Math 1010

I am scheduled to teach an intermediate algebra course next semester with 200 students enrolled. I am not even going to talk about the fact that I don't think these classes should be taught at the university level. All students should be ready to take college classes when they get to college (yeah, I know). But since we do offer them, I would imagine our goal is that students learn the material so they can take college algebra (no idea where the name came from since that course shouldn't be taught in college either). All that aside, I vehemently oppose large sections of anything really, but especially of low level classes. The students who are in those classes are not there because mathematics comes easily to them. If they can learn it by listening to lectures given by an instructor, they would have already done so. I know why these courses are taught like that: money. And I can't stand that we continue to do it although we know that a failure rate in those classes is much larger than in any other. In fact, some people go as far to claim that the failure rate is large because students do not show up for classes (as if the learning happens by osmosis, so all we have to make sure is that they are there). I actually do think that the students should attend classes, but I don't know how I feel about policing students who should by now have some sense of responsibility for their own actions and choices. Also, one must admit that taking roll in a room with 200 students would not be time efficient. That is, it wouldn't if clickers (also known as audience response system) didn't exist. Each student has their little clicking device that emits signal (when a button on it is pressed) that is received by a little antenna hooked up to a laptop with an appropriate software installed. You can ask a multiple choice, T/F, Y/N questions that are projected on the wall, and I believe actually received on students' pads, they answer it and the program records each answer (each student is identified by their student id #) and can immediately project what answers are given. In any event, taking roll becomes extremely easy, asking questions that can help direct the class are easily implemented and quizzes are quickly graded. Which brought me to my next problem. I never give multiple choice quizzes. I was trying to think of a way to actually look over students work, and I came up with the idea of giving a clicker quizzes, but after each quiz I would list 20 or 30 students (chosen randomly, but so that over the course of the semester I see each student's quiz at least once) whose work I will actually grade. Haven't quite figured out what to do with that grade. Maybe have an extra quiz grade which would really be only 0/1: 0 if the work doesn't correspond to the answers given, and 1 if it does. Anyway, needs more thought. But then I just ran into this multiple choice tests post, and I like the idea. I feel a little better about these quizzes. Anyway, better go work on that syllabus. And if you have great ideas about how to teach 200 students in a chunk of hour and a half, please send them my way.

Never been tagged before...

I was catching up with JD's blog and was very surprised to find my name at the bottom of this post. I was curious if I could even come up with seven things. They might be fairly random. Not too weird, I don't think. Here we go:

1. Quit smoking about three years ago, cold turkey. Speaking of which I believe I only ate turkey once shortly before becoming vegetarian (don't think two were related). Quitting smoking and eating meat are both surprising if you know anything about country I come from: Bosnia and Herzegovina. Still drink coffee, though.

2. I've started many a thing in which I lost interest shortly thereafter. It worries me still. I wonder what I else I will give up.

3. I've twirled my hair since I had it, is what my parents say. It is so me, that I haven't even noticed that my husband, then boyfriend, was doing it as well until our parents cracked up about it. My hair is now less than an inch short which makes twirling really hard. I steal his now and then for twirling purposes.

4. I don't understand people's obsession with procreating.

5. I used to be tolerant of people's religious beliefs. I still think to each their own, but I am not very concerned with not being perceived as offensive. I realize that I do not respect nor tolerate ignorance, nor unfounded statements in any other realm of life, so I don't see why I should when it comes to religion. And I don't mind saying it.

6. I have never been afraid of flying. Well, never until two years ago. Particularly turbulent flight from Zagreb to Sarajevo has made me really nervous about flying. I've been trying to ask as many questions as I can about planes and flying. I am particularly sad that I hadn't been paying attention in my physics classes. I just realized that I forgot to ask if a wing can fall off and what would happen then.

7. I can talk pretty much with anybody about pretty much anything. Except for my dad. He has a particular gift of upsetting me almost as soon as he opens his mouth. All right, a slight exaggeration, but only slight. My sister says that I am much more critical of my parents than of other people. I think I ought to be. They are my parents.

Here are the rules:
- Link to the person that tagged you and post the rules on your blog.
- Share 7 random and or weird things about yourself.
- Tag 7 random people at the end of your post and include links to their blogs.
- Let each person know that they have been tagged by leaving a comment on their blog.

Tagging 7 other people would be hard. I'm afraid the meme dies here.

Safe on internet, at home?

My students are worried that they can't let their students on line. Because it's not safe. I'm currently more worried about being safe in my house. What follows is a text message conversation between me and Mark:

e: Sweetie, I just got a bit worried. I did hand a key to our house to a person we don't know!
**
m: Don't worry about it. It will be fine. Besides, what are they going to do? Steal all my photo shit in the basement?
**
e: She told me she used to dog sit for people in park city and how nuts they all are :)
**
**
m: Going to bed now. What are you up to?
**
e: Grading! Is there anything else? You don't think she'll come back and kill me?
**
m: Sweetie! The homework will get you first!
**
e: Are you sure? I feel like I may be on fox news tomorrow "a gullible math geek hands the keys of her house to a known criminal. Stay tuned, more after the weather"
**
m: Sweetie, don't be ridiculous. They always show the weather last :)

Mondays are hard

I teach my elementary teachers on Mondays, 4:30 to 6:30. Last night I spent another hour after that with a crying student. I found out I was supposed to apologize for having high standards! Joy. Came home, walked the dogs, ate, passed out.

Kucinich rocks!

That's all.

ADHD

I have a hard time with this one. I think it's children's natural state to have attention deficit and to be hyperactive and I do not think we should drug them because they are. That is not to say that I doubt disorder's existence, I just doubt that it is as common as it is diagnosed. I am listening to Talk of the nation and they are talking about the latest study that shows that an individual may grow out of the ADHD, that is that the part of brain responsible may just take little longer to develop in ADHD individuals, but once it does these people no longer have ADHD. A woman calls in says she was told she had ADHD, so if she doesn't have it anymore, how would she know!?!

Vouchers, again

From Salt Lake Tribune: With 85 percent of the state's precincts reporting, 62.1 percent of Utahns voted against school vouchers and 37.8 percent voted for them.

More to follow.

Chalk?

Has anyone seen Chalk? What did you make of it?

On a different note, I asked my students "If you had to think of something that is as different from mathematics as possible what would it be?" I got a fair number of "I can't think of anything different".... What would be your answer?

And something well, maybe not as unrelated as it may appear. The other day (10/18 show) I listened to The Bryant Park Project and they had two people who were going to some sort of right conference. They were asked for three most important issues that would decide who they would vote for. The answers were, in this order: the abortion issue, the marriage issue, and the judges that the candidate would appoint. When asked "Why these?" the woman answered "I'm a mother of two children, and I'm concerned about their future." So am I.

