Fraction Catch

Sometimes it surprises me what I haven't written about here. Fraction Catch is one of my favorite games and it's been pretty successful in implementation from third grade to ninth grade. Partly the game, and partly the fraction cards.



I'm quite happy with the rectangle representation for the cards, as I have seen younger students use it a lot to do reasoning, and get a better sense of what the fraction is.

I often think of number sense having several parts:

• understanding the number as a quantity
• being able to flexibly represent the number
• being able to compare two or more numbers
• being able to compose and decompose the number flexibly

Whether the numbers in question are whole, integral, rational, real, radical, complex, or matrix. (Matrical?) The last bullet is, of course, the key to powerful computational fluency. Often I see students who have too little experience with the number as an actual quantity as opposed to a symbol, and it's positively frequent that students have no or limited ability to represent numbers other than symbolically.

It's very possible that the last two bullets are not actually part of understanding the number, so much as they are activities that deepen the first two characteristics, but I don't see the point in distinguishing them.

The game is very simple. Each player has a hand of three cards, plays a fraction onto the line of cards arranged least to greatest, and captures the lower adjacent card if they were able to play in between. Here's the rules and an example:


Playing this week with Mr. Schiller's class, I thought maybe this would be a chance to focus on the rules aspect of a game.  I asked what might make the rules for a game good, or understandable, and they had no idea. It took a bit of rephrasing just to get across my question.  I got the sense that I was not starting at the beginning, and switched tacks. Instead of demonstrating it first, I asked them to read the rules and then tell Mr. Schiller and I how to play.

It was challenging. Not very engaging, switch from the normal routine, and really communicated to me that I have to or maybe just should do some equipping to get them to be independent game players before working on teaching them designing. The class leaders got the idea, and taught the game to us and the rest of class. Mr. Schiller trounced me which they very much enjoyed. It was the terrible draws, I'm telling you.

Actually playing the game, though, students played pretty intently, though playing a couple games was enough for some. One interesting thing was what they went on to play. We suggested war or high-low-war for some, but one group used the cards to play their own version of Flower Power, a rational number ordering game from MangaHigh. (A lot of my favorite free computer math games are there; teachers register students and can track their progress.) Another group just wanted to put all the cards in order to make as long a streak as possible.  Then they were noticing patterns about which cards were in the set and which weren't.

After the game, there was one good suggestion for a new rule: if you have two no-play turns in a row, you can swap in your whole hand for a new one.

Students made lots of good connections with the representations, and used them to compare fractions. Not too many got to the point where they were developing a strategy on where to play, but far enough that they would choose scoring plays over easy plays.

To summarize I put up some of the comparisons I had seen. All the students did well on comparing like denominators, like 3/8 and 5/8. They also were mostly solid on comparing like numerators, like 2/5 and 2/3. We talked for a bit about 2/3 and 3/4, and they used the nice strategy of how far from a unit they were, but the class couldn't figure out together how 7/10 compared to those two.  Mr. Schiller let them know they'd keep the cards so they could play again later. He was impressed how well they played even though they had covered little of this in class beforehand.

Game Evaluation:

1. Goal(s) -spot on. Really addresses important ideas.
2. Structure - representation, ordering for the comparison, and some strategic depth that requires the numeric understanding.
3. Strategy - present.
4. Interaction - high in interaction, as what you are able to play and what you choose to play are both influenced by the opponent.
5. Surprise - the card game aspect helps with this and with catch up.
6. Catch-Up - check.
   
7. Inertia - the game ends quickly enough that most students want to continue playing. Because strategy deepens and fact knowledge increases with more play, it has pretty good replay value.
8. Rules - seem clear enough for the students to make sense of, but it was better modeled than read.
9. Context: Fun-Flavor-Hook. No context, not sure if it would help. More professional cards would be something; I had some paper decks and some cardstock, and the students preferred the cardstock. Talking about the rectangles as brownie pans was interesting to them... so maybe you could contextualize it. I think it's better as playing cards.

Strong in the yellow, green and blue makes this a good learning game. Give it a go and let me know what you think!

