Tag Archives: geometric mean

Opening Day as Opening Night

I really like our late start; why the hell are so many school districts kicking off in early August? (They want higher test scores, Ed.)

Anyway, I’m teaching trigonometry for the first time. In every course, I assess my kids on algebra I, varying the difficulty of the approach based on the level of math. What to do with trig? My precalc assessment was too hard, my normal algebra assessment too easy—or was it? I didn’t want to discourage them on the first day, but I also didn’t want to give a test that gave them the wrong idea about the class’s difficulty level. After much internal debate, I created a simplified version of an early algebra 2/trig quiz. I dropped the quadratics (we only had 45 minutes). Then, just to be safe, I made backup copies of my algebra pre-assessment. If the kids squawked and gave too much of the “this is too hard” whine, I’d be ready.

And so in they came, 23 guys, many of them burly, a few of them black, none of them both, and 11 girls. Fully half the students I’d taught before, two of them I was teaching for the third time. (one poor junior has only had one high school math teacher.) Perhaps their familiarity with me helped, but for whatever reason they charged right in and demonstrated understanding of linear equations, systems, a shaky understanding of inequalities, and willingness to think through a simple word problem. Good enough. Great class—rambunctious, enthusiastic, way too talkative, but mostly getting the job done.

I’m still not much of a planner, which is why I gave no thought to my trig sequencing until I saw how they did with the assessments. If they’d tanked, I would have done a simple geometry activity to give me time to regroup, start after the weekend with some algebra. But they didn’t tank, so how did I want to start?

Special Rights. Definitely. I would use special rights to lead to right triangle trig. All clear. But how to get to special rights? Algebraic proof of the ratios. But why special rights? It seems random to start there. As long as I’m going to be random, and since trigonometry has something to do around the edges with right triangles, why not start with right triangles? At that moment, this image popped into my head:

Hey. One step back to geometric mean, and I’ve got a nice intro unit all set up.

So the next day, I started with this:

geomeanquestion

Note: I told them the questions were separate—that is, the square was equal only to the area in #1 and only to the perimeter in #2.

I wasn’t happy with the questions. They gave too much away. But every rewrite I tried was even more confusing, and in a couple cases I wasn’t sure it was an accurate question. Besides, on the second day of school, you want to release to something achievable. Better too straightforward than have the kids feel helpless this early.

And it went great. Top kids finished in under five minutes; I had them test out the process for cubes vs rectangular prisms. All the rest completed the work in 15 minutes or less, with some needing a bit of reassurance.

I had to prompt them to recognize that the perimeter to side relationship is the “average” algorithm (that is, the arithmetic mean). “If I add two numbers and divide by 2, what is the result?” I think I noodged for a few minutes before someone ventured a guess.

I followed with a brief description of geometric mean, reminded them of the various measures of central tendencies, pointed out that now they all knew why the SAT followed “average” with arithmetic mean. Finished up with practice problems.

I was stumped briefly when a student noticed that the arithmetic mean always seemed larger. Argghh, I’d mean to look that up. I told them I’d look up the answer and get back to them. Meanwhile, I wondered, could the two means ever be equal? I made the stronger kids do some algebra, and let the others just talk it through.

Great lesson, not so much from the content, but from the energy. Look, I was winging it. I do that when I have a good idea that isn’t fully fleshed out. I cut back goals, keep things very simple, and watch for opportunities. I always advise new teachers to avoid mapping things out—they are often wasting time, because things will go off the rails early in some cases. Keep it broad, tell the kids that you’ll adjust if needed, and go.

The rest of the opening “unit”: a brief review of similarity and then use of geometric means in right triangles, leading to my favorite of the Pythagorean proofs. Then onto special right triangles, deriving the ratios algebraically. This puts things nicely in position for introduction of right triangle trig and I can drop in a quiz. Well, I’ll probably put in a day of word problems first.

After school today I ran into a group of football players waiting for practice to start, many of them previous students and two of them currently in that trig class. After hearing what they were all up to, how their summers had panned out, what the team’s chances were, Ronnie, one of the two current students, said, “I’m glad I have you; I would hate to be dumped for low grades my senior year.”

“Ah, yes, that’s my claim to fame. I’m not a great teacher, but by golly, I give passing grades.”

Shoney, the other of the two, a big, burly, not black senior, was laying along a school bench calmly watching the conversation, and spoke for the first time.

“You know. Trig was….fun today. It really was.”

Ronnie nodded.

The point is not oh, gosh, Ed is a fabulous teacher who makes kids love math. That’s never my goal, and it’s not what Shoney meant.

Recently, Steve Sailer writes that “school teaching can be thought of as a very unglamorous form of show biz, which involves stand-up performers (teachers) trying to make powerful connections with their audiences (students)”. He’s right. Education and entertainment are both, ultimately, forms of information transmission.

His next paragraph is dead on, too:

We are not surprised that some entertainers are better than other entertainers, nor are we surprised that some entertainers connect best with certain audiences, nor that entertainers go in and out of fashion in terms of influencing audiences. Moreover, the performances are sensitive to all the supporting infrastructure that performers may or may not need, such as good scripts, good publicity, and general social attitudes about their kind of performance.

People tend to construe the “education as entertainment” paradigm as “show the kids movies all day” or “keep the kids laughing”, but just as all entertainment isn’t comedy and happy endings, so too is education more than just giving the kids what they want.

I’m a teacher. I create learning events. I convince my audience to suspend disbelief, to engage. Learning happens in that moment. Some of the knowledge sticks. Other times, only the memory of learning remains, and I’m starting to count that as a win.

And so the year begins.