Nothing to do with math

I couldn't resist:

Vouchers

I am really not too huge on advertising. I've ever only worn one shirt ever (well, two, but one is really warm and fuzzy and free and unfortunately has word MICHIGAN on it, I guess you could think state, not the university, although highly unlikely) that had any sort of writing or picture on it that is larger that half a centimeter by 2 centimeters. Anyhow, I have put up two Stop vouchers signs on Friday, and taken one For vouchers sign down, not personally of course, so I was pretty proud of my accomplishments. I also explained the whole thing to a person who claimed not to have known much about it and he said he was going to vote against them, but it could be that was said only because he thought his tip depended on it (he was cutting my hair). Anyway, for those of you who do not know, the voters will be deciding on vouchers for private schools: will the parents who decide to send their children to private schools be allowed to receive up to $3000 towards tuition from state education funds? Anyway, you can see the whole bill here, but some my favorite parts are (italics was added by me):

Eligible private schools:
....
(g) employ or contract with teachers who:
(i) hold baccalaureate or higher degrees; or
(ii) have special skills, knowledge, or expertise that qualifies them to provide instruction in the subjects taught;
(h) provide to parents the teaching credentials of the school's teachers; and
(i) provide, upon request to any person, a statement indicating which, if any,
organizations have accredited the private school.
...

The scholarship application form shall contain the following statement:
"I acknowledge that:

(1) A private school may not provide the same level of services that are provided in a public school.
(2) The private school in which I have chosen to enroll my child has disclosed to me
the teaching credentials of the school's teachers and the school's accreditation status.
(3) I will assume full financial responsibility for the education of my scholarship
student if I accept this scholarship.
(4) Acceptance of this scholarship has the same effect as a parental refusal to consent to services pursuant to Section 614(a)(1) of the Individuals with Disabilities Education Act, 20 U.S.C. Sec. 1400 et seq."

The polls seem to be saying that the vouchers will be voted down. I hope they are right.

Further development with unsatisfied students

I had just written a whole paragraph that got lost. I do not like when that happens.

I was going to tell you more about these two students. We have a 2 hour class, so I usually try to give them a break half way through, so we can all take a breather. Well, on Monday these two students left during the break, without telling me that they would, what is what most students do if they need to leave. So I sent them a message that said something like "I noticed you left, I hope everything is ok, and that you'll be able to catch up. Let me know if I can do anything." Only one of them bothered to respond, and said something to the effect that she doesn't feel comfortable in the classroom, that assigned groups make her feel irresponsible and she already has issues with group work. "The main reason I left was simply becuase of this discomfort. I know you are trying to get us to work with other people, but there are more effective ways to achieve this goal." Of course I emailed back and said that I feel saddened by this and that we should meet in person and talk. I also asked about these more effective ways. She responded only to tell me that she can't make it before class because she had another class right before and not a word about the more effective ways. I responded that she should name any time and I'll meet with her. Hadn't heard back.

Anyway, it appears that they since emailed the chair of the department to say that nothing has been done about this and to complain some more. The chair politely directed them back to the associate chair whom they had spoken to already. The thing is that it is not clear what it is that they would like to happen. Would they like me fired? Would they like to get a new instructor? Would they like me to let them do whatever they want? What is it that I am supposed to do? There are 2 students who are fuming, out of 43. Do I ignore them? What would you do?

Funny thing: Another student came by to talk to me and told me that a while back some sort of petition had been written, and that he had signed, but under pressure, and that if anybody ever mentions it he would gladly go talk to them on my behalf. I have no idea what they asked for in the petition. The funny thing is that the student who started the petition apparently decided to take the sequel of this course next semester. From me.

First time for everything

We were on fall break for a week. After spending a week with the in-laws (just to clarify: that is not a great thing), I have come to school to the first official complaint about me ever. The associate chair came to speak to me because 2(!) of my (elementary) students came to talk to him about me. Granted, one of them said things were getting better, but that is hardly making me happy. I don't feel terrible though. Well, I won't cry this time :)

Opposite of recruiting?

Jonathan over at JD2718 is asking how we retain teachers. Although a very good question (I think he probably means how we retain good teachers) to which I would love to get an answer, I have a slightly different question: how do we prevent some people from going into the teaching profession? Some of the students I see who are studying to be teachers are mysteriously passing their courses and will get a degree, but from what I can tell though they are not ready mathematically to teach, and I am not convinced that they enjoy mathematics. I have had conversations about this and the argument I most often hear is that we need to let them pass if they are not totally bad. Because if we don't then we will have such a shortage of teachers that the people who will end up in the classroom will have even less training (read: they'll be much worse) than the ones I'm worried about. However, this problem feeds into the one Jonathan is talking about (well one version of it anyway, the one that encompasses all teachers). If people who come out of our teachers' programs are not ready to be teachers or should not be teachers at all, then they are likely to quit pretty quickly. But that is not necessarily a bad thing. Unfortunately, once again, I do not have a solution.

Hurt feelings

Today is the third (fourth? who's counting) time in two weeks I cried because of things my students said. I guess I am starting to doubt that I can even teach. But what I wanted to leave a note about is this: we always talk about how we have to be careful when we talk to our students, but nobody ever tells students they should be careful about the way they talk to their teachers. Teachers have feelings, too. I suppose it's fine of the time, but it's the students who want to be teachers that I am sad about. It just seems that there is a serious case of split personality. When you're a student you're a student only, and when you're a teacher you forget you were a student. I'm just gonna go mop around some more.

Division

So far I have largely ignored the way Americans write their division: divisor first then dividend. I learned it differently. Mind you, the algorithm is exactly the same, it's just the way it is written that differs: dividend, divisor and quotient are all in the same line, in that order, and the rest is the same. Now that I teach elementary math I have to train myself to do it the "wrong" way. Can anybody explain to me why the order in which these numbers are written in the algorithm seems to contradict what is being done, or is that just my perception influenced by years of drill I was subjected to? Also, whenever I show "my way" to people they say it doesn't make sense even though it is exactly the same process. Does that show the lack of understanding or lack of willingness to understand?

Insane

Ok, so I have no idea how other people do it, but I feel like I am on the verge of drowning. Getting ready for classes, grading constantly, now we'll start math circles for teachers, need to write grants. Yikes.

So, we've been going for almost three weeks now. Apart from feeling like there should be 36 hours in a day, I'm enjoying both my classes a lot. Since I have some grading to do and I have to get ready for tomorrow's class I'll only write two observations.

El. math teachers course: From what I can observe (and this class isn't exception, I don't think) my students believe that in order to learn mathematics they only need to solve problems. But how can you find a complement of a set if you don't know what a complement of a set is. It is hard work to make them learn definitions. Or even to see the value of definitions.

