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.

Math 093 Boot Camp

I have made it my goal to put together a functional textbook for Math 093 and have it ready for the Fall 2009 semester. I'm quite disappointed with our current book. It's poorly organized and not well presented. It also does not have nearly as many problems as there should be. I ran a draft outline past Russel today, and he likes what I've put together. Now I need to think through the layout and come to a decision about how I'm going to teach presentation through examples.

The plan is to start by developing a workbook. I'm not going to spend a lot of time working on writing up explanations right now because that's not as important as laying out the problems in the right order, having the right number of them, and other such issues.

If I get far enough in the next month, I might even be able to pilot some of the worksheets in my Math 093 classes next semester. The first two chapters (arithmetic and fractions) are likely going to be the most difficult ones to put together.

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.

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.

A chance to refelct on the first half of my first semester

I'm taking some time right now to think about the first 8 weeks of the semester. I'm doing this now, right before the big onslaught of graind that is to come next week. There are certainly a number of things I want to do differently next semester.

#1 - Homework: I'm going to try something new next semester. I'm going to make part of the homework for my algebra classes simply copying the examples out of the book. This is first to force them to read the examples and work through them. Secondly, I want to give them the opportunity to circle parts of the examples that doesn't make sense to them that they would like clearified. I also want to force them to compare the examples to their own work that follows. Hopefully, this will help them get into a good routine of using the book effectively instead of just a source of homework problems. Thirdly, I think giving students a full week to do homework is simply an excuse for them to put it off. Only a couple students ever start the homework ahead of time like I ask, and so I need to force the issue a little with them. Finally, I need to assign problems from previous sections with much more regularity than I have been. This will hopefully reinforce the necessity of not forgeting past concepts.

#2 - Grading: I made the mistake of making homework out of too many points. It really only needs to be worth 3 or 4 points. I had a student ask me why she lost 5 points out of 20 when I only marked one error on her homework. My response was that her error demonstrated that she didn't understand a particular concept very well. She argued that she did lots of other problems correctly, and I agreed with her that she did. However, the point is that she failed to grasp one of four or five concepts for that section (which she agreed didn't make sense), and that her grade reflected that reality. However, there's a psychological difference between 15/20 and 3/4 that she had a hard time getting over.

#3 - Quizzes: I might do the quizzes at the beginning of class and schedule class start time 10 minutes after the start of the class. It's a much more regular routine that way. I might also cut it to one quiz per week.

#4 - I want to buy "The Inner Game of Tennis" and read it so that I can potentially send some of that message to the students on the first day of class. Many of the students have already defeated themselves by coming in with an attitude of failure. I've heard this sports psychology book is very good at discussing how players lose before they even begin. I think I'll jump to Amazon right now to do that because I'm done posting.

Presentation matters

Next semester, I'm going to start off by placing a much higher emphasis on presentation to my algebra classes. My experiences this quarter indicate that this is actually a foundational part of their math education. It has a number of benefits:

1. It sets the standard of reading and following directions carefully
2. It forces students to think about what they're doing and exposes their lack of thought
3. It makes the papers significantly easier to grade
4. It outlines a style of thinking that encourages logical and organized thought
5. It prepares them for more difficult and more complicated algebraic maneuvers down the line

I've made some adjustments and most of my Math 097 students are on board with the presentation (and I hope they're seeing their current material of adding/subtracting/multiplying polynomials as a clear extension of these things). Jason is considering writing up a document about these algebraic presentation things, and I might help him out if I can.

My first stumbling block

I'm now just about two weeks into the semester, and I've hit that first bump that I've been anticipating. What is the problem? It's complicated, and I think it's the same problem that everyone who teaches developmental/college algebra runs into. The students don't get it. Simply stated, the students are underperforming by a wide margin. Why? I'm not entirely sure. I have some conjectures:

1. Bad teachers: I can probably blame other teachers for teaching sloppy, lazy math to students and probably be right. But that doesn't really help anyone.


2. Bad habits: Consider the following musical analogy -- Students of music who are classically trained from the beginning learn their fundamentals right away and develop the good habits early. Students who sort of pick it up as they go along develop bad habits because they simply do not know better; it's a matter of making it work however it does. Classically trained musicians who stick with the program all come out consistently good, with some very high caliber exceptions. The self-trained musicians are generally not so good, but those with natural talent and musical instinct still come out of it playing exceptionally well. Those musicians who are self-trained and are not doing so well have a difficult time when put into a formalized classical setting because their bad habits prevent them from doing better. It takes considerably more effort to break bad habits than to form new ones from scratch.


