Blog of a math teacher

I got copies of textbooks for the classes which I'll be teaching later in November. For the 4-semester Algebra 1 course, they use McDougal Littell's Algebra text, and looking over the book, I find it a bit disturbing that they give the quadratic formula before they even do polynomial multiplication, let alone factoring. The end result is that the quadratic formula ends up seeming like it comes out of nowhere.

On the other hand, they defer talking about percentages until they deal with ratios, direct variance and rational expressions, which has some benefits, I think.

I had a stroke of genius today. I'm not assigning homework for the Saturday class this week. That will save me writing an assignment for Saturday (the primary goal), and will also let me adjust for the fact that we're spending too little time on new material in class by eliminating homework review for that week (and hopefully allow me to avoid the problem in the future).

I will have a quiz on the 15th: It will be a randomly selected subset of questions from the 8th so that the students will have some incentive to make sure that they understand the material on the test that they may have gotten wrong. I'll put up a copy of the test on the class website after it's been administered.

Here's what I've found in my first week of occasional playing (but no manual reading):

• Sketchpad wants to give lengths with units. I don't want lengths with units. I want to tell sketchpad, segment AB has length 1, and have all other lengths given in terms of that. As near as I can tell, this isn't possible.
• Some sketchpad objects don't seem to get written to the HTML/Java pages.
• It doesn't seem willing to give me angle measurements greater than 180 degrees.
• It doesn't have a graphic export. OK, I'll do a screen grab and save a bitmap, but it would be nice to have export as EPS or WMF or something.

I'm almost inclined to work with this thing a bit and write my own equivalent program. It's really not that complicated a piece of software...

Another grad student needed some help understanding factor groups and Lagrange's theorem earlier today. I found that explaining this to her helped me deepen my own understanding. Very handy as I keep pushing through the 500-level abstract homework that I'm so far behind on.

From Steve Harvey's column:

The World Series is over but not the speculation about which jinxes prompted the dramatic collapses of the Red Sox and Cubs in post-season play. I'm surprised that no one has mentioned the dreaded O.C. Curse.

It befell manager Dusty Baker, whose teams blew big leads in the World Series last year and the league playoffs this year.

Baker has had nothing but bad luck ever since he declared on the eve of the 2002 Giants-Angels World Series: "Our focus is 'Beat L.A.' " When it was pointed out that Anaheim isn't in L.A., Baker insisted, "It's Southern L.A." You insult Anaheim that way, you pay for it.

RCML Musings has an interesting article that I found recently about the curent reform movement in mathematics. I'm posting this mostly as a bookmarker. I'll update this post later and move it back to the top when I've had a chance to really read it and respond to it. Probably sometime next week. I'm in heavy-duty test prep mode (I'm way behind in my Abstract class and have to average 12 problems per day, every day, to catch up).

So last night was my group's turn to teach a lesson in my math teaching class.

Do you see the problem yet?

My group's turn.

What a completely unrealistic experience. What are the odds of group teaching a lesson in my career? Close to zero. And if that group is all math teachers? I'd have to say even closer to zero. I'd say zero without qualification, but I suppose there is some tiny chance that it could happen. In bizarro world.

OK, sure we have 30 students in the class so each person can't have a whole class period, but the way I figure it, if we have each student teach a 40 minute lesson, we can in 10 weeks have each person teach. My organization plan for the class would be to have each person teach for 40 minutes, then we'd take fifteen minutes to discuss and critique that student's lesson plan. Now that would be a valuable learning experience. And it's not like this idea comes out of nowhere: There's a program called Lesson Study that iirc originated in Japan, and provides a means for teachers to engage in a continuous professional development and develop their lesson planning and implementation techniques.

So here's an interesting idea: Teachers trained with this sort of lesson study, and given an opportunity to engage in it as part of their training, might well evangelize for it when they're actually out training. So we'd not only be training teachers, but we'd be producing teachers who would have a goal of improving the overall quality of teaching in their schools and tools that they could use to implement this.

Here's yet another story about an experienced teacher being deemed unqualified by NCLB standards. Does Michigan not have a test option to determine qualification? It would seem that this would solve the problem at least partially.

And it seems to have worked a little less well this time around... I suppose that a big part of its success was novelty the first time around. I guess it's not that it didn't work, just that I'm less excited about it. So maybe it's just me. I have to be wary of my desire to always try new things and avoid repetition.

Well not just here, but in the blogosphere. I'm realizing that there's a handful of teachers who are blogging and reading each other's blogs that makes for a nice supportive community, especially since so many of us are new teachers. One of the more interesting things popped up in the comments on Eric Gjovaag's blog where first, Katie revealed that she had used one of Eric's other sites in a paper she wrote in high school. I took a quick look and discovered, that I too had visited his site before I ever saw his blog. It's neat how these things work out, and while in the grand scheme of things it might be insignificant, I like to think that this is a sign that we've got something very special going here.

