Thursday, January 31, 2013

Intellectually Needy

"So does this bore the heck out of you?" a student asked me.

The problem is that this was after two days of doing the Barbie Bungee Jump activity. The fabulous Barbie Bungee Jump. (Cf. Julie and Fawn) I was assisting a very nice and competent substitute teacher.

Coming on the heels of Christopher Danielson's and Chris Lusto's #globalmath session on building intellectual need, it was clear that most of these students did not have it. Nor were they looking for it.

This is a good school with good students and good teachers. What's going on, or not going on? My first brief observations:
  1. Students were given all the steps to follow. Being told to do a, b and c and then doing a, b and c is not engaging.
  2. There was no hook. How much of a hook depends on the lesson. This one could have used the video, a discussion about bungee jumping, etc. Going straight into 'here's what you do' gives no chance for wondering. Even if it's what the students want or are asking for.
  3. There was no expectation that this was worth their time or could be interesting. There is always time to start, but this might also be about developing a culture of inquiry. Students need to learn that this is what math is, and this is what math class is like or could be like.
Day 1: Students were given a worksheet with a table, told how to assemble the rubber bands and washers and to collect data for 1 to 6 rubber bands. Then graph all of their data and freehand a line of best fit. This is the beginning of a functions unit that will end with linear functions. Then they were asked to make a prediction for how many rubber bands they would need for the drop. We didn't have the actual heights, so they predicted for 3 m. Mostly, their prediction method was pick a number that was bigger than 6. 20 seemed nice to them, though some went with 18, since 6 rubber bands was close to a meter. Two groups found the average increase per rubber band.

We weren't clear about how to do the drop in the stairwell. We didn't have a set (or maybe even one) tape measure for long distances. Two teachers wound up determining 3 drop spots and measuring the distances, between 3 and 3.5 m.

Day 2: (After a snow day and a PD day.)
The students coming back were not much more enthralled than they were Day 1. I shared how this was the start of a functions unit, the math idea of having a rule to go from input to output. I tried to phrase the question as given the input of how many rubber bands, could they predict how far it would drop. (Not much traction, as there were already instructions on the screen.) The substitute gave each group their drop height. The two groups that had figured out the averages used this to make quite specific predictions, and one group made the complete table that this would generate. At the last second they cut 2 rubberbands off of their total, to allow for the length of the disk and acceleration. They were worried that it would be traveling faster at the bottom and that would make it stretch more. Two other groups adjusted their number a bit, but without reasoning that they could share.

We proceeded to the stairwell and groups took turns making their single drop. 2 hit the floor, including one of the more mathemaical groups, 2 got about 70 cm away, 1 was more than 1 m, and the group that had made the table got to within 10 cm. Went back to the classroom, shared the results and had the winning group describe their efforts while few listened. I talked with the mathy group that hit the floor about what went wrong. Basically they felt math failed them. Double checking their work I saw the problem was that they were computing for the wrong drop height! Their calculation would have put them quite close.

So What?
The students were pretty happy. Better than a typical math class, playing with rubber bands, leaving the classroom. The sub was okay with it, as students were mostly in control and made it through all of the steps. I felt like we missed an opportunity.

So what would you change about the lesson? What would add/create/inspire intellectual need in what is a (potentially) great activity?

Post Script:
Excellent discussion! I just want a few of the shared links to be more visible here. But many people put great thinking below so don't skimp on the comment reading.

Curse You, Nate Silver

OK, not really. I am a big fan of the Silver one, and his application of quantfied reasoning to realms rife with superstition.

But his Super Bowl Analysis got me wondering about the data. He uses a stat called SRS, Simple Ranking System, from They have a nice linear algebra explanation of it on their blog. Mr. Silver used this stat to consider whether good defense or good offense was more conducive to winning. By looking at the top 20 in each category, he concludes defense is the key. Or more of a key than offense.