Secondary: There was a homework problem that I assigned that asked "How would you want your students to respond to question: Explain why the sum of the digits of any multiple of 3 is itself divisible by 3." The question came form Principles and Standards, so one might imagine that this is material that these students will be expected to teach once they become teachers. I wonder what response you think I got. I'll tell you about three that I found most surprising: they said that they wouldn't expect their students to know the answer since they themselves don't know it. What was surprising, or rather what made me sad, was the fact that not only did they not know, but they didn't express any desire to find out. They didn't say "We'll look it up" nor "Can you tell us?", and they definitely didn't say "I'll work it out." I guess it's only a beginning of the semester :) Maybe we'll see some improvements as the class goes on.

New school year

Where did the summer go? Well, I do know. Moving takes time and energy, and settling in is usually fun. And frustrating. In any event, the school starts tomorrow. Crazy. I have two classes: Math for elementary teachers and Methods for secondary. I never really feel ready for any of the classes. But I'm sure it'll be fun.

I almost forgot. Mark and I were looking for dual flush toilets. As a friend of our would say "a peepee and a poopoo". We went all over the place looking and asking and we happened to be in Lowe's for something else, so I decided to see what they can order. The man in the plumbing was most helpful. He was looking them up and we ran into one made by kohler, I think, and it read 1.4/1 gpf. He read "This one uses one and a quarter gallons and one gallon per flush".

Long time no write

Well, here I am, with few minutes to write. In case anybody missed me, and they don't know where I've been, here is a little update. We moved. Well, most of us moved. We now live in Salt Lake City, and life is good! Our new house is starting to feel like home. There are few more boxes that need to be dealt with, but it looks decent. Three days after we moved in, I had to go to another mathy thing, this time at AIM in Palo Alto. We were shipped out to learn how to run Teacher's circles. In case anybody heard of Math circles, that's basically it, just for teachers. So anyway, we're starting that in September and if any Salt Lake middle/high school teachers happen to see this and are interested they should let me know. But I'm sure I'll find them either way.

Then PCMI started at Park City. I have expected a lot more from it, and am rather disappointed. PCMI organizes a three week event that is supposed to do what administration thinks is vertical integration. There are several programs running at the same time, and people are supposed to interact and learn. There is usually a mathematics topic around which the whole program is structured. This year it is statistical mechanics. There are research program, graduate seminar, undergraduate seminar, mathematics education research seminar, secondary school teacher prof. development, and professional developers' seminar. I am there as a part of prof. development program, but that only meets about 1hr a day. The rest we hang out with high school teachers and observe what they do. The teachers' schedule is something like this:
8:20-8:45 there is one or the other administrative thing that needs to be addressed.
8:45-11 teachers work in groups on some math problems, statistics and probability. Good idea, but I think poor implementation and not great problems.
11-12 research teaching topics. This as far as I can tell was just a disaster. We were given a choice of several topics, groups were formed based on a topic, we got a list of few research articles that we were supposed to read and write a little report about. However, at least in my group there wasn't much discussion. We each wrote up a blurb on the article we read, but when the time was to write it up, they had the prof. development session running parallel, and I think half of us were gone. I think I am likely to miss tomorrow's presentations.
12-1 lunch
1-3 working group. I went to geometry, but that too was rather disappointing. It mostly appears to be Geometer's Sketchpad advertisement.
3:15-4:15 cross program. This is where we listen to talks whether they're given by research mathematicians, high school teachers, math ed people.... or there is nothing.
4:30-5:30, 6 is when prof. development group meets and so far this mostly consisted of them giving us math problems to work on. Hello! Half the people have Ph.D. is math. It just seemed like not a good use of time.

In the meantime, my preparation for my classes has stalled. My preparation for the session of ThEMaT has stalled as well, and I am supposed to be presenting it in less than three weeks (this is another potential PD that involves geometry). Also, since I am back in the most fun Salt Lake there are plenty of things to do that do not involve sitting at a computer. However, if the temperatures continue the current trend I may be writing more :)

Nice!!

Small world

The second day of the conference is over. First of all I have to say that everybody was very civil so far, although there were some gentle pokes at the "other camp" and allusions to math wars, and who does what and how. Clearly, we do it right and they do it wrong, but the tone was leaning towards "we can still learn from each other". The last point actually brings me to some cultural differences I find in the two fields: mathematics and education. In my experience mathematics is very generous field. People do their work and are extremely open to sharing it, providing their written work, talking about it in private even if it's not really finished, sharing ideas and collaborating with others. Yes, there are some isolated cases that got press coverage lately that would imply otherwise, but that's what they are: isolated cases. My short experiences with education people was somewhat different. Maybe I have run into their isolated cases, you tell me. I'll mention two that happened today. We had a session in which University of Michigan people talked about their content course for elementary teachers masters program, and University of Delaware people talked about their preservice elementary mathematics courses. They seem to have been very similar to each other, and both seemed like very good ideas. So I said something to the effect that it wouldn't make sense for all of us who might be teaching these types of courses to reinvent a wheel, and would they be willing to share their materials. There was a noticeable silence before I got an answer that in short said "Yes, but only a couple of lessons". After the session I was approached by a person from, I believe, San Diego State, who said that they are about to publish some materials, and I should come to their session. That was great, but I suppose this is not a community that appreciates freebies :)

But to go back to my title. Half way through the day I ran into Yvonne (you will have to pardon the lack of linkage in this post), a graduate student from UCD whom I met at some math conferences couple of years ago. Turns out she's organizing Math Circles in Davis, and I will be helping with Teacher's Circles in Salt Lake City. She and her friend Brandy went out to dinner with us. Brandy turns out to be a grad student at UCD as well, but in math biology. So I mention that I know a professor there who is math bio, and she immediately tells me a name of another professor who used to be in Salt Lake when I was a grad student there who I remember moved to Canada. I guess it was too cold up there for him. With us at dinner were Sendhil, a math teacher, and John, a former high school math teacher who is now with Math for America, both from NYC. Sendhil was at the Math and Social Justice conference about which I first read at mrc's blog. Then we started talking about blogging teachers, so I mentioned Jonathan. Do I have to say that John knew him? Sendhil knew about Darren's class blogs. The world is small.

math-mathed conference

I kept meaning to mention that I was going to another conference. Well, here I am, in Berkeley, sitting in my room, about to fall asleep (and it's only 9:50, but I guess really 12:50, so I feel little better for being exhausted). The title of the conference is Teaching Teachers Mathematics, and it is being held at MSRI. We had a short day today: 2 big talks and several little ones that could hardly be really called talks. There are two days ahead, and multiple parallel sessions most of the time. I am little worried that it will all turn into "us vs. them", since we got some whiffs of it already. I'll keep my fingers crossed that I'm wrong.

Feel good story

Some feel good stories leave me feeling little queezy. You know, when things are just too good to be true. This one, although with a predictable ending, made me happy, then sad. It would read completely differently if the kids in question were from MIT. In fact, it probably wouldn't be much of a story at all.

