I want to share what might be the best lesson I’ve created and a few reasons why.
I actually wrote about this a couple years ago but since we’re doing it right now I thought it useful to reflect on and share it again.
This lesson came from the following problem I was struggling with:
I had spent a lot of time thinking about how to help students understand the connection between trig ratios on the unit circle and the graphs of trig functions on the Cartesian plane. Despite a couple activities and practice I was convinced, mainly through questioning, that they didn’t fully understand it.
My solution to this was to make a giant unit circle and cartesian plane and have students use them to work out problems. This would allow us to literally walk to specific angles and equivalent places on the cartesian plane. The hope was that this would help students solidify the connection between the two.
The details of how the activity works are in the original post and the materials are linked at the end of this post, so I want to emphasize the aspects of the lesson that really make it effective, in a convenient list.
The activity is broken into two parts. The practice portion and the assessment portion. The assessment requires students, working in pairs, to come into the hallway and work through five problems (like these). This portion is vital for the following reasons.
Like any assessment, it helps me know what they know.
It makes students take the practice seriously. The assessment mirrors what the practice rounds were like. They take it seriously and practice until they’re confident.
Students work in pairs, sometimes disagree, and then must convince each other of their reasoning. Tremendous mathematical conversations come from this time.
It puts students in a position in which the teacher is there, but can’t help. This is true of assessments in general, but the format of this one means students must convince themselves and each other that their answer is “their final answer”.
No calculator. No notes.
On the assessment, and likewise on the practice, students cannot use a calculator, notes, their unit circle, or anything besides their brains and a whiteboard. This means students don’t have any crutches with which to rely on. These problems are not algorithmic. Each one is slightly different from the other ones. This means that the only way to be successful is to truly understand what is going on in the math.
This is my third year doing this activity and every year there’s nearly full engagement. Now, this is precalc and while I wouldn’t say that all of these students want to be there, it is an elective. But it’s difficult for me to get this level of engagement from them.
This is, in part, because they know there’s a test coming after they’ve practiced. But I think it’s also because each problem sparks at least a little bit of curiosity. “How do we figure this out?” Initially many students don’t have a clue about how to approach something like sec(2pi/3) with only their brains and a whiteboard. But with a good understanding of trig they can figure it out.
And figuring it out is satisfying. Students are proud of themselves when they solve one of these problems correctly. I love seeing high fives in my classes, and this is one of those activities where they happen.
I’ve written about embodied cognition before so I won’t go into too much detail, except to say that it’s incredibly valuable if you can incorporate it effectively. There is something fundamentally different from paper and pencil when you can stand there with a student inside of a unit circle and discuss these problems. It’s something that is hard to describe, but once you’ve tried it you clearly see the value.
Purposeful practice without a book assignment
A few weeks ago students initially learned how to do these problems via a lesson and practice problems. If that was effective, then I wouldn’t have needed to do this activity. What ends up happening in this activity is that students end up doing a bunch of practice problems, that I never assigned! I just tell them they can do as many practice rounds as they feel they need. Then they work until they have convinced themselves they’ve mastered it.
The test and practice require students to work in pairs. This is incredibly valuable as students are constantly conversing and helping each other understand. Once again, the knowledge that there’s an assessment plays into this, but who cares? From my observations students are rarely begrudgingly woking through these problems. They seem to enjoy them.
I probably see more learning and teaching happening between the students in this activity than any other lesson I do, for any class.
I understand that without seeing it happen it might be difficult for you to implement this. I’ve included some images below to give you an idea of the set up. Feel free to contact me with any questions you have. I’d encourage you to look for opportunities to use embodied cognition in your classes as I think it can be an incredibly useful teaching tool.
Here are the resources for doing the activity
Assessment Cards (Yeah, I’m not posting these on the web. I, shockingly, sometimes have students read my blog. But if you reach out to me I would be happy to email them to you and save you the time of making them.)