Always fabulous and frequently mathy Foxtrot.
I got to wondering about the differences. Maybe what mattered was your offense to opponents defense, or vice versa. I started putting his data into a spreadsheet, but because teams are rarely exceptionally good at both (exceptions 68 Colts, 96 Packers), there wasn't enough data. I was surprised to find that Pro Football Reference is free and an awesome resource for data. So I started filling in a spreadsheet, figured I'd tumblr it and people could fill in the rest if they were interested.

I got hooked, which is why I say, "Curse you, Nate Silver."

Turns out my idea about the differences was only mildy predictive. I took NFC Offense - AFC Defense and compared to AFC Offense - NFC Defense. ("How is that different from NFC Total SRS  - AFC Total SRS?" I would ask students if I had them this semester.) Here's the Google spreadsheet.

There's definitely some things to notice there. My next thoughts were to compare defenses, offenses, and finally to look where those two agree.

Bad news for Baltimore. When a team is favored in defense and offense, they've won about 3/4 of the time. So Raven fans you should be getting 3:1 odds when most people are feeling it's pretty close. Looking at these numbers, you could consider it the 4th or 5th biggest upset should Baltimore win. No 68 Jets or 07 Giants, but significant.

What do you notice about the data?

Sunday, January 20, 2013

Too Puzzling

The content objective: adding and subtracting fractions with unlike denominators, for 5th grade students.

What game? On short notice... I have a couple race games and Mr. Schiller has pattern blocks - but those use the fraction cards which I like, but were also at my office.I was thinking about a game where students start at one, and one team tries to race to 2, while the other team takes away and races to zero.  That would be good for building intuition, and learning to record fraction number sentences, but this is their last day with fraction operations before moving on to something else. What could serve as a review?

I thought about a game you could play with a small number of cards (one sheet), or a game where they would be doing review problems and checking each other as they play. Somehow I was reminded of a tarsia puzzle. (Here's a bunch at tes, among other places. Not sure who tes is, really.) Mostly those are triangular, but I decided on a square pattern, more familiar to the students. Here's what I wound up with. Made the grid in GeoGebra, of course.

A lot of thought went into the puzzle. I repeated a couple values to make it so that a match did not conclusively mean that two pieces went together (harder). I made all of them doable with 24ths (easier).  I decided on a 3x3 vs a 4x4 (easier). I made almost patterns on borders (harder and easier). Made sure some of those were nonmatches (easier). Related problems, like 1/4+1/2 and 3/4-1/2 (easier). Some operations where the common denominator is one of the present denominators, 2/3-1/6. And one mistake. (Crazy harder. Fixed version above.) I tried to put in some common characteristics on adjacent tiles. (Hmmm?) If they've been doing these problems in general, I thought that this would be enough support for everybody, especially working in teams.

To differentiate, then, I was mostly thinking upward. On the original puzzle they could make 3 more squares to make it a 12 piece puzzle. I made one with some triangles blank for the students to work out sums, differences or make up a problem, and then an entirely blank one for them to make up their own Tarsia.

I launched the puzle by showing cut out pieces, telling them it was a puzzle and asking how it might go together. Through whole group discussion they figured out that the sums and differences matched some of the fractions. I compared it to the puzzles that are all squares that divide up pictures, which are pretty tough. After finding a few of the matches together, they formed pairs and came up to pick up their choice of puzzle. Only one group took an option on the partially filled in puzzle.

They did not like it. Found it too hard, or didn't know how to start. Mr. Schiller and I circulated and helped people get started. They found lots of matches, but before our time was up were moving on to other pursuits. Not interested in making their own.

To wrap up, we came back together and did some together to get a firmer idea of how to do it. They confirmed that it was beyond them. When I asked for words of wisdom, one student volunteered: "You might want to try it yourself, first. If it's too easy, add stuff to make it tougher. If it's too hard, make it easier."

Wise words, indeed.

To use this in fifth grade again, I think I might concentrate on first getting a square of four made, and then try to grow it. Mr. Schiller recommended either a fraction equivalence puzzle or a fraction-decimal equivalence puzzle to get it launched. I still love these kinds of puzzles, but fee like I learned something about introducing and using them with younger learners.

