The MTBoS, Week 3 (Day 91)

Before we get to Lecture 17, it's time for Week 3 of the MTBoS Blogging Initiative. This week's topic comes from Sam Shah, of Continuous Everywhere but Differentiable Nowhere:

https://exploremtbos.wordpress.com/2016/01/24/week-3-of-the-2016-blogging-initiative/

This week, the blogging prompt is going to be around questioning.

Ages ago (in internet time, that would be 2010), Dan Meyer showed us an infamous example of a bad textbook question.

(Of course, this refers to Dan Meyer, the King of the MTBoS. Whenever anyone talks about the MTBoS, the name Dan Meyer will inevitably appear.)

This blogging prompt is designed to help all of us think a bit more about our own questioning. Pick one of the following and blog about it!

• A student asked a question that really got other students thinking. What was the question?

That's the one I want, so let's get started:

For me, it's interesting that this weeks topic would be about questions, because the teacher whose class I've been subbing for the past three days is a strong believer in having the students ask each other critical thinking questions. This is evident from the way his classroom is organized.

The desks are divided into clusters of four. Each group has a small plastic box containing pencils, markers, and other types of supplies. On top of each box is taped a green card on which 13 critical thinking questions have been printed:

1) Does your answer make sense? Is it reasonable?
2) How did you reach that conclusion?
3) Does that always work? Why or why not?
4) How does this relate to __________?
5) How can you prove that?
6) Explain the pattern, if any, that you see in the problem.
7) What is alike and what is different about your method of solution and his/hers?
8) What model can be drawn to illustrate your method or solution?
9) What ideas, that we have learned before, were useful in solving this problem?
10) What made you perform that step?
11) Explain what the number in your solution means.
12) What would be a good estimate in the problem?
13) What would happen if __________?

Inside each plastic box are several raffle tickets, on which is printed:

You have been caught using an open-ended critical thinking question. You are entered into a monthly drawing for In-N-Out Lunch with Mr. K. Per. _____
QUESTIONER ____________________
RESPONDENT ____________________

So as we can see, the students are strongly encouraged to ask each other these questions. How fitting, then, is it that I'd be assigned to this particular class during this week of the MTBoS Challenge? In fact, I thought I even heard some of the sixth graders asking each other these questions during my first two days in the classroom!

But it's today's lesson for which asking these questions -- the first one in particular -- is a required part of the assignment. On Monday and Tuesday, the students have been working on converting from language to math, such as from "the cost of a hamburger and regular fries" to h + r. But today, the sixth graders must go the other way -- they are given h + r and must come up with a story or skit which can be modeled by that expression. Students can get very creative with this -- so they can say "the cost of a hamburger and regular fries," "the cost of a hippo and a rhino," "the total number of hits and runs," and so on. And yes, while most students stuck to menu items, a few got creative and really did include animals or sports in their examples.

Now during the last ten minutes of class, the students are to ask each other the first question on the list, "Does my answer make sense? Is my story/skit reasonable?" And so I gave students raffle tickets for asking and answering this question about their dialog, just as their regular teacher would do.

In some ways today's lesson was successful. As I said, the students were creative with their skits, and the tickets encouraged them to ask and answer thoughtful questions. This occurred in one class:

Student 1: Does my answer make sense?
Student 2: Yeah, because it's two of everything.
Me: In other words, because the coefficient is two.

Student 1: Does my answer make sense?
Student 2: Um, yes, I guess.
Me: Ask her what variable she used for a small drink.
Student 2: OK. What variable did you use for a small drink?
Student 1: I used s ... and x ... oh ...

There were a few problems, of course. Some students entered the room hoping that we would play the Conjectures/"Who Am I?" game (as described in my MTBoS Week 2 post) again. Others weren't swayed by the raffle tickets reward -- recall that the tickets are in a plastic box on every group's desk, meaning that their teacher basically used an honor system (since he can't hear all conversations that happen in the classroom).

Still, I believe that today's activity and the ticket incentive encourages the students to be creative and ask each other better questions than they normally would.

To make sure that this post isn't too long, let me give links to everything else that I want to discuss in today's post. David Kung's Lecture 17 (Yes, we're 2/3 of the way there!) is called "Bending Space and Time," and one of the paradoxes he discusses involves spherical geometry -- including the "What color was the bear?" problem. I've already devoted several posts on this very blog to these topics:

What color was the bear?
Spherical Geometry: AAA and Triangle Sum

The Quick Conundrum also involves geography. Kung asks, what U.S. city lies in one state A, yet when you travel either north, south, east, or west, you end up in the same state B? Here's a link to one of my favorite geography sites, Twelve Mile Circle, with a discussion of the answer:

http://www.howderfamily.com/blog/cartalk-trivia-samford/

Here are links to the two blogs that are right above mine on the MTBoS page (and skipping up to the next Blogspot for my third comment):

https://fractionfanatic.wordpress.com/2016/01/27/better-questions/
http://forbetterproblems.blogspot.com/ (Marisa Aoki -- I've commented on hers earlier!)
http://www.8ismyluckynumber.blogspot.com/2016/01/a-better-question-for-quadratics-mtbos.html


Finally, here are the Geometry worksheets for today. They are based on Lesson 12-3, with an extra page for the proof of the Dilation Distance Theorem -- this proof comes directly from PARCC: