Friday, March 1, 2013

It almost looks like I know what I'm doing

I made some huge changes in how I did trig functions this year, and I am pretty sure it was several orders of magnitude better. It's only now that it's over, and I'm summarizing it here, that I realize that a lot of what made it so much better is kind of flukey. I made some decisions early on, or rather I just did some random things, that ended up making it all fit together perfectly. If I had tried to make it this perfect, it probably never would have happened.

Here's the skinny:
  1. The very first thing they saw, and did stuff with, was this ferris wheel.
  2. I based EVERYTHING that followed on that concrete reference.
  3. I gave almost no homework, or mindless practice examples, or whatever you want to call it.
  4. I insisted on daily blogging. 
  5. I rearranged the sequence of lessons by moving one lesson later.
  6. The very last task brought them full circle back to the first activity. 
Details/Observations/Reflections about these changes:

1. The ferris wheel: I posted the details of this activity here.
    Guess how much I like heights.
  • So all they saw was a circle with a little car that they could move around themselves. I didn't say anything about waves, or angles, or sin or cos. But I did show them a picture of me the very last time I was on a ferris wheel >>>>
  • They did not know what the graphs would look like ahead of time. That added a kind of suspense to the whole operation, and I heard feedback like "I really didn't think it would look like that." That is a good thing. It made them look FOR something, instead of look AT something I gave them.
  • I didn't say anything about angles until I was asked that magical question - how can we be sure we're putting the car in the right place on the ferris wheel for 5 seconds, or 10 seconds? Then and only then did I tell them about the special secret tool hidden in the ggb file. And by the way, that tool was initially not great, it was more about sectors than angles, but I fixed it so that it's all about the angle. I did manage to bring their thoughts in to realm of angles nevertheless, but only once they had asked that question.
  • Assumptions about the overall shape of the graphs were made based on very few points. It was funny to see how many people not only assumed that it was a series of jagged teeth, but also who followed other peoples' lead and stopped doing their own thinking. I revealed the answer to them in this series of slides, and when I added the last few points, right around slide 6, I got a big "OOOHHH!!!"
  • Once they had the visual, concrete connections between height & time, and distance from wall & time, it was easy to go from there to height & angle, and distance & angle. The real key, for me,  was to connect time with angle, to get why we even bother with angles at all for circular motion.
2. I based EVERYTHING that followed on that activity.

  • The unit circle became known as a much smaller, and weirder ferris wheel that was able to go underground (negative height) and behind walls (negative distance from wall). Once they saw the corresponding graphs of the weird ferris wheel's car, they recognized it from the first activity - same graphs, different sizes. THEY asked ME what kind of a rule would give a graph shaped like these crazy waves that they got. I mean, that was huge! And in answering that, I actually got to show the connection between sohcahtoa and unit circle NOW, when it makes a lot more sense. I used to put it in at the end of the whole unit, as a little bonus interesting tidbit for those few who were still with me and able to grasp it. Which last year was I think one kid.
  • In each voicethread from then on, the graphs they drew on day one kept coming up, so that they slowly but surely chipped away at its parameters. All except for one. Parameter b. My nemesis. See below.
3. I gave almost no homework.
  • Instead, I packed as much practice, and as many class activities as I could while still leaving them time to work and get help during class.
  • This was important, not only because I wanted them to have time to blog (see below), but I wanted to see if I could actually do this - get enough done in class that homework wouldn't be necessary. It was a gauntlet I threw down for myself. It was time to teacher up.
  • I say almost because they did have to do the blog posts, and they did have a few voicethreads to watch. Some practice came from a few minutes of warmup on the board, like converting degrees to radians, or finding amplitude from a graph. But most of it, I think, came from USING what they were learning right away, sharing it, and reflecting about it.
  • I didn't even give them a checklist, which I got some complaints about. And I found that hard not to have, because some of them have really taken to letting me know important stuff privately that way.
4. Daily blogging:
  • This was a real slow starter. On the second day, when I should have had 22 blog posts, I had 4. Third day was better, but I had to keep at it. I had to remind them that if they wanted the option of doing a final blog post instead of a test, they had to blog everyday. This may seem like a bribe, but to me it was more of a natural consequence. I knew that if they did this everyday, it would help them understand, and would therefore improve their results on any assessment.
  • I had an aha moment when I realized that at least some of that reluctance to blog came from their likely misconception that they had to get everything right the first time.
  • At the end of the week, some told me they found that doing this helped them. But I wanted more authenticity than that. And it came in the second week, when I said "You don't have to blog tonight." and I had 2 ask me if they could anyway, and 2 who just did it.
  • At the end of the first week I put this together, posted it on the classblog, and had them read it. LOTS of positive feedback from them, that it was fun to see what others had said, fun to see their own names and words. Also I think it made them see the value of reading each others' posts.
  • It took a long time to do that snippet post, but it greased the wheels for the next week, when I almost never had to remind people to blog.
5. Change in sequence:
  • The lesson I moved was the one on the exact values of sin and cos for angles like 45°, or 210°. I decided to skip over this as a result of the momentum we had going at the end of the first week. I knew that I would lose a lot of them if I went from the concrete to the abstract this soon, even for a day, so I waited, and when I did circle back to it, again, instead of homework, I had them do this, in groups, during class, to help them memorize. Feedback - Miss this was so much fun! - actual words.
  • I now realize, after many many years, that this is likely where I lost many of them in the past. It used to come just before the basic functions lesson, just in time to make them get all squirrel-ly about numbers like √3/2 and 1/√2. Anything after that must have been all static to most of them. Sorry, students of my past.
6. The very last task - this was today. It was a thing of beauty. It was the most beautiful, intuitive, logical, wonderful thing, and it was all a fluke.
  • This time they had to use what they had learned about sin, cos, amplitude, etc to regenerate the hand-made graphs they started with, but this time, by typing a rule into our beloved geogebra and getting it to match these original ones, for which they still had their own paper versions, plus these snapshots:
Height vs time

Distance from wall vs time
  • Did I know I was going to do this when I started? NO!
  • Unfortunately, it being the last day before March break, many kids weren't there. But those that were zoomed right in on the one thing that they were missing, the only parameter that I hadn't brought up in any of the voicethreads, parameter b. And why hadn't I brought it up? BECAUSE I FORGOT TO!
  • But it was perfect, because one by one, they came to me and asked how to get parameter b, whereupon I had the privilege of watching them get the final piece of the puzzle, which is that parameter b takes the variable "time", and changes it into an angle. How? By multiplying radians per second by seconds. thereby giving you radians, which is an angle.
Want to know when I got that? When the teacher figured it out? Today. At about 10:35 am. 


  1. I have a question, more than a comment. How did you switch from this wonderful, creative type of learning to the sad fact that eventually the students are going to have to sit very different, formal exams where their performance "on the day" has a significant impact on their entire futures. How do we, as teachers, balance creative learning and formal assessment when the two measure very different things.

    1. I worry about that exact thing myself, Sinead, and I don't really have an answer for you, except that while I do all this, I keep the baby in the bathwater. They still write tests, and they practice on old formal exams. My hope is that this way of learning brings them further than the old way ever did, so that the formal assessment is small potatos to them compared to what they have to do to write a good summary on their blog.