Out of Bread

Out of bread this morning so ... tortillas for the kids' sandwiches. (Making tight little appetizer style rolls.) The first piece of salami for my Ysabela's funroll (marketing considerations) instantly prompted my math curiosity.

Zero

Any questions?

But then I wondered about what photo would be best for #anyqs?

One

Two

Three

And then just because...

Mmmm, ellipses. Did you see that eccentricity comic recently? (Xavier won't eat salami.)

I think the original photo (Zero) is the best for getting at the question I like here - how many pieces of salami to cover the tortilla? One gets at diameter comparison, Two does that even more literally, and Three might help create some dissonance. Which would you use?

I neglected to take pictures of the fun rolls (TM) despite my recent interest in spirals. May have to make a jelly roll.

Game Evaluation via NCTM

The Product Game (pdf)
aka Times Square (online)
still my choice for best math game ever.

This is a quick post comparing NCTM's math game evaluation criteria with my design framework, based on Mark Rosewater's game design criteria.

I just stumbled across the NCTM's Tips for Teachers page on Math Games. Definitely check it out, they have solid links to games at the end.  After a brief motivation about why to use games, they give their criteria for evaluating a game. I've reworded and reorganized them here, as the lack of any organization and structure was a wee maddening to me.

Mathematics:
M1. Does the game reward engaging in mathematical processes? (They connect with strategy.  NCTM’s Process standards)
M2. Does the game's structure or context support the mathematics?
M3. Does the game promote conceptual understanding?

Game features:

G1. Does the game have a random component or choices to make with clear outcomes? Are students empowered?
G2. Does the game reward replay? (Variety in tasks or different pathways to the end.) Does the game have clear scoring?

Teacher and Student:
S1. Does the game give feedback throughout?
S2. Does the game support players through the most challenging parts? (Can they get stuck?)
S3. Does the game have teacher support for classroom use? (Extensions, connected lessons, chance to track students' progress.)

Learning environment:

L1. Does the game promote positive competition and a safe learning environment?
L2. Does the game promote social play? (Competition, collaboration, and communication. )

I want to compare it to the framework I've been using lately. Both to see

1. Goal(s) - M1-processes, M3-concepts
2. Structure - M1-processes (representation), M2-game mathematics, S3-extensions.
3. Strategy - M1-processes (problem solving)
4. Interaction - G1-choices, L2-social play
5. Surprise - G1-randomness
6. Catch-Up - G1-randomness
   
7. Inertia - G2-replay, S2-support as needed.
8. Rules - S3-teacher use.
9. Context: Fun-Flavor-Hook. G2-replay, L1-positive,

What the framework handles well: 

• Math goals.
• A lot more clarity on gameplay.
• Covers their characteristics in a usable format.

Do I need more?

• I've got to think about the feedback throughout (S1). That feels important for an educational game. In some games it's just your success.
• Similarly, support through the challenging parts (S2).
• Classroom support (S3). 

These are basically the education specific characteristics, though S1 is worth thinking about for games in general. It makes sense to me that if the framework is lacking it's in the context of educational games, since its origin was more general.

Note also the slides Maria Droujkova captured from Keith Devlin's math game webinar. His principles have a lot of overlap with the NCTM checks, but are expanded and better suited to multiple platforms. I think there is a place for skills mastery, though, as I would much rather have that in the context of a game than in drill and practice.

Spiral - So-So

I've never seen a spiral board... wow!

Since Vi Hart released her Christmas time spiral celebration, I've been digging spirals again. I made a GeoGebra sketch to go with her video (link includes a link to her video), that I quite like. My tumblog is where I post one-off math and reblog other Tumblr math. So when I got the word from Mr. Schiller that this gameday the "Topic will be polygons/angles/rotational symmetry," it didn't take me too long to get to the idea of a spiral game.  The way it worked out, though, has me wondering: how good does an educational game have to be?

Click for full size

There are lots of things to recommend it: kids think spirals are cool, it makes a nice race track, it allows you to see circle connections to angle, and that angles have the same measure whether small or big in size. I like race games for practicing with quantities, because it gives some repeated experience with a variety of the quantity, and gives you a reason to talk about the quantities. The GeoGebra I used to make the Archimedean Spiral (as opposed to Vi's logarithmic spirals) is posted on Tumblr, too. Then I just used GeoGebra's export as image to get the track into Word.

The problem with race games is that many of them devolve into chutes and ladders (American; snakes and ladders elsewhere).  This one definitely did. I thought a one die game might be easiest, and after some practice settled on moving 15º times the die roll. It included right angles and gave some nice opportunity for mental multiplication. I justified the simplicity of the game to myself by adding a game-design objective. The winner adds a rule; that rule has to help with the catch-up characteristic of the game.


In addition - since the game was simple - I wanted to have another option. I brought some triangle grid paper and an eightfold diagram (links to Google docs) to support the students in making an art project with rotational symmetry.  As I told the students, math art is as close to my heart as math games. I explained how to color a piece and then imagine it turning, or on the 8-fold to color in 1, 2, or 4 wedges and then copy it.

The students gave the game a good try, and they seemed to meet many of the objectives quickly. Watching them play, the game seemed a bit too long. One student who had gotten disengaged was willing to collect data for me on how long a game took. His results: 27 turns for a full game. 16 turns for 1.5 loops shorter.  After one try, some people played on, many moved on to the math art, and a few pulled out the games they made in December.  Mr. Schiller and I agreed it was too long, though the class was just barely in favor of okay-as-is.  In terms of new rules, some students modified it to have 2 dice. Others added rules for if you land on someone +15º, an "if ahead, out one loop line," a -30º spot and similar. Several students were proud to share their art. Not many tried the triangle paper except for making free designs. 

A revised, shorter game is at the end of this post. In general, it was so-so. The whole experience really raised for me the question of how good does an educational game have to be. These students have played some really good games so far, and I think they were disappointed that this one was more regular.  I've definitely thought that educational games have a lower bar, since you're not interested in replay on many of them once your objective is met.  My experience has been that any sort of game is a welcome change.  But maybe if games are regularly played, the bar rises. I'd be interested in your opinions below, by twitter or email.

In terms of the game design framework I've been trying, and my rating of Spiral:

1. Goal(s) - good concrete objectives.
2. Structure - the spiral really fit the objectives well.My main question here is if the board should have angle measures on it. (Definitely, if polar coordinates are the objective.)
3. Strategy - no real strategy. Real room for improvement here.
4. Interaction - with no choices the interaction is limited to the racing.  It's a hook, but no way to effect your opponent.
5. Surprise - not really relevant to this game.
6. Catch-Up - this game has it, both through randomization of the die and the board structure; but it's of the candyland/chutes and ladders variety.
   
7. Inertia - main reason for shortening the game. Overstayed it's welcomed. I think race games, in particular, probably need to be mindful of pace.
8. Rules - clean, simple. The add-a-rule rule was a big hit. I'll be using that again.
9. Context: Fun-Flavor-Hook. The spiral is a start to this, but some context for the spiral might have helped here. With all the spirals in nature, it shouldn't be too hard to add something. Maybe birds flying to the eye of a tornado? Hurricane?

The warning sign for this game is being weak in the green characteristics. Mr. Schiller and I were excited about it because of the strength in the yellow areas. If I was thinking about a commercial game,  I think I'd make a deck of cards for movement, that would give more strategy and interaction. But that's a lot of printing for a one-off classroom game. Maybe you could simulate that with multiple dice? Make the rules a bit more complex, but worth it for gains in the green. Maybe roll three dice, pick one to use that you'll reroll next time. Trade for an opponent's die with one of yours that is higher.  Worth a try! It will even increase angle use.

The modified game is up at Google docs.


Snakes & Ladders Image: Smabs Sputzer @ Flickr

Luddites on Facebook

From London Permaculture @ Flickr

I tweet, blog, tumbl and I'm trying Facebook for a class this semester. I'm a huge advocate of Wolfram|Alpha and GeoGebra for math. But I'm secretly a Luddite. I only have an emergency cell phone for the car. I text once a month on my iPod just to keep the service, and I only got the iPod because I was trying to figure out how to use the technology in class. (Though I have grown fond of... my precious. Wait, what?)

Calvin College has a January series each year, accompanying a mini-term when students take enrichment classes. The talks are free to attend, some stream, and there are many remote locations at which you can catch the talks. (Around Michigan, across the country, Canada and ... Lithuania?)  Today's speaker was Sherry Turkle. (Talk description, Dr. Turkle's website and infrequent Twitter.) Her new book on technology is Alone Together, which really captures the essence of her talk in a nutshell.

An outline of the problem:

• Because of mobile devices we can bail out of reality at any time and moreso, we want to.
• Technology is children's competition for their parent's attention, and now it is their turn to be distracted.
• It's dangerous when technology's affordances meet our vulnerabilities. For example, if we're lonely, but afraid of intimacy, or so self-critical that we want to construct our own image.
• Technology can turn distraction into busyness. Interviewed business people discussed being too busy to think or create, and ironically, too busy with communication to communicate.
• People are comforted by keeping in touch with a lot of people but keeping them at a distance. Technology enables companionship without the burdens of friendship.
• What was once collegial is now considered an interruption, but this also becomes a rationalization for avoiding real communication.
• The essential paradox: the world is growing more complex, but - through social media - we're training ourselves to ask simpler and simpler questions.
• Technology has interfered with the traditional separation process of adolescence. Separating from parents is harder, and forming your own identity in comparison to friends is harder. She captures this with a phrase: "I share therefore I am." Her subjects have made the validation of a feeling part of having it. A side effect of this is reducing people to something you use for your own purposes.

 What's missing and what to do with it?

• We are not cultivating the ability to be alone.  They are missing the solitude that refreshes and restores. Creativity demands solitude.
• This may be especially significant for teachers, who now have the burden of teaching students to work independently and how to be alone.
• We need to be able to discuss the costs of new technology, without over-reacting or condemning critics as Luddites.
• We need to ask the question: "does it serve our purposes?"
• There's a tendency for us to oversimplify by assuming maturity of technology, when in reality it is still changing and malleable. In particular, because we grew up with the internet, we assume the internet is grown up.
• Everything feels like it's about ramping up volume and velocity. Teaching might need to help students be able to slow down, reflect and isolate.
• The language of addiction is not helpful, because it frames the discussion as addicted or cold turkey. Instead, analyze for how the technology is meeting your purposes.

Privacy

• Young people have all but given up their right to privacy.
• Mark Zuckerberg said: "People have really gotten comfortable not only sharing more information and different kinds, but more openly and with more people. That social norm is just something that has evolved over time.  We view it as our role in the system to constantly be innovating and be updating what our system is to reflect what the current social norms are."
• Intimacy requires privacy.
• Democracy requires privacy.

Crux of the matter

• Does the technology encourage us to inappropriately substitute fantasy for reality?
• She contrasted immersion in a novel, which we know ends, and does not have relationships, with Second Life, where you build a home and can have a sort of imaginary family.

So what did I take from this?

From Intersection Consulting @ Flickr

It made me think about some of the progress I've seen with students accepting and participating in group work. I attributed this to shifting school experiences, but maybe it's due as much or more to developing interdependence.  And was my frustration with students not doing homework this past semester in part caused by increasing difficulty in doing individual tasks?

My fear is the ease of distraction making it harder for students to learn to engage.  People texting at stop signs shows the blurred boundaries. How much easier to text or update your status when you're really stuck on a problem?  I don't want all solitude, but I want to help students find a balance.  I'm definitely considering more solitary work as a precursor to group work.  The power of our technology to support discussion and collaboration would have Vygotsky dancing, so I'm not giving up on it in any way.

I like that Dr. Turkle was against blind opposition, but, rather, in favor of intelligent use.  The problem for me is that I am so different in my use of these things, that it's hard for me to know. My use of Twitter helps me get some insight, though, into the idea of monitoring to see if the tech is serving my purposes.