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Promoting Student Ownership and Engagement in Math

Checking for Understanding

Teacher Karen Crounse at Codman Academy in Boston, MA checks for understanding in her eleventh-grade math class to help all students discover the patterns behind trigonometric ratios.

This video accompanies the book Leaders of Their Own Learning: Transforming Schools through Student-Engaged Assessment.


- Okay silence right now, working on the do now. We are doing an expedition in math class called traveling through space. The problem we’ve been working on is how do you find the height of the rocket as it travels through the air? So we had some ideas, some brainstorming about how to figure out how high the rocket went, and I’ve realized that this problem is really, really hard. So instead of starting with that, we focused instead on finding the height of a church, something that doesn’t move, and investigating the height of the church, the kids realized they needed to know more about right triangles.

- Because that’s 90, you can find out what the other two angles are by using the protractor.

- I think that’s a fine hypothesis, okay? And maybe we need to know a little more about these ratios... The main learning target was to investigate ratios and right triangles. The second learning target is I can state the trig ratios and apply them and the goal is to do the investigation to lead to the second learning target which we will work on today and hopefully tomorrow and in the future. The opening of class always begins with a do now. The purpose of the do now is to review what we did yesterday and to also lead us into the work for today. Most of the activity in class is done in groups, but the do now is always done independently. I wanna now assess what a specific student knows about that content. Not what everyone else knows, but what they got out of the previous class and what they know that they’re gonna bring to this class today. And then what? I make it a point to check on certain students that I know in general struggle with the material. And so maybe talk about the ratios... Every student has a binder where they keep all their math materials organized by major learning targets. By having students write things down, I can see exactly where they’re having problems, what mistakes they’re making, what the errors in their thinking are, and if they’re asking me a question, I can right then and there see where the problem is. While walking around and seeing what students did for the do now, I noticed that they were talking about math terms but not specifically using them. Mary, Chantell has a question for you.

- What do you call something where it repeats more than once?

- Repetition?

- When you’re talking about a list of elements and data, if there’s a number that repeats?

- The mode?

- Mode, that’s it?

- So that’s interesting that the way you’re saying... So during the debrief of the do now I made sure that I brought up that vocabulary so they can talk about these ideas and use the correct vocabulary. How would you get a typical value for that?

- Add them all up.

- I feel like you had a good idea.

- And then divided it by the number of students.

- Yeah, and what’d you get?

- .67.

- .67? Okay, so you found the mean. So you’ve got the mean, the mean and the mode, two ways to measure with typical values. Anybody use a different method for finding a typical value?

- I got one. For any column, well you’re supposed to put the numbers in order, then you just basically cross them out until you find the middle but if there’s two numbers in the middle you have to add them then divide them by two.

- Okay.

- That’s how you find your median.

- Right, oh okay, nice job. Often times when I’m at the board, I’m just recording what the groups are telling me. To me, it’s okay if wrong things, errors are put on the board, because we’re all gonna be looking at it. Malik, you have the mode, what is it?

- .74.

- Mode is .74, nice.

- And .77.

- [Karen] Say that again?

- And .77.

- [Karen] And .77. And it makes it even, it makes everybody engaged trying to figure out if what’s up there is right and makes sense with the patterns that we’re finding.

- [Girl] .77 only comes up three times.

- [Student] It only comes up three times.

- And does .74 come up more?

- [Students] Yeah.

- Okay. Sorry, Truth. Doing these kind of activities in math or discovery in math, every student is engaged. There’s no student that can sit out or opt out of doing math or understanding the math.

- Okay I’m doing the easy one, I’m doing range.

- [Student] There is no range, mean, median, mode.

- [Karen] Every student is actively building our resources so that we all understand the problem.

- Adjacent hypotenuse.

- .68.

- Without naming it, kids were exploring the ratios and sides of right triangles, they were actually deriving the trig ratios sine, cosine, and tangent. So sine of an angle is defined as opposite over hypotenuse. Cosine of an angle is the adjacent over the hypotenuse and tangent of an angle is opposite over adjacent. Okay so take this okay see this. So using your handy dandy calculator...

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