3. Bad self-assessment: I asked my students on the first day of class to rate their mathematical ability. Most students rated themselves in the 5-8 range out of 10. However, the work that I see puts them in the 3-6 range or even a little lower. Why do they think they're better than they are? Probably because they don't have a full vision of what the range of mathematical talent is. Most of them have probably only been in classes where they were average or above average. They have probably only seen students as good as a 7, and so they see their talent as about 50%-80% of that (3.5-5.6) and so they are working on an broken scale.


4. Bad expectations: I think they just don't know what is expected of them. Because they don't know what's expected, how can they reach that goal? I think this is where I need to begin. I don't know how just yet, but I'm working on it.

First class.... sort of.

I filled in for Jason yesterday in his precalculus class. I didn't do much lecturing, but instead spent the whole time answering questions. As a result, it felt a whole lot like teaching section all over again.

I didn't do anything different from teaching sections back at UCSD. I asked them what question from the homework they had problems with, and then I proceeded to explain the concept and solution to them. However, I did notice a distinctive "desire" to learn, mostly from the students who seemed to be non-traditional (I originally typed "older", but I feel that it might be received as derogative to some).

There was one student in particular who struggled with something involving absolute values, but could not find the right question to ask to solve her difficulty. I had to ask her a couple times to try to rephrase her question because I couldn't figure out what was tripping her up. But she politely and patiently persisted in trying to wrap her mind around it until she was finally able to put it together. I've seen that sort of thing happen before at UCSD, but I also remember the student simply say "never mind" after struggling with it for a couple minutes.

I was a bit amused by her response after she put it all together: "Thanks for humoring me." That's a new one. I hope to see more of this from my evening classes this Fall.

Spring 2007 Math 20D Evaluation Results

If you've looked at my poker blog, then you would know that I like record-keeping and data. I've compiled the results from my TA Evaulations and put them in a this document for future reference.

What must I get on the final to earn an A?

I bumped into a professor in the math department the other day on the way to the math department party and during the course of the conversation he shared the following little anecdote that I think is worth sharing:

Why is it that students ask, "How well must I do on the final to earn ...?" If you them that they must get 80% on the final to earn a B, are they somehow more capable of getting an 80%?

Non-rhetorically, why do students ask this absurd question? I don't know, but I think it's all an illusion and I dislike it. Instead of the "do the best you can" mentality I hope for students to have, the exam is seen as a game of grades. Yes, the assessment leads to a grade, but the grade is the afterthought to the educational value of the content being taught. The right question is "What can I learn?" not "What must I do?"

Thinking about this question has made me contemplate the type of response I should have to students who ask that question. I have gone through a few which I've listed here along with some commentary:

• "Your best chance of getting the best possible grade is to get a perfect score." - I'm not happy with this because it brings the grade into focus, and not the act of learning.
• "Just do the best you can." - While this is better, I don't think it demonstrates to the students that I care. Instead, this sort of sidesteps the question and doesn't give a helpful response (of course they will do the best they can).
• "Does it matter?" - It's getting closer. A question puts the ball back in their court to try to justify why they care. I'm not quite happy with it because I know the answer. It matters because they care about their grade.
  
• "Would it change anything if I told you?" - This is the best answer I have so far. It puts the responsibility on them to look at their question in a different light. I'm not denying the importance that they have put on their grade (as misplaced as I believe it is), but I am leading them to realize that this question is ultimately unhelpful to them.
  

I won't have to face this question for about 5-6 months, so I'll probably put these thoughts on the back-burner for now.

The Standard Syllabus

Every class deserves its own syllabus, but a lot of the content in a syllabus remains static from one class to the next. I spent much of this afternoon reflecting on the common content and pulling it together into a single document. I call it the "Standard Syllabus" and you can find it by following this link.

There are three components to the syllabus.

The first is a set of pithy comments that address the basic philosophical approach to teaching and assessment.

The second piece is a short essay to help students think about how they actually learn math. Just as problems are broken into two types on the first page (computation and concepts), their main structure for learning is also broken into these two components. Homework builds up the computational skill and classroom time will be spent trying to build up the conceptual framework. I think what I have now is an acceptable form, but it may be changed later.

The third piece is a layout of the expectations that the students should have of me and what they should expect from themselves (it's also what I expect of them). I think it's vital to the students' education that they actually own their own education. They must be responsible for it and I've made that explicit in having the space for them to sign their name. I admit it feels a little contrived, but I will gladly exchange that for being able to make this point absolutely clear to them.

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.