No time

I haven't written in a while. One reason is that I've been too preoccupied with making various decisions. And thinking about the courses I'll be teaching next semester. I'll teach two and neither I taught before, and I sort of have books for them. I definitely have a book for one, which is the math for elementary teachers course. But the second, which is the methods for secondary teachers, they have used Principles and Standards for years, supplemented with various other texts and articles. I don't actually know what they've done and how they've done it, so I've been thinking what to do and how to do it. I'm almost panicking. Luckily there is lots of time left, but also a move and three trips and a three week workshop, and then another workshop. Lots to do. But first things first: a trip to Hawaii. It will be fun!

As for the pledge: they recited it in a middle school I visited as well. And I talked to a friend's mom who is a second grade teacher. She says that she makes her students stand. She gave three reasons: to keep them form making noise and fuss during, to teach them that they should be grateful for living in the us (as opposed to unlucky kids without freedom that we enjoy here) and, if I remember correctly, to teach them respect. I am fairly certain that all of those things could be done without standing and reciting the pledge of allegiance. I am surprised that this is still going on and that people are not complaining.

Pledge of Allegiance

I've visited a high school and was surprised to hear the Pledge of Allegiance first thing in the morning. I thought this was an elementary school phenomenon, and am wondering how common it is.

Next observation is that fair number of kids did not bother to stand up and recite. I wonder how many had reasons other than laziness.

Equity in the classroom

I've been thinking about how best to discuss the issues of equity in the classroom with pre-service teachers. Some people aren't aware of myriad of issues that exist when one is placed in a diverse population of students (some people still grow up in homogeneous environments). What would be the best way to discuss this in a methods course? I have a classroom episode that involved ESL students which I think would be good to discuss, but there are plenty of other things I would like them to think about, such as cultural and socio-economic backgrounds, gender, disability, different learners.... So, do I just try to weave that into the course as it goes on and not talk about it in one spot concentrated, rather keep coming back to it throughout the year?

Electronic portfolios?

I've been searching around for documents about teaching portfolios. Most of the stuff I've found talks about what should be in there, which is good. However, it seems that largely favorable form of presentation is a paper copy. Do I need to say that that seems boring? I ran into one commercial software that helps teachers build one, although I know nothing about it: Teachers' Portfolio. Now, free is the word, so I look for other stuff. Should html be the answer? Hence, questions for math teachers out there. How do you make your portfolio? What would you suggest getting ready to come out of school not quite yet teachers use to make their own and what you suggest they put in there?

What is mathematics education?

This was a title of the talk given by Hung Hsi Wu in the Mathematics Department at MSU. He happens to be one of the two mathematicians (I hear there is a third appointed recently) who are on the National Mathematics Advisory Panel, but about that in a second. I have gone to the talk not quite sure what to expect. People I know have given me opposing views of this person's work, or rather of what they thought he represented. Wu has tried to give a definition of mathematics education that I haven't heard before. He said: "Mathematics education is mathematical engineering". From what I understand he views engineering as a discipline that customizes abstract notions so that they are usable by wide populations. Therefore mathematics educations should be taking mathematics and turning it into a user-friendly product, i.e. into a product usable by a population under consideration. I thought this was an interesting way to look at it. Otherwise, what I learned is this: this nation is in crisis, W. is the best president as far as the education goes, it is against constitution to have national curriculum (I asked about this), in 2007 mathematical engineering urgently needs close collaborations of mathematicians and educators, there are no mathematical engineers yet. Few of these, I must say, came as a huge surprise to me. But let's not dwell on the politics. After the talk a smaller group retired to a smaller room to talk to the speaker some more. As people would walk in they wold introduce themselves and inevitably would say "I'm from the math department". It felt as if this was a secret handshake, or a a secret club. They were surprised to hear that I was as well. The comment I got was "You were laying so hard that we thought you were from the education side". Needless to say that nobody from education side was there, in this smaller, more intimate setting. Anyhow, I've heard, yet again, what I heard from teachers: It is all somebody else's fault. These future teachers don't know enough math, they don't want to learn, it's high school fault, it's their old teacher's fault... Maybe, just maybe, we aren't doing a good job either! How novel idea. I asked about NMAP, what the goal is, and what he though of the panel's composition. The goal is apparently to make recommendations about algebra. And the panel could be better, but it could be worse. Couple of us agreed that that seemed like a fairly lame answer, and since he thought it was perfectly fine, I decided to ask what he thought in particular of the fact that there is only ONE mathematics teacher on the panel. He agreed that that was a shame. I asked why they didn't ask that that be changed. He said that these things don't work that way. I think my words were:"I thought your job was to make recommendations. Couldn't you have made a recommendation to add a teacher to the Panel?" He said no. But here is the kicker: apparently they've just added three new members: one mathematician, one cognitive psychologist and mathematics education researcher (elementary math), I believe.

Update A friend of mine emailed and was asking "What's up with tallying mathematicians?" Once again, I fail make myself clear (or clear enough in a single post). I have talked about this before: it's not the lack of mathematicians that I find troubling as is the lack of mathematics teachers.

news and blogs

Following blogs has reminded me of following news. I read news and I get upset by them, then I wonder if it wouldn't be easier just not to pay attention. But of course, that would be an easy way out, and completely irresponsible. So I keep reading. I've been getting upset by a lot of blogs that I read. At first I was very excited by the numbers of math teachers who were writing and I started reading their blogs. A lot of them talked about very interesting topics within math and teaching and policy. But then I realized that for a fair number mathematics is a rather marginal interest. I've made some comments about mathematics that I've seen here and there, some people write it off to nit picking. Makes me sad. Then, of course, there are people who seem to think that mathematics classroom should be divorced from issues whose consequences students encounter every day, such as (in)equity, and social (in)justice, and poverty. Others may be well in tune with those, but forget that they are supposed to be teaching mathematics. So lately, I've removed a whole bunch of blogs form my bloglines to save me some time, energy and stress. I've come down to about 10. I wonder if this is as irresponsible as quitting the news. Maybe I should stick to reading them, so that I can get to know these people better, what they think and how they feel about various topics.

Really cool

It's amazing to me how many things, one might want to say "easy" things, in mathematics I never really thought about. I was reading an ed article today, which reminded me of something I saw a while back about how people explain why a product of two negative numbers is a positive number, so I tried searching for it. Needless to say I wasn't able to find it, but I ran into something else which was even better! Here is the source, and an excerpt follows:


We take any Euclidean line, mark point 0 on it, choose a unit length and mark point 1, and then the non-negative "numbers" are associated (identified) with all the points on the ray from 0 through 1in the usual way. The negative numbers are then identified with the remaining points on the opposite ray in the usual way so that we have now a Cartesian coordinate number line L. Now choose a second line M through 0 and mark a 1 on it at the same unit distance as we used on L, and complete this so M is also a Cartesian coordinate line. Now for any number b (point) on L, and any nonzero number c on M, the product point b*c can be found: let J be the line from 1 on L to c on M, and K the line parallel to J through b on L, then b*c is the intersection of K and M.


Here is a picture that shows L as x-axis, M as y-axis, and we are looking for 4*3.




The only problem with this explanation is that the students need to know about similar triangles! Regardless, this was way cool. If I had thought about this on time, I would have given it to my students on their final that had them think about similar triangles. Next time!

If I can't be an actor

Lately I've heard lots of talks about equity in education, and in mathematics education in particular. Jeannie Oaks from UCLA gave a very informative talk at UofM last week where she talked about the consequences of Proposition 209 (banning affirmative action) in CA. The reason this was relevant is that almost identical proposition passed last year in Michigan. As a side note, she gave some frightening statistics about math teachers' qualifications, or the lack thereof, in CA schools. Mrc just yesterday said he was going to Math and Social Justice conference. You might want to check out a very good film The Boys of Baraka. It is a documentary about boys from Baltimore's projects who are sent to boarding school in Kenya. A tidbit from the special features, just for math teachers:

I'll go to LA to go to college to become an actor. If the whole actor thing doesn't pan out, I can always be a math teacher,

said one of the boys from the movie.

New: Coincidentally, I just saw this article in NY Times:

A scathing 18-month evaluation of California’s public schools has concluded that the state’s educational system is “broken,” crippled by a complex bureaucracy, flawed teacher policies and misspent school money, leaving it in need of sweeping reforms that could cost billions of dollars.

Assigning unworked problems?

I taught geometry for teachers course last semester. Towards the end I showed them an animation which depicted a geometry classroom in which students were given a problem to work on, to come up with conjectures and to prove them. The way I went about this was to give my students exactly the same problem before they saw the movie. They came up with conjectures, but didn't have to prove them immediately. Then they saw a movie. I guess I should point out that the students in the movie gave a partial solution, but there were loads of misunderstandings both on the side of students and the teacher. The goals were many, but one of them was for my students to be able to follow what students are saying at a pace at which things happen in reality and to try to determine what is correct, what is incorrect, what they should follow up and what they should leave alone. Things went fairly well, and now I am thinking whether to stick with the same plan or to alter it slightly. One questions is: Is it fair to show them the movie and expect them to be able to follow the action live :) When I saw the movie for the first time, although I have not worked out the problem before hand, I could see where the students were going, and why, and what was causing misunderstandings. Is it fair to put my students into the same position? Should I expect them to be able to do the same? Will the benefit for them be as big as if they had enough time to think about the problem themselves? Why the title above? If teachers have a tendency to assign problems they have not worked out beforehand (I've been guilty of that) then this could potentially show them dangers in doing so. Is that a valuable lesson? On the other hand, if they see the movie first then they loose the perspective of the student trying to solve the problem for themselves. A solution is thrown at them and will they try to come up with something different from what they've already seen? What say you?

P.S. Jonathan, you still didn't tell me why you hate teaching geometry :)

Back in town

The reason I haven't written in a while should be obvious from the title. Not only did I not write, but I wasn't even following what others wrote (skiing was way too good), so I am trying to catch up. For now, just a link to the third carnival of mathematics. It's not linked here anywhere, so I am doing it now: I loved the second .

Mistakes

Algebra class. Teacher writes a list of 5 problems under the heading
Solve and graph each compound inequality
The teachers solves 4 problems, during which time the students are reminded that the compound inequality means that both inequalities need to be satisfied (not in those words), how to graph solutions on the number line, various notation, and so forth. They arrive at problem numbered 4 that reads:
-5 < -5x ≤ -20
While working out the problem the teacher says: "Now listen very carefully. We have to pay attention to our signs". They arrive at the following:
1 > x ≥ 4
and proceed to graph it:

Students make no comment, they copy down the solution and the class proceeds to the next problem. A conversation similar to this happens as the next problem is being solved:

Student: "So, if the bigger one is on the left then they go out, and if it's on the right then they go in?"
Teacher: "That's right"

After the class, I brought problem number 4 to the teacher's attention and got the following response:

Teacher: "I realized it as I was writing it down. I'll fix it by the fifth hour. I didn't want to confuse them."
I, dumbfounded: "?!"

How and when do we teach our teachers that making a mistake in front of the classroom happens, and is not something we should hide and sweep under the carpet. I remember making mistakes in classes I teach; we all do, sooner or later. I apologize every time I do as if I had wasted their time, and not taught them something of value. We make mistakes, but we need to deal with them in a responsible manner. Go back, fix them, show your students that even when you know something it is not shameful to make a mistake, but it is to hide it. Show your students that we are learning all the time. And that we should not think that we ever learned it all. I think I wrote about this before, but it seems that our teachers think that there is nothing more they need to learn once they have their teaching certificate. They are ready. Are we really?

Mental?

I am impressed by lots of people who manage to write regularly. I supposed I havne't quite learned how to manage my time. That's why my posts are short. Not that that's a bad thing.

I am still observing. Yesterday and today I saw something that seems recurrent, so I have to comment/ask. In a process of solving a problem my teacher gets to:



She writes



Or, let's have even simpler example:



She proceeds to multiply 6 and 50, and then divide by 3. I saw identical process in the first classroom that I observed. In fact, kids (9 grade, I believe, possibly even 10) there couldn't do things like -4-2 or 7*3 without a calculator (the latter we figured out, and then two minutes later we had 8*3 in a problem and the student couldn't do it even after I reminded her that she knew 7*3). The teacher there said "Their mental math skills are terrible"?! I wouldn't call that mental math. Should I? Regardless of what we may call it though, I was terrified. I still am.

Back to my examples, I can understand using the long process when teaching it, but this was not an introductory lesson. Why do the teachers not use the "shortcuts"? Is it out of some consideration for the students? Do they think the students can't handle it? Or they'd be confused? Or is there some other explanation?

Tags: shortcuts+math

Mathematics and lyrics

I just saw this on Darren's blog, and just in case there are people here who read my blog, I had to put it here. I would have anyways, because it's awesome. And, yes, I guess I'm a nerd. For this kids out there who may think that's bad: It's not really! Enjoy:

I have to say it all sounds sligthly less impressive when you know that the leader was a math graduate student , who in the mean time finished his PhD, but still. Pretty neat. I have to go continue investigation, and enjoy my nerdom.

Tags: math+lyrics

Geometry teachers out there?

Here is a problem:

Prove that the midpoint quadrilateral (a quadrialteral obtained by connecting consecutive midpoints of sides) of an isosceles trapezoid is a rhombus.

I know of three different proofs. Would love to know if there are more.

Update Here is one remaining proof that I know of. Well, sort of proof :) more of an idea.

Funding for what?

I've been reading a lot about NCLB lately. This morning it was Washington Post's turn to enlighten me further:

The budget would add about $1 billion for the education law, most of it directed toward high schools to help pay for a proposed expansion of testing and other programs.

This reminded me of somebody's comment about testing agencies and how they might be profiting from NCLB (can't find the actual comment now) that I read while following the discussion on Dan Meyer's blog about NCLB.

