Special Seminar

Yesterday, I found myself surprised while sitting in a math department in-service. Not only did the training extended beyond our contract day, but it also overlapped the time of a funeral many teachers desired to attend. Even the location of the meeting changed at the last minute from our convenient high school campus to the downtown district offices. Oh, and did I mention that we were going to be "taught" how to "teach" to the TAKS test by someone who has had proven results "TAKS teaching."

With all these seemingly negative circumstances going into the meeting, you would think that I would be silently resistant and mentally aloof through it all (my MO for such circumstances), but I was surprisingly open-minded and receptive to the presenter's well-researched and proven insights. Because I teach the PreAP and AP math students, the TAKS test really isn't part of my solicitude. Of the juniors I do have, I expect all of them to pass (and to get commended). Seniors don't even take the exam. Instead, I concern myself with my curriculum and in preparing my students for the rigors of the AP exams and universities.

Perhaps is was the combination of the presenter's professionalism, her sense of humor, and awareness that she was "preaching to the choir," she did a great job of disarming us, rather than angering us, and I was drawn it. In fact, in all my years of teaching, hearing about the TAKS test, and listening to all the petty squabbling over ratings, subgroups, data this, AYP that, etc., yesterday was the first time I got a comprehensive, carefully dissected view of what ALL that means. I even learned something new about the way the test was actually constructed, with the first and last quarter being the easiest questions to answer. This is in stark contrast to the SAT, which gets progressively harder with each increasing integer problem number, and the AP Calculus exam, which has no pattern of level of difficulty.

This knowledge alone arms teachers with a great strategy to pass on to students (without actually having to "teach" specifically to the test questions--of which I am trenchantly morally opposed.) Even the best students tend to work in numerical sequence and can get bogged down, frustrated, or even disinterested near the end of a 72 question test. This means they could potentially shut down before reaching the tantalizingly easy ones on the last 25% of it.

So for the first time in a long, I felt that I actually learned something valuable at an in-service. But just as I thought about calling the local newspaper with a headline, the presenter posed a question to the group: "What makes special triangles so special?" Having prefaced the question by saying she asks that ONE question to all prospective math teachers in her district, I knew she would be looking for an answer that went beyond to obvious or trivial. As many chimed in with what I thought were great characteristics of the special triangles i.e. consequences of them being special. As expected, the presenter quickly dismissed each of the responses. "Thank God this isn't a job seminar," I thought.

I felt that I was in a unique situation to answer the question with what I thought she wanted to hear. Because I have taught geometry, trigonometry, and calculus, I have seen and used many principles that are direct applications of the two special triangles (30-60-90 & 45-45-90) including the unit circle and trigonometric identities. I'm also an avid reader of math history, so I felt I could add a little of that in as well. I thought carefully how I could summarize the underpinnings of my colleagues responses in as concise a statement as possible. Never liking to "sound off" in a group setting at the risk of appearing almighty, self-righteous, or idiotic (besides, I enjoy concealing my knowledge a bit), I had mentally selected my words. I spoke with quite clarity and self-assurance: "It's because they are two building blocks, from which we can find information about any angle in increments of fifteen degrees when used with trig identities."

Proud of the fact that I didn't get into the details, but only alluded to them, I awaited my accolade . . . . only to be off-handedly dismissed as the others had. I immediately got angry. "She must have not heard me," I thought. "Perhaps she didn't realize the depth of my statement," I rationalized. I thought that perhaps I should have conveyed that since the measure of a full rotation is 360 degrees, preserved from the ancient Babylonians sexagecimal (base 60) number system, the angles of 30, 45, and 60 degrees were factors of 360, so that combinations of their angles can be used to develop other angles, though not all. It is for the same reason we only memorize multiplication tables up to, say 12. At some point, for other products, we develop algorithms that allow us to use our knowledge of the "fundamental" products. Sure, we can develop unit side lengths for other right triangles, but it is mentally and mathematically more efficient to use the fundamental "special" angles to develop the other angles as we need them. In essence, they are easy to remember.

