Friday was one of those days when I was happy to put a big X through the date on the calendar. No disasters, just a steady stream of annoyances and minor frustrations, all of which I played some active role in creating.

In my second geometry class, I made the mistake of sending a student to the board despite the fact he missed class the day before and was not completely up to speed on the material. He is a very bright student who consistently earns the top score in the class, so I suppose I have a tendency to assume that he understands whatever is going on. We were using the Triangle Side Inequality Theorem to write an inequality describing the possible values of one side given the lengths of the other two. The student got the basic information correct, but got lost when I tried to generate discussion about why the “or equals to” was not necessary in his answer and why “3 < x < 10″ might be a better way to write the answer than “10 > x > 3″. Not sure what the exact combination of lack of understanding by the student, poor wording of questions by me, and general tiredness was that made a top student unable to answer simple questions, but things went south somewhat quickly. I should have remembered the student had been out the day before, apologized to him for putting him on the spot about material he’d missed, and asked someone else to take over at the board. Naturally, I didn’t think of that until a hour later.

After school, one of my AP Calculus students came by to practice related rates problems. He did quite well with basic problems — things like “Find dx/dt for this relation given values for dy/dt and x” or “Find the rate of change of volume of a cube given the rate of change of side length”. When he came in he told me as much, and said that his difficulty lay in writing appropriate equations to represent a word problem. He worked on a number of these and had me check his work at various points. With one exception, every single mistake he made was in material covered in 9th grade (known as “Integrated Algebra” here in New York). Some of these, like forgetting to square the coefficient of a term in parentheses, were minor enough that I could write them off to end-of-the-week exhaustion. Others, like drawing a complete blank when faced with a situation that called for simple right-triangle trigonometry, made me worry about what a disservice we might be doing to students by ramming them through a state curriculum that touches lightly on many, many topics without going into depth on many.

Later on, I thought more about the AP Calc student’s struggles and realized that I had made a similar error as I did earlier in the day. At some point, I should have told him that maybe he needs to step away from math for the time being, as due to some combination of being tired and being frustrated, he was making too many easy mistakes that would only lead to him being more frustrated with the class.

Knowing when to walk away from a difficult problem in order to come back to it feeling refreshed and thinking more clearly is a useful and important skill to learn, and it’s one I need to remember to teach when necessary.

One of the tasks I have outside the classroom is being part of the selection committee for the National Honors Society at our school. Once a year, I get to read over a number of applications from a select group of students and offer opinions on who best meets the standards of scholarship and service the NHS is meant to represent.

An interesting side to this process is that I’m given an opportunity to read recommendation letters written by other teachers. The good part of this is that several teachers not only write well, but write distinctively enough that I can hear the teacher’s voice in my head as I read. The bad part of it is when it becomes very obvious that a teacher merely modifies an existing letter for a different student.

To be fair, I am definitely not one who writes groundbreaking works of literature every time I sit down to pen a college recommendation letter, and despite my efforts to personalize each letter, there are always a few key phrases that nearly always appear with slight modifications at most. I also understand why writing letters for an in-school organization may be treated with less gravity than writing for someone outside the school.

However, if you are reusing an old letter for a new student, it is definitely a good idea to be sure you replace every instance of the old student’s name with the new student’s name.

I can’t decide if the first four days of class have gone by ridiculously quickly or ridiculously slowly. For several years now, I habitually X the days on the school calendar in my classroom, and I was a bit surprised to see only one open square left this week. The first week back after any break is always a bit of a challenge for students and faculty alike, with everyone adjusting from a long streak of empty days back to living on a regimented schedule.

Tuesday I collected a set of first semester reflection papers from all my classes. Positive for the most part, with the usual handful of asskissers and proclaimers of vague generalities thrown in for good measure. I specifically asked the kids for feedback on standards based grading, and it was overwhelmingly positive. They particularly appreciate the change from counting the last score on a standard to counting the highest score, even if they can only receive a maximum of 3 out of 4 on the first assessment. I am happier with it as well, as it did bother me that one bad quiz could sink an individual grade for students.

Not too surprisingly, the fact homework is not graded came up a lot. Some students seem to grasp its role as practice and like that they get the choice of how much homework they need to do. Many simply noted this as a major difference between this class and their previous math classes. What surprised me is that there were four or five who wanted homework to be counted towards their grade! None of these students are ones whose homework is done consistently or thoroughly enough to merit a good grade. Perhaps they think they need the added motivation of points? I see where they are coming from, but I think they are missing the point of practice. I also think some of them may be a bit self-deluded w/r/t how much more homework they would do well if it had points attached.

On the whole, the positive vibe of the year continues. I hope that continues through midterm exams and beyond.