I'd still keep a journal of my teaching, even without the internet, but the way that I can let people read my journal, without risking too much of myself, and get their feedback makes this an especially great way to reflect on the process of teaching.

I received my copy of geometer's sketchpad in the mail today. Installed it immediately and it looks reasonably intuitive, although I will spend some time with the manual. Perhaps on one of many trips over the coming months (the joys of a long-distance relationship). My most immediate purpose is to be able to quickly create illustrations for handouts/quizzes, etc. But beyond that, the java export looks like it could be very handy indeed.

I think that between students beginning to "get" the material that we're going over and a realization that complaining about the workload doesn't change things has resulted in us spending more time learning and less time complaining. Third period was a lot better behaved, I think partly because of the above and partly because of being scolded earlier in the day by one of the administrators.

Next week is a shortened day which I hadn't been really given sufficient advance warning of. Now I have to decide how much of the class we'll be able to do. The geometry kids I had told that my plan was to not lecture on the material at all, but rather to give them a reading assignment and homework based on that. I also have parent conferences next week so I'll want to get grade reports for all students assembled so that they can see how their students are doing and in what areas they're falling behind.

I can't find the original blog posting (so if it's yours, leave me a comment and I'll properly link it), but I read something someone wrote about using textbooks in science classes that I think can be translated really well to math classes as well:

The teacher in question talked about how reading science texts is an alien experience for most students (the same thing applies to math), and so she has her students read through passages in the textbook together and she teaches them how to read these dense texts and how to pick out the important information.

Talking about this with classmates today and on Wednesday, the consensus seemed to be that this was a really good idea. In many math classes that we've seen (and experienced) the textbook is little more than a repository for homework problems.

I suspect that the ability to read a math textbook can make the difference between a student being a student who "gets" math and a student who does not.

So if you're that original blogger and you see this, please leave me some comments. I'd love to discuss this some more.

Update: I heard back from Ms Frizzle who was the teacher to whom I was referring.

Note to self: When I'm teaching full-time, I need to stagger quizzes and tests in classes so that I don't have 150 tests or quizzes to grade all at once. I've only done 9 tests and already I'm sick of grading. At least this time around the students are doing a lot better. Plus they tend to come out as all or nothing on getting the question right. It's quite a challenge sometimes to follow some students' work to figure out where they went wrong and decide how much partial credit to give them.

This Chicago Tribune article details one of the most stunning screw-ups imaginable in education:

State law mandates that beginning teachers get special training and mentoring programs to obtain a standard teaching certificate required after four years on the job. But those programs don't exist.

And it's not just that some school districts lack the program. Or that many school programs lack the programs. There are no programs at all.

There's also the question of, what sort of mentoring program makes sense for a teacher who has been teaching for four years. Implementing a one-size-fits-all program is a recipe for disaster, as those teachers who have more experience will be doubtless annoyed if they have to go over the same "here's how to write a lesson plan" material that beginning teachers implement.

Once again, one of those situations where a well-meaning regulation at the state level causes havoc because it's not thought through.

So you want to be a science teacher makes the comment, "If I'm lucky enough to get a good physics placement, I'll be able to teach without worrying quite so much about the standards."

I've got two minds on standards. On the one hand, I think that it's a very good idea, to come up with a set of topics that should be addressed in a standard high school course (there are too many classes in math, at least, in urban areas which go by high-level names but offer low-level content). But on the other hand, the standards have to be properly written and designed. Illinois' standards have whole sections which don't correspond to the high school curriculum at all, while other portions of the curriculum are underrepresented (Trigonometry) or omitted entirely (Calculus/Analytic Geometry). Even those areas "addressed" are inadequately addressed.

Then there's the question of inadequately addressing the whole matter of how standards should be used. I think a big part of this comes down to the faculty preparing teachers (and a lot of teachers themselves) haven't had to deal with standards, so they tend to viewed as "look through the list to find something that kinda sorta matches what I'm teaching today and list that on my lesson plan" kind of thing. Ideally, the standards, lesson plans and curricular materials should all automatically flow out of each other. I've not looked too closely, but it looks like the California standards do a better job of this (at least in terms of having the standards more closely align with the standard course offerings).

This is the question that AssortedStuff asks in his blog today, and while I agree partly with him, I find myself at least partially agreeing (how strange) with NCLB in that teachers must demonstrate competence in their subject matter. There are too many teachers out there that just don't know the material that they're supposed to teach. If you're in a math classroom, and you don't know how to add fractions, then you should consider a career selling insurance instead.

I'll go a step farther and say that math teachers need to know not only the subject matter that they teach but at least two years of progress in the subject area beyond what they teach. It's not enough to know what's in the book, you need to know what the book is preparing students to do in their next class.

