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

Tuesday, September 30, 2003

15:09
Test review strategy tried--it's a success 

I decided to use the post-test review strategy that I saw last week while observing today, and I found that it worked very well. Since the discussion section is all about reviewing questions, I think that I will use this as a template for future classes.

15:05
Bilingual classroom 

I spent today's observations in a bilingual classroom. It was quite interesting. The teacher was reasonably good, although not quite to my tastes (I'm finding that I'm very particular about how I want to see classrooms run, although I don't live up to that either). It left me thinking that perhaps I will pursue the CLAD/BCLAD certification. The one thing that I would do differently if it were my classroom is that everything I said in Spanish, I would repeat in English since I think that part of the purpose of a bilingual classroom is to teach English.

Sunday, September 28, 2003

10:33
First week of observations over 

I'm overdue for an update, and readers are actually stopping by now. I've had, at this point, a total of five different class rooms to observe. Friday's was the first that I really had any positive feelings about her teaching approach, although that may have been the students that she had. I liked her approach to reviewing the quiz that the students had and I think that I'll apply that approach to what I do in my TA sessions.

I've given a list of classes that I'd like to see to the department head. I've decided that I want to see at least one section of every class that they offer from the most remedial level to AP calculus, so that means I won't do any longitudinal observations this week.

I've been laid up this weekend with a nasty cold, which is unfortunate since I'm behind in my homework already and I've got a big stack of midterms to grade for Tuesday.

Wednesday, September 24, 2003

19:55
Another NCLB horror story 

Read this, then go here.

Monday, September 22, 2003

15:01
Observation begins 

I began my final set of observations before student teaching in the spring today. My plan for this week is to observe a different teacher each day for at least 3 classes. Today's teacher was Mr K who taught 2 sections of remedial algebra and 2 sections of advanced algebra.

Having been reading and talking about observations, I was looking at the classroom with fresh eyes, paying close attention to how the teacher approached the class. I think viewing a class as "a class from hell" altered how the teacher dealt with one of the remedial classes. I was also struck by how lecture-driven his class was. My style is much more of a questioning style. Or at least I hope it is. My plan is to videotape as much of my teaching as possible so I can reflect on what I do (and if I've got some good lessons, I can include copies of the tape with my portfolio).

Wednesday, September 17, 2003

12:19
Do Not Learn Not Important 

I was flipping through the old college trig textbook that I found in my parents' basement and I happened to find on page 12, next to where the axiom of induction was introduced, the scrawled note: "do Not LearN NOT Imp."

Well, it amused me.

11:07
Getting ready for teaching on Saturdays 

I finally finished off my syllabus for the fall semester of my saturday teaching job. I'm really excited about my ideas for teaching. Since we only get the students for about an hour each week, and we're going to be teaching them new material that they won't get M-F until a year later at the earliest, I've decided that the way to manage this is to have part of the instruction take place through discovery assignments to be completed as homework. Students, for example, will be guided through the geometric construction of sin(a+b) to derive the formula for sin(a+b) in terms of sin(a)+sin(b).

My syllabus is also heavily weighted away from graded homework assignments. The bulk of the grade will be determined by two tests and a journal (which will be, effectively, the students creating their own textbook). The journaling will provide the students to get used to the idea of writing about mathematics and help fulfill the goal of teaching writing across the curriculum.

Tuesday, September 16, 2003

12:36
Cultural bias 

A question in the Algebra I textbook:
The perimeter of a football field is 962 feet not including the endzones. How wide is the football field.
A simple enough question, assuming that the reader knows that a football field is 100 yards long. The illustration accompanying the question almost provides that information, but the yard line markers are not identified with units, so there's no way for the reader to know that they are yard markers and not, say foot markers. Just one of those little cultural things that we can easily be blind to.

Sunday, September 14, 2003

16:43
More on No Child Left Behind 

This article gives a pretty good overview of exactly what's wrong with NCLB. The standards for what makes a school "failing" are set so that even a school with 80+% of students demonstrating proficiency or advanced proficiency can be labelled as a failing school.

If you're as outraged as I am about this, I suggest going here and making your views known to congress.

Friday, September 12, 2003

13:42
Inside Secrets of Finding a Teaching Job 

I stopped at Border's this afternoon and decided to take a look at the book Inside Secrets of Finding a Teaching Job. I ended up spending about an hour in the store skimming the book and taking notes. This is definitely a book that I want to have to use as a resource while I'm putting together my portfolio and preparing for my interviews. I mentioned this on the phone with my fiancée and apparently she had the same reaction and was glad that I hadn't bought the book yet because she had ordered it for me as a surprise. That's two teaching books that she's bought me now. I'm so lucky.

Thursday, September 11, 2003

10:16
I'm a morning person now 

I woke up this morning worried that I might have overslept: My alarm had turned off in the night. And fortunately, it was still only 6.40. No problem getting in to teach my 8a class on time.

The 9a class is beginning to distinguish themselves from the 8a class. The later class has more students asking questions than the earlier class, and glancing over the quizzes, I think that they did better on the quiz. I did make an error and covered one more section on the quiz than I should have, so I'll possibly add an adjustment to scores. I'm thinking taking the class average on the first three questions, then adding the difference between that and the average on the fourth question which went beyond the standard material. This will also compensate for some differences in what was discussed before the quiz began.

Wednesday, September 10, 2003

20:10
Fall observation assignments are here 

We got our fall observation assignments today. They aren't official yet, but it's almost a certainty that this is where I'll be. Unfortunately, I was placed at the school which is the top of my list for student teaching. I had really wanted to get a diversity of settings between fall observation and student teaching and this messes things up for me a bit. I have to decide whether to keep my student teaching list, or switch to a different school which will be a bit less convenient to get to.

10:36
My little mathematical breakthrough 

Nothing major, nothing new, but at the beginning of the semester I set out a challenge for myself to find a group with infinite order, but with all of its elements having finite order.

For readers who may not have studied abstract algebra, let me back up and explain what these terms mean.

First, a group consists of a set and a binary operation (for example, the set of integers and the operation +) with the following requirements:

  1. There exists an identity (we'll call it e) such that e+a=a+e=a for all a in the set (note that I'm using + as my operation here).
  2. The associative law applies.
  3. For each a in the set there exists an element a-1 such that a+a-1=a-1+a=e
For integers and addition, the identity is 0 and we get the associative law axiomatically and the inverse will be the negative of a number (so, for example, 5-1=-5).

Note that we do not require a commutative law, and there are plenty of groups which meet the above requirements but are not commutative (one of the most notable being the set of invertible 2x2 matrices with the operation being binary multiplication).

When we talk about the order of a group, it's the number of elements in the group. The order of the group defined by integers and addition is infinite. On the other hand, if we choose the set of complex numbers {1, -1, i, -i} and the operation multiplication, the order of this group will be 4.

When we talk about the order of an element we mean the smallest positive integer k such that ak=e. The order of the identity is always 1, and with addition and integers all non-0 integers have infinite order (note that in this instance when we talk about ak, we really mean ka.

For our second example, the order of -1 will be 2 ((-1)2=1), the order of i will be 4 (i4=1), and the order of -i will also be 4.

One of the easiest ways to verify that you have a group is to construct the multiplication table and if you can have each element of the set appear exactly once in each row and column, then you have a group. I played around with some ideas using the set of whole numbers for a while trying to make a2=e for all elements and this morning I finally hit upon an arrangement which worked. Looking at patterns I realized that my binary operation was a bitwise binary exclusive or. For non-CS types reading this, the operation works by expressing the two non-negative integers in binary. In each bit position, the result will be 1 if only one of the two integers has that bit position on, 0 otherwise. So, for example, 5$3 will have the bits 101 and 011 and will result in 110 or 6 as its result. For any number a 0$a=a and a$a=0.

In fact, we can extend this to an infinite number of groups by taking the breakdown of an integer in base n and having a$b being the sum of each digit position being taken mod n. This is actually the additive group of polynomials Z/nZ[x] but without the upper limit on the power of polynomials (although it seems to me that the upper limit of the exponents should be n-1. In this instance, the order of each element will be no greater than n.

After coming up with this, I also realized that if we take the set of complex numbers with the property that xk=1 for some integer k, we also have a finite order for every element in the group, although here it's not bounded.

I'm afraid this might not be as clear as I'd like it to be, but it should be possible for someone with some basic group theory to follow this argument at least in outline.

Tuesday, September 09, 2003

20:51
My sin is pride 

Today's classes went reasonably well. No one wanted to offer questions on homework, so I went with plan B, I picked problems from the text for the students to put on the board. 9a went more smoothly than 8a since that time everything was going through on the second try.

During the tutoring hour, one of my former students came in for help. She had failed during my semester teaching, but retaking the class over the summer she had made the breakthrough to being able to do algebra. She's in very good shape now from what I saw. So why the headline? Because there's a big part of me that is disappointed that she didn't make the breakthrough under my watch. I have to remember that not every accomplishment has to be by ME.

She's also a great exemplar of the student who may have difficulty with arithmetic but who can still do algebra. She couldn't do 20-8 without her calculator, but she could do algebraic manipulation with no difficulty.

Saturday, September 06, 2003

19:34
My Saturday job 

We had an orientation meeting today for my Saturday teaching job. It looks like things will be more organized this year than last, which is good, and I'm really looking forward to the upcoming year. I'll be teaching trigonometry and calculus to my students, most of whom will not be taking either subject Monday through Friday. I need to dig through my books here to see what I've got in terms of Calc and Trig texts.

One of my goals will be to make sure that they're learning an adequate amount of information to be able to handle the university's flawed math placement test.

Speaking of which, it looks like many of my algebra II students are trying to get placed into more advanced math classes.

Thursday, September 04, 2003

14:33
No Child Left Behind 

I'm personally of the opinion that NCLB is one of the biggest disasters to hit public education in this country. It consists of an overemphasis on testing (which is itself problematic), unfunded mandates and a set of standards for failing that will rank nearly every school in the country as failing as NCLB continues wreaking its havoc on public education.

Read more about this and sign a petition opposing it here

08:34
Note to self 

Ask, don't tell. We were going over some absolute value questions. And overall, I think I did a good job with it (even incorporating some of the ideas from my ed class reading), but I still had a tendency to tell the students things rather than letting them work them out for themselves.

But overall, I was pretty happy with the results of the class. It looks like a few students are taking my advice and trying to move into higher-level classes.

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