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Vito Prosciutto: Teaching community college math on the road to a PhD.

Monday, June 27, 2005

16:57
How to make sure that college students don't learn Algebra 

Maybe other people can do this, but I'm finding that a once a week format for Algebra I students is a very bad idea. The students need frequent reinforcement of the ideas that they're learning and having them get 4 hours of class time once a week is not doing the trick.

And I feel like I'm banging my head against a brick wall with this self-directed pre-algebra class. Actually Mondays and Wednesdays aren't too bad because I usually have a small enough number of students self-direct their way into the class that I can actually attend to them.

Here's today's gem: I've started a practice of having the sign in sheet next to me so that as each student comes in, I can ask what they're working on during this class. Today, a student who hasn't been around for about a week or so responds with, "math." Well, what in particular, I ask? Chapter 2. I check my gradebook and see that he failed the chapter 1 test. I point out to him that he has to pass the chapter 1 test and that I never got his chapter 1 homework. He says he left it at home. Along with his book. Then he decides he's going to go ahead and retake the chapter 1 test. He's spent most of the class period on the computer. Yeah, this is a really successful class structure.

14:52
Theories 

The following movies from my netflix queue have availability issues. I'm offering my theories as to why they're hard to get a hold of.

The kids are home from school for the summer. They can watch cartoons. Preferably Japanese cartoons.

Ooh, that's out on DVD, let's watch it now

The Halo Effect

Damned if I know

14:51
The template issue 

This seems to have affected just about everyone who uses this template or something close to it. So I'm guessing it's a blogger issue and I'm not going to expend effort on finding a workaround. If anyone else turns up a workaround though, I'd be glad to hear about it.

Update: I found a work-around.

div { clear: none !important; }
in the CSS part of the template.

It seems that Blogger's inserting a <div style="clear:both"> at the beginning of each post and messing up my formatting. This kind of fixes it, but I still have an ugly line-break after my posting times.

14:45
Now with pictures 

While checking the blogger dashboard (yucky name, I know), I discovered that they've launched google pictures which unlike the old system doesn't discriminate against us Mac types. So to celebrate, I offer a genuine picture of an evil gas station just outside Yosemite. They advertise last as before Yosemite, but not the fact that not too far into Yosemite, you can buy gasoline for a buck cheaper per gallon.

So if you're ever coming into Yosemite from the west side, stop off here and expel bodily fluids on their evil little gas station.

Thursday, June 23, 2005

12:57
Weird dream 

I dreamt last night that I was back at the nerd school which occupied the bulk of my undergrad career.

Teaching fractions.

That's it. This is my career goal: I plan to one day to never have to teach anything about fractions again. OK, maybe continued fractions. But I want to teach at a school where every single one of my students will have mastered fractions long before they ever see me. And Algebra too.

Teaching at the nerd school would be nice. Math 1 there is Calculus. Math 2 is multi-variable calc (the stuff that's normally third semester elsewhere). That's my goal.

Wednesday, June 22, 2005

17:22
The Book Meme 

This is one that I've thought would be interesting to tackle, so here goes:

  1. Number of Books I Own Approximately speaking, I'd guess about 1200. Last time I did a count I had a bit over a thousand, I've bought a bunch since then, but also sold/donated some. Add in my wife's books, and we're probably pretty easily at 1500. She's not quite as inclined as I am to keep books that she's read. If she were we'd need even more bookcases than we have (currently 8 plus some shelves I installed in the living room).
  2. Last Book I Bought Depends how you count. I just ordered Counterexamples in Topology and Rational Points on Elliptical Curves from amazon, but the last one to enter my greedy little hands is Complex Functions of a Single Variable.
  3. Last Book I Read The last one I finished was A Theology Primer. Currently reading all sorts of stuff: See the side bar for most of the math stuff, but I'm also in the midst of Ulysses and some Ovid and Allende (the latter in Spanish).
  4. Five Books that are Important to Me This is a tough one. What would I pick. I'm thinking probably Graham Greene's Monsignor Quixote which opened a lot of new intellectual vistas to me when I read it at age 14. And the dictionary, because I always have enjoyed opening up the dictionary at random points and checking out words (if I were to be specific, I'd go with the OED which is really the greatest dictionary of them all). Hmm, need 3 more. How about Descartes Meditations on First Philosophy which lead me to an existential precipice, pushed me over the edge and never really rescued me (Descartes' reasoning to believe in reality never really convinced me, so I really don't believe that I can know that my experiences are real). Let's see two more: Non-fiction book whose title I won't reveal which lead to a particular business endeavor on my part which occupied a decade of my life (I won't reveal the title since it could be enough to identify me). And I guess the last would be Richard Rohr's book on the enneagram which inspired an interest in psychology (but also lead me down a bit of a personal rabbit hole at the same time).

