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Old-fashioned trig table

I’ve been using Andrew Stadel’s estimation180 with my 9th grade geometry students regularly this semester. Yesterday, I gave them an “estimation challenge” of my own.

This picture was taken on Sunday. At what time of day did I take the picture?

shadow

Since this was so different from other estimates they’ve encountered so far, we worked as a class to establish a window of reasonable answers. Someone noted the picture was taken during daylight hours, so between 6am and 7pm seemed like a good place to start. Someone else observed that shadows appear much longer very early and very late in the day. This was probably some time in the middle of the day. Someone remembered that the sun rises in the East and sets in the West, so which direction is he facing? Isn’t that how a sundial works (yes)?

When pressed, I conceded that the left-hand side of the picture was “West-ish.”

After deciding on a window of 1 pm-5:30pm, students wrote down their estimates. And then I showed them this:sun angle

If they could determine the angle of the sun, they could determine the time the picture was taken!

The students have been introduced to definitions of sine, cosine, and tangent, and they have been putting those definitions to use solving for unknown right triangle parts using an old-fashioned trig table.trig table

Why a table? Well, it seems to reinforce the idea that “sin 75” is a number and not just some fancy piece of calculator magic… And, then on the day (today) I show them a picture of my son and his shadow, I can also ask them to figure out what time the picture was taken and have them discover inverse trig functions for themselves. And they did! They measured my son’s height, they measured his shadow, they determined the angle of the sun, and they determined the time the photo was taken. They can see so clearly on a table the relationship between (in this case) “tan” and “tan-1” and, in fact, define it for themselves. Beautiful. Sometimes that old technology (paper) is the best teaching tool.

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Perseverance

Two weeks ago my advisory students participated in a challenge program. Back when I was a student I found these kinds of programs corny and uncomfortable and I never bought into them. Never. At all. How surprising that now as a teacher I found the experience to be pure gold. It gave me insights into my students that will help me as their advisor over the next four years, and as their math teacher this year.

Working together outside of a school context my students set (and met) goals and completed tasks that challenged them. Three who had seemed quiet and reserved demonstrated leadership qualities I did not expect or anticipate. One who quickly lost focus in the classroom was sometimes able to stay focused on these tasks (and sometimes not). One who had seemed “too cool for school” was among the most eager to get involved in the problem-solving and the action. They all had a willingness to stick with tasks even after repeated attempts did not seem to get them any closer to a solution.

The task that surprised me the most was called “Ramble Road.” It consisted of a zig-zagging balance beam built from a set of 2 x 4s that included both inclines and horizontal components. Our facilitator told the group that this represented the school year and that we needed to get everyone from the beginning to the end. The catch: you could not move on the balance beam unless you were connected to (touching) another person who was also on the beam with you. The first pair of students made substantial progress before one fell off the beam, and a second pair also managed to traverse a significant distance before one fell off. However, after this initial success, no students managed to stay on the beam beyond the first short segment – for at least 20 minutes. In spite of this, they kept climbing back up and trying. Over, and over, and over again. They were persistent.

Which brings me back to my classroom. The CCSS mathematical practices are a big part of my thinking about instructional design these days, and number one on the list is:

“Make sense of problems and persevere in solving them.”

What are the characteristics of a task that makes students not just willing, but eager, to persevere? How can I teach them to persevere? I can’t claim this as a hallmark of my classroom yet – but it’s where I want to get.

I think the noticing & wondering that Max Ray and the folks at The MathForum describe can help students feel they “own” the mathematics. Ownership makes people want to persevere. I’m trying to do more of this.

I also think Dan Meyer’s approach with the hook in Act 1 of his 3 Acts creates buy-in from students that leads them to stick with challenges. (Maybe someday I’ll even invent my own…)

In my own experience, I’ve noticed that students are willing to persist when

  • they feel empowered and capable
  • they feel they are making progress
  • they have a framework for how to proceed when they are stuck
  • they feel it is safe to try new things
  • they are having fun or because they enjoy a challenge
  • they are well-matched to the challenges the task offers (zone of proximal development)

I’ve also noticed that some students are more willing to persist than others. Is this more because of students’ mindsets, or more because the tasks I’ve created are better suited to some than others? (Probably a combination, and further incentive to become better at differentiation.)

What else have you noticed leads students to persevere? And how do we as teachers help them get there?

Back at the challenge program… In the end, my students ignored the time limit our facilitator had set for the task (anticipating frustration and eventual disengagement). They refused to quit and they rose to the challenge, managing to get everyone, including me, across Ramble Road. I’ll be thinking about this experience this year as I design instruction. Here’s hoping for a year that I’m better able to help keep my students getting back up. Again, and again, and again.

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