Anyway, my anger subsided as I realized there was no "right" answer, any more than being told that vanilla is not my favorite flavor of ice cream (which it is, even if you say "no"). I sat quietly, calming myself, shoring up my argument mentally, and eagerly awaiting the "correct" answer. Second seemed like minutes. The presenter went on and on about "thinking outside the box," "creative answers," and other platitudinous commentary. When she drew a breath, I awaited the answer. Out came information about bugs and spiders--she had moved on to an unrelated activity about scatterplots. Never mentioning why she thought special triangles were special, she left the entire group hanging in mathematical suspense. But to my surprise, no one else WAS! I figured she must have answered it while I was preoccupied with mentally beating myself up. At the risk of making a bigger fool of myself, I bit my tongue and tried to forget about it, figuring I'd ask a colleague later.

Well, she NEVER told us. It wasn't just my imagination. I was apparently the only one that took offense to being "shot down." I wanted to know. That's why I read those damn boring math history books to begin with. If I think some little insight, regardless of how small or insignificant, will make me understand the big picture better, I know it will ultimately make me a better teacher--I want to know. Now I NEEDED to know.

I emailed the presenter the next morning (this morning) requesting the answer, to which she replied that I would have to wait until our next seminar with her in late February. She only hinted that it had something to do with "where" they came from, rather than "how" they are used. My suspicions exactly.

I guess I have no choice but to be content for now with my over-analysis argument. But as my wife reminded me, in today's educational setting, it's really unfair to single out any one group. ALL triangles are special, not just the two. To treat any triangle any differently from all the others is to show favoritism. She's right. In just a few years, every triangle will be "college-ready."

Homework is Hardwork

Each year, I struggle more with my homework policy more than anything else. This is really the most important part of a math course, the individual practice that will allow the student to master the concepts and algorithms; however, it is not the “big game.” The tests are where the real assessment comes. This is where students must "stand and deliver" or "sit" and deliver, whatever the case may be. Many college math courses’ grades come solely from 2 or 3 tests each semester. Homework often counts very little as part of the final assessment. In some classes, it counts for nothing.

This is why I really do not like the current grading system enforced unilaterally for our high school, which not only puts such little emphasis on homework, but also allows students to turn it in late (imagining practicing for a football game AFTER you've lost the game!). When students arrive at the university, professors expect that the student knows the importance of homework as a means to mastery, and that they have the prerequisite skills through the practice of homework. Although this is obviously not the case for all upper-level high school students, we should be preparing them for such an assessment system.

However ideal, it becomes an overwhelming burden to personally grade each student’s homework daily and provide critical feedback. To have each student grade his own in class allows them to immediately see their errors and correct them, but this is a serious chunk of time, meaning less time for the new material, to play with the mathematical kittens, so to speak. I have found that this creates a downward spiral: the more time we spend covering homework, the less time we have for the lesson, the less they understand the new information, the more questions they have the next class period, the less time we have for the new material, ad infinitum. The difficulty comes in finding the balance between holding their feet to the fire, getting them to do the homework everyday and not spending a lot of time going over things they should have picked up on their own. The challenge is to create an upward spiral.

I have found that a combination of completion and accuracy grades, depending on the difficulty of the lesson, the complexity of the solutions, etc., works the best for me. We have a homework assignment daily, without fail. The homework grade should be a reward for the student who exerts the daily discipline required to master the subject. In addition, homework is due at the beginning of class, as soon as the bell rings. Quick quizzes every other class period give me a good idea of who is doing quality individual practice, and who is copying their homework from another student. I never permit students to use notes or homework on quizzes (even though they lobby for this consistently.) This forces students, those that are motivated my grades, at least, not only to go through the mechanics of the homework, but to do it for comprehension as well. When done right, math homework is a very long, arduous, personal triumph of a student over his own limitations, and he walks away from the session sweaty and in possession of something he will never lose.

This is something that is ostensibly lacking in classrooms across the world. I have found that students go through the routine of completing the assignment as if the completion itself was the goal, rather than an opportunity for them to get a better return on their investment of time, energy, and resources. In the end, students reap what they sow. Good grades do not equal learning.

Pedagogical Pizza

Obtaining an education has been compared to the filling of a vessel and contrasted with the lighting of a candle. Without both, however, education would be incomplete. The acquisition of a knowledge base is essential to be a productive member of society and to be culturally literate. But schools should be more that just little learning factories, making pedagogical pizza; they must help students to use their mind well and to put that knowledge into practice.

