The fifth element

Going through the National Board process is helping me to keep focus on a goal of being mindful of lots of different elements, when designing lessons:
What am I doing to ensure that this lesson is mathematically meaningful?
That it's addressing equity and is accessible to all of MY students?
That students will be able and willing to be responsible for their learning?
That I'm prepared to offer meaningful help to struggling students, and meaningful extensions to rock stars?

And, today, I came to recognize the fifth piece of this puzzle:

What am I doing to ensure that this lesson will be entertaining?


Yes, I'm thinking now in terms of entertainment, rather than engagement. Because it's not always necessary that the lesson is engaging, itself: if a student is bored, then the lesson won't engage them. If the student is having fun being in class, then (in my recent experience) they'll happily tackle a worksheet. Also, it's a heck of a lot more fun for me...

Project Plans

Next week is the last before the holiday break. We're pretty much done with the current unit. So, today I caught up with the other teacher who teaches juniors, and we agreed to start the new unit after the holidays, and spend next week on extensions and reteaching other topics.

This gives me a great opportunity for another "fun project". I had in mind a miniature "automatic free-throw machine", using rubber bands or springs to launch a ball into a cup. It offers a lot of opportunities for good math content, and I definitely think it'll be worth doing, some time-- but for now, my students weren't really into that idea. In 1st period, one girl said, "We've been doing this thing with cars, so why don't we stay with cars?" A few minutes of discussion brought them around to this:

Build an obstacle course, including ramps and curves. Two radio controlled cars will race, as students keep time through different points on the course. Prizes for the fastest overall run, and the highest top speed (which will require some calculation of distance and time, unlike the "fastest overall")


Second period also wanted to stay with cars, but they have a different project in mind: Each group will build their own car, powered by an engine of their own design (some have ideas, already, for rubber-bands, balloons, or springs). Prizes for the car that goes furthest on a single "wind up" of the engine, for the fastest top speed, for the one that goes furthest up a steep ramp, and for the car that travels the straightest.


We'll see what 7th period comes up with, and I'll update this post then.

Repetitititition

Started cracking the door on sinusoidal functions this week (yesterday, actually), with the calculus class. Some good progress, but it is a bit slow going. I'm taking the plan of doing short lectures with short practice problems. The initial push was pretty good: I used a Javascript animation of a point moving around a unit circle, with sin and cos developing alongside it as functions of the angle measure.

That was enough to develop a sense of the shape of the sinusoidal function, and to give a somewhat weak but present justification of d/dx(sinx)=cosx and d/dx(cosx)=-sinx. Today, we covered the same ground again, without the animation, but with me effectively reconstructing it by hand, which allowed me to also make some basic linkage back to right-triangle trig, and to wave my hands a bit around "cos is the horizontal, sin is the vertical."

The students are very clear about the fact that they aren't solid on this, yet, and I'm surprised at how okay with that I am-- it comes from experience, I guess. At this point, I just expect that I'm going to show them this thing, every day for the rest of this week, and then next week, I'm going to get them to show each other this thing. Because, it's a tough concept to wrap your head around, and it's appropriate that it'll take a bunch of seeing it before it clicks.

This is also good, because it gives me an opportunity to develop that "trig wheel" project that's been kicking around the back of my head (which, if done right, would make quite a good Entry 3 for the National Board portfolio...)

We haven't touched radians yet at all. It's a long row to how, this year, especially having lost 5 weeks at the beginning. I'm pretty sure we're not going to get everything in before the finish line, which means that I'd better start deciding what to de-emphasize. I don't plan to cut anything, but there are some fiddly bits around the ends that I won't be able to spend this kind of time on, to make them solid. Still, I think it's better to have something solid, and something else not at all, rather than two things shaky...

Professional Development

I'm in a professional development session right now, with a couple of faculty from National Louis-- we were asked to write about the worst teacher we've ever had (the session itself is really about using writing in the classroom-- a lot of modeling of how to create a positive writing experience, sharing the writing using an "author's chair" and a set of ground-rules for discussing the writing).