Hmmm…I seem to have neglected this blog for nearly two months. Oops.

Today was the last teaching day of the calendar year. Everyone must report tomorrow, though it’s only for a Christmas assembly and an early dismissal.

AP Calculus ended 2010 on a positive note. We talked about the first derivative test when reasoning from graphical or tabular data, and not only did the majority of the class do well when presented with old AP exam questions that clearly stated “This is the graph of f’, not the graph of f”, those who didn’t quickly figured out why they were wrong. From time to time, I ask my class the same question one of my grad school professors asked us — “What’s the most common wrong answer?” In grad school, it was useful for us to think about the misconceptions students would have for a specific problem, as those usually indicated some more profound lack of understanding. In my classes, I think it’s useful in two ways — primarily as a way of checking understanding and secondly as a form of test prep. The same students who gave the common wrong answers originally were the same ones who explained why their answers were the most common incorrect ones — they mistook the graph of the derivative for the graph of the function. Hopefully those students will be more careful about what they read in future problems.

I also told the calculus group that they were over a week ahead of the students from last year. This was meant to be a bit of positive reinforcement, and I believe it was effective, if only because of the surreptitious fist bump I glimpsed.

The geometry classes ended 2010 on a more mixed note. I had to lay into them a bit about not practicing the material through homework, and I got two very different sets of reactions. The students in the earlier period laughed, made jokes about one another, or complained about having to do homework. The students in the later period got very quiet and somewhat bashful.

As we begin the new calendar year, I am going to stop stressing myself out by checking homework for completion every day. Right now, homework is not factored into the report card grade, and I am not changing that. In the past, I took points off for missing homework and that didn’t get me much more than copied homework or half-assed work with things like a list of problem numbers and a list of answers. No way I’m returning to that. Instead, I’m going to make fully completed homework a strict requirement for reassessment. Today, I put together a few SMART Board slides to convey my vision of what well-done homework looks like (with captions ranging from “EXCELLENT!” to “UTTER CRAP!”). On our first day back in January, I will discuss these with my classes and see what they have to say. I imagine that very little will change in terms of frequency of afterschool visits and students who visit me.

One of my big experiments this year was to implement Standards Based Grading for the first time. For anyone who hasn’t been following the discussion over many teaching blogs across the web, here’s a horribly simplified explanation (or at least a brief summary of my understanding) of what SBG is all about:

• Grades should reflect learning and nothing else. What we put on a student’s report card should indicate how well the students have mastered the material for the class rather than be a mishmash of snapshots of their understanding of various topics at specific points plus grades for homework, preparedness, attentiveness and other nonacademic issues.
• Course content should be organized as lists of standards, which are essentially the ideas a student should know and the skills a student should master as a result of the class. A teacher then assesses these standards in a variety of ways and grades a student’s mastery of each standard using a uniform rubric.
• Grades on individual standards should always be changeable as a student develops his or her understanding of the topic. One of the big ideas is that understanding a topic is what matters most, not the speed with which a student understands. Students should be given the opportunity to reassess individual standards if they did poorly on in-class assessments but have put forth the extra effort to get the material.

One of the best summaries is found here, at Think Thank Thunk, an excellent blog by a math and science teacher named Shawn Cornally. While he’s certainly not the first or only teacher blogging about SBG online, he’s a bit of an evangelist who has posted quite a lot worth reading on the topic.

My initial setup looks like this:

• Standards are graded on a 0-4 point rubric, and the grade on the last assessment counts towards the marking period grade. I assess each standard at least twice during class and student may see me outside of the normal school day to reassess individual standards. The average of the standards grades makes up roughly 75% of the report card grade (I say “roughly” because I have slightly different final grade setups for geometry and calculus).
• I give one test per marking period which covers all the material from the marking period and is worth a little under 25% of the grade. There are no makeups, bonus points, or curves.
• Each class is given one writing assignment per marking period because I believe that not only do students need to get as much practice with writing as they can, but they should get comfortable with the idea that writing is not a skillset reserved exclusively for English class. This counts as 5% of the report card grade.

Student reaction has been mixed. Some appreciate the fact that they get multiple opportunities to show me they understand the material and are not necessarily stuck with silly mistakes or delayed understanding factoring into their class grade. Many failed to actually read the syllabus or pay attention on the occasions when I detailed the grade breakdown and didn’t really get how important the standards are to their overall grade.

I find the actual grading process to be far more easy with SBG than with the general points approach. Judging the level of a student’s mastery is not too challenging, so I am able to go through a stack of quizzes in very little time. Of course, it still takes me just as long to grade papers as it did when I had to figure just how many points a student lost for various errors as I am much more inclined to write out commentary on a student’s work. I can’t tell how much of an impact this has had yet, but I do see students reading what I write, which is reassuring.