And this applies to the people writing the books too. I've ranted before (and I'll rant again) about "geometry" problems like:

Angles 1 and 2 are supplementary. If angle 1 has measure 2x+1 and angle 2 has measure x-2, what is the measure of angle 1 in degrees.

That's not a geometry problem and as far as I know it doesn't correspond to any actual use of geometry. It's an algebra review pretending to be a geometry problem. Maybe if we dumped problems like this we wouldn't have to offer geometry-without-proofs classes.

So I would argue that we need to have educated people writing the standards and we need to have teachers who know the material inside and out. Frankly, I think that the tests that I've seen are in fact inadequate. The test that I took for certification only verified that I can do high school mathematics. It didn't test any sort of understanding or ability to connect knowledge within different subtopics in the discipline or across disciplines. There's no point, for example, asking Calculus questions unless you also address how calculus and physics are related.

But there is the very real issue of ability to teach, and this is an ability that is gained only in one way: By actually being in front of a classroom and teaching. And I have yet to see any multiple-choice test that can measure that. Apparently, at one point in California, there was an actual simulated classroom test for credentialing, with college students pretending to be the (unruly) high school students a teacher might face. Now there's an idea that's worth resurrecting.

So bottom line, I agree with Assorted Stuff that the standards for "highly qualified" are insufficient, but I disagree in that I think that demonstration of subject area competence is necessary.

... are working a bit more smoothly. I'm still talking too much, but it's a good discipline for me. It seems like the students are also getting used to this structure and rolling with it.

There allegedly is a better correspondence between the sample test and the midterm tomorrow so I'm cautiously optimistic that this will see higher exam scores on the test this week.

I did have the mystery student who showed up for the test but has missed every quiz this semester appear in class today, so I can't just delete his row in Excel. He is in fact real.

I'm working on my philosophy of education for my portfolio and I've managed to make my response to the "how does this relate to my life?" question into a nice bullet point for the philosophy:

• ... the question will not be “how does this relate to my life?” but “how do I make my life relate to this?”

In this New York Times article about improving test scores, there's an interesting observation:

The suggestion that city schools are on the upswing put Chancellor Joel I. Klein, who is overhauling them, in a tricky position. While the chancellor's critics pounced upon the higher scores as evidence that the school system did not need such an overhaul, some of his allies acknowledged that he would now be under even more pressure to show gains next spring.

So now Klein is forced to argue that the test scores are not necessarily reliable, which raises the question of whether the whole concept of using test scores to evaluate schools is a good idea to begin with.

Eric Gjovaag lets us know today that the teachers in his district have voted to return to work on Wednesday, so his teaching blog will finally be about teaching rather than about striking.

Meanwhile, in Chicago, the teacher's union has voted to reject the current contract offer and has entered mediation. A strike may result, but the hope is that the district and union can come to an agreement.

So, I would say that today's teaching with the notecards and round-robin questioning worked well, although I need to work on refining it a bit: I don't really want to take volunteered answers, but in doing that, I may miss out on additional questions that come up. I have one student who I can't seem to reach. She shows up every class (or mostly), but she consistently scores around 50% on quizzes and homework and her attitude in class today seemed to be as if it was my problem that she wasn't able to answer even the simplified questions that I asked. I'll continue the practice on Thursday and see how it goes, but I think that it shows good promise.

So I think that I'm going to really work towards getting everyone thinking during class tomorrow, so one thing that I've done is that I've made up index cards for each class with each student's name on it. I'll work through problems typical of the test using the cards to call on individual students and get them to think about the problems. Maybe it will work. I'll know tomorrow.

I'm about halfwy through grading last week's algebra quizzes and the current average is 10/25. The week before the average was 15/25. They've got a midterm on Friday so there's a lot to take care of in the interim. I'm thinking of making the two discussion sections this week a boot camp-style review session.

But I think I'm getting better. I know that there are things that I'm not doing well, and I need to work on keeping myself from getting angry with the students, but I've found that not talking over the students, but rather just stopping talking and, if necessary, walking up to the talking students and standing in front of them, does wonders.

The perennial question from students. And the answer I wish I'd given:

Perhaps never. At least not with the everyday life that you lead now. But what you should aspire to, is to lead a life that calls on you to use your mind in new ways. To learn not only what we're doing today, but to want to explore beyond what you know now, perhaps because you use it, but perhaps because you enjoy learning and expaning your mind.

Or maybe I'm just a hopeless dreamer.