Amateur survey of mathematics teachers

There is a lot of talk about what kind of knowledge mathematics teachers need in order to teach well. I am curious as to what practicing teachers think about this. Here are few questions for those of you who may stumble upon this page:

1. Do you think that mathematics courses you took as part of your preparation program (whether it be undergraduate, or certification, or anything else) are relevant to what you do in the classroom? In other words, do you think that what you learned there is directly or indirectly applicable to your profession?
   
   
2. Same questions about methods courses you my have taken.
   
   
3. In light of your experience in the classroom, if you had a say in what should be taught to future teachers as a part of their preparation, what would it be? Or, what do you wish you learned before you started teaching?
   

It is possible that I am asking wrong questions. If you think that is the case, then offer your own questions. And answers :)

Doing the steps and not going anywhere

If I made an actual step every time my teacher said "step" in two classes I observed today I could have had a nice walk.
I don't see them learning, I just see them stepping. Everything is done for them and when they make mistakes and do procedure incorrectly, she says "Oh, you just forgot to ______________". No, they didn't forget. In order to forget, you must have known it at some point.

It seems to be a common practice to have students work on their homework in class. This, I believe, is an American construct. I find it useless, but people feel attached to it (also, which part of the word homework says that the work is supposed to be done in class?). I saw a teacher give kids and upward of 20 minutes to do their homework. About 75% of the students, if they even bother to attempt the homework, are out of their seats and ready to head out the door within 5 minutes. Now, there is all this talk about not having enough time to go through the material that one is required to cover, but there seem to be so much wasted time. Am I observing an extreme case, or is this practice common? Here is a questions for all the teachers out there: How do you feel about homework time, how much do you give and is it really beneficial?

Research vs. Practice

I just spent a few days in to me foreign environment: mathematics teacher educators' conference. Until few months ago I didn't even know that these things existed and now I know that there is a professional development for professional developers!

I don't think I made this clear so far, but I am in between two worlds right now: mathematics and mathematics education. I feel I'm in a no man's land, and there aren't too many of us hanging out there (or we just don't know each other?). Until a year ago I was a semihappy postdoc working on my own little projects, proving my little lemmas, was happy and felt guilty when I taught. Why semihappy? I guess I didn't feel that my contributions to the field or the world at large would be big enough to justify spending my life picking at the mathematics pie and hanging out with people who think that what they do is earth shattering (especially since most of the time I disagreed). Where I did think I could be more useful was with my teaching. I like to think that I am a pretty good teacher, I certainly love doing it, although I may not see myself in an objective light (see the post and post below, and I plan on revisiting this later). In any event, I taught a course for teachers this past semester, and as I was getting ready for it, I started realizing that I don't really know much about teaching, or methods, or pedagogy, or anything really. I decided I ought to learn and started talking to people over in education. It didn't take me too long to realize that they are more similar to mathematicians than either group would like to admit. It seems to be a widespread opinion, especially in education circles, that mathematicians don't care about teaching. That may be, but I would venture to claim that education faculty do not care about teaching any more than mathematicians do. This may need clarification, apart from the one where I say that I haven't actually conducted any research on this and everything I say are observations only and the samples aren't too huge :)

Educators as well as mathematicians largely think that teaching takes the time away from scholarly work. From the professional development programs that I saw, I can not claim that I am convinced that the benefits for teachers are what is on the education faculty's mind, but rather data that they may collect along the way. There also seem to be a big issue of who teaches content courses. Education stance: people who teach it (a.k.a. mathematicians) know the content, but not in the way that is relevant to teachers, so it should be taught by education people. Mathematics stance: the teachers need to know lots more content than they do, and all they get from them (a.k.a. math educators) is this touchy-feely, hold each other hand and reflect nonsense, and consequently they should never be allowed to teach the content courses. Am I exaggerating? Maybe, but only slightly. So what is is missing in this picture? Could it be conversation ? But who is to start it when everybody is busy with their research??? The obvious answer to me is the people who do not want to do research, but want to practice. If they are diplomatic enough (hmm, guess that rules me out) they could bridge the divide between the two worlds. Mathematicians who are also knowledgeable about results of research by mathematics educators (slight misnomer: mathematics education researchers would be more precise) seem to be a natural choice. So, if you are out there, and are reading this, or know somebody who is, let me know.

Too much information?

All thing considered ran an interview with Rafe Esquith, a fifth grade teacher in an LA public school. He was talking about his book Teach like your hair is on fire. The title sounded great (hmm, should it be "don't judge the book by its cover, nor its title"?), so I thought I'd google it. First clicked link was a review from Teacher magazine. I got little taken aback by the second sentence in the article:

“I’m only here,” Esquith announces at the outset, “to share some of the ideas I have found useful.” But most of the things he shares aren’t all that useful and barely qualify as ideas.

The rest follows similar pattern. I have to admit that most of the time I don't doubt NPR much, so this came as a surprise. I decided to look some more. Next was a blog entry , and I list it only because it pointed me to Washington Post's article America's best classroom teacher ?! Its first sentence could not be further from the above quote:

Rafe Esquith is the most interesting and influential classroom teacher in the country, but he is not getting nearly as much glory as he deserves.

I read Washington Post regularly. Along with NPR and BBC, it's one of my favorite news sources. I expect more from them than just stating that somebody is the most interesting and influential classroom teacher in the country. What is the criterion for "interesting teacher"? Or "influential"? Have they visited every single classroom in the country? How many people have participated in determining that he should get the title?

I suppose after seeing such opposite opinions there is no other option but to read the book myself, and make my own judgment. I was going to say that I'd prefer my news to be less grandiose and more specific, but after paying little more attention to where this article appeared, I realize that it was written in a weekly education column Class struggle, so I guess there is no reason to complain.

Observations cont'd

I visited a new classroom and a new teacher. Although there are great similarities between the two classroom, there are some major differences as well. One of the things that I liked immensely in the new classroom was teacher's constant inquiry and/or revelation of alternate methods for solving problems. The beginning of the class was spent by going through some problems from the homework. 4-5 students were on the board at the same time giving their solutions to the problems that other had asked to see. Once they were done writing, each student had to explain their solution to the whole class. Whether the solution was correct or needed fixing, the question that would be asked was whether somebody had a different solution. Sometimes there was, and they would go to the board and write it out. If nobody offered anything, the teacher gave her solution. I am pretty sure that on all but one (sample of 10) they have shown more than one strategy.

One of the things that bothers me and seem to be common across the board: each class was interrupted by something/somebody from the outside anywhere between one time and three times during the class. I am not talking here about student's random talking, but PA/phone/students walking in or out/student council announcements/teacher aids.... This is something I do not remember ever being done when I went to high school. I don't even know if we had AP!

Observations

There are two things that I'd like to mention today. I finally finished The Teaching Gap, and I find it full of very good observations and recommendations. It is sad that it doesn't seem to have been noticed sufficiently. I would like to give one more quote that is on a similar in spirit to one below that compares teaching to acting.