Monday, June 20, 2005

19:50
Dots 

Hmm, who'da thunk it. I found some place which had no "don't link to these color dot gifs" images on their website, so I figured it was safe to do so. I got a threatening e-mail from the webmaster. I guess their website must be served by a TI-83 since serving up a couple 35 byte files roughly 50 times a day was apparently a HUGE strain on their resources.

I've found the dots somewhere else which, from the disclaimer, looks like linking is just fine. As long as I say that open source software is cool.

Open source software is cool.

OK, that should take care of it.

I've not bothered dealing with dots in the archives (beyond the one post below where they're in the body). So the progress bars might disappear from archived posts in the future. Oh well. Not a big concern to me. I'm not going to republish the whole damn blog (not to mention that I kind of want the progress reports to be unupdated for the sake of people who are curious about my progress rate on the projects (what little progress there is to follow).

14:33
Enough is enough 

OK, after finding yet another blatant calculus error in the Cain Complex Analysis book, I'm officially dumping it. Erasing the PDFs off of my computer and pretending that I've never seen it. It seems more than a little elementary and I'm finding that Needham is a better coverage of the same material (albeit in a different order, which is fine). I've also got my grad text for this fall's course, so I'm not going to do Cain any more. Once I get to the end of section 32 on Munkres, I'm going to move it down the priority list and put Needham on the task list because at least that's a fun project.

Friday, June 17, 2005

12:19
So much for my day of math 

My wife had a day-long workshop today and I offered to drive her to the other side of the megalopolis for it. My plan being to spend the time I was waiting hanging out at Borders and doing math.

But there were a couple complications: First, not every Borders has finished the conversion to Seattle's Best. And second, and more important, I left my Topology book and notebook at home. So much for spending the day doing math. I do have the Cain Complex Analysis e-text on my laptop, at least as well as Kaplan and Kaplan so I can do some serious work on planning for the fall semester. And a BIG pile of grading. But I was really looking forward to doing some Topology today. Maybe I'll find some time over the weekend.

Wednesday, June 15, 2005

17:23
Topology question (does this question make sense?) 

Munkres, p.92, question 5:

Let X and X' denote a single set in the topologies T and T' respectively. Let Y and Y' denote a single set in the topologies U and U' respectively. Assume that these sets are nonempty.

(a) show that if T' is a superset of T and U' is a superset of U, then the product topology in X'xY' is finer than the product topology on XxY.

So here's my question: Are we to assume that the X and X' differ only in the topologies that they belong to? i.e., that we might be talking about the open set (0,1) in R and RK? Because otherwise, if they're two arbitrary sets in the topologies then there's no reason to assume that the subspace topologies of the product topology are even comparable. That's how I'm interpreting the question and it seems that the statement is true and easily provable, but I'd like to get some confirmation on this.

11:23
Topology update 

No questions at the moment. Mostly just wanted to bask in the glory of feeling like I'm learning things (actually not just feeling like, I actually am learning). Right now, the decision to work towards the PhD in math is clearly the right choice.

11:20
Latest follies 

The pre-algebra student who had a temper tantrum when I pointed out to her that it might be better to focus on math during math class.

The unbelievably long day courtesy of the subbing I did in the morning (8a to 10p, with, at least, a 4 hour break mid-day, the bulk of which I spent watching The Interpretter which is a serviceable thriller although it really was a bit too predictable and pointless at times).

The students who asked for a fraction review after class making my long day even longer.

The administrator who seemed to have forgotten that I was at the end of a 14 hour day when he asked to talk to me about assorted administrative matters at the end of that long day.

On the plus side, just before he spoke to me, he'd gotten some positive feedback on me from one of my students and he'd had a chance to see the general success of my students on the fraction review. So on balance, yesterday wasn't so bad.