Another good Open Source Politics article on math education. It seems a lot of critics of math education are caught up in the idea that math should be about doing computation. This is a silly view of what math is about. After all, few people, even in mathematics, do a lot of computation by hand on a daily basis. Why bother when computers and calculators will do the job much better. What mathematics is really about is a way of thinking. It's a thought provoking article, but doesn't really address the big issue in contemporary mathematics education: That we don't really do a good job of teaching computation or mathematical thinking at the secondary level. Part of that is the testing. We're stuck about a generation behind in how we test mathematical understanding (e.g., when I taught remedial algebra I last year, the first quiz was supposed to be completed without calculators. Why? Is it really essential that students be able to do, e.g., 0.0004/0.2 without a calculator to do well in math? Hell, I can do that, but I'll as often as not whip out a calculator for something like that just to make sure that I don't inadvertently misplace my decimal point. Much of the standardized testing falls into the same trap. If it were up to me, the math sections of standardized tests would be based entirely on finding the underlying formula for word problems (or very close to it).

On a related front, something that I've seen in geometry textbooks which leaves me perplexed are questions along the lines of:

Angles 1 and 2 are supplementary. If angle 1 has measure x and angle 2 has measure 2x + 3, what is the measure of angle 2?

Is there any case outside of a contrived problem like this where this will come up? I have a vague notion that I saw questions like this in one of the ACT prep books that I looked at, but this has no relation to how someone actually works with supplementary angles. It's not a geometry problem, it's an algebra problem, and while students should be able to solve this, the contrived nature of the problem is such that its ex nihilo nature outweighs the benefit of the algebra review, I think.

The third part of the Open Source Politics series on NCLB has been published. Voice your outrage once you've finished reading it.

Hmm, I actually happened to look at the links which appear at the top of this page and noticed this one which promises "Homework problems solved at reasonable prices."

Visiting the site, I found:

We know that doing homework isn't a lot of fun, so that's why we'll do it for you!

Mark T~ "This service is great. I had 10 problems in math and they solved all of them in a few hours. Now I have a shot at an A."

Howard S~ "Tutors are great when I need someone to teach me things about the subject.  But when I just want the answers, Homework Solver gives me them fast! PayPal makes it soooo easy!"

What a horrible idea.

Or not.

Personally, I think that it's a good idea to assign homework problems for which the answers are available by some means. Homework should not be an end in itself, but rather a route to learning. If a student does 10 problems incorrectly, they won't learn. If they do the problems and get the answers so they can check whether they have the answers correct, that's a bit better.

But of course, there's the issue of grades and measuring whether students have actually learned. On that front, I'm not entirely convinced that grading homework is a good or useful thing, and services like this Homework Solver group actually will ultimately force teachers to be a bit more thoughtful in how learning through homework is evaluated. My approach is to grade homework on a "was it done?" basis, but then have quizzes that pull from the homework to verify that students managed to learn something from the homework. Ideally, the quizzes would call for students to synthesize ideas from the homework, but that might be asking a bit much at the secondary level. Or perhaps I could ease them into it after a few weeks...

We got feedback on our presentations today. I got great reviews. I love getting affirmation. There's a lesson as a teacher in that too: I don't think that I do as good of a job as I should in giving affirmation to my students. In fact while grading the quiz, I found myself "shouting on the paper" at some of the errors the kids made.

So for my math teaching methods class, we're doing group presentations. My group is presenting on rational and irrational numbers. I came up with the (I thought) brilliant idea of organizing the lesson so that we would each teach the subject as if it were being brought up in a different class: Algebra, Geometry, PreCalc and Calculus. I had a great lesson where I would introduce Taylor's Theorem as a motivation towards doing the Taylor expansion of e^x, thence finding an infinite series definition of e and then finally proving that e was irrational.

The prof vetos this idea. Suddenly we're teaching a single lesson to a 10th grade class. Everyone else's material can be easily adapted to this change, but mine not so much. I've got five minutes to come up with an alternative.

Which I do.

Instead of that proof, I decide that we can look at countability of infinite sets. The Math Circle in Boston does this with 6th graders so it's certainly within the grasp of 10th graders (although I intentionally move the proof that the cardinality of the set of reals is not equal to the cardinality of rationals into the "if time" section of the lesson--I'm not at the level of over-learning with this concept to be as comfortable with it as I'd like).

So I'm pretty happy with how I managed to dodge the bullet, but generally annoyed with how the class as a whole is being run. I'll be so happy to be done with all this education stuff and finally start teaching.

Time time time... I've been running a bit short. So a backlog of thoughts will follow forthwith

Thanks to Katherine Fielding for the link to Pursuing Happiness, Through Hard Work (washingtonpost.com).

I agree that the idea behind education should be to challenge the students. It's a matter of really pushing the envelope, making them do things that they think are beyond them. When I told the 10th graders on Saturday that we'd be doing trig, there were (predictable) protests, but I hope that as they work through their assignments, they'll see that they can do it. Frankly, I think that of all the secondary math curriculum, trig is the easiest stuff. All you need to know is the Pythagorean Theorem, some stuff about similar triangles and one simple diagram (which I can't show you because I don't particularly want to pay for hosting), and you're good to go.

