a terminal course, meaning that most students who take this will have this as their last math experience. Unfortunately, the content of the class doesn't really climax very well as a mathematical exeperience, and just leaves students with the feeling that math is a bunch of nonsense that doesn't go anywhere or have any real application. That's all very disappointing because it continues to perpetuate the sense of how hard and weird math is.

As far as how I might change how I teach this class, I'm really not sure. The difficulty is that this class has less time than the other one, so I can't break it up into lots of little pieces and have time for things. However, I still like the idea of breaking it into a lecture/lab combo, but I need to spend time working out how much information I can put in front of them in a 20 minute span and have them make sense of it.

Math 283

This was my fun class all semester long. I hope my number theory class next semester goes just as well if not better than this one. I don't really have too much to say about this one just because it was the one that I could relax the most in (even though it was the most difficult content) and everything that happened in that class just seemed to be very enjoyable.

"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...

Daily Quizzes

As I've been thinking about my Math 097 class in the Fall, I seem to have broken the class into two components:

1. The development of computational proficiency
2. The understanding of mathematical reasoning

On the first point, I believe it is a reasonable task to have the students complete a 5-10 minute daily quiz to emphasize the importance of being able to compute things correctly in a reasonable amount of time. My immediate guess is that giving a student 7-10 seconds to complete a one or two step algebra problem is a perfectly reasonable amount of time to give them. This means that a 5 minute test will be at least 30 problems long. Does that seem reasonable? I think it does.

I would give this quiz at the end of the class so that they can leave when they finish and not have to sit around and wait. Also, doing it at the end of class instead of before a break means that students who compute more slowly do not have their break time penalized.

I can also see how this can also be an instructive tool. For example:

• 85 + 74 = ???
• (80 + 5) + (70 + 4) = ???
• (80 + 70) + (5 + 4) = ???

It should not be hard to imagine doing this for the distributive property of multiplication over addition and for common errors involving fractions.

By giving problems that are suggestively sequential, I can introduce various aspects of arithmetic that will become relevant to their future algebraic manipulations. It could also be used as a starting point for a discussion for the next class period. I'll definitely have to take a closer look at the structure of the textbook to see how effectively such a scheme could be woven into the material.