One thing we’re working on in the classrooms I coach in is how lessons might be formatted differently when you’re helping kids learn a new skill vs. having them practice something they know. Lots of classrooms have the warm-up, work on your own/in a group, go over, exit ticket model with very few variations, whether the “work in a group” phase is completely new learning or a review of a concept you’ve mastered. That can make it hard for kids to:

• realize what you expect. They get in the, “the teacher asked me to work on this so he must think I know what to do. I don’t. Augh!” mindset which leads to blind guessing, giving up, and defeat.
• use appropriate tools. They try to apply algorithms when you want them to guess, draw, make lists, or reason. And they guess, draw, make lists, or reason when you want them to practice algorithms.
• store new learning in the appropriate category. If you think kids are going to discover a new tool from your activity, and kids think they’re rehearsing something they’re supposed to know how to do, the experience will be filed in the “hard stuff I can’t remember” file instead of the “cool things I discovered and want to use later” file.

So… here’s a possibly illustrative pair of lessons.

Lesson 1: Rehearsing old stuff

The background: Students had learned the Pythagorean theorem in middle school, and most had mastered it, even finding missing legs given the hypotenuse and other leg.

The lesson:

• 2-minute drill: students have two minutes to solve a problem or list 3 things they notice about it on an index card. The problem is finding the missing side of a right triangle. Students turn in their index card and the teacher does a formative assessment while the students do the following:
• Go over the 2-minute drill problem:
  • Students hand off the whiteboard pen to one another as they take turns putting “noticings” on the board.
  • When all noticings are recorded, a student who has an idea to begin the problem solving writes their idea on the board.
  • If other students disagree or can add to the problem solving, they raise their hands, get the pen, and add to what’s on the board.
  • When the problem is solved to everyone’s satisfaction, and the teacher is done assessing students’ index cards, he shares their 2-minute drill stats and asks students about any aspects of the solution that he finds unclear (or thinks other students may find unclear).
• Give students a worksheet of “finding missing sides in right triangles” to work on in their small groups. During that time, meet with each group to make sure they are having success. Pay special attention to finding missing legs problems.
• Go over a finding missing legs problem as a class if multiple groups are struggling on it. Ask students, “what’s different about this problem?”
• When students finish, give them a word problem in which the Pythagorean theorem would be helpful but no diagram of a right triangle is provided. Solve as a class, in a teacher-led discussion.

Lesson 2: Learning new stuff

The background: After the prior lesson was implemented, the teacher and I discussed it and he mentioned that the students struggled a lot with the word problem with no diagram given. Given a right triangle, their Pythagorean skills were triggered, but looking for right triangles (say in squares, kites, isosceles triangles) was a real struggle.

I wondered what strategies students can use to help them with that specific skill, and he decided drawing a careful diagram would help.

I noted that this was a chance to try a lesson format for getting good at a strategy and applying that strategy to a particular content area (word problems about right triangles). The format I suggested was a Smartboard version of a “gallery walk”

The Lesson:

• 2-minute drill on a Pythagorean word problem.
• Students go over the 2-minute drill question, generating together a picture and the work with the Pythagorean theorem.
• Students are given another Pythagorean word problem, but this time are asked to work in their small groups to “come up with a picture that could help you solve the problem”
• As students work, teacher circulates and lets groups know, “I would like you to put your picture on a new slide on the Smartboard.”
• Once each different picture is on the board, the teacher asks students to compare and contrast the pictures. Student ideas include:
  • That one has a lot of labels
  • Those three are the same, just some have more labels
  • That one’s the best because it has a, b, and c labeled already.
• Finally, the teacher says, pick one picture that makes sense to you and use it to solve the problem.
• As groups get answers, they compare their solution paths with other groups, comparing both pictures and work.
• Each group eventually comes to the correct answer through comparison with other groups and prompting from the teacher to check, “is that answer reasonable? Does it make sense in the story?”

The Comparison:

The main difference between the lessons that I noticed were:

• In the recall-based lesson, skills weren’t broken down for the students. They were expected to know and apply a whole bundle of skills, a reasonable expectation given their middle-school curriculum, but which got hard when the students didn’t have the skills (e.g. when students couldn’t draw a picture for the word problem).
• In the lesson for learning, students were asked to explicitly compare multiple approaches to the same task.
  • A consensus emerged, that labels are good in mathematical diagrams, that will probably have more oomph than hearing the same reminder from the teacher.
  • Generating and comparing multiple approaches sets the tone “I don’t expect you all to do or think about this in the same way… but I do expect you to compare and make use of different ideas.” Students are in a learning, not just doing, mindset.
• In both lessons, students participated (and led the 2-minute drill process), but only in the lesson for learning was work from all students solicited and made public for comparison; and only in the lesson for learning were the relative merits of multiple types of drawings discussed.
• In the lesson for learning, much less teacher input was needed. The students drew all the pictures, decided which to use, and decided whether or not their answer made sense. When students struggled in the recall-based lesson, the teacher led students through the problem. He realized that students needed a different experience than being led through the problem to master creating diagrams for right-triangle word problems.
• Overall, the students were more on-task and engaged, and used more resources available to them, in the hard parts of the lesson-for-learning. That makes sense because when students are in the “I should know this but I don’t” mindset, it’s easy to disengage when they don’t know. That students stayed engaged and tried different ideas in the lesson-for-learning shows the teacher was effective in setting a tone of learning, not practicing, for this portion of the lesson.

