Who WIns?

 I'm teaching our Statistics and Probability for K-8 Teachers for the first time. Had excellent support and suggestions from Jenna Laib, colleagues Jon Hasenbank, David Coffey and Hope Gerson, and from Stephanie Casey and ESTEEM folks.

Working out bit by bit what we're going to do. Luckily I get to teach it again next year...

We just finished our first week, and I loved how the Day 2 lesson worked out, so I wanted to share it and think about it a bit. I miss the reflection of blog writing about my teaching!

On Day 1 we had started exploring measures of typical. We got out the unifix cubes, and did a bit on how could we make the distribution more fair. It took several rounds of give aways, but we got there. Some tubs of 252 and some with 251. Then each person built a stack as long as their first name. We organized from shortest to longest and thought about how to answer "How long is a 323 student's first name?" We had initial estimates, then talk went to median and mean. We retooled and did full legal names. Much bigger range, but a surprisingly dominant mode. So there was another consideration. Our emphasis was not the number you said to answer the question, but why you would say so.

Each day one of the teachers leads a Slow Reveal Graph, and Day 2 Tessa started us out on Disney Princess Baby Names, so energy was pretty high. If they had to name kids with Disney princess names they'd go with Belle and Aurora. Yvette's great question was what can you say about this without calculating?

Good discussion. I added in that since all of them had 7 elements, we could just look for a total instead of dividing. Some good call backs to Day 1's discussion of mean and balancing or distributing.

Then the main activity. They used NRICH's great millisecond timer tool and each collected 5 points of data for trying to hit 10 seconds exactly. They each found their median and mean, and thought about what would make a better measurement of who was best at estimating 10 seconds. Uniformly, each table decided on the mean. I raised up the idea of how some sporting events use your best score. They decided that they wanted to include outliers, that consistency matters.

The NRICH page that got me started on this idea had a set of data for discussion, which they credited to the great Don Steward, from his awesome Median blog.

Anna	Ben	Charlie
31	36	37
26	19	32
32	39	24
27	36	32
29	20	30

Typically great prompt from Don. They all found Ben had a mean of 30, though they all thought he was the worst of the three. People were divided over whether Anna or Charlie should win. People liked Anna's accuracy, but others were compelled by Charlie's spot on 30 seconds. They had time to propose a new summary statistic to answer the question of who is best.

One table proposed |mean-30|+|median-30|+range, low score wins. Another table proposed just the best score, tie broken by 2nd best time. Voting was mostly in favor of the more complicated statistic. But definitely the Charlie fans wanted best time. 

And then the breakthrough moment! Someone said 'What if we just totaled up how far each guess was from the target?' Absolute deviation! I tried to play calm and cool, and gave them time to think about it.

Revoted, and people went 22/24 for the absolute deviation. My inner teacher was screaming.

They reevaluated their own times using this metric, and decided on a table representative for the class champion. Then, a class championship, three rounds. Very exciting, started close, and then someone ran away with it.

In their summary, lots of great discussion about variation, mean, median and their limitations.

Making Whoopee at TMC15

Whoopee means games, of course. (For the song, Ella or Ray are the best options. Though I suppose I should go with Dr. John - no relation, despite the resemblance.)

Spoiler: I got rambly here. If you're going to not read this, here's two quick takeaways: the four new games and James' start of a game/curriculum alignment.

I was delighted when James Cleveland asked if I'd do a Twitter Math Camp morning session (meaning three 2 hour sessions) on math games with him. He'd led a one hour session at TMC13, and Sebastian Speer led another at TMC14, but this would be the first one with time to really make new games.

The format seemed pretty natural and intuitive: look at some games together, introduce a few principles, and get making. Just in case, we each had some ideas for games in case people didn't have any urges, and James had a couple of neat statistics games ideas burning a hole in his pocket in particular.  I had a mechanic; my family recently discovered Sushi Go, which has a simple and elegant drafting mechanic. There was also this Tug of War that I had been discussing with Nora Oswald  based on a Daniel Solis idea.

The plan:

• Day 1 - play good games and discuss.
• Day 2 - start design.
• Day 3 - playtest.

Writes itself, eh? Materials from the first day are mostly here on the TMC wiki, and the 2nd & 3rd days in a Google doc. (Including rules for the developed games.) 

Day 1 Games
I brought Linear War  , but we didn't actually play! I considered it for day 1 because I like how the students make the cards for the game (learning part one) and then play (learning part two) working on vocabulary, concept, quick recognition and computation. We did play: (in order of complexity)

• Product Game: Illuminations, handout (original & integers), post (decimal). My nominee for best math game ever. Comes at the content from multiple ways, amazing replay value due to the deep strategy, quick to learn, structure supports students in learning the content, adaptable... The only thing missing is context, but this would distract from it as a strategy game. Teachers thought of several different uses for this immediately.
• Quod Game/Metasquares (app not currently in US app store) All you need is a grid, and the strategy is deceptively deep. Subtle approach to content, though, as there is a great mathematical structure, but it's more about noticing it than learning it.
• Factor Draft, James' great game. Interesting in that you can parallel play or interact a lot. Really requires the mathematics that it concerns. Needs its own pieces manufactured, but they can be used for multiple purposes. Great example of development in balancing the pieces for interesting play. High cognitive load game, lots of challenge.
• Domain Ranger, post 1 and post 2. Norah's serious game. It's an intense strategy game, for which you need the math ideas of domain and range, and the ability to compare different graphs. Participants had awesome suggestions about this. Recognizing the difficulties in learning such a complex game, they thought about doing a 1-dimensional board set up learning game. And also the great idea of doing a preset first game, Settlers of Catan style. I'll try to work one up the next chance I have to use the game.

It became clear pretty quickly that this was going to be a good couple of days. We picked these games because they all are content focused. I do not have any problem with review games that fit any content. (In fact, here's my list.) Instead of that, we were looking to design games for specific content. Where the game play was the learning activity. Day 1 was promising because the group as a whole was really able to focus on what aspect of the content the games addressed, and where in the lesson/unit on that  content it would be appropriate.

Harvey Mudd had CHALK boards. Deja view.

Day 2 Design
We started off this day with a look at Decimal Pickle, maybe my best game, with a focus on desing thinking, the mathematical goal, and how the mechanic works in the game. One of the most interesting parts about preparing with James was thinking about classic game mechanics are use of them in math games.

Classic Game	Math Game version: K-7	8-16
Apples to Apples, Dixit		Graphles to Graphles
Blackjack	Zero!
Connect 4/Tic tac toe		Exponent & factor block game
Guess Who		Polygraph
Racko	Decimal Pickle
Rummy/Concentration	Unit Rummy	Trig Rummy
Taboo		Algebra Taboo
Uno		like terms UNO
War		Log War, Trig War,
Wits & Wagers		Ideas for Use

 
There's a lot of room for addition in there. I'd also like to hear your thoughts about what's missing. Even just writing this I got thinking of Farkle and Yahtzee. (King of Tokyo is an example of a tabletop game that uses that great Yahtzee mechanic. I have an upper el math game that's a direct rip off adaptation of Yahtzee, too.)

For design principles, I have this goofy list of 9 I use as a framework (adapted from Mark Rosewater of Magic fame). We emphasized just a few-

1. Goal(s). Design starts with objectives. (Whole point of Day 1.)

4. Interaction.

5. Surprise.

6. Catch-Up. As you start to playtest, these two are important to attend to for good design.

 
What's really promising from the prep for this day, though, is James' start of a spreadsheet for curriculum aligned games. Here's  a Google spreadsheet version - open for editing. If you know of things to fill in, PLEASE DO. If you have a hole you especially want addressed, let us know on Twitter.

People got designing pretty quickly. We divided into 4 groups, working on statistics (James was in this group), Fraction operations, Arithmetic Sequences, and Unit Conversion. I floated amongst the groups. This was a bit of a breakthrough for me. I design mostly in isolation. But (like for most things) collaboration was energizing, powerful and fast. Between this part of day 2 and some wrap up on day 3, we finished four good games. 2-3 hours of work. My contributions floating were questions, connections with other game experiences, and the occasional idea.

Day 3 Playing
Also today, we took time to do some rule writing:

1. Rules. For me this comes late; kind of a synthesis step as you think about how to communicate the game. It will often result in design revision, though.

 James knew of a good blank template for rules writing.

People needed a little time to finish. We had a good Skype chat with Nora, who shared her experiences playtesting, took people's questions, then discussed some of their interesting feedback on Domain Ranger. Dave Chamberlain (participant) shared his published game of Team Up! which is a 4-12 common core review board game, and some of his process. Also what it took to get it in commercial finished form. 

James gives a good write up of the statistics game, Fighting for the Center. Use playing cards, players build a data set that meets some goals (measures of central tendency) hidden from the other players. It's great at making players think about how changing a data point affects those measures. Lots of interaction, since you both are playing on the same data set, catch up is not an issue, and students will find more means and medians than they ever would in a homework set.

The fraction game is about addition and subtraction, modeled a bit on the Connect 4/Product Game framework. The board is really interesting, by asking for ranges, which really leads students to using representation (on the fraction cards we had). The teacher may want a way to get students to add precisely. I think there's some more playtesting to do here, too, as the placement of the various squares was more about coming up with them. 








Honeycomb Madness, unit conversion game. This is a classic board game, and the closest to being a general review game. You start in the outer ring, and are trying to get to the inner ring by answering questions. The ring level serves as a kind of rubric, though, and might support some kind of awareness in students as to different levels of understanding of the material. There's a nice bit of randomness that's reminiscent of Trivial Pursuit. I liked that it is not the kind of game I might design; I think it might be quite popular with students, too.

Arithmetic Sequence Game. This one is right up my alley, though. Deck of playing cards. Deal three: starting value, common difference, step number. That determines a target. Each player is dealt 5 cards, and tries to get as close as possible. Then the idea that complete changes it: you bet on your play, 1-6 points. Closest gets the points everyone bet. 2nd closest gets their bet back.  Wow. Plays great. I'll be trying this out, next algebra class for sure. I made a GeoGebra sketch to help with the calculation and to practice. 

Thanks to James and all the participants. I feel like I learned a lot about collaborating in game design, and broadened my tastes a bit, too. This more than ever makes me want to get students designing games, so if you're in the area and wouldn't mind a mathematician in the room...

 

My Oreo Lesson

Finally... my chance to do the oreo lesson!

I'm teaching one of our math for elementary education courses and the content includes measurement and statistics. I love measurement as a context which needs statistical understanding. Measurement introduces variability, and has a strong need for producing a number to represent typical. If the question is, "How tall is the ceiling?" then 2.60, 2.7, 2.725, 2.73, 2.735, 2.735, 2.74, or 2.76 meters is not a satisfying answer.

The oreo lesson, if you are unimaginably unfamiliar with it, is the brainchild of Christopher Danielson, aka @Trianglemancsd, the purveyor of much fine snack food mathematics. (All the oreo posts; this one is sort of a wrap up.)

Previous to this lesson, we investigated measurement, did an introduction to statistical typicals, and worked on statistical displays. (Two of those covered in a previous blogpost.) On the day before oreo day, I brought three packages of oreos to class: regular, double stuf, and mega stuf. Their interest was definitely piqued; it was like they could smell the sugar. Not much mathematical interest, though. So I prompted - what might a mathematician wonder about this? They immediately jumped to the idea of is it really double, and what is mega. Then we brainstormed together - what do we need to gather data on for the next day?

Their list:

OREO: data to collect

weight of the whole cookie

weight of white stuf in each cookie

height of each cookie (mm)

diameter of each cookie

weight of cookie/black sides

height of black cookie

height of white stuf

diameter of white stuf

how many of each size fit in a specific container/height

volume by displacement

compare deliciousness of different types

nutrition information

stuffing v serving size

calorie content (burning)

Not bad. Calorie burning turned out not to be viable with that short of a notice... but I'd like to see it! I made a data sheet, so that we'd have a whole class worth of data, and a google spreadsheet to share.


The points about measuring like a scientist (half of the smallest unit) and recording to show the accuracy measured are obviously still in progress. Also the statistical thinking of gathering and using data need more development - most were happy to answer the main question with just their measurement. "It's double." "It's more than double." "It's less than double." No one used the measurements, they went entirely by weight.

That wasn't what bothered me. I expect those kind of goals to take time.

What bothered me was that they weren't into it.

They were excited about the cookies, and figuring how many each person got, and eating the cookies afterward. But they weren't into the math.

Dave Coffey sometimes recounts (or makes fun of me for) how I want to be obsolete. Sitting back and watching students direct themselves at the end of the semester. I always want to hand off to the students. Have them make it their lesson. Look, here's a pile of data! On something interesting! What can you do with it? What else could we look into? How many ways can we come at the question?

But on this day, they said no.

My personal metaphor for this is a Smothers Brothers routine. (That's how old I am.)


(The whole brilliant bit... the show was amazing. Steve Martin got his start there as a writer, for example. They used their folk singing to make the show safe for sharp political commentary. Like we use math class as a ruse to get students problem solving and thinking critically. They were cancelled and replaced with Hee Haw.)

So this lesson felt like, "Take it, class!"
"No."

My response was to ask them to make sure they got all their group's data, and to write about the measuring and their answer to the question for a standards based grading assessment. And this is a compliant class, so they did, and did a good job on it. But that's far from the peak experience for which I was hoping with this lesson.

Part of the problem, I think, was in my desire for efficiency. By introducing the problem in the previous class and then making a record sheet, I took the initiative from them. They went into fill in the blank mode, from long habit in math class. Another part of the problem was lack of a focus, in the workshop sense. I think I should have discussed statistical thinking with them, and how that's different from single measurement thinking. It's all about the data! This is very reminiscent of the Barbie Bust. It was my problem and my lesson. "My" doesn't help me be a better teacher. (Gollum.)

Reflecting afterward, I think my high expectations helped create the sense of disappointment, like an overhyped movie.  And it led me to rush into a lesson instead of building suspense and anticipation.  I think this kind of experience contributes to teachers who "tried that once" and that was enough to turn them off of inquiry-based learning.  In the end it is the learning that needs to be the center of engagement, not the cookie.

Good Data Day

Nothing revolutionary here. Just what turned out to be a nice sequence of experiences with a group of preservice elementary teachers. No handouts, just a story and pictures of what happened.

Previously... we measured some distances, from a whiteboard marker to an outdoor commons. We discussed technique and accuracy, and I shared a bit on with what elementary students struggle. They went off and measured, and then we shared data in a Google doc. (New to some, somehow.) Choice 1 we agreed on as a class. Choice 2 was up to their group. Distance to the bathroom, trash bins, and one group was obsessed with how do they measure a bus. Then we talked about how do we go from all those different numbers to a specific measurement. How tall is a desk? We want a single number! Of course, they first go to the average.



I always push on that. One, because you need to justify, man! Always. The other because the mean is so inappropriate for low sample size because of the power of outliers.

Then last class was a day to explore  measures of central tendency. I love human statistical displays and it was a gorgeous day. (October in Michigan.)

First sort: height.

They naturally lined up this way. We found the median height, with some good discussion about 19 students, which was the middle and what if it was between two people. When the middle height person stepped forward, we talked about if she was a good representation of the typical height of student in our class. People were a little uncomfortable about that. Especially at the tall end of the line.

Second sort: age.
First take, they lined up like they did for height. We found the median and discussed how well she represented typical. People were curious about the twins, who were gracious to answer age questions. I asked if it was true that boys tended to be older than girls, which led to a comparison of the significance of their position in the height chart vs their position in the age chart.

Then I asked them to line up spaced for their age. Start at 19, each sidewalk section was 4 months. "What if we're the same?" "Just line up." "Oh, it's a line plot!"   Students sorted themselves within a month by age. So when we determined the median, we talked about whether to count from the front or back of that month's line.  They thought that this showed the data much better because of the gaps, and were surprised how old it made that ancient 22 year old look.

Third sort: name length.

Finally split up the twins! For this one, we added unifix cubes. They made a stack of cubes the height of their name, first and last as it is on their birth certificate, and used that for comparison. The new question - could we use the cubes to figure out the mean? Would the mean be higher or lower than the median? "We could lay them all down in a big line and divide it by 19!" (Literal adaption of the arithmetic rule is pretty typical.) "What does that do?" "Could we all just share to have the same?" "We have these left over... what do we do with them?" Then we discussed if that meant the mean was closer to 13 or 14. Why was it higher than the median?

Fourth sort: shoe size.

I gave them the option of what to sort, but they were too shy to say anything. When I say shoe size, they jumped at it, and several said they were thinking about that. I like the ambiguity between men and women's, and the apparent diversity from shoe size not varying perfectly with height. I wanted to use the cubes again, but how do we do that with half sizes? Two cubes for each half size... that works! This time they sorted themselves into a line plot without being asked, working out their own spacing.

Again the mean was higher than the median. Will that always happen? How would it be different if we added 4 cubes to the men's shoes?

Starbursts
As they came back inside, I distributed Starbursts, very unfairly. Enter your data on the board before you sit down. DON'T EAT THEM YET. (Only one person did. Not bad.)

We discussed the challenges of using candy in class. Sugar, red dye, communicated values. (Why are prunes the only individually wrapped healthy snack?)

I asked them to make a line plot and pie chart of this data. The group that shared had a very nice way to make their pie chart:take the 80 starbursts, and think about 20 per quarter. Then half that is 10, half that is 5. "But there's 81?" someone asked. The class agreed that this wouldn't change the pie chart much.

Next was the hard discussion. If we want to be fair, how many should each person get? 2, since that's what most people got? But if people contributed, then there would be enough for 3 each. Wait a second, there's enough for four for each of us. With some left over! I tried to make the connection between this idea of equal distribution and the mean. Here's where we want to take outliers into account.

(Unfortunately, I showed them a graph of US income distribution at this point. Probably should have skipped that. It is one of the reasons I think statistical education is important - we can't understand our world and society without it. What do you think?)

Finally we looked at one more way to make a pie chart.

You can see two wrappers there. We compared this to the pie chart on the board, and they said that the lines really helped and they would add them. But they agreed that the drawn chart was very close.

After that we discussed how to share them fairly. I shared that this was real problem solving. The textbook problem is 81 starbursts, 19 students, how many do each get. The real life problem includes messy things like people wanting particular colors.   We did a quick check of color preferences by going to the corners of the room. One corner each for red (30%), pink (25%), orange (15%) and yellow (5%).  Center of the room for no preference (25%). A few methods were suggested, then someone said we should just do rounds of taking one. People agreed and we started. They wanted to make the pie chart for each one and who am I to argue.

So it was a good data day. Statistics and sunshine, who could ask for more?

Next class: starting the glyphs project!

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 pro-football-reference.com. 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?