If Lincoln and Douglas Had Been Math Teachers

"Punches are no longer being pulled." -MSNBC
"At each other throats." -CNN
"Ugly as a wedding dress inside a slaughterhouse inside a divorced vegetarian." -Fox News

As you can see, the 2012 debate season is heating up. And by debate season, I am obviously referring to the debate of how best to pose the following math problem to students:

Draw a rectangle on a square grid. An example 9 by 3 rectangle is drawn for you below. Draw one diagonal. How many squares does the diagonal pass through? Develop a rule to determine the number of squares a diagonal passes through for any rectangle of any size.

First, the back story.

The players

On August 12, 2011 Kate Nowak posted a seemingly innocent post on her blog sharing this problem. On par with my general google reader habits, I read this post about six weeks later. I liked the problem. Liked it very much. When I got the call from Gallup, it got my vote for being a good problem.

I worked on the problem myself. A pattern eluded me. Worked some more. Had a particularly helpful insight. Now a pattern emerged. Proving this pattern now eluded me. A few false steps and insights later, I was satisfied with my argument for why this pattern would in fact continue.

The set-up

On October 29, 2011 I used this problem in an Escape from the Textbook workshop as the context for a meta-discussion about mathematical habits of mind. From some perspectives, it was a great discussion. From other perspectives, it was a lovely way to spend a Saturday morning. Alas, history is complicated.

The tale

The following week I posed the below "problem" to my sixth graders.



But where's the problem, you ask? Exactly. By now my kiddos are used to seeing things like this. They know that the first step is to come up with an interesting mathematical question, what some might call a #anyqs or 100qs or "what can you do with this" or WCYDWT or WWJDWT. For now I'll ignore the barks of "Who cares!" My students bought in and came up with a great list of questions. Here's a selection (edited for mathematical fanciness) for your viewing pleasure:

1. What is the length of the diagonal?
2. If I change the size of the rectangle, how will the length of the diagonal change?
3. What's the area of the rectangle?
4. How many squares are in the rectangle?
5. What's the area of the triangles?
6. What's the perimeter of the rectangle?
7. Does the red line cut the rectangle in half?
8. Can I walk around the blue and red lines without crossing my path?
9. How many times does the diagonal go though an intersection of the grid (a vertex)?
10. How many squares does the diagonal pass through?

I gave them 20 minutes to work on whatever problem(s) interested them. I then shared with them that I found #10 to be particularly intriguing and had everyone work on this (if they weren't already) for the remainder of the period. Oh, and again the systems were already in place, but I didn't have to remind anyone that they weren't finished after counting the nine squares in the above shape. They were all over trying to find the general solution. Anyway, conjectures were made, simpler problems were solved, but no one solved the problem completely.

The wire

On November 5, 2011--a warm and rainy day in Palm Springs, CA--I presented this problem as part of my workshop Making the Process Standards More than an Afterthought at CMC-S. As I am known to do, I blogrecapped and blogflected on this presentation soon after. Dan Meyer was at this workshop and, being the Brad Pitt of the math ed world, picked up on a subtle but important choice (actually, I would call it a mistake) I made in my presentation.

In short, I posed the same problem I had earlier posed to my students to a group of teachers and asked them what questions they had. The problem is that I didn't explicitly tell them that while I appreciated their answers, we were going to focus on one question in particular. Another problem is no one asked the problem I had set up on the next slide. Anyway, you can read the details on Dan's blog if this recap wasn't enough.

I am indebted to Dan for pushing me to be more careful about how I talk about posing problems with both students and teachers. That said, the piece that fascinated me the most was the assumptions in the comments about why this #epicfail occurred.

Part of the problem, of course, is grounded in teachers’ fear of what would happen if a truly open question led to something that exposes teacher ignorance.

Expecting students to play mind-reading games and then disrespecting their ideas when they fail to come up with exactly the “correct” question/answer seems like an error to me. 

I don't think this is just me being defensive, but I think these comments are way off the mark (at least in my particular case). I have zero fear of exposing my own ignorance in front of my students. Feel free to ask them if you don't believe me. :) And I did not have a "correct" answer in mind, I just had a next slide. A later comment by Maria Droujkova actually did an excellent job of describing what was in fact going on and how this could easily be resolved.

[Y]ou have to tell people, explicitly, what it is you are doing. Here is how I would plan it:
1. What questions would you ask about this diagram?
2. Thank you! If we had an extra hour or five, students could solve these fine questions and more. Hopefully, forming the questions helped you to get to know the diagram more personally. The next task: ask just questions about things you can count.
3. Thank you! So, again, we could pursue these questions with students. I prepared a question of this type ahead of time to demonstrate the next stage of the process. I like it because it recently inspired a good discussion in math teacher blogs, and has some history in Olympiads.

The sting

Fast forward to today. I posed this problem again, this time using the updated version of:

I made sure to preface with the fact that I valued their questions and would love to have the time to explore each and every question, but had a particular question in mind. Once again, teachers came up with some questions (nowhere near the number or diversity of questions my sixth graders came up with, but my sixth graders had more time and more training). Once again, no one asked the question about counting squares (which, I will remind you, was a question one of my sixth graders asked).

So I'll leave you with this question. Was this another #epicfail? Should I give up with this particular problem and just include the question? Does it really matter? Anyone at today's session reading this feel cheated? Maybe I'll see what 101qs thinks.