Post Script:
Jeff says: The equivalent fraction version of that worked a lot better.  I sort of sneakily encouraged them to also incorporate equivalent decimals too.  In the version we played in the afternoon,  they created their game board in pairs and then they matched up with another team and exchanged puzzles and it became a race to finish the puzzle first.  I sort of mentioned in an offhanded way - oh yeah, and if you wanted to make it a little more challenging for your opponent, you might include some equivalent decimals, too.  (That did the trick)

Tuesday, January 15, 2013

#globalmath Math Games

Tonight there's a #globalmath session on games where I'm one of the three panelists. I have loved the #globalmath sessions I've attended or watched afterwards so far, so I'm excited to be a panelist. EDIT here's the recording!

Here are my slides:

All the links are gathered together in a, Global Math Games. I tried to add other links that came up during the presentation. Unfortunately one of the panelists, Elizabeth, @cheesemonkeySF, couldn't join us because of school issues. She was going to talk about the Life on the Number Line game. But James Cleveland, frequent writer about games on his blog, was there.

James talked about his basic rule for analyzing a math game: does the math action have anything to do with the game action? He gave a couple of examples where the math is like a pause on the game, and you have to do some math to continue playing. He shared Ice Ice Maybe (MangaHigh) and Dragon Box as examples of good games. (I like one of those better than Dragon Box.) And then he talked about his game Totally Radical, which seems very cool, and is a great example of the math action being the game action. There's a homemade version to print up or you can even by a slick commercial version.

In planning my part of the presentation, I wanted to get across:
  • what makes a good math game
  • what I like about my best games
 Hey, how much should you shoot for in 20 min?

  1. I like games because they're fun and engaging. I love playing games myself, so it's not crazy to want to do them with students. But I also think that games are mathematical, as math is game like. The actual process of being a mathematician is inherently playful. Trying things out, intentional variation, strategizing. I am actively jealous of literacy teachers, who have all these stories about the book that turned this student onto reading (Potter, Bridge to Terabithia, Desperaux, etc.) and we math teachers never get that.  For me games are our best shot. Watching kids play Pokemon or Yu-Gi-Oh you see them doing amazing problem solving.
  2. I'm in favor of gamification, but see that as being different from a math game. There's been some fun discussion of the Ninja board on Twitter with Jim Pai (his ninja board), Jeff Brenneman and @algebraniac1. That's just fun. I think how video games present challenges for which you need new skills and giving you notice for things you've accomplished (level up!) are worthwhile for teachers to think about. And review games are a good use of gamification as well. (Here's my list of other people's review games.)
  3. The Product Game (NCTM online, my copy) is to what I aspire. It's a great game, with the strategy of a Connect Four (which is a real game, if by some chance you haven't played since a kid), but all the game actions are math actions. Furthermore, the encourages you to organize multiplication facts by family, and use patterns to find products you don't know. Evenfurthermore, it encourages reverse thinking preparing students to think about factoring. Plus it is just fun. The game structure is so good it lends itself to adaptations (decimals, fractions, integers, exponents...)
  4. Decimal Point Pickle is one of my favorite games that I made. For students who have worked on decimal representation, it encourages better number sense and place value before moving on to decimal operations. It has a blackjack feel, and never fails to generate excitement. Mathematically, the comparisons of decimals in tenths, hundredths and thousandths have been invaluable.
  5.  Eleusis Express. This is an adaptation I made of a deep and difficult game by Robert Abbott. He captured an essential mathematical process, though: making and detecting patterns. It is an amazing game to play that affords opportunities to reason, discuss arguments, and problem-solve. It really helps me communicate with students the value of capturing thinking. Because it doesn't feel like mathematical content, it is safer for them to struggle. It's just a hard game. Handouts for this are at the top of my games page.
    Plug: I also love Robert's endlessly clever mazes.
  6. Where I'm heading now is trying to get the students to be the game designers. This is done over several sessions, releasing more and more of the design task to them, or by working through the different aspects of design and having them work on the piece of a game. Even students who don't get into the playing of games find this to be rewarding and engaging. Plus, I know from making games that a lot of the math is in the design and getting things to work out correctly.
  7. The framework:
    Adapted from Magic: the Gathering's lead designer's thoughts on design, this framework has helped me to better understand what I'm doing and to make better games.
People were nice, but I did not make a good presentation. Good thing I had cartoons.