AssortedStuff links to an editorial by a bilingual teacher in Oregon who, despite being a finalist for Teacher of the Year, and winning a National Geographic Society grant, was declared under-qualified by NCLB regulations. And he's not alone. There are at least two national board certified teachers who also got the same categorization. Read it yourself then voice your outrage.

... comes courtesy of Apt. 11D.

Despite my best efforts, I was a bit late this morning courtesy of bad weather and worse traffic. I noticed today that the questions asked by the 8a section are almost completely orthogonal to those asked by the 9a section. Today was largely working with radicals. There was one problem (cube root of 4 times the 4th root of 3) that I didn't have an immediate solution for, but I think that this was a good thing, in that it helped the students see how to think about the problem rather than just putting the problem into a box that says, this questions belongs in category A, so we use algorithim A to find the answer.

During my tutoring hour, I had a student come by with a questions that I could not answer (nor could I see from the chapter how it would be solved): The problem was to find the remainder when x¹⁰+x⁸ is divided by x-1, without using synthetic or long division. From the chapter, it would seem that it is somehow related to the fact that the polynomial has zeroes of 0, i and -i, but I was baffled at how to solve it. I sent her off to find a precalc TA. I think that I might stop by the bookstore and spend $50 on my own copy of the book. The way that precalc is taught is so different now, it would behoove me to re-learn it myself.

I've noticed that my traffic has gotten high enough that I've cracked into the top 50 for Nedstat's Secondary School rankings (at #26 today). I doubt that I can ever make #1 (my traffic would need to go about by about 2000%!), but cracking the top 20, or even top 10 might well be do-able. So thanks to everyone who stops by. Feel free to keep it up.

From Behind the Teacher's Desk links to this article at the Washington Post on grading behavior. My approach is based partly on a grading scale from an article in Mathematics Teacher in an article called, "Oral Quiz" (I don't have the citation handy, unfortunately). For my Saturday class which meets once a week, I have four points that are given out at each class meeting:

• Was the student on time for class?
• Did the student have all required materials for class?
• Did the student ask a question, answer a question, or otherwise contribute to the class discussion?
• Did the student behave appropriately in class?

This is based on the things that I value in my student's participation in class. There's one point for each item. The total points for class participation add up to 10% of the grade. Enough to make a difference of one letter grade, but not enough to pass a failing student, or make an otherwise dramatic difference in class. I'm not necessarily crazy about the fact that half the points are available just for showing up on time with class materials, and I'm using this system for the first time this year, so it remains to be seen how effective it will be, but I'm hoping that with students aware that their participation in class does effect their grade, it will positively shape their participation.

AssortedStuff tells a story about contradictory requirements in NCLB that would be funny if it weren't our kids' futures on the line here. Once you begin to feel the outrage, follow this link to get the message to Washington.

I had hoped to post this yesterday, but I ended up having home internet problems. Still do, so I'm sitting at Border's using a hotspot to get on the net (it is, however, pleasantly fast).

So after all the last-minute schedule changes and problems (the classroom that I should have had at the university was locked, so I ended up in the room next door), I managed to get things moving. The first class was trig with the 10th graders. I was unable to do demonstrations of how to do the constructions of the diagram on the overhead since none of my markers worked in that context (I really need to get one of those blackboard-style compasses), and we ended up spending more time constructing the diagram of the six trig functions than I really wanted to. Next time I teach this lesson, I'll just let them draw the lines freehand. I hope I managed to cover everything adequately for them to understand, but I'll need to get the online notes up soon.

Then it was advanced algebra/geometry for two classes of 9th graders. The first class was good, and there were some who actually knew their way around the Boolean algebra that we did already (one asked about the two-way implication, which we'll do more with next week). The second class seemed a bit of a nightmare. They were harder to keep quiet, and harder to keep interested.

I've not had a chance to review the videotape of the lessons. I'm guessing that I'll do that tomorrow afternoon or evening. My general sense, though, is that I lectured far too much and didn't have enough in-class work on the material. I fear that our curriculum is too ambitious for the time that we have.

As if this article weren't scary enough on its own merits, skim down to the end of the story. It's always the math teacher. I don't think I'll bring food or drink to my classroom.

My classroom schedule for Saturday has changed again. Now I've got three classes instead of two. So much for having a free period to observe other teachers. But on the plus side, my class size has decreased. It looks like I'll have to buy my own sets of compasses and protractors for the classroom as I've not heard anything back from the school, but now my total is quite a bit smaller. I can get away with buying 20 sets rather than 30. Maybe I can work out a bulk purchase deal with the manager at Target or Walgreen's.