Much of the specific knowledge, such as important dates or mathematical formulas, will lie dormant after schooling careers. The lighting of the candle, then, is the more valuable gift of a public education. The primary goal of schooling should be to produce graduates possessing informed respectful skepticism. Many schools include qualities similar to this in their ideal portrait of a graduate, but it is a goal that is difficult to assess, hard to measure, and therefore usually left as a hopeful by-product of education. Under the current system, the emphasis is on coverage, rather than depth, on the quantitatively measurable, not necessarily the long-term qualitative results. Due to pragmatics, the emphasis has been on exposing students to a broad spectrum of ideas, rather than a deep, rich investigation of fewer concepts. Which method produces a better graduate? By my definition, the second accomplishes the goal.

There is such a plentiful supply of cheap information in our society that it is more important than ever to teach students how to interpret what is useful and what is not. By honing respectful skepticism, post-graduates and citizens can wend their way through the false platitudes and sophistries spun by politicians. They can discriminate information given by a pandering media. They can sift through the intentions of deceitful advertisements and clever marketing schemes. They can interpret the true intentions of disingenuous officials. Schools should be about the life of the mind. Only the well-trained mind can tolerate the rather paradoxical task of respecting a wide range of ideas, while yet remaining skeptical of their veracity. Skepticism without respect becomes cynicism; respect without skepticism becomes perilous absolutism. Both are needed for a healthy democracy. As Wilson Mizner says, “I respect faith, but doubt is what gets you an education.”

I am a teacher of students, first. My MO happens to be mathematics. We teachers are at the mercy of the current educational trends that so heavily influence the bureaucracy which set the guidelines under which we perform, which can sometimes be frustrating, if not creatively inhibiting. We as teachers and parents can only do our very best under the circumstances. We can find motivation, however, in our students, whose youthful naiveté, uniqueness and diversity, beautiful but awkward dance, and energetic optimism, provide an endless supply of drama, inspiration, and reassurance.

The challenge of today’s teacher will continue to grow as our world becomes more populated, more complex, more regulated, more violent, and more refractory. We cannot afford to take our eyes off the horizon.

Abandon ship?!?!?!

For the past nine years, I have dealt with the usual downside to any job, namely all the crap you have to put up with. A job without crap to deal with is a job that doesn't pay anything. A PAYING job is one whereby one gets paid to deal with crap. But there comes a time where the crap exceeds the reward of being monetarily compensated for it. Call it the "break even" point, but "breaking" point seems more appropriate.

In the past, as a matter of self preservation, I've been able to close my classroom door and take care of my little 20 foot by 20 foot square of the educational edifice. I weathered the monthly, even weekly, bombardment of parent complaints, and administrative mandates. Most of these "disruptions" to my pedagogical paradise came out of hyperbolic, knee-jerk reactions to trivial issues. Others were short-sighted manifestos designed to quell a particular inferno and to foster a genuine sense of cynicism and alienation among those upon who it was forced, i.e. the teachers.

Considering myself a Mathematician first and a teacher second, I let my love of each carry me through the occasional negative aspects that I know any job entails (I spent almost three years prior to teaching a successful homebuilder--believe me, IT'S EASIER TO MANAGE NON-COMPLIANT STUDENTS AND ANGRY PARENTS THAN IT IS TO MANAGE UNDERPAID, OVERWORKED SUBCONTRACTORS.) But slowly over the nine year of my teaching career, I have seen the defined roles of the teacher "devolve" to include more and more things that should be a parent's role. I have seen the burden of education transfer from that of primarily student's responsibility to student AND teacher, to the current status of teacher responsibility. Teachers how now become the stopgap for every social ill. We get the blame and none of the credit. We are viewed as whiny and expendable (what other professional would complain when they have summer's off?)

New educational edicts empower students to be at their worst. We are implicitly enabling students to ride the tide of mediocrity. Without and real standards of conduct, individual responsibility, or work ethic, we are producing a generation of students who are not "left behind" because they are spoon-fed and given piggy back rides across the finish line with no REAL, I mean REAL, DEEP, AND SUBSTANTIVE concern for how they will land beyond the finish line of high school and that standardized test.