SUM Conference: Day 2

Here's slides for today's session which will focus on doing some math and reflecting on the habits you use. We'll then create a lesson and/or an assessment around mathematical habits of mind.

Careful, there are a number of really good problems here. :)

Saskatchewan Understands Mathematics Conference

This weekend I'm up in North Country working with teachers from all over Saskatchewan at the Saskatchewan Understands Mathematics Conference. Random? Sure. Thank you twitter and thank you @park_star for organizing a fantastic two days. After a fantastic opening talk by James Tanton, I led a 2 hour sessions this morning titled Beyond Polya: Making Mathematical Habits of Mind an Integral Part of the Classroom. Here are the Powerpoint slides (I will warn you that a few of the animations did not export correctly, but it's readable). 


In short, we walked through my ten mathematical habits of mind, brainstormed what this meant to us and how we could see and teach this in our classroom, and did some math.

Starting with a personal reflection, I felt that the session went well overall. We did, however, go too quickly and we didn't spend enough time doing math. In retrospect, I should have focused on some of the habits which would have allowed for more time to engage more deeply. That said, hopefully some of the participants will come back for day two where I hope to give more space and time for participants to work. I am also finding it hard after every conference talk I do to get a real sense of whether or not my session was at all useful to people. It's one thing for people to like a session (which even that is sometimes hard to gauge across the board). It's a whole other animal for people to change or evolve. I was super jealous when I heard that The Space Between the Numbers got an email from a workshop participant at her NCTM talk on proof through logic puzzles saying that he had created a lesson around this talk the very next week. Anyway, I'm learning that conference workshops are very different from the classroom.

Technology Through Time

A visual of digital technology my 5th and 6th grade students have learned, used, explored, and/or benefited from this year. I'm proud to say that I'm continuing to increase my ratio of

using technology because it helps solve an existing problem : using technology because it's available

By the way, I created this as part of a Technology Through Time presentation to parents given by ten teachers intended to give a brief overview and scope of how technology is used at different ages and in different classes.

Escape From the Textbook Winter Meeting

Below are some lightly edited notes from the most recent Escape meeting at Willard Middle School in Berkeley. Our next meeting is tentatively set for Saturday, June 2nd somewhere in San Francisco.

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Escape from the Textbook meeting: March 24, 2012

Introductions

We spent the first part of the session exploring the game of Set. Over the course of the morning there were a few other games/ideas mentioned including Eleusis, Attributes Blocks (similar to set except with physical blocks), Spot It and Ricochet Robot.

We played the game for a bit, but then worked on solving and creating mathematical puzzles. Below are some examples:

* What is the maximum number of cards you can have without a set?

* Can you make a strip (a train) where any three cards next to one another are share the same number of characteristics?

* Games could be created with these cards similar to dominoes

* Start with some number of cards in your hand. Continue to play cards that do NOT make a set with the cards on the table.

* How many cards are in the deck?

* How many different sets can one card be a part of?

* How many different sets can 2 cards be part of? What if only 3 characteristics need to match (or be different)?

* Is it possible for all the cards to be used? What is the probability of this happening?

* Can a subset of the deck be organized in a way to make sets with no extra cards left over?

Our group then began working on "magic square" problems. We determined that if cards are on the four corners of a magic square, the entire square is determined. We then saw that this was also the case with 3 corners (and that it would be impossible to create the entire magic square if the 3 starting cards made a set).

We then shared our work with other tables and talked about. Some comments of note:

*Students who do well anticipate teacher questions and understand why they're being asked.

Ways SET can be used:

*the big picture of "Here's our universe. What is true in this universe?" Parallels other axiomatic systems.

*sorted attributes, categorizing

*union/intersection

Interesting to maybe think about how the cards are exactly the same mathematically as just having the 4-tuple (0,2,1,1) with four dimensions (four different attributes) and values that can be 0, 1 or 2 (three different options for each attribute).

Kim then shared (and asked us to reflect on how this could be done with students in a more meaningful way than just telling them) a way to determine which card is missing after playing a full game and ending up with a number of remaining cards that is not a multiple of three.

Break

Shared some summer opportunities:

Promoting Algebraic Thinking Summer Institute: Lawrence Hall of Science

Center For Innovative Teaching: Hands on Geometry

Assessment

How do we assess content learning? (I would add habits learning too)

Rigor?

How do we evaluate assessments?

Am I testing what I think I'm testing?

How can we use assessment to understand what sense students are making of instruction?

What are we looking for and how do we measure it?

Targeted, selected observation...

Engaging students in self-assessment

Some assessment ideas:

* Give assessments with lots of questions and then at the end tell students that their top scores on 5 (or whatever) problems will be the only ones graded.

* Ask students to solve 2 of 4 and give different point values for more complicated problems

* Have small skills, concepts, and pushing quizzes that students take weekly (with a choice of which quiz they take) and can retest until they show proficiency

Big question: assessing what?

*students making good choices

*have you learned this skill
*access

Next time, talk about different kinds of assessment. Thinking about bringing assessment questions around a particular topic/skill/concept (probably one middle school topic, maybe proportions, and one high school topic, maybe systems of equations).

Brainstorm ideas on how to make assessment a better tool for learning? What do you do after assessment?