Blog of a math teacher

I spent some time after school offering additional assistance to one of my students. I think that his biggest problem is language, he doesn't seem comfortable with English, but talking with him and watching him work through problems, he's really amazingly smart. This is a kid who should definitely be going to college, but I'm afraid that it won't happen. Linguistic issues and tracking have him well off the college prep curriculum, I fear.

After school another student amused himself by repeatedly saying, "Mr Prosciutto! Do we have homework?" to which I always responded, "There's always homework!" It amused him to no end.

My gym class is officially over as of today. I've got my 98 points so the class is passed. The only real difference in my life is that the one class of the three remaining for the semester that I wasn't planning on skipping, I can now skip.

Which leave me wondering whether what I have to say is always interesting. I hope so. But I can see a time when things become sufficiently routine that writing about my teaching day won't be that interesting to me or anyone else.

Tomorrow (today, actually), is my last day of supervised teaching. I'm done with the geometry class, but the algebra kids have their unit running until Wednesday. Since I'm going out of town starting Tuesday (I have to help my fiancée move into our new place), I'll be handing the class back to their regular teacher for the review and test.

After that it's the last week of regular classes, then finals week (although I only have one actual final scheduled for that time, although I'm proctoring the college algebra final as well). Christmas break will be much welcomed.

The question where students did the worst was distinguishing between altitudes and perpendicular bisectors. Best question performance: All students correctly identified that an equilateral triangle had 3 60^o angles.

Once upon a time I had considered becoming a psychologist so I've done a bit more reading than most have on this sort of topic. One thing that I found (through a footnote in Listening to Prozac) was a study done at UCLA where they found that in treatment of OCD, cognitive behavioral therapy resulted in nearly identical changes to brain chemistry (as measured through PET scans) as did treatment through medication (fluoxetine).

With depression, studies have repeatedly shown that nearly all treatments are equally effective, whether its traditional psychotherapy, cognitive-behavioral therapy, medication or even just talking. Another successful treatment was physical activity. A study done with college freshmen found that daily aerobic exercise was highly effective at reducing and preventing depression. This is probably a key thing for teachers as it's a career that tends to lead towards a sedentary lifestyle. It's hard to get the energy at the end of a day of teaching to go for a 30 minute walk in the evening, but it may well be a vital thing for our sanity.

Geometry had their unit test. The students tore through the exam (although I had a lot of absences). Since there was about 15 minutes left over before the end of the day, I plugged in my mac to the monitor and showed them a few things about centers of triangles that we didn't cover in class, but that might have been interesting things to see. I think they were more entertained by the animation in Geometer's Sketchpad then by the amazing qualities of the Euler line, but it was a start.

Before he left, I pointed out to him that he could still pass the class with a C if he scores 80.2% on the final (the lowest test score is dropped). Afterwards, I thought about this, and I'm not entirely sure that it would be a good thing for him to do this. I have no doubt that he could cram and do well enough on the final to get the C, but would he really be prepared to move on to the next math class then? After all, the reason that we require a C to move on at the university is that the classes are meant to build on each other.

For that matter, why do high schools allow students to advance to the next class in a sequence with a D? I think that this is really doing students a disservice. On the flip side, if we required a C to move on, I fear that this could result in a bit of grade inflation.

It does relieve me to think that this teaching assignment is almost at an end. With Geometry, I'm giving them a test tomorrow and that'll be it for me. Algebra I'm teaching tomorrow and Monday, then handing them back to the teacher for review on Tuesday and a test on Wednesday (I'll be out of town early for Thanksgiving and to help my fiancée move to our new apartment).

I decided today that when I start teaching next year, I'm not going to do anything else for the first semester at least. No music ministry, no night classes, no band, nothing that obligates me to commit any time. It'll be rough enough doing all the teaching stuff that will be necessary. I want to have some time to spend with my wife too.

My view is that teacher salaries should be tied to the median home price in the district where they work. I would require that the median teacher salary would be high enough so that a teacher making the median salary could purchase a median-priced home in the district. In some areas, this is already the case, but in most districts (urban districts especially), the salary would need to be increased quite a bit. In the areas where I'm doing job search investigations, the median salary would need to be where the current maximum salary is.

With geometry we did some computer exercises. I gave them an 8 page worksheet to do some constructions with Geometer's Sketchpad. It was intentionally too long for them to complete so I wouldn't have to worry about anyone finishing up at 2:00 and not having anything to do. There were some difficulties because of differences in the versions.