Under the guise of "the best interest of the student," we adopt policies that help the student "feel good" about turning in late work. We reinforce that it is "Okay" to fail several times, as long as they eventually try, whenever they feel like it. We are sending a message that the world revolves around THEM, and that the safety net will always BE there. How far from the truth that is, but by the time they realize it, if they ever do, they are no longer part of our "data," so . . . . good luck to them after that.

Take for instance the scary, threatening comments from my "Tackle TAKS" video (see it here) I did last year in an effort to "boost" student moral as they went into their state standardized test. I busted my butt in the classroom preparing students for that darn test, and even got my butt busted in my efforts to be a part of the game in which we all play. Here are the comments of one viewer who came to the realization that public education has come down to "passing one test" and not so much about espousing real learning. He responded three times within 24 hours.

• fu@$ the taks I faild math part and dint graduate cuz of it why dont you fu@$s show us how to make money taks dosent make money it just makes familyes worry more
• my teachers was to petrafide in class to teach
• I know you teacher put all these thumbs down we are just speaking the truth you dont need taks I make about 500 a day from my own online bussines its not wat you know its who you know fu@$ school they left me I didnt leave thim im from the dfw were lots of theachers see bad thing so they are perafide every year i think teachers in the hood are one of the best cuz they see a young kids strugle

Or take this comment:

• I think TAKS is a waste of our time!!!I think that teachers are too busy preparing children for the TAKS and not for college!!! Also children are still being left behind. I think that we shouldnt have the TAKS test but instead just Final exams to see where we stand. TAKS testing puts alot of pressure on us students. We want to go to school to learn something thats going to carry us through life, not to prepare us for the TAKS!!! Thats my honest oppinion!!!

Unfortunately, we are going the other direction. Currently, the AP exam is a cumulative, comprehensive, rigorous exam that requires a students to demonstrate real understanding of, in my case, calculus, for which they can earn up to 8 hours of college credit for passing (BC calculus.) It is nearly impossible for someone to pass the test by merely being "taught the test" itself. I have had great results in both teaching this subject, preparing students for college, AND, coincidentally, getting them to pass this exam. I pour my heart and soul and every energy into doing this effectively. I have been certified by the College Board to teach both AB and BC calculus, I have attended week-long summer institutes training to be more effective, and I have even been selected as an AP Calculus Reader, a prestigious appointment where I get to actually grade the AP exams.

The current, back-breaking, decree on table is that the most intense, most rigorous, fastest-past math class on campus, BC Calculus, a class that I started at the request of students in the top 10 of their class (not just top 10 percent--students who are applying for Harvard, Princeton, Yale, MIT, UT, Rice, etc) and their parents has been suddenly mandated to be a dual-credit class. This means that I, not having a Master's degree, will be unable to teach these awesome kids with whom I feel I connect so easily with, but more importantly, it means that a small, local, private college with mediocre status, will accredit the course, whereby students can simultaneously earn college credit. It's essentially "take a high school class, and get college credit, as if it were a college class." Unfortunately, the accrediting college does not have much control over the quality in which it's delivered. They, instead, rely on the "professionalism" of the high school to maintain the college standards. This really equates to a crap-shoot on the dedication and qualifications of the teacher, with at least a Master's degree, teaching the class. It's not surprising that few major Universities actually accept these hours towards a degree. Some might offer "general elective credit," but the students who will be taking BC calculus, they will get NOTHING, but instead will have to start back at ground one: Cal I

Compare this to the 9 students out of 14 last year that I had earn 5s (the highest score possible) on the BC Calculus exam. These students went off to UT, Villanova, University of Virginia, and MIT with 8 hours of RECOGNIZED college credit. Some retook Calculus II (not I), and several went directly into Calculus III, where I can happily report that they are doing superbly.

I can't help but take a step back and think of how the captain (or captains) commanding the ship are steering it into such shallow, shallow waters. I believe it's either time for mutiny on the grand seas, or as I fear, it's time to abandon the ship altogether.

Arrrrrrrrrrrrrrrrrrrrrrrrg!