Just in case anyone's wondering what we do in these "days off". For the record, here's my writing.

"Many of the most frustrating experiences I've had as a student came during my sophomore year of college, particularly in physical chemistry class. The professor was the sort of serious researcher who saw teaching undergraduates as a necessary evil. The class itself was a large lecture, with around 75 second-years, all struggling to get past the hardest year of engineering school. That in itself wasn't a problem-- I'd had bigger classes-- but it didn't help the professor to develop any understand of nor empathy toward his charges. After the first examination, he announced with equal measures of rage and disappointment that the average score was around 35%. Initially, he posted failing grades for the entire class, but later curved the tests (an action which he attributed to "pressure from the dean"). By all accounts from other students, this was standard operating practice for him-- somehow, he'd managed to spend at least 10 years being shocked that his students didn't understand his class. Isn't that one definition of madness-- expecting different results from the same input?"

One of the presenters makes a point about writing along with your students, as a way of modeling the activity, and establishing your belief in the legitimate importance of the act. As one presenter says, "I'm part of the writing community." It occurs to me that this is a lot like how I'm handling my calculus openers: creating a problem on the fly, on the way to class, and then solving it while the students are solving it. I'm part of the calculus club-- and I should probably emphasize how cool it is that they sometimes beat me to the answer. I mean, how cool is that, that a novice calculus student can beat me to the answer?


I like the "observation of detail" exercise, too: Describe a blueberry, without using the words "blue", "berry" or "blueberry". The purpose of the exercise is to develop the ability to include descriptive detail (which is a challenge for most students), which is definitely a thing that I struggle with with my math students. It'll be a challenge to adapt it to math projects, but I like the idea of that sort of restriction-- and the obvious starting point is "you can't use the words 'this' or 'that'."

Testing strategy

My colleague and I collaborated on a mid-term exam, which ended up being pretty enormous (five pages, free response). Some of the questions were very straight-forward (what's the slope of the line between these two points? What's the y-intercept of the equation y=3x-2), but many more were multi-part, rich questions.

As the day went on, I realized that the majority of students weren't going to come anywhere near to finishing this beast of a test. So, as I was grading the 2nd period tests, I developed a points system (different questions, or parts of questions, worth different numbers of points), such that the entire test ended up having 400 points. That done, I gave the last class a rubric (including the points-value of each question): 100 points to get a D, 125 for a C, 160 for a B, and 200 for an A (yes, a non-linear function, which tops out at 50%).

Seems to be pretty well calibrated: there's a LOT of different content, and a LOT MORE different presentation of similar content, on this test. By providing an unrealistic number of opportunities to demonstrate what a kid knows, and indicating that I don't expect anyone to finish the entire thing, I ended up providing options to students, so that they can really show me what they know well (by choosing the problems that "look easy", whether that's the visualization, or the abstract, or the verbal, and focusing their best efforts on that...) To get an A, a kid will have to correctly cover at least some ground in each modality (so, it connects up to the Rule of 4...)

In all, I'm pretty intrigued by the happy accident-- and I'm definitely looking forward to designing my next test in a more straightforward way, to work on the same principles.

Natural language processing

I've been searching for a couple of weeks for a way to handle dimensional analysis with my juniors. It was clear from the first presentation of the word "units" that using that particular vocabulary was going to be an uphill battle. For the past couple of weeks, I've been struggling to find a way to explain what units are, and why they're useful.

Today, in a moment of synchronicity during first period, I happened to say "So, what thing does that number represent? What label can we put on it?" And, viola: the terminology is found: "Label" as a proxy/synonym for "unit". Worked well all day-- suddenly, students were comfortable with using the (ahem) "label" for every number they put on the paper.

Later, of course, I'll have to spiral back through and explain that "label" isn't really Proper Mathematical Terminology, and that Real Mathematicians And Scientists use the word "unit", instead. But at that point, the "what it is and why it's useful" part will be handled, and it'll just be a little mini-lesson in linguistics (unit, from "one", as in "one of these types of measures.")