There's a newly linked blog on the left, assigned seat. It's a captivating read, although more than a little frightening to me as someone soon to have his own classroom. But a big part of why I started doing all of this was to be able to get other peoples' experiences and to share my own.

The news-based postings here, by the way, will doubtless subside as my classroom time increases.

Listening to NPR last night, I caught a story about the garbage strike in Chicago being about to end. And for the first time, I heard an accurate statement of what garbagemen make. Previously I had heard that they were currently getting $21/hour. WTF? that works out to about $42k/year, which is more than beginning teachers in Chicago make.

But the NPR story said that garbagemen make $12-21/hour. Well, that makes a lot more sense all of a sudden. At the top of your career as a garbageman you might make more than a beginning teacher, but the teacher who's been working as long as you have will doubtless still be making more.

The other disturbing part of the story was the mention that the mayor of Chicago was considering sueing the union for the costs that were incurred because of the strike. But shouldn't the private garbage collection firms also share part of the blame for this? This really bugs me because it's happening in other spheres as well. Eric Gjovaag reports in his blog about a plan by parents in his district to sue the striking teachers, ignoring the culpability of the district in that strike.

Not that there were that many to begin with, but I've switched to Haloscan for my commenting. It's much nicer than the old documentroot commenting system. All the old comments are gone, but as I said there weren't that many and most of the commented posts have scrolled off anyway.

AssortedStuff has written an interesting commentary on part 3 and part 4 of the Milwaukee Journal Sentinel's series on math education (his comments on parts 1 and 2 are here. I agree that not everyone needs Calculus, although I think that I would redirect the curriculum a bit differently than he has. In particular, I think that discrete mathematics would be a much better culmination to the high school math curricula for almost all students. This is the sort of mathematics that students need to have much more frequently, whether its in being able to estimate the length of a roll given the inner and outer radii plus sheet thickness, or understanding that the lottery is a sucker's game (although if we succeed in the latter, we could jeopardize education funding in an awful lot of states).

I would also (as I noted below), like to see more statistics in the high school curriculum. I think that it's not really necessary for most students to be able to manually calculate the slope and intercept of a regression line, but they really should be able to understand what it means and do it in Excel. Actually, a good course in math for spreadsheets would be an excellent addition to almost any school curriculum K-12 and undergrad.

I've got my Saturday schedule now. One nice thing is that I have first period free. I'll probably come in early anyway and watch some of the other teachers during that period. I should head over to the other building today or tomorrow and scope out my room in advance. I realyl hope I don't have those nasty interlocking desks which make moving around the classroom so difficult. It looks like it'll be one class each of advanced algebra/geometry and trig/calc. About 24 students per class.

I think the discussion part of today's classes went reasonably well: I've managed to get the kids to at least offer enough questions to fill the pre-quiz period. The secret, of course, is to wait for them to volunteer the questions (and wait and wait, as necessary). The first few quizzes that I've graded have been a bit disappointing, although I think that a lot of these are the kids who skipped Tuesday's class. I wonder if they're making it to the lectures...

That's the title of today's exercise in my Ed class. We were to do a presentation of our first five minutes in front of the class. The content of my presentation doesn't seem so noteworthy in the summation: I have two class rules:

1. Respect
2. Responsibility

and I ask students to help me understand what these demand of them and of me.

But I was on tonight. It was like I was channeling Edward James Olmos or something. I had great classroom presence. I hope I can do it again on Saturday morning when the room will be filled with high school students.

There were some good uses of powerpoint (for the introduction of a foreign language class), and some ultimately empty (for a geometry class). My favorite intro had the (freshman English) teacher confessing to a reading addiction and playing the Moxy Fruvous song, "My Baby Loves a Bunch of Authors."

Take a look at the link list here and see if you can figure out why I'm so honored. Maybe I don't write so clear.

The New York Times (free registration required) talks about some revisions in the Math A Regents exam in New York City. I've been a bit interested in some of this myself as I've been looking at standards for math while putting together lesson plans. There's a whole category in my state (statistics) which isn't neatly addressed by the standard Algebra-Geometry-Precalc-Calculus college prep sequence. The ACT exam similarly covers this material which isn't addressed in the standard curriculum. I agree that this is important material, but it seems like the curriculum and the standards are far from being aligned, even though the standards I'm looking at have been in place for nearly a decade. I may be wrong, but I suspect that the curricular materials don't adequately address the statistical knowledge listed in the state standards or the ACT exam.

The Times article is a bit sparse on the details, but it seems to me that the problem here is similar to what I'm seeing with statistics here: The curriculum and the standards are still not adequately aligned. I fear that what might be happening in New York is that they've decided to adjust the standards to match the curriculum rather than looking at both and adjusting the curriculum if necessary.