Doctors don't try to figure out a new technique or procedure for every patient who comes into their office; they begin by using the standard techniques and procedures based on the experience of many doctors over the years. Nobody considers this a way of doctor-proofing medicine, although they do have a name for the failure to use standard practices -- it's malpractice. The standard practices that doctors use contain the wisdom of the profession.

Last few days I have been observing a class in a near by high school. In this country teachers are not often observed, and from what I can tell even when they are, they do not seem to consider it an opportunity to improve their teaching. I asked the teacher in question today whether she would like or mind me making a couple of comments about what I've seen. Her immediate reaction was less than welcoming and she told me that she gets little defensive about it. On one hand, I suppose there is really no reason for me to think that what I have to say would improve student learning or her teaching. On the other hand, if I were her, I would at least want to know, so that I can judge myself whether the comments were warranted or not.

Another thing that book mentions, and that I am becoming more and more aware of is the following. A lot of teachers can explain, in theory, how they should be teaching. They also may think that they are doing what they know they should be doing. However, their practice is far from their words. I guess what I am trying to say is that how can one expect teachers to apply their theoretical knowledge if they don't know what the application should look like.

Technorati tags: teaching, math, observation

Over-achievers

We listen about under achieving students, poor quality of education, inadequate preparation and low test scores so much, that it was quite refreshing to read this New York Times article. However, (I think I may have already mentioned this somewhere below) moderation isn't bad at all. A high school kid with a working day 6 am to 1 am is little scary. When do they get to relax and have fun?

Technorati tags: education, students, content quality

Self-critisicm

It is not common for a teacher in this country to be observed by other teachers. It is a shame, as I think, and apparently lot of others as well, that it is a great practice. Here is another excerpt from The Teaching Gap:

As researcher Catherine Lewis found, teacher collaboration can create a profound motivation to improve. A young teacher she interviewed recalled that after watching a lesson by her fellow first-grade teacher, she burst into tears: "I felt so sorry for my own students. I thought their lives would have been so much better if they'd been in the other teacher's class."

Past semester it has become apparent to me that not only majority of people are not critical of themselves, but are not critical of others either. There is a constant fear of hurting other people's feelings. Criticism can be expressed in a manner that is not offensive (ok, maybe I am not the best example, but I am sure it can be done :) ), but rather constructive. Not to be blunt, but I'd rather have one person's feelings hurt, than thousands of children not learning what they are capable of.

In any event, this book is truly amazing, and I recommend it to everybody who aspires to be a teacher (or is one). One of the major points in my view is that majority of U.S. teachers believe that having finished their studies they are ready to teach and the only area in which they might seek improvement is in HOW they teach. But not in WHAT they teach.

Technorati tags: improvement, math, teaching

Michigan Radio and probability

I walk my dogs twice a day. The dogs are not known to be the best conversational partners, and they usually forgive lack of intellectual stimulation from me, so I almost always listen to Michigan radio (local NPR station) as we walk. Half way through our walk there was a weather blurb that, among other things, said: "And there is 70% chance of rain this evening in southeast Michigan including Ann Arbor and Detroit". I look around me and think: It is 5:20 pm, which during this season I believe qualifies as evening since it is pretty dark out here. I am in Ann Arbor. True, it's not pouring rain, but something is falling, and trust me it's not cats and dogs. Last time I checked they called it rain. Now, what needs to happen before they say that the chances of rain are 100%? Or even better, why can't they say "It's raining, and it doesn't look like it's going to quit any time soon"?

Oh, and one more thing: Ann Arbor and Detroit are in southeast Michigan. No need to include them just for the weather report.

More on typesetting

I tried to write more about this mimeTeX, but the frustrations with blogger made me erase the whole thing. Turns out that you can use a public server that was generously provided by Mr. Forkosh to generate gifs of LaTeX code. Here is an example


< img src="http://www.forkosh.com/mimetex.cgi?c=\sqrt{a^2+b^2}"
alt="" border=0 align=middle >



Unfortunately, I think one needs very light background, for these to be visible, so I need to make that happen (I'm guessing it'd be enough to make a white box around the gif). Maybe this'll work:




Gotta go walk the dogs. I'll try to make it work later :)

Gifted students

It appears that nothing in this country can just be average, which makes me wonder where this phrase "average Joe" came from. About the gifted, from Washington Post. It is good to hear about the gifted every once in a while :)


Technorati tags: elementary education, math, gifted

Frustrations with blogger

I've been trying forever to write a post, but this stupid thing keeps popping all sorts of junk when I try to type. I go between "Compose" screen, "Edit HTML" and "Preview" and I haven't been paying attention to patterns, but every once in a while it totally junks my html code with all sorts of additions of stupid symbols and it's a pain to recover it (and every once in a while it also delets parts of what I wrote). Incredibly annoying!!! If you know what I'm doing wrong, let me know.

The Teaching Gap

I have often heard the following statements from teachers:

"There is no way my students could handle this."
"This is too hard for my students."
"Good luck teaching that. That's way above their heads."

If I complained that I didn't think that particular topics under discussion were that hard, I would get these answers:

"You don't have experience with (insert your favorite k-12 grade) students."
"You haven't been in our school."

These statements are correct. I do not have yet that much experience with K-12, but I am fairly positive that students respond to unstated expectations very well (and stated ones even better). What I mean by that is: if your students believe that you think they can not learn math or that the math is just too hard for them to grasp, then they do not have much incentive to prove you wrong. After all, you are the expert. However, if you set the bar high for them, their performance and in the process you let them know that YOU think they are able to become good in math, then they will try harder to prove you're right. I do think that the attitude teachers have influences greatly their students. The reason I am writing about this is the book I just started reading: The Teaching Gap by Stigler and Hiebert. I have a couple of quotes that I'd like to share:

One of the most striking impressions when watching the videotapes is that students in teh United States encounter a different kind of mathematics from that encountered by their peers in Germany and Japan. The content appears to be less advanced and is presented in a more piecemeal and prescriptive way.

As it turns out there were NO mathematical proofs in U.S. lessons. In contrast, there were proofs in 53 percent of Japanese lessons and 10 percent of German lessons.

Incidentally, German students did not perform significantly better on the achievement test then American students. The following figures also made quite an impression (I hope they will look decent):

Average percentage of topics in eight-grade mathematics lessons that contained topics that were DEVELOPED or STATED.

Percentage of lessons rated as having low, medium, and high quality of mathematical content (as rated by a team of mathematicians who did not know which lessons came from which countries).