One thing I am not sure about is the process of keeping the most recent grade for each standard. I do like that this encourages retention and gives everyone at least two chances to demonstrate understanding. I do not like that it invites comments like what I heard in geometry class today, when one student’s reaction to doing poorly on today’s quiz led another student to say, “That’s ok, because this one didn’t count anyway!” I have two ideas for addressing this, which I’ll be thinking about as possible changes for a new marking period:

1. Switch to using the highest grade only, but make the first assessment worth a maximum of 3 out of 4 on the scoring rubric (i.e. make it an assessment that judges basic understanding only, not the ability to apply ideas to novel situations and make connections between different topics). Definitely easier to set up the grading system for this method, though I can foresee some challenges with explaining the 3/4 vs. 4/4 assessments to students.
2. Use a decaying average that makes all assessments count towards the final score, with more recent assessments carrying more weight. For example, a student who scored 1, 3, and 4 on the same standard may have the final score determined by something like (1 + 2(3) + 3(4))/6 = 3.2. I would definitely need to work out an appropriate, scalable method for averaging the individual assessment grades. The big challenges for this would be setting up my gradebook to compute these values for me (we have a web-based system at school) and explaining to my students how to compute their grades.

We shall see where this goes. I suppose that one of the advantages I have is I could experiment with both of these approaches in class this year, though I do not think it is very fair to the students to keep changing the grading system for the class, even if the fundamental principles and weighting structure stay the same.

My overall impression of SBG is quite positive. The emphasis on understanding is quite attractive. Figuring out how to best incorporate individual assessments into an overall grading scheme remains a tricky spot. However, no grading policy is perfect and I’m happy to work towards improving one that keeps student learning as the focus.

We started the year in AP Calculus with a few explorations so the students could get an early grasp on the big ideas of the class, such as the derivative being a slope or rate of change and the definite integral being the area under a curve.

Yesterday, after handing back the first test of the year, I tried to explain that I would not be curving tests or report cards in this class, as they have multiple opportunities to reassess standards and improve their grades by showing me they understand the material. I didn’t say it quite so eloquently. Instead, I said “There are no curves in this class.”

To this, one of the quicker thinking kids asked, “Well, then how are we going to study definite integrals?”

Totally nerdy, yet pretty awesome for someone a month into calculus.

It seems like just yesterday that I was worrying about setting up my classroom and making copies of my syllabus for the first day of school, and now there’s postseason baseball on TV and the first set of grades due in just over a week.

The good:

• Started the year in AP Calculus off with six days of group work based on Paul Foerster’s Calculus Explorations. It went fairly well, and it gave a chance for my students to get right into the two big ideas of first year calculus without slogging through a precalculus review first. I snuck some algebra review and calculator skills lessons into the explorations as well.
• Overall, I get a positive impression about all of my students and the classes in general. I can sense potential issues with a handful of students, but overall I got the feeling that this will be a good group.
• One of my AP calculus students came up with the slogan “There is no ‘I’ in calculus — there is only ‘us’.”

The bad:

• It took a mere seven days before I felt burned out and ready for June to arrive.
• I have felt more overwhelmed than usual this fall, although I do not have any extra work or responsibilities. I’m not sure where this feeling originates, but it has had a negative impact on my motivation to do important things like plan good lessons and effectively implement standards based grading.
• I very nearly overreacted and got personal in response to a “When will I use this in real life?” question during class. Caught myself in time and gave a general answer that may not have been the most effective ever but avoided any sort of personal confrontation. At that moment I realized how thankful I was for a three day weekend.

The ugly:

• It took all of four weeks or so for me to come down with my first cold of the year.

Hopefully it won’t take a month to post another entry.

The title of this post is certainly not strictly true for all teachers, but it sounds so much better in the first person plural than in the first person singular.

At any rate, I once again left work for the last minute and am now setting about revising my first day of school presentation at 9:24 pm the night before the first day of classes. It is roughly 80% complete, but that final 20% is the oh so crucial difference between “I said everything I wanted to say and made it worth my students time” and “Not only did I forget multiple key points, but I inspired half the class to attempt blinding themselves after witnessing multiple atrocities of graphic design.”

As I was sitting here typing this I suddenly remembered that I haven’t made seating charts for two classes. Perhaps it’s time to switch windows and get this business out of the way.

Oh well, it’s not like I was going to sleep well tonight anyway.