AssortedStuff has a good reflection on attrition in teaching. My own stance on mentoring is that, should the district that I end up working at next year not have a mentoring program, I will seek out a mentor anyway.

In a discussion about the golden mean in my math teaching class, it was brought up that the golden mean had the wonderful property that phi=1+1/phi=1+1/(1+1/phi) which the student instructor followed up by saying, "you can check this on your calculators".

And to which, I say, wrong. This isn't a good use of calculators. Instead, we should use substitution to get from phi=1+1/phi to phi=1+1/(1+1/phi) (and of course extending into infinity).

Especially, since we then went into the series of continued fractions:

1+1/1=2
1+1/(1+1/1)=3/2
1+(1+1/(1+1/1))=5/3
1+(1+(1+1/(1+1/1)))=8/5
etc. (can anyone recommend a good free, anonymous site for in-line image hosting so that I can replace nasty in-text equations with better-quality gif files?)

Anyway, this sequence will eventually converge on phi, and the limit proof is easier to get to if you use substitution (as is the proof that each successive continued fraction simplifies to a ratio of adjacent Fibonacci numbers).

I finally managed to make contact with the people who are organizing a graduate student union on campus. Being the pro-union type that I am, I immediately offered to help (much to the surprise and delight of the organizer who I spoke with). That's one good thing about having a monster shared office... I get a higher than average number of opportunities to interact with other TAs.

And to forestall any anti-union folks who will doubtless comment (assuming I get the critical mass of readers to make that happen), why am I pro-union? Because I believe that unions have played a critical role in allowing working people to live in dignity. It's worth noting that the decline of union membership since the 70s directly parallels the decline in real wages and employer-provided benefits as well as the increase in the length of the work week.

Like not working 80 hours a week? Thank unions for that. Like having insurance? Unions. Safe working conditions? Unions.

Working America, by the way, is a worthwhile organization to join for those of you who are ineligible to join unions but care about these sorts of issues. Current lobbying efforts include preserving overtime pay and preventing the gutting of medicare.

I've been reading, bit by bit, Jerome C. Yanoff's The Excellent Teacher's Handbook. While it's more prescriptive at times than I think is justified, it's by and large a very thought-provoking, and I think, helpful book. As I read this, I'll pick occasional questions that I found interesting. The first that I'll reflect on is question 3, "A Teacher's Primary Goal."

The choices given were:

1. to make a living
2. to retire with a good pension
3. to touch the lives of many children
4. to feel good about the work being done

my choice was "to touch the lives of many children", but Yanoff suggests instead, "to retire with a good pension". He gives a good justification for this: The teacher must always be aware that he is in the career for the long haul. Teachers who don't make it to retirement, "use up their energy fighting losing battles, become unhappy with their work, and burn out."

Of course, in a job interview situation, it would be wise to phrase this more diplomatically.

Are there any public school teachers or administrators out there who are supportive of No Child Left Behind? Most of the pro-NCLB commentary I've seen comes from people outside the education system, and I would estimate that half of that tends to be more a critique of administrative attempts to work aroudn NCLB rather than outright endorsement.

For that matter, how do the NCLB supporters rationalize the widespread opposition to NCLB by those in education? Do you simply assume that all educators are more concerned with protecting the status quo than improving education? That those on the front lines have less understanding of the issues facing schools than do politicians in Washington?

There are schools desparately in need of reform. Only a fool would argue otherwise. The question really is how best to accomplish this reform, and I've not seen any indication that NCLB is anything close to the way to do it.

AssortedStuff has a nice overview of a pair of articles from the Milwaukee Journal Sentinel about math instruction which mirrors a lot of what I've found to be true: That both Mathematically Correct and their opponents are wrong. There's a balance that needs to be achieved, much like the balance between phonics and whole language in reading instruction, for math instruction to be top quality. The Mathematically Correct people, for example, rail against the use of manipulatives and discovery learning, failing to see their value in math education (in my experience, many MC people tend to have the view that they learned math without these things, so why can't everyone else?). In the other corner, there are too many incompetent math teachers who don't move beyond the use of manipulatives or make sure that discovery learning ends with, well, discovering something. A classroom that I watched last week had the students using algebra tiles to find some strategies for factoring trinomials, but a lack of real guidance for the students hindered this somewhat. No possible strategies were offered for working with the tiles (like arranging the x² tiles and the constant tiles into rectangles), so many of the students spent time aimlessly rearranging the tiles. One might argue that what I'm suggesting is awfully close to the ac method for factoring, and in fact it is. That's sort of the point. The value in the algebra tiles is to give students an idea of what factoring means, and, that's going to expand beyond polynomial factorization to also factoring integers.