Technorati tags: education, math, content quality

Typing mathematics in blogs

I was talking to couple of friends about blogs in mathematics classrooms and they both asked "How do you typeset math in HTML". I had to say I had no idea. We use LaTeX, but that doesn't translate to HTML, unfortunately. So, instead of asking around, I tried to look it up. The is the first thing I discovered, courtesy of Clark Grubb , and I guess it should be accessible to high school and college students. Scroll down to find LaTeX lover's solution :)

The character entities can all be invoked via their numeric or alphabetic names. For example, both

x &#8712; A and x ∈ A

will display

x ∈ A

A radical can be done. The HTML

&radic;<span style="text-decoration: overline">a + b</span>

will display as

√a + b



Below is a partial list of the character entities specificied in the W3C Character Entity References for HTML 4. It includes most of the characters of interest to the mathematician.

Generic Character Entities

#	alpha	sym	IE6	FF	description
38	amp	&	Y	Y	ampersand
160	nbsp		Y	Y	no-break space = non-breaking space
8194	ensp		N	?	en space
8195	emsp		N	?	em space
8201	thinsp		N	?	thin space
8204	zwnj	‌	?	?	zero width non-joiner
8205	zwj	‍	?	?	zero width joiner
8211	ndash	–	Y	Y	en dash
8212	mdash	—	Y	Y	em dash
8230	hellip	…	Y	Y	horizontal ellipsis = three dot leader

Mathematical Character Entities

#	alpha	sym	IE6	FF	description
60	lt	<	Y	Y	less-than sign
62	gt	>	Y	Y	greater-than sign
176	deg	°	Y	Y	degree sign
177	plusmn	±	Y	Y	plus-minus sign = plus-or-minus sign
215	times	×	Y	Y	multiplication sign
216	Oslash	Ø	Y	Y	latin capital letter O with stroke = latin capital letter O slash
247	divide	÷	Y	Y	division sign
8226	bull	•	Y	Y	bullet = black small circle
8465	image	ℑ	N	Y	blackletter capital I = imaginary part
8472	weierp	℘	N	Y	script capital P = power set = Weierstrass p
8476	real	ℜ	N	Y	blackletter capital R = real part symbol
8501	alefsym	ℵ	N	Y	alef symbol = first transfinite cardinal
8592	larr	←	Y	Y	leftwards arrow
8593	uarr	↑	Y	Y	upwards arrow
8594	rarr	→	Y	Y	rightwards arrow
8595	darr	↓	Y	Y	downwards arrow
8596	harr	↔	Y	Y	left right arrow
8709	empty	∅	N	Y	empty set = null set = diameter
8711	nabla	∇	Y	Y	nabla = backward difference
8712	isin	∈	Y	Y	element of
8713	notin	∉	N	Y	not an element of
8715	ni	∋	Y	Y	contains as member
8719	prod	∏	Y	Y	n-ary product = product sign
8721	sum	∑	Y	Y	n-ary sumation
8722	minus	−	Y	Y	minus sign
8730	radic	√	Y	Y	square root = radical sign
8734	infin	∞	Y	Y	infinity
8736	ang	∠	Y	Y	angle
8746	cup	∪	Y	Y	union = cup
8747	int	∫	Y	Y	integral
8764	sim	∼	Y	Y	tilde operator = varies with = similar to
8773	cong	≅	N	Y	approximately equal to
8776	asymp	≈	Y	Y	almost equal to = asymptotic to
8800	ne	≠	Y	Y	not equal to
8801	equiv	≡	Y	Y	identical to
8804	le	≤	Y	Y	less-than or equal to
8805	ge	≥	Y	Y	greater-than or equal to
8834	sub	⊂	Y	Y	subset of
8835	sup	⊃	Y	Y	superset of
8836	nsub	⊄	N	Y	not a subset of
8838	sube	⊆	Y	Y	subset of or equal to
8839	supe	⊇	Y	Y	superset of or equal to
8853	oplus	⊕	Y	Y	circled plus = direct sum
8855	otimes	⊗	N	Y	circled times = vector product
8869	perp	⊥	Y	Y	up tack = orthogonal to = perpendicular
8901	sdot	⋅	N	Y	dot operator
8968	lceil	⌈	N	Y	left ceiling = apl upstile
8969	rceil	⌉	N	Y	right ceiling
8970	lfloor	⌊	N	Y	left floor = apl downstile
8971	rfloor	⌋	N	Y	right floor
9001	lang	〈	N	Y	left-pointing angle bracket = bra
9002	rang	〉	N	Y	right-pointing angle bracket = ket

Greek Letter Character Entities

Both IE6 and Firefox implement all of these characters.

#	alpha	sym	description
913	Alpha	Α	greek capital letter alpha
914	Beta	Β	greek capital letter beta
915	Gamma	Γ	greek capital letter gamma
916	Delta	Δ	greek capital letter delta
917	Epsilon	Ε	greek capital letter epsilon
918	Zeta	Ζ	greek capital letter zeta
919	Eta	Η	greek capital letter eta
920	Theta	Θ	greek capital letter theta
921	Iota	Ι	greek capital letter iota
922	Kappa	Κ	greek capital letter kappa
923	Lambda	Λ	greek capital letter lambda
924	Mu	Μ	greek capital letter mu
925	Nu	Ν	greek capital letter nu
926	Xi	Ξ	greek capital letter xi
927	Omicron	Ο	greek capital letter omicron
928	Pi	Π	greek capital letter pi
929	Rho	Ρ	greek capital letter rho there is no Sigmaf, and no U+03A2 character either
931	Sigma	Σ	greek capital letter sigma
932	Tau	Τ	greek capital letter tau
933	Upsilon	Υ	greek capital letter upsilon
934	Phi	Φ	greek capital letter phi
935	Chi	Χ	greek capital letter chi
936	Psi	Ψ	greek capital letter psi
937	Omega	Ω	greek capital letter omega
945	alpha	α	greek small letter alpha
946	beta	β	greek small letter beta
947	gamma	γ	greek small letter gamma
948	delta	δ	greek small letter delta
949	epsilon	ε	greek small letter epsilon
950	zeta	ζ	greek small letter zeta
951	eta	η	greek small letter eta
952	theta	θ	greek small letter theta
953	iota	ι	greek small letter iota
954	kappa	κ	greek small letter kappa
955	lambda	λ	greek small letter lambda
956	mu	μ	greek small letter mu
957	nu	ν	greek small letter nu
958	xi	ξ	greek small letter xi
959	omicron	ο	greek small letter omicron
960	pi	π	greek small letter pi
961	rho	ρ	greek small letter rho
962	sigmaf	ς	greek small letter final sigma
963	sigma	σ	greek small letter sigma
964	tau	τ	greek small letter tau
965	upsilon	υ	greek small letter upsilon
966	phi	φ	greek small letter phi
967	chi	χ	greek small letter chi
968	psi	ψ	greek small letter psi
969	omega	ω	greek small letter omega

Immediately after stealing all this code from Clark (thanks a bunch), I ran into Flip Tomato's Blog. This was an absolute gem, as it contained the following link: mimeTeX. Score! I better get this running on my computers. If anybody has better solutions, let me know.

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