I came upstairs and my wife asked me how it went. "I was too vague," I said.

"I'm not sure what you mean," she  says.


In retrospect, I should have just talked about the three games, and given some of the play of them. Don't spend time talking about what you're not talking about! But it was a good learning experience, nonetheless, and I love being a part of "PD you like." If you haven't peeked in at #globalmath, give it a try. Most Tuesdays at 9 pm ET. Next week looks good already: Building Intellectual Need.

Tuesday, January 8, 2013

Skemp and the Baseball Coach

I love discussing the Richard Skemp article, “Relational Understanding and Instrumental Understanding,”  (Mathematics Teaching in the Middle School, September 2006) with preservice secondary teachers. Not first thing in a course, but after some experiences with deep mathematics learning, and some groundwork in thinking about teaching, it is amazing. I've recorded two blogposts about it already with my reading guide and with video of the novice teachers' summaries.

The article discusses how there are two views of understanding that makes discussing learning difficult. Instrumental understanding is associated with rote earning and mastering, and relational understanding is connected, transferable learning. Skemp puts it better, thank goodness he wrote about it.

This past semester, one student in particular made a strong connection that seemed to also make a lot of sense to the other teachers. He was willing to share it on video, so, here it is! Thanks to Ryan for his work in making sense of the article and his willingness to share.

Monday, January 7, 2013

Family Math: Origami

My son Xavier did an origami project before break for an AIMS class. (Integrated Math-Science, their website has some free samples.) He was quite proud of the cube he made, and really learned how to make the basic shape. He was happy to teach me, but it took until this last day of break to get to it.

Here's the basic shape:
 A parallelogram when laid flat. The center square has flaps for tucking in tabs to build.

Xavi was upset a bit when I started experimenting to get the angles and shapes. "No extra folds!"

First: In half, then each half in half to get four parallel quarters. Then with outside edges folded in, fold the short edge to the long edge.

Second: open sheet, then fold in the corner of the right quarter, and the corner of the left half.

Third: fold in the right quarter, with the corner tucked in.

Fourth: repeat with the opposite corner. This picture shows the first corner fold. Remember to tuck in the small corner dogear.

Fifth: Once both corners are done, fold them in to the center square. Xavi had a good test for if the pieces would fit together - see how well they nest.

Sixth: Assemble into a cube.

We had no instructions for that. We built one cube, but there were gaps at the edges, which was not the case with the one he had brought home. The teacher put together the cube for anyone who was having troubles with it in class. When we noticed that each edge had a square 'covering' it, that gave us the clue to get it put together. Every tab goes in a slot turned out to be an important characteristic.

But where's the math? There was some in recognizing and naming the shapes, and some problem-solving in figuring out the cube.  But then...?

I was interested in how it fit together. So I decorated our plain white cube by tracing those edge-squares that turned out to be a good clue. I added the dots because I thought it would help make the cube into a puzzle, and it would be interesting to help study how it fit together. I just asked Xavi what he thought the pieces would look like and he took off with it. Great work, and with the barest of nudges, he wrote it down.

Can you make all the faces have a 3-1 pattern? 2-2 pattern? 2-1-1? 1-1-1-1 - each face with 4 different colors?

This was a great context for talking about conjectures and proof. He repeatedly talked about how fun this was, and was very keen on sharing the results. If only he knew someone with a math blog...

While you might think the child of a mathematician is naturally interested in school mathematics, but that's not the case. At some point, teachers started to have more authority than the father did about how things could be. But maybe that's a post for another day.