Kimberly Swygert has an interesting post about College Summit, an organization dedicated towards getting middle-of-the-pack students to college. Looking at their web site, it appears that they're largely about getting students who are college material, but don't act on it to apply. It's interesting, but more interesting is a group that I've just recently learned about, AVID, which is geared more towards raising the academic performance of the middle-of-the-pack students. I think the two groups are rather complementary, and well worth checking out.

This is exactly the sort of thing that I have as a goal as a teacher: I want to move kids who don't realize that college is an option into the college track. My big concern on this front is that the hyperinflation of college tuition combined with poor state support for public higher education will make college an impossible dream for these students regardless of what they can do. Students who can make it into top-tier schools will be able to access generous financial aid (that was my secret to success for my undergrad), but for those who can't get into Harvard, University of Chicago or the Claremont Colleges, what are their options?

I'm in the midst of getting my first two weeks' worth of lesson plans, handouts and homework assignments written for my Saturday teaching job. I've got my topic lists written for the two classes (trigonometry and algebra/geometry), so now it's on to the detail work. When I first started working on this, I was thinking that the trig class was going to be much more fun than the algebra/geometry. After all I've practically taught nothing but algebra for the past year and the geometry is going to be a bit too ACT-based for my tastes, but I would do my best to work around that.

But at this point, I'm thinking that algebra/geometry will be a lot of fun. I've made a conscious decision that the students will do proofs, even though this can't be justified within the ACT-centric framework of the program. I just think that the ability to do proofs is far too important to neglect. In fact, in general, I think that proof-writing is an essential aspect of geometry and schools which teach a proof-less geometry are clearly short-changing their students. It would be far better to spend two years on the material (or better yet two periods over one year), than to leave out the proofs.

The other thing that I've done to revise the class from the original outline is to move much of the algebra out of the lecture and into the homework. Part of each homework assignment will be a daily assignment in which the students will do an ACT-style word problem which employs their abilities to do algebra. I'll make these a big part of the in-class quizzes and exams to make sure that they don't just skip those problems.

The trig class will still be fun. The homework assignments will be big on discovery and will have them learn some important formulas and rules by working out in a scaffolded matter the bases behind those formulas and rules. For example, finding the formula sin a+b= sin a cos b+cos a sin b can be done using some basic rules of right triangle trig that the students will have worked with in the first week and some fairly simple geometry.

... but I'll post something later today about prepping for my Saturday teaching job (which starts in the classroom this Saturday.

Instead, I noticed that there's a requirement for "winning" a new blog showcase that I've entered that I link to three other entries. Fair enough. I do, after all already have some links of my own.

So who to link to? Here goes (in no particular order):

• Quigley's On-line Dating Blog To me, the most interesting blogs are those that recount personal experiences and it doesn't get more personal than this. Way back in the days before blogs when people would update the pages by hand, I remember reading a college student's diary of his account trying to find a girlfriend and thought that it was especially interesting. The entries I've read here also seemed interesting.
• Owensoft Another personal experience blog, the entry here is about watching the closing credits of movies, something I generally make a point of doing, being a big-time movie freak (although grad school looks like it has me trimmed back to only about 30 movies a year).
• Djangowire And just because the top-ranked blog at the moment has as its tagline, "a nice place to dream about no more liberals", and the post referred to (scroll down to September 28th, his permalinks seem broken) is a nice antidote to that attitude.

The second try of the review technique didn't work out so well. It was very difficult to get people to volunteer to come to the board. I think, though, that I'll keep pushing this approach and see if I can get the kids used to it. I do want to see greater participation in this respect.

Here's the second part of Jay Bullock's series on NCLB. Some interesting points, although most striking to me is something that bothered me when I first read the NCLB documents at whitehouse.gov: The idea that threats of punishment will somehow make schools better.

The big problem I see is that taking money away from a failing school can't possibly improve it. There's no mechanism given to improve failing schools, just threats of punishment. Giving more money to a school won't necessarily make it better, but taking money away from a school will almost certainly make it worse.

And one more thing: "to sanction" means to give approval to. "a sanction" is a punishment. The NCLB documents from bush get it wrong, and they speak about failing schools being sanctioned.

My main purpose in writing this blog was to give me a forum to reflect on my teaching experiences. That said, I should get some reflecting in.

I have, at this point in my high school observations, had a chance to see nine teachers in action. It's interesting to note that teachers tend to attribute student's behavior in the classrooms primarily to the students, but what I've found watching the teachers is that student behavior levels are pretty consistent across classes for a given teacher, but do vary for different teachers. I haven't seen enough to know why, for example, teacher A has consistent talking in class while teacher B does not. I imagine a big part of it comes from the tone set at the beginning of the year by the teacher. My own experience has been that when I had classes start with a relatively social activity (like a non-academic ice breaker), it resulted in excessive social talking for the whole class. I wonder if that had something to do with how things went?

Any teachers out there reading this, I'd love to have your comments on this.