What's a Problem?

We had a fun class in the elementary math course today. Introducing SMP1 - problem solving, we got to an interesting question: what's a problem?

Here's the story as told by the residue on my whiteboards.

Schema Activation: jot down about a time you solved a real problem in your life. You won't have to share with anyone if you don't want to - this can be private.

After a few minutes to write, I shared how one of the big justifications for teaching and prioritizing math in school, other than the jobs to which it gives access, is that it teaches problem solving.

 Actually more yes than I expected!

People asked if I meant did math class help with the problem that they journaled about or in general. I said we want to know about problem solving in general, but they could use their instance as a specific.







Next question: is it possible that math class could help teach problem solving? Short time to discuss at table.








These were definite yesses, with a lot of confidence.




One of the reasons I like teaching teaching math is teaching is so much like math to begin with. So rephrasing: our problem is how to go from the current situation to get what we want.

So the next prompt was to quickly brainstorm. Ideas for making this happen.

















I shared that I liked how many of these were things that were up to the teacher. And I paraphrased Marzano, about how there is bad news: a small percentage of factors affecting student learning are under the teachers control; good news: that percent still makes a big difference.

What did they like?

• The emphasis on real life. This brought out uniform hatred for unlikely impractical story problems.
• Logic problems: one of the students shared how engaging and powerful these were for here. I asked about the contrast with real world, but people were comfortable with the crazy logic problems because the emphasis was on how did you do it.
• Teachers can ask for multiple ways and have students compare them.

Okay. So if we're going to teach problem solving, we need problems. I asked a question, then asked them to think about if that was a problem or not. There was a tub of square tiles on each set of two tables.

After they all had answers, I asked them to think about the second part. After a short time to discuss, we went to +cheesemonkeysf 's talking points structure for the statement.

This is a problem.

We're still working on the structure, so some of my feedback was about that. The no comments idea is hard.

Pretty strong agreement. People were willing to share their reasoning with the whole class ...  even the small minority. Maybe especially the small minority?

I was really happy to see the "depends on what the teacher does" idea come up. And I added the "depends on students" too. This is not a problem for me. For first graders, a serious problem - there aren't enough tiles in the tub to cover! For them... well, they talked about methods, chose how to do it, discussed results; these are problem indicators to me.

Then we went back to the question. What answers did they find?

 There was shock at the diversity.

One question I ask a lot is: is this a question where different answers could all be right?

They discussed and...

There was concern about the largest answer being too big, but that table and an adjacent table figured out the first group's table is actually larger.

I shared how measurement is one of my favorite content areas, exactly for this reason that a diversity of answers is to be expected. It can be culture setting.

So, with all this, definitely time for a problem. One of my favorites. How many pentominoes are there?


We played with domino patterns last class, so that's a natural starting point. We agreed that there was only one way to make a domino with two squares. If they touch, they have to share an edge.

With three squares, the issue came up about putting them "in the middle." That's solved by the edge rule. What about the elbows in a different direction? No, the class agrees, if you can turn it to match, it's the same shape.

So what I want to know: is how many pentominoes?



People got to work with tiles and graph paper. No group found all the tetrominoes as a step, which I was trying to suggest. A couple of times I had tables report on how many they had and were they expecting more. 8 more, 9 more, 12 probably, etc. Next time, 12, 14, 15 and a mix of have them all and think there are more. When our time was up, they sent emissaries to the board to draw what they had:

Now the best part: math fight! Are flips the same or not?

They divided up into groups based on their answer: 14 flips matter, 9 flips are the same. They shared reasoning. Flips matter because these are flat things and a flip requires another dimension. And if you try to match them up they do not line up. Flips don't matter because you can get them to line up, and what's the difference between a flip and a turn, really?

I refuse to settle the issue, and ask them to make a complete set before Monday.

If you care to comment or tweet a response, I'll share your answers with them:

• How do you recognize what is a problem for your students?
• Are flipped pentominoes the same or different? Why?

Clap Hands - a motion pattern game

This game must exist in some form elsewhere, but it came to me yesterday and we worked out a good version of it with my preservice teachers this morning.

It starts with getting to do some of Malke Rosenfeld's Math in Your Feet this summer at Twitter Math Camp, and then subsequent discussions with her that have me thinking a lot about embodied cognition. The example of this in Math in Your Feet was knowing what I needed to do but the challenge of getting my body to do. Move left foot, move! In discussions, she connects this powerfully to research and writing of Seymour Papert. She said something like:

embodied vs “non-embodied” from the research: there is no non-embodied math. Either we’re pulling from previous lived-in-the-world experience to learn, or we’re actively constructing our understanding of self moving in space. We can harness that to give students an understanding of the world.

She's deep that way.

On our first day of class, one of the things we did was watch Ken Robinson's Do Schools Kill Creativity?  (If you haven't watched it, give it a go. He's a powerful speaker on creativity, and as close to Ricky Gervais as we're going to get in academia.) The student response from my class was really focused on movement. The Gillian Lynne story especially seemed to resonate; good omens for some of the learning I hope to do this semester.

So today we're studying patterns. First activity was pulling out the pattern blocks. We used that to model how to introduce math manipulatives to elementary students, and introduce the principle that with a new manipulative you need free play. Either immediately or promise the students specifically when they will get it. (Good management meets good pedagogy.) We used free play to introduce the question: is this free play doing math? Which we discussed in Elizabeth's Talking Points structure. (Fabulous, even the first time out.) Then in whole group used our examples to discuss the difference between a design and a pattern. (Is there a difference to you? I'd love to know what you think about that.)

Then it was time to go outside...

Clap Hands

groups of 4 to 7 people

Arrange people in circles of about 6. The game is pretty simple:

Building

• One player starts, introducing a motion. Like, for example, a simple clap. Going around the circle, each player does the motion.
• After the starting player does the motion, the next player adds a motion. Clap hands, raise right hand. Each player does the 2 part sequence.
• After the second player does their two, the next player adds a motion. Clap hands, raise right hand, turn around clockwise.
• And so on, until each player has added a motion and it has gone around. Clap hands, raise right, spin right, jump, snap fingers, shake right foot twice.

Survivor

• The goal is to get the pattern to go around twice more. When it does, that pattern is complete!
• If a player messes up the sequence, they step out. Try to get twice around from there.
• If you get down to two people, the pattern is done.

I didn't get video because I needed to play this! Thanks to Jordan and other students who had great suggestions. Reaction to the game was very positive, and people were quite engaged. There was much laughter, too. Keeper!

I'm interested in your feedback on the game, and how you present patterns. So if you have time to tweet or comment, let me know.