The trig students gushed unforeseen excitement about seeing me derive the Law of Sines today. They loved it! My plain old plan for it had been this:
Go through derivation clearly but quickly so students see a connection between SOHCAHTOA and sinA/a=sinB/b=sinC/c
Demonstrate using the LoS to solve a triangle
Have students practice solving a triangle using LoS (make sure they do the practice problem)
Little did I know that step one would be so energizing and fascinating to them! Although I told them they would not have to reproduce the derivation, students who normally forget the proper function of notebook paper were furiously copying down the steps, asking questions (“In the area equations, what represents the height?” “What’s the pattern of the letters in the different equations?” “Why is it that way?”), and answering other students’ questions.
My favorite moment was when I asked the class, “Does everyone understand how I went from this to this?” A pedagogically poor question, it has traditionally been answered by silence, a dull or uncertain “Yeah,” or even a perturbed “We get it.” Today in my class it was answered by, “Yeah, you broke off that part and everything cancelled out except the a on the bottom.” The language is not terribly specific or mathematical, but it’s far more so than any of the traditional replies.
I wonder whether all derivations would strike these students as forcefully, or whether this one was especially suited to the task. Either way, I slapped a sticky note down on my lesson plan to remind myself to hit it out of the park again next year.
Yesterday my trig class worked on a genuinely interesting problem, but as I described what we were trying to find, one student asked that hated question, “Why would anyone want to do that? They could just ________________[insert non-math route to the same information, perhaps reasonable, perhaps not].” In hindsight I see that the student was actually enjoying class, that he asked the question not from pure frustration, but playfully, hoping to elicit fun banter from myself and the other students. And yet. And yet it just plain irked me to have prepared and presented an intriguing problem only to encounter this response in the key of whine.
Tell me, how should I have responded? Should I welcome the question because it’s an opportunity to make the case for mathematics? Should I establish a class culture that eschews that question in favor of more productive ones? Or should I put up with the question, acknowledging that I work in a high school and must occasionally partake of the culture of complaint?
Along the same lines: As a support teacher I sometimes step in as a substitute, which I recently did in the geometry class I “support.” This time when I convened the group they were attentive and engaged rather than giving me the wake-me-when-we-have-a-real-teacher-again attitude I’ve received from them in the past. But almost instantly upon deciding they respect me, they delivered the question: “Why do we have to learn geometry?” which exploded into, “I don’t get why we have to learn history,” “Yeah, and why do we have to learn physical science?” Fortunately, one student held everyone at the level of a respectable discussion instead of descending into pure lamentation (oddly, the student who led the wake-me-when-we-have-a-real-teacher movement is the same one who led the come-on-guys-let’s-keep-this-respectable movement), but really? How do I defend the worth of their entire education to students (all of whom have some kind of learning disability, remember) when my only instructions were to review the triangle congruence postulates? Sincerely seeking input.
Last Friday I took advantage of a teacher work day to try to formulate some cool math problems based on the law of sines. I ended up on the soccer field with a partner, a camera, a 100 ft. measuring tape, and a compass.
The value of accurate tools was revealed when I remeasured the angles multiple times with the low-quality compass and got multiple measurements. Lacking any better resources, I resorted to choosing whichever of the measurements led me to the correct calculation.
Now, I’m not fabricating the data, since the numbers I settled on were actual measurements I took. And since I know the law of sines to be reliable, using it to judge the truth of uncertain data is defensible. But… it just feels wrong somehow. Maybe I can assuage my conscience by making the judging of data part of the lesson. Or maybe I should find a compass with sighting guides.
*Interesting but irrelevant note: Despite our best intentions to create a non-right triangle, my data-gathering partner and I ended up with one angle that measured almost exactly 90° (ahem, somewhere between 88° and 95°). Subconscious determination?
Students at my school are preparing for parent-teacher conferences, for which they (the students) prepare a slide presentation about how they’re doing, where they can improve, and what techniques they’ll use. In view of that, one student came this morning to ask how he’s doing in my trig class. As I collected my thoughts I asked him to sit down and tell me how he thinks he’s doing in the class. He then delivered this unexpected simile: “It’s like we were thrown out of an airplane,” he said, “and into an ocean.”
“Before it was like we were in a boat and we’d go scuba diving, but yesterday it was like we were thrown out of an airplane into the ocean.”
The before that he mentions I assume refers to our most recent topic, trig functions on the unit circle. The thrown-from-an-airplane-and-into-an-ocean feeling comes from moving into trig functions based on triangle side ratios (SOHCAHTOA).
I’m always glad to hear students describe their understanding but, to date, this is my favorite description. The sentiment was so unexpected and the image so expressive! It is also strangely satisfying—not because I want my students to feel lost at sea, but because he could have said something far less heartening. If he’d given me a tired idiom (“It’s like we jumped from the frying pan into the fire”) or something just plain tired (“I’m lost”) the message I would have received would be that he’s not confident with the material. That’s a valuable message, but what this student communicated to me is that although he is not yet confident with the material, he’s engaging with it enough to come up with an original and vivid analogy for his struggle, and to compare it with an original and vivid analogy for a previous unit.