Monday, June 13, 2005

16:07
The REAL reason I blog anonymously 

It would be unwise to put this quote on anything associated with my name:
A chorus of voices exhorts kids to study science. No one stops to ask whether it is inhumane to force adolescents to spend the bulk of their time studying subjects most of them hate."
-- Unabomber Manifesto, Ted Kaczynski

14:09
Wonderful math books 

Since Becky has blogged about her wonderful math book, I figure I ought to mention my own: Tristan Needham's Visual Complex Analysis. I really want to teach a class from this book. I've been having fun deriving trig identities from complex arithmetic as I work through the first chapter. This is so much fun!

09:05
Today's gripes 

I'm subbing this morning. And the teacher that I'm subbing for has no conception of a lesson plan. He's written in 1/2 a page what I would spend at least 4 pages on.

And here's the latest education folly: I have a student who has complained to me that what I'm teaching is not anything that she's seen before.

Seriously. As if it was all about making sure that there was no danger of any new information coming before her eyes.

On the other hand, my Monday afternoon algebra class is being a bit enjoyable. Part of it is due to the fact that we only have 3 students today, so I ended up doing a lot more, check to see if they know how to do X, they do? OK, let's move on! There wasn't a lot that we needed to do. A quick refresher on finding common denominators for one student, and more out of habit than necessity I taught the last couple of sections of the planned lesson, but now it's all over.

Meanwhile, I've wrapped my head sufficiently around the basis for a topology so that I can start the section 13 exercises in Munkres. That time spent making sure I understood things has made the homework a piece of cake to do so far.

Saturday, June 11, 2005

19:04
Movie watching progress 

So on the 11th of the month, I'll post something like this which will give some accounting of my progress on the IMDB top 250.

May list on 6/11 June list on 6/11
Top 50
Top 100
Top 150
Top 200
Top 250

Highest ranked unwatched movie: "Paths of Glory (1957)."
Most recent unwatched movie: "Der Untergang (2004)."

Friday, June 10, 2005

14:33
Speaking of easy 

My darling wife has finally updated her resume, and courtesy of linkedin.com (which seems pretty useless for academic types: I can (and have) had e-mail conversations with textbook authors and Donald Knuth himself explained to me about Turing Completeness in an exchange when I was too young to vote. But in the business world, there's not quite the openness to contacts from out of the blue so I was kind enough to connect my wife through me to what's a surprisingly large network (I'm within 4 degrees of separation from the CEOs of every company at which she's interested in working). She got an application in for a job on the other side of the megalopolis this morning and we're hoping that this will lead to an interview and job offer and thence a move to the far side of the megalopolis. This will make my life a bit harder (mid-level University is on the way to the far side of the megalopolis but it will be farther away and my current teaching assignments are in the opposite direction). This does impact on where I'm going to focus my adjunct search.

Thursday, June 09, 2005

16:23
Easier than expected 

Computers are handy things: My on-line app for graduate classes for this fall was already pre-filled because of my aborted application for last spring (stupid residency requirements). Plus the University still had my transcripts on file and the department had my application packet on file so everything is in with only about 10 minutes of actual work (plus the application fee charged to our credit card).

So that's one less thing to worry about for the rest of this month.

I did let slip early my plan to skip a semester of topology. I should have held off until the work was a fait accompli so that there would be less argument. It won't make a real difference. I'm still working through Munkres and I'll still go to the class and do the homework no matter what else happens, no doubt to the confusion and consternation of my advisor.

Wednesday, June 08, 2005

08:47
Why I am not conservative part two 

So this is what the conservative voice says.

Sunday, June 05, 2005

20:00
Ten myths of mathematics education 

Courtesy of MathForge, I've found out about the budding debate on mathematics education between Mathematically Correct and NCTM.

I have to confess to a bit of prejudice against the Mathematically Correct people, less on substance than on the fact that many of the advocates lack people skills and tend towards absolutism.

And it's really that last part that really gets to me: It's the overwhelming belief on the part of the Mathematically Correct that there is one right way to teach mathematics. Perhaps this isn't really what the membership believes, but it does seem to be how many of the members come across.

The other part of Mathematically Correct (as I've been lead to perceive it) which tends to prejudice me against its stance is that they seem to want to return to a mythical golden age of mathematics education (much like there is a desire to return to some sort of mythical golden age of American society among many conservatives). The problem is that the golden age wasn't so golden and mathematical education has always been problematic. The existence of a class called "College Algebra" at my high school 20 years ago testifies to a point in time when Algebra was indeed a college freshman-level subject (albeit one which was a bit more in depth than the algebra that I teach now). Yes, mathematics education has been trend-driven but then so has just about everything else in the history of civilization. Don't let anyone try and tell you otherwise. Doubtless the ancient Mesopotamian elders decried the spread of whatever the latest trend in clay tablet baking was among the younger clay tablet bakers.