And also…

In an ideal world, there would have been more time and attention and the end of the lesson for learning in reflecting on the learning. A discussion on what makes a math diagram useful, what people noticed that led to certain effective diagrams, what students want to remember for the next time they have a word problem with no picture, etc. would have helped cement the “this is a learning moment” idea and also the “file this under cool stuff to use again” practice.

A lot of talk on the math-ed-web-o-sphere has focused on effective techniques for engaging students in solving problems (e.g. Dan Meyer’s 3 Acts) and especially on perfecting the hook: the juicy image, movie, story, question, etc. that gets students wondering and conjecturing.

In the part of my job in which I am a math coach, we’ve been having a lot of conversations about structuring the curriculum to make big ideas and connections more prominent. One hypothesis is that students in “math class mode” focus on solving this problem and that other problem and never “how do these problems fit together and help me learn about interesting stuff?”

If that’s the case, a good hook for a unit can help students learn in service of a question that’s important to them, which is a powerful and sticky and organized kind of learning. But how do you plan for that? What does it look like?

Another way to ask that is, “if a good question is Act 1, and solving it is Acts 1 – 3, what Acts might be the rest of the unit? How do they relate to Acts 1 – 3?”

Leaving aside completely for now the challenge of picking good questions, I tried to make a lesson planning framework that would help me and the teachers I work with use the good questions we come up with to make coherent unit plans that hang together around an interesting (dare we say “essential”) question.

Here’s a blank-ish version and a sample version having to do with statistics. The sample is pretty long because I used the same question to do three parts of an entire month+ long unit on statistics (displaying data, measures of center, and measures of spread).

At ATMOPAV session 6, we compared what we noticed, what we wondered, and how we worked on some of the math questions we had about this story:

Feeding Birds: You have 7 cups of birdseed. You use 2/3 of a cup of seeds each week.

Lots of people agreed to try the problem with their colleagues and/or students and share noticings, wonderings, and strategies. Check the comments to see what the results of our experiments are!

1) Students ask mathematical questions (that in turn drive their learning)
2) Students and teachers seem happy to be in class and proud of their contributions
3) Students and teachers relate what they learn to what they already know (in math and in their lived experience) and to what they want to know more about/do better (in math and in their lived experience)
4) Students and teachers collaborate to learn and make sure everyone is learning
5) Students and teachers communicate their ideas fluently out loud and in writing
6) Students solve novel problems with a variety of strategies, tools, and representations
7) Students and teachers ask not just “am I right?” but “how do I know?”
8) Students and teachers ask, “what other math/patterns/generalizations can I discover from this?”
9) Students and teachers ask, “what are we learning about? what’s the big idea? what do we need to practice to get better?”

What’s missing? What’s superfluous/not fundamental? How do you know that your classroom looks like you want it to? How does this relate to what you assess?

Working on this problem:

100 people are in a room. In the next room, there are 100 identical boxes. Each box contains one person’s name (and each person’s name is in exactly one box). One at a time, each person can go into the box room and open up to 50 boxes. Then, they must return the box room to its original state (no re-ordering boxes, no marking boxes in any way). They leave the room and don’t communicate with anyone else. Individually, each person is asked, “which box had your name.” Fabulous riches are showered down on the group if all 100 people answer correctly. Are there strategies they can use to improve the odds that they answer right?

Since my by-hand attempt at archiving a Twitter conversation, I was told about Storify by the folks at Chatterblast Media who are helping the Math Forum learn their way around social media.

Anyway, Storify saved me tons of time capturing screen shots of tweets and made it easy to annotate the conversation. I wish it didn’t have to be quite so linear (I ended up re-ordering some tweets and breaking my one picture into two stories). But I guess stories tend to be linear.

Here is the finished product: http://storify.com/maxmathforum/the-three-acts-of-an-anyqs

If you are ever engaged in a Twitter conversation you think would interest others, or want to refer back to, just favorite the Tweets. Then whenever you get a chance, log in to Storify and search for your favorited tweets. They’ll be there waiting for you, and you can easily drag them into a story, re-order them, and annotate. Storify also collects Facebook posts, RSS feed items… all of the ephemera of the web. It’s pretty awesome!

I so thoroughly enjoyed James Tanton’s videos on new breakthroughs in partitions that I inspired to play around with my own patterns in the partitions. In particular, I’m pretty familiar with the dragon fractal but hadn’t seen the way you can generate it until Tanton showed the binary representation of the dragon fractal (in Video 4).

I started to play around with coloring bits of the partitions that appeared “self-similar.” I can see know how they are quite a bit more complex than the dragon fractal, and I see some intriguing patterns. I wonder if I play around long enough if I’ll recreate Euler’s method for generating the number of unordered partitions?

What’s gotten me hooked, I think, is that Ono’s work (Ono is the guy who came up with an explicit formula for the number of partitions of N) was inspired by visualization and partitions are so visualizable, and so I would like to see what the partition numbers look like to him. I don’t know anything about -adic spaces (or whatever they are called) but I do know some things about fractals and primes… I wonder if there is some representation that would allow me to see the fractal structure in the partition numbers.

I’ve wondered how one might do that before, and then in the middle of a really interesting conversation with Greg (@sarcasymptote), Dan (@ddmeyer), Dave (@dcox21), and John (@thescamdog) Greg requested that someone archive the conversation. It was a sprawling conversation with no hashtags, so I took a very rudimentary stab at flowcharting it.

Next blog post will be my thoughts on the content, but for now I’m intrigued by the format of the archive itself. Click the thumbnail image below for a “full-size” copy, and then zoom in to read when you have the lay of the land.