When I was a kid, I probably thought the classrooms we walked into at the start of every September were designed that way from the beginning, with all the posters and colorful borders and alphabets and number lines carefully installed according to the teachers’ guidelines by whoever built the building. As a grade school kid it was often hard to imagine any of my teachers having ever done something other than show up to school everyday. I do remember the classroom decorations complementing the personality of each teacher, from Mrs. Q’s “Texas Fly Swatter” in 7th grade to the skull dubbed “Mister Morehead” (more head than hair!) Mr. K kept on his desk in 9th and 10th grade history.

My first September as a teacher my decorating involved making a panicked shopping trip to the teacher supply store on Livingston Street in downtown Brooklyn the day before classes started and loading up on colored paper so I could hide all the ugly corkboard in the classroom. The other thing I remember buying that day was a chalkboard compass, even though I wasn’t sure I would ever need to use it. It struck me as something a math teacher should have in his classroom.

The main decoration in my classroom is a print of M. C. Escher’s Metamorphosis II that stretches across the entire top of the blackboard at the front of my room. I also hang a couple Jasper Johns posters in the back of the room (Flag and Map) whenever I get my hands on enough tape to hold them up. Most of the decorations in my room are things like plastic penguins, toy soldiers and a Yoda action figure that are stashed about the room for the wandering eyes of students to discover. I am not entirely sure what this says about me, but I think those semi-hidden items reflect my personality reasonably well.

The standard math classroom decorations on sale in most teacher supply stores seem to be packs of posters that feature a dense, uninspired layout of figures and formulas. I understand the possible benefits of these types of posters, as they provide students with an opportunity to reinforce things they learned in class when their eyes go wandering about the room. Yet I am not a fan of them. First, the majority of the ones I have seen are artless at best and ugly at worst. I don’t want to look at them everyday! Secondly, part of me believes that posting a bunch of out-of-context formulas on the walls reinforces many students’ idea that math is all about memorizing formulas and picking the right one for a problem. Lastly, the posters seem like a generic item tacked up by a teacher who was hounded by the administration to decorate his room. The one pack of geometry posters I bought 6 or 7 years ago after administrator mentioned decorations in three consecutive reviews has been languishing in my classroom closet for 5 or 6 years now, ever since I went and picked up the art posters I mentioned earlier. The connections of Escher and Johns to math may not be immediately recognizable, but they make the room feel more like my place, which strikes me as something more important than having a few more formulas for students to gaze upon.

Today was the official first day of school for this year, and that seemed like a good a reason as any to actually start writing here. All we did at school was move from meeting to meeting, and we don’t actually meet with our classes until Friday. I still honored my two first-day-of-school traditions:

1 – I didn’t sleep well last night. Ever since I was a little kid I have had restless nights before school starts. It is a mix of excitement, nervousness, and general anxiety. I’m not sure I will ever get over this and I’m not sure I want to do so.

2 – I listened to the album 8 A.M. All Day by the band Chisel on the ride to work. They were a campus band at my university, and I’ve been a fan since I first heard them play at a “Farewell to Bush” concert in October of my freshman year.  (And that was the first President Bush, by the way.)  8 A.M. All Day came out a year or so after they graduated, and among my friends it had the power to generate spontaneous dance parties when the first chords poured forth from stereo speakers. When I started teaching 8 years ago, I decided that would be my first day music as it was positive, upbeat, got me moving, and brought to mind positive memories of being in school. This turned out to be a habit that stuck.

This year I hope to generate a few habits that stick for my students. The two big ones are that homework is worthwhile practice and that it is okay to make mistakes while learning. The former is inspired by a number of other math teacher bloggers writing about standards based grading and the latter comes from something one of my AP Calculus students wrote in his end-of-the-year reflection last May.

As for me, I figure that after 8 years of trying to teach well, getting frustrated with falling short in various areas, and jotting down the occasional journal entry in my handwriting that frequently toes the line between recognizable letters and abstract line art, I should start organizing my thoughts by writing in a more formal venue. True, a blog is about as formal as a tucked-in polo shirt, but it’s better than the dirty t-shirt and flip-flop world of scattered bits of inked paper. One of the highlights of my time earning a Masters in Math Education was being required to write weekly reflections for my practicum course. My main motivation in starting this blog is to get myself back into that habit.

And so we come back to habits. I named two habits I want to stick with my students. Writing regularly is a habit I hope to make stick with me as I move through what will hopefully be a challenging and rewarding year. This year I plan to implement SBG, be more thoughtful in my lesson planning, get the students doing more of the work in the classroom, and try to wrench myself out of the rut of routine that it is much too easy to slip into after close to a decade in one school. To steal a phrase from the end of the story Basil the Dog by Frances Sherwood, it is time to “learn to do serious battle.”