But let's get on to the myths. Here are the myths as presented by Mathematically Correct and NYCHold. I'll be talking about each of these in a separate post (with manipulated dates and times so that they appear each in their own post in a sensible order below). Doing a long multi-part series like this is a bit exhausting, and I'm not too happy with some of the distortions that ecto is introducing into my formatting, so I'm going to do this 2 at a time over the next week or so.

19:36
Myth 1: Discovery Learning 

Only what students discover for themselves is truly learned.

OK, here's another problem with the Mathematically Correct people. They seem to set up straw men to attack. I don't claim to be intimately acquainted with everything in the NCTM Standards, but given that my MST was highly influenced by NCTM perspectives, particularly in the math ed classes, I've got a pretty good sense of what's emphasized. And frankly discovery learning was not that central a focus of what we did. I had one course in number theory where we used Holt and Jones Discovering Number Theory as the text and it was set up on a discovery learning structure (as dictated by the book) and we found (through discovery!) that a pure discovery approach doesn't work. If you peek at the NCTM reply in Jay Mathew's article, you'll see that the NCTM doesn't argue this either.

That said, there's a lot to be said for using discovery learning as part of the educational process. Pretty much what the MC people say, and not surprisingly what the NCTM says as well.

19:35
Myth 2: Inventing your own algorithms versus learning the standards 

Children develop a deeper understanding of mathematics and a greater sense of ownership when they are expected to invent and use their own methods for performing the basic arithmetical operations, rather than study, understand and practice the standard algorithms.

Here's another one that I seem to have missed in the ed program. From my own personal experience, this can be closely linked with the discovery learning approach. Consider, for example the Lizzie Method for factoring a trinomial in the form ax2+bx+c. If Lizzie hadn't experimented with her own algorithms, she wouldn't have (re)discovered this approach to factoring.

As for standard algorithms, the Mathematically Correct people bring up long division and I agree (to an extent) that this is really important thing for students to learn, not just because it leads into polynomial long division but because it provides good practice in addition/subtraction/multiplication. But there's more to this than they seem to realize. When I do divisibility testing by hand, for example, I often work right to left in long division instead of left to right because it allows me to more quickly rule out possible factors.

Saturday, June 04, 2005

10:13
Some thoughts on calculators and teaching students things that they should already know 

My current teaching load consists of pre-algebra and introductory algebra.

These are college students.

My policy is that I allow the algebra students to use calculators, but not the pre-algebra students. Part of that is I've decided that if my students don't know how to do arithmetic by the time I have them as algebra students it's too late. I'm not really interested in spending my time teaching them arithmetic at this point. If they need a calculator to do 2+3, then let them use the calculator.
But for the pre-algebra students, I think that the arithmetic skills are an essential part of what we're learning. I need for them to develop some intuition about what happens when they multiply numbers and what factors really mean. We'll come back to this in algebra and I'll let my students use their calculators to look for factors, but those who have had more extensive practice working with this by hand will have the advantage, but we just don't have the time to do this in the algebra class, at least not as a class.

At the high school level (and below), however, I think it's an entirely different story. During my month of teaching high school math, I had a no calculators policy. Part of this was dictated by the fact that the annual standards tests that the students would be taking didn't allow calculators. And if they couldn't use calculators on those tests, then they shouldn't use calculators in the class which presumably is preparing them for that test.

Part of what's inspired me to write this is looking over some of the test materials left behind by my predecessor for the pre-algebra class. While I think that drilling students on arithmetic is important in the pre-algebra class, having them add and subtract 3 digit numbers (and, for that matter 2 digit numbers) is kind of behind the point. It would be nice if they could do this, but to be honest I'm more than happy to pull out a calculator for this sort of thing. Not that I can't do the calculation, but let's be honest, I'm going to be more accurate if I use the calculator where appropriate. For the purposes of this class, I can more effectively test my students' knowledge with 2-5 than I can with 234-531. After all, what I want to see is whether they know that the answer should be a negative number. Allowing the use of a calculator if this is my objective doesn't let me check this.

On the other hand, in algebra, I'm much less concerned with whether they can get this result by hand (although I did teach them the rules). I just want to make sure that they can get the answer. There are more important things before us that I need to make sure that they know.

Friday, June 03, 2005

17:01
Playing with ecto 

A pretty content-free post. I've decided to give a shot at using Ecto to post entries on the blog here. It looks like it might be a good improvement of blogger's web-based interface.

Update: I'm not yet sold on Ecto yet, but it does seem to have some nice features. Alas, it doesn't provide a nice integrated approach for posting pictures, so I'll likely remain rather picture-free for the time being. Expect most posts for the next 20 days to be written using Ecto.

Update (6/4): Hmm, there seems to be an annoying level of this where it wants me to manually mark all paragraphs in the rich text mode. Feh. There's a setting somewhere for this, but I can't find it. Found it. I'm off-line at the moment, though, so I can't easily check to see if it will work.

00:32
Definition of a topology: A question 

I've been puzzling over the definition of a topology from Munkres (section 12, p. 76):
A topology on a set X is a collection T of subsets of X having the following properties:
  1. The empty set and X are in T.
  2. The union of the elements of any subcollection of T is in T.
  3. The intersection of the elements of any finite subcollection of T is in T.
It's that third item. Why must it be a finite subcollection? Can anyone give me a straightforward counterexample where the intersection of elements of an infinite subcollection doesn't work?

I suppose my other problem, is that Munkres hasn't really explained what a topology is for yet. I kind of feel like I'm looking at the abstract definition of a geometry without ever having seen any geometry before.

Thursday, June 02, 2005

22:46
You will not pass this class without adding fractions 

That's what I wrote on a quiz today. My student was making a point of not paying attention while I went over how to add and subtract fractions (I think I did a good job of connecting it with combining like terms, incidentally), and then demonstrated her inability to do this when we had guided practice. On the quiz she pretty much skipped the questions on adding and subtracting fractions.

"Can you really do that?" my wife asked after class.

Sure, I replied. I'll just make sure that there's enough questions about adding and subtracting fractions for the rest of the semester that she won't be able to pass unless she can do it.

Because sometimes a math teacher has to draw the line.

And when did fractions become a part of algebra anyway? Granted, I took algebra I in 1981-82 and Algebra II in 1982-83 (and knocked off the final semester of HS Algebra in the first few weeks of the fall of 1983), but I seem to remember us having put fractions behind us by the time we started algebra.

In other student news, I had a student threaten to complain to the education office because I wouldn't allow calculators in the pre-algebra class. "But the last instructor did!" "I'm not the last instructor."

08:54
Why I Am Not A Conservative or A New Reading List 

Human Events, which bills itself as "The National Conservative Newsweekly" has published a list of the Ten Most Harmful Books of the 19th and 20th Centuries. There's the predictable (Marx, Engels, Mao, Lenin). But come on, Dewey's Democracy and Education? You've got to be kidding. And of course the key indication of where these people are coming from intellectually shows up in their honorable mentions in which we learn that just slightly less dangerous are Silent Spring (silly liberals, you'll rue the day when you decided that you'd rather have nature than pesticides!), Darwin (evolution, after all is only a theory, not a scientific fact!), B. F. Skinner (have these people even read the books that they're criticizing or did they decide that the provocative title of Beyond Freedom and Dignity was enough to make it harmful), Betty Friedan (woman, get back in the kitchen and fix me some dinner!) and Unsafe at Any Speed (consumer safety? That's just code for stripping corporations of profits. Besides it was written by Ralph Nader and we all know what a dangerous force he is).

Not to mention the whole idea of a list of harmful books which at a gut level just seems so wrong to me. So should we remove these books from our schools and libraries. Perhaps burn them?

And yet somehow they left Catcher in the Rye out of the list. After all it has the word "fuck" in it. And sexual situations. And high school students are forced to read it. And it killed John Lennon! (Oh wait, I guess that's what spared J. D. Salinger. Conservatives hate John Lennon. Or is that V. I. Lenin? Who can keep them straight?)

For the record, I've read 2 of their top 10 and 1 of the honorable mentions.

Wednesday, June 01, 2005

15:14
Nihil sub sola novum est 

08:58
Another movie list 

Apparently Time has put out there own top 100 movies of all time list. I've seen 36 of them. I'm guessing about half aren't even in the IMDB top 250. For comparison purposes, I've seen 80 of the top 100 at IMDB.com.

Of course Time's list also cheats a bit by counting The Lord of the Rings and Godfather I & II as one movie each.

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