Tomorrow is a half day with no instruction time so I get a break from daily lesson plan writing. I'll be happy for the break, although I still have a lot of work to do today and tomorrow.

Geometry went a bit better, although students were not correctly drawing altitudes. More inert knowledge. I hope that tomorrow's computer-based lesson will help a great deal.

Trig, on the other hand, requires that we have to go back and re-learn a lot of the material. I've decided to dramatically scale back the scope of the class from where we started. We spent the day focusing on trig identities. By connecting some of what we need to do explicitly with algebra that they already know, we're getting a better understanding of this. I had hoped that the students could have engaged in self-directed learning as part of the homework, but that wasn't happening. I guess my expectations were too high. Still, I think that it's better to have too high of expectations than too low.

So I went over the questions and picked those for which the class average was 70% or more. Those were covering material that the class as a whole apparently learned. I gave extra credit for each of those questions equal to the points that each student learned (so students who didn't learn the basics that everyone else did were effectively punished in their point total). Then I took those questions which had a class average under 20% and added the points earned for those questions as extra credit to reward those students who had learned material that their classmates had not. This improved the distribution although it was still not where I wanted it to be: My final grade distribution for the first class:

The teaching itself went reasonably well. The one problem that I did run into was when I realized I'd forgotten to print the geometry worksheets at the university this morning. No problem, I figured. I'll just burn a CD and print them off on the classroom computer.

Oops: The classroom computer doesn't have adobe acrobat. No problem, I'll download and install it.

Oops: Windows 2000 won't let me do that because I'm not the administrator.

So then it was run down and find the technology people and have them take me to a computer with Acrobat installed on it. Fortunately, I was able to do that and get it copied in the 25 minute lunch period I had available to do the copying.

Geometry, my Ed professor came in to observe me and for the first time in 6 classes that I taught at the high school I had too little material rather than too much. Oops. Class ended at 2:20 and the students had all largely finished their homework by 2:05. So I improvised a bit. We were learning about perpendicular bisectors, so I linked what we did with the perpendiculars today with yesterday's medians and Tuesday's isosceles triangles, plus last week's right angle trig. Since they haven't had triangle congruence yet, I was able to show them that we can see that a median is only a perpendicular bisector if the triangle is isosceles as well as the converse.

The professor was very pleased with my teaching. It was a nice remedy to yesterday's despair.

I'm thinking that if I were to come up with the alternative version of this course where the lowest-level math students still did proofs, my curricular materials would start with the triangle congruences and work their way into more interesting proofs from there.

I think that the algebra kids are doing ok, despite my failing to keep my planning where I wanted it to be. I'm actually a touch ahead, assuming that they do as well with today's homework as they did with yesterday's. I think that I may re-insert a section which I had pulled on simplifying radicals since it seems like we're in pretty good shape and actually a day ahead of schedule.

Geometry, I'm still struggling with making it all a coherent whole. Part of that was a result of my being very disorganized with the start of class. Doing a paper-folding exercise while I was checking homework turned out to not be a good idea. Tomorrow, I think that we'll do some reinforcement of some ideas with fractions that were evidently weak in today's exercise. They weren't sure how to find, for example, 1/3 of 12. But how to write that in a lesson plan? Perhaps we can do that at the review time later in the unit? But that's no good, it seems then like I'm letting the lesson plan be too much of a dictator of what I'm going to do in the class. I'm not even sure what section I'm doing tomorrow. Probably bisectors since I can do more with the fraction stuff at that point. I'll put it in a worksheet for homework.

I feel awfully stressed right now. I should have done much more work over the weekend.

There are some rather startling observations in Jones's teaching. The one which catches my attention most immediately is the "hopeless hand raiser." That student who requires the teacher's attention during seatwork and effectively gets the teacher to re-teach the less on to him individually. I hadn't looked at it this way, but that is an issue. Jones's advice: "Be positive, be brief, and be gone."

Jones's approach to discouraging misbehavior:

• Arrange the room so that the teacher can easily move among the students.
• Have general rules defining the guidelines for the class and specific rules describing procedures and routines.
• Assign classroom management chores to as many students as possible.
• Begin classes with "bell work" which does not require active instruction from the teacher. "Jones says that, on average, about five to eight minutes are wasted in most classrooms immediately after the bell rings." (!)

"Say, see, do" teaching is used to keep students actively involved in the class.

Responsibility training is managed through incentive systems. The reward is generally in the form of preferred activity time (PAT) and must follow the requested behavior. For chronically misbehaving students, Jones suggests omission training in which a misbehaving student can earn PAT for the class as a whole by omitting the undesirable behavior.

There are last resort back-up systems for when the PAT incentive system doesn't work, with backups ranging from low-key whispered messages to those punishments we all remember from our school days (loss of privileges, detentions, suspensions, etc.)

The final matter, and, I think, one of Jones's most valuable contributions, is in the handling of one-on-one interactions with students. I think this an area where many teachers (myself included) fail and think that they're succeeding. He suggests the following three-point approach to working with students:

1. (Optional) praise the student for anything positive in the work thusfar.
2. Give a straightforward response that will enable the student to get to the next step. It's probably a good idea to have some sort of set of instructions or guidelines that students can use on the task at hand so that students can be directed to this.
3. Move on. Jones has an optimal goal of 10 seconds at a student's side giving assistance. One could go up to a maximum of 20 seconds if necessary.

The primary text for the Canter discipline system is Assertive Discipline but this is just the tip of the iceberg. I do find myself a bit put off by how much of an industry the Canters seem to have developed around their ideas, although perhaps that's just a sign of how much demand there is for this sort of thing.

But on to the system (and apologies for being a bit disorganized): The Canters have built their system on humane behavior management through attending to student needs, formalized classroom rules, positive attention for students, talking helpfully with misbehaving students and establishing an atmosphere of trust and respect.

The Canters divide teachers into three broad categories:

• Hostile teachers. These are the teachers who view students as adversaries. I'm sure that those of us who have been in front of the classroom can relate. Not surprisingly, they consider this to be the least effective aproach to classroom management.
• Nonassertive teachers This is where I tend to fall at my worst. The teachers who are not sufficiently consistent in discipline. There's often some oscillation between being nonassertive and hostile.
• Assertive teachers And as you might guess from the title of their book, this is where the Canters feel that teachers should be. Clearly and confidently express classroom expectations with clear and plain consequences for not living up to those expectations.

One of the most valuable things I see in the Canter philosophy is the idea that proper classroom behavior must be taught. Ideally this should be happening at the primary level, but if it's not, that doesn't mean that I, as a secondary teacher, can ignore that. Much like if I have students who don't know how to work with fractions, then I need to remedy that problem.

The fact that different teachers have different discipline expectations means that even if students have been taught proper behavior, a teachers' expectations must still be taught to the students. After all, if hand-raising is required in some classrooms and not others, it needs to be made clear that in this classroom, hand-raising is expected.

The Canters have good ideas on redirecting off-task behavior, and ideas that I've used myself: "The look", moving towards the misbehaving student, and mentioning the student's name have all been strategies that I've used. Proximity praise, where on-task students near the off-task student are praised for their behavior is something that I've not tried, and I think might be a bit less effective at the secondary level where praising students by name for appropriate behavior could be an embarrassment rather than an incentive.

Dealing with difficult students is the topic of another book by the Canters. Their strategy can be summarized as identifying students' needs and working to meet them. Needs are identified by examining how students react to your responses to their behavior (or lack of work) and using that as a basis for evaluation.

Charles indicates that the Canter system can be implemented fairly easily. The system evolves in response to feedback from users so it's probably worth getting the current editions of books rather than trying to save a few dollars and getting previous editions.

The bulk of the book is a set of chapters on each of a number of different classroom discipline systems:

• Canter and Canter: Assertive Discipline
• Jones: Positive Classroom Discipline
• Albert: Cooperative Discipline
• Gordon: Discipline as Self-Control
• Nelson, Lott and Glen: Positive Discipline in the Classroom
• Glasser: Non-coercive Discipline
• Curwin and Mendler: Discipline with Dignity
• Coloroso: Inner Discipline
• Kyle, Kagan and Scott: Win-win Discipline
• Kohn: Beyond Discipline
• Charles: Synergistic Discipline

The book opens with some interesting background in psychology and history of classroom discipline. My own orientation in psychological thought tends to be a mix of behavioralism and cognitive psych. I mention this primarily because as I talk about each of the above chapters, it's worth revealing where my own biases lie (and knowing that I'm a math teacher may be inadequate towards that end).

You are mRNA. You're brilliant, full of important,
interesting information and you're a great
friend to the people you care about. You may
have sides to you that no one understands. But
while you understand more than most people,
you're only half-there most of the time.

Which Biological Molecule Are You?
brought to you by Quizilla

So why is there geometry without proofs? The ACT and SAT I imagine. You can't test proofs with a multiple choice test. And as we all know, knowledge that can't be tested with a multiple choice test does not exist.

I've found my mission in life: To eliminate, always and forever, the teaching of geometry without proofs in all schools. And if I succeed, my next step will be to reincorporate proofs into calculus (at both the secondary and college level).

• Isosceles triangles
• Medians
• Angle bisectors
• Perpendicular bisectors
• Altitudes (but not area!)

What too many educators realize is that Dewey's question, "is this learning vital" doesn't necessarily require that the learning connect to something outside the classroom. But it does have to connect to something.

And that there's a tendency for math ed people to get a little over-focused on manipulatives. When we were prepping for my group's lesson last week, there was this struggle to find manipulatives to illustrate rational and irrational numbers. What we did were, I think, a bit of a disaster. Some times there's no need for a manipulative. As I'm prepping for my teaching next week and the week after, the geometry classes will have a paper-folding exercise to begin nearly every class (except for one where we'll be doing computer work). But the algebra classes will be manipulative-free (although I think I will use Geometer's Sketchpad to put some graphs on the big monitor at the front of the room when we get to graphing.

It was kind of gratifying to see that the students that I helped yesterday during seatwork time managed to retain the help long enough to correctly complete their homework.

I'm not sure what I'm going to do about student work product since my mentor teacher's normal protocol is to not collect homework. Fortunately there's a scanner in the room, so I may just scan a few homework assignments while the students are in the room. I would prefer scanned assignments since those can be more easily duplicated. I would like to be able to leave copies of my portfolio with principals at schools where I'm interviewing.

Earl Gunnerson, a Huntington Beach resident, said he has tried to find out why the trees and other vegetation have been allowed to grow on the riverbanks. "The birds? Well, let me ask you, where were the birds before all this? Let them fly back to where they came from," he said.

What would make you change the way that you teach your class? That is, if some new educational idea comes along about how a class that you teach should be taught, what would it take for you to adopt that idea? Does it matter who presents the idea? How it's presented?

This is mostly to satisfy my own curiosity. If you respond, please indicate your subject area and how long you've been teaching.

I'm not sure I really approve of how seatwork is handled by my mentor teacher: For most students it's a chance to try and finish the homework so that they don't have to do homework at home (but then, it's not actually homework, is it).

The other problem I have is that she has a tendency to just not teach things that the kids aren't getting. For example, the class was looking at graphs for exponential decay (a^x where a<1). She wanted them to put the bases in ascending order, but decided that since they were all expressed as fractions, that since they had difficulty ordering fractions, she'd have them convert to decimals.

Argggh! If they have difficulty ordering fractions, then they need to do it more, not less! She actually had a day to fill in her schedule, so if she slipped by a day, then it wasn't going to be a problem.

Then there was the issue of the crappy ordering of information in the book. I spent some time looking at the book and determined that doing the chapters out of order wasn't really a big problem: We could do chapter 10, slip in graphing quadratics from 9.3 part way through, then after 10 was finished, go back to chapter 9.

But the official school curriculum puts chapter 9 in the fall semester and 10 in the spring. We can't do the switch. What kind of idiocy is that?

Geometry? It's geometry without proofs. This should be banned to the deepest pits of hell. You may as well have English class without writing assignments. Or science class without experiments. But we play the hands we're dealt, I suppose.

3rd period today spent the whole class period arguing class policy with me. Arg. It was a shortened period to begin with so we didn't have that much time to go over this week's homework. In 2nd period (same class, different kids), we managed to do some group work going over the proofs but the planned quiz had to be cancelled. I'll give it to them in two weeks (after the test) instead.

The trig kids disappointed me by not getting the homework assignment at all. I had hoped that they would be able to use their graphing calculators to do things like graphic sinθ without too much difficulty, but apparently it just escaped them. It doesn't help that they have teachers telling them that this is beyond them. The hell it is. Sophomores, especially sophomores taking algebra/trig at one of the prime college prep schools, should be